Algebra i - Bourbaki

NICOLAS BOURBAKI Elements of Mathematics Algebra I Chapters 1-3 HERMANN, PUBLISHERS IN ARTS AND SCIENCE 293 rue Lecourbe, 75015 Paris, France A vv ADDISON -WESLEY PUBLISHING COMPANY Advanced Book Program Reading, Massachusetts Originally published as ELEMENTS DE MATHEMATIQUE, ALGEBRE 0 1943, 1947, 1948, 1971 by Hermann, Paris ISBN 2-7056-5675-8 (Hermann) ISBN 0-201-00639-1 (Addison -Wesley) Library of Congress catalog card number LC 72-5558 American Mathematical Society (MOS) Subject Classification Scheme (1970) : 15-A03, 15-A69, 15-A75, 15-A78 Printed in Great Britain 0 1974 by Hermann. All rights reserved This book, or parts thereof, may not be reproduced in any form without the publisher’s written permission TO THE READER 1. This series of volumes, a list of which is given on pages ix and x, takes up mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledgc of mathematics on the rcaders’ part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course. 2. The method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular. This choice has been dictated by the main purpose of the treatise, which is to provide a solid foundation for the whole body of modern mathematics. For this it is indis- pensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands of proof impose a rigorously fixed order on the subject matter. It follows that the utility of certain considerations will not be immediately apparent to the reader unless he has already a fairly extended knowledge of mathematics; otherwise he must have the patience to suspend judgment until the occasion arises. 3. In order to mitigate this disadvantage we have frequently inserted examples in the text which refer to facts the reader may already know but which have not yet been discussed in the series. Such examples are always placed between two asterisks: * . . *. Most readers will undoubtedly find that these examples will help them to understand the text, and will prefer not to leave them out, even at a first reading. Their omission would of course have no disadvantage, from a purely logical point of view. 4. This series is divided into volumes (here called “Books”). The first six Books are numbered and, in general, every statement in the text assumes as known only those results which have already been discussed in the preceding V TOTHEREADER TO THE READER volumes. This rule holds good within each Book, but for convenience of expo- sition these Books are no longer arranged in a consecutive order. At the begin - ning of each of these Books (or of these chapters), the reader will find a precise indication of its logical relationship to the other Books and he will thus be able to satisfy himself of the absence of any vicious circle. 5. The logical framework of each chapter consists of the definitions, the axioms, and the theorems of the chapter. These are the parts that have mainly to be borne in mind for subsequent use. Less important results and those which can easily be deduced from the theorems are labelled as “propositions”, “lemmas”, “corollaries”, “remarks”, etc. Those which may be omitted at a first reading are printed in small type. A commentary on a particularly important theorem appears occasionally under the name of “scholium~’. To avoid tedious repetitions it is sometimes convenient to introduce nota- tions or abbreviations which are in force only within a certain chapter or a certain section of a chapter (for example, in a chapter which is concerned only with commutative rings, the word “ring” would always signify “commutative ring”). Such conventions are always explicitly mentioned, generally at the beginning of the chapter in which they occur. 6. Some passages in the text are designed to forewarn the reader against serious errors. These passages are signposted in the margin with the sign 2 (“dangerous bend”). 7. The Exercises are designed both to enable the reader to satisfy himself that he has digested the text and to bring to his notice results which have no place in the text but which are nonetheless of interest. The most difficult exercises bear the sign 8. 8. In general, we have adhered to the commonly accepted terminology, except where there appeared to be good reasons for deviating from it. 9. We have made a particular effort always to use rigorously correct language. without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability. 10. Since in principle the text consists of the dogmatic exposition of a theory, it contains in general no references to the literature. Bibliographical references are gathered together in Historical Notes, usually at the end of each chapter. These notes also contain indications, where appropriate, of the unsolved problems of the theory. The bibliography which follows each historical note contains in general only those books and original memoirs which have been of the greatest impor - tance in the evolution of the theory under discussion. It makes no sort of pre- tence to completeness; in particular, references which serve only to determine questions of priority are almost always omitted. Asto the exercises, we have not thought it worthwhile in general to indicate their origins, since they have been taken from many different sources (original papers, textbooks, collections of exercises). 11. References to a part of this series are given as follows : a) If reference is made to theorems, axioms, or definitions presented in the same section, they are quoted by their number. b) If they occur in another section of the same chapter, this section is also quoted in the reference. C) If they occur in another chapter in the same Book, the chapter and section are quoted. d) If they occur in another Book, this Book is first quoted by its title. signifies “Summary of Results of the Theory of Sets”. The Summaries of Results are quoted by the letter R; thus Set Theory, R vi vii CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES LIE GROUPS AND LIE ALGEBRAS 1 . Lie algebras . 2 . Free Lie algebras . 3 . Lie groups . 4 . Coxeter groups and Tits systems . 5 . Groups generated by reflections . 6 . Root systems . COMMUTATIVE ALGEBRA 1 . Flat modules . 2 . Localization . 3 . Graduations, filtrations. and topo- logies . 4 . Associated prime ideals and primary decomposition . 5 . Integers . " ~~ 6 . Valuations . 7 . Divisors . SPECTRALTHEORY 1 . Normed algebras . 2 . Locally compact groups . DIFFERENTIAL AND ANALYTIC MANIFOLDS Summary of results . CONTENTS ............................................... To THE READER CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES ................ INTRODUC~ON ............................................... CHAPTER I . ALGEBRAIC STRUCTURES .............................. $ 1 . Laws of composition ; associativity; commutativity .......... 1 . Laws of composition ............................... 2 . Composition of an ordered sequence of elements ........ 3 . Associative laws .................................... 4 . Stable subsets . Induced laws ......................... 5 . Permutable elements . Commutative laws .............. 6 . Quotient laws ..................................... $2 . Identity element; cancellable elements; invertible elements . . 1 . Identity element ................................... 2 . Cancellable elements ............................... 3 . Invertible elements ................................. 4 . Monoid of fractions of a commutative monoid .......... 5 . Applications : I . Rational integers ..................... 6 . Applications: I1 . Multiplication of rational integers ..... 7 . Applications: I11 . Generalized powers ................ 8 . Notation ......................................... $3 . Actions ............................................... 1 . Actions ........................................... 2 . Subsets stable under an action . Induced action ......... 3 . Quotient action ................................... 4 . Distributivity ...................................... 5 . Distributivity of one internal law with respect to another . V ix xxi 1 1 1 3 4 6 7 11 12 12 14 15 17 20 22 23 23 24 24 26 26 27 29 xi X CONTENTS CONTENTS S 4 . Groups and groups with operators ....................... 3 . Subgroups ........................................ 4 . Quotient groups ................................... 6 . Subgroups of a quotient group ....................... 8 . Products and fibre products .......................... 1 . Groups ........................................... 2 Groups with operators . .............................. 5 . Decomposition of a homomorphism .................. 7 The Jordan-Holder theorem 9 . Restricted sums .................................... 10 . Monogenous groups ............................... . ......................... $ 5 . Groups operating on a set ............................... 1 . Monoid operating on a set ........................... 2 . Stabilizer, fixer .................................... 3 . Inner automorphisms ............................... 5 . Homogeneous sets ................................. 6 . Homogeneous principal sets .......................... 7 . Permutation groups of a finite set .................... 4 . Orbits ............................................ $ 6 . Extensions, solvable groups, nilpotent groups ............... 1 . Extensions ........................................ 2 . Commutators ...................................... 3 . Lower central series, nilpotent groups ................. 4 . Derived series, solvable groups ....................... 5 . p-groups .......................................... 6 . Sylow subgroups ................................... 7 . Finite nilpotent groups .............................. .......................... 1 . Free magmas ...................................... 2 . Free monoids ..................................... 3 . Amalgamated sum of monoids ....................... 4 . Application to free monoids ......................... 5 . Free groups ....................................... 7 . Free commutative groups and monoids ............... $ 7 . Free monoids, free groups 6 . Presentations of a group ............................ 8 . Exponential notation ............................... 9 . Relations between the various free objects ............. $8 . Rings ................................................ 2 . Consequences of distributivity ........................ 1 . Rings ............................................. 30 30 31 32 34 37 38 41 45 47 48 52 52 54 55 56 58 60 61 65 65 68 71 74 76 78 80 81 81 82 84 88 89 90 92 94 95 96 96 98 3 . Examples of rings .................................. 5 . Subrings .......................................... 103 6 . Ideals ............................................ 103 7 . Quotient rings 105 8 . Subrings and ideals in a quotient ring ................. 9 . Multiplication of ideals ............................. 10 . Product ofrings ................................... 11 . Direct decomposition of a ring ....................... 101 4 . Ring homomorphisms 102 ............................... ..................................... 106 107 108 110 12 .................................. . Rings of fractions 112 ................................................ $ 9 . Fields 114 1 . Fields ............................................ 114 2 . Integral domains ................................... 116 3 . Prime ideals 116 4 . The field of rational numbers ........................ 117 ...................................... ................................ $ 10 . Inverse and direct limits 118 118 119 120 123 1 . Inverse systems of magmas .......................... 2 . Inverse limits of actions ............................. 3 . Direct systems of magmas ........................... 4 . Direct limit of actions ............................... Exercises for 9 1 ............................................ 124 Exercises for $2 ............................................ 126 Exercises for $3 ............................................ 129 Exercises for $ 4 ............................................ 132 Exercises for $5 ........................................... 140 Exercisesfor$6 ........................................... 147 Exercises for $ 7 ........................................... 159 Exercisesfor$8 ........................................... 171 Exercises for $ 9 ........................................... 174 Exercises for $ 10 .......................................... 179 Historical note ............................................ 180 CHAPTER I1 . LINEAR ALGEBRA .................................. 191 1 . Modules; vector spaces; linear combinations ........... 191 2 . Linear mappings .................................. 194 3 . Submodules; quotient modules ...................... 196 $1 . Modules .............................................. 191 xii ... Xlll CONTENTS CONTENTS 4 . Exact sequences ................................... 5 . Products of modules ................................ 6 . Direct sum of modules .............................. 7 . Intersection and sum of submodules ................... 8 . Direct sums of submodules ........................... 9 . Supplementary submodules ......................... 10 . Modules of finite length 11 . Free families . Bases 12 . Annihilators . Faithful modules . Monogenous modules ... 13 . Change of the ring of scalars ......................... 14 . Multimodules ...................................... ............................ ................................ fj 2 . Modules of linear mappings . Duality ..................... 1 . Properties of Hom,(E. F) relative to exact sequences .... 2 . Projective modules ................................. 3 . Linear forms; dual of a module ...................... 4 . Orthogonality ..................................... 5 . Transpose of a linear mapping ....................... 6 . Dual of a quotient module . Dual of a direct sum . Dual bases 7 . Bidual ........................................... 8 . Linear equations ................................... fj 3 . Tensor products ....................................... 1 . Tensor product of two modules ....................... 2 . Tensor product of two linear mappings ................ 3 . Change of ring ..................................... 4 . Operators on a tensor product; tensor products as multi- modules ......................................... 5 . Tensor product of two modules over a commutative ring 6 . Properties of E @A F relative to exact sequences ........ 7 . Tensor products of products and direct sums ........... 8 . Associativity of the tensor product .................... 9 . Tensor product of families of multimodules ............ 197 200 202 205 208 210 212 214 219 22 1 224 227 227 23 1 232 234 234 236 239 240 243 243 245 246 247 249 25 1 254 258 259 fj 4 . Relations between tensor products and homomorphism modules 267 1 . The isomorphisms Horn, (E @A F. G) -?. Hom,(F. Hom,(E. G)) Horn, (E @A F. G) --f HomA(E. Hom,(F. G)) and 267 268 273 2 . The homomorphism E* @A F -+ Hom,(E. F) .......... 3 . Trace of an endomorphism .......................... + Homc(E1 @c E.. Fi @c Fz) 4 . The homomorphism HOmC(E1. F. ) @Oc Homc(E.. FZ) 274 . .......................... $5 Extension of the ring of scalars .......... 1 . Extension of the ring of scalars of a module 2 . Relations between restriction and extension of the ring of scalars ......................................... 3 . Extension of the ring of operators of a homomorphism module ........................................ ...... 4 . Dual of a module obtained by extension of scalars . ............................. 5 A criterion for finiteness ...................... fj 6 . Inverse and direct limits of modules . ............................ 1 Inverse limits of modules 2 Direct limits of modules 3 Tensor product of direct limits . ............................ . ...................... $ 7 . Vector spaces ......................................... 1 . Bases of a vector space .............................. 2 Dimension of vector spaces 3 . Dimension and codimension of a subspace of a vector space 4 Rank of a linear mapping 5 Dualofavectorspace 6 Linear equations in vector spaces 7 Tensor product of vector spaces 9 Extension of scalars for a vector space . ......................... . ........................... . .............................. . .................... . ...................... ............... 8 . Rank of an element of a tensor product 10 . Modules over integral domains . ................ ....................... ............. fj 8 . Restriction of the field ofscalars in vector spaces 1 . Definition of K’-structures .......................... 2 . Rationality for a subspace ........................... 3 . Rationality for a linear mapping ..................... 4 . Rational linear forms ............................... 5 . Application to linear systems 6 . Smallest field of rationality .......................... 7 . Criteria for rationality .............................. 3 9 . Affine spaces and projective spaces ....................... 1 . Definition of affine spaces ........................... 2 . Barycentric calculus ................................ 3 . Affine linear varieties ............................... 4 . Affine linear mappings ............................. 5 . Definition of projective spaces ........................ 6 . Homogeneous coordinates ........................... 7 . Projective linear varieties ............................ 8 . Projective completion of an affine space ............... 9 . Extension of rational functions ....................... ........................ 277 277 280 282 283 284 284 284 286 289 292 292 293 295 298 299 304 306 309 310 312 317 317 318 319 320 321 322 323 325 325 326 327 329 331 331 332 333 334 xiv xv CONTENTS ........................... ............................ 10 . Projective linear mappings 11 . Projective space structure 335 337 CONTENTS $ 10 . Matrices ............................................. .............................. 1 . Definition of matrices 4 . Matrices and linear mappings 6 . Matrix of a semi-linear mapping 7 . Square matrices 8 . Change ofbases ................................... 9 . Equivalent matrices ; similar matrices 10 . Tensor product of matrices over a commutative ring .... 13 . Equivalence of matrices over a field ................... 2 . Matrices over a commutative group 3 . Matrices over a ring 5 . Block products ..................................... ............................... ....................... ..................... .................................... ................. .................................. ................................ 11 . Trace of a matrix 12 . Matrices over a field .................. 338 338 339 34 1 342 346 347 349 352 354 356 358 359 360 .............................. ........................ .......................... ................................ tj 11 . Graded modules and rings 363 363 364 371 374 375 Appendix . Pseudomodules 378 378 2 . Pseudomodules 378 Exercisesfor$l ........................................... 380 Exercisesforfj2 ........................................... 386 Exercisesfor $3 ........................................... 395 Exercises for $ 4 396 Exercises for $ 5 398 Exercises for $ 6 399 Exercises for $7 ........................................... 400 1 . Graded commutative groups 2 . Graded rings and modules 3 . Graded submodules 367 4 . Case of an ordered group of degrees .................. 5 . Graded tensor product of graded modules ............. 6 . Graded modules of graded homomorphisms 1 . Adjunction of a unit element to a pseudo-ring ......... ........... ................................. .................................... ........................................... ........................................... ........................................... Exercisesfor§8 ........................................... 409 Exercises for 3 10 ........................................... 417 Exercises for $9 ........................................... 410 Exercises for$ 11 .......................................... 424 Exercise for the Appendix .................................. 425 CHAPTER 111 . TENSOR ALGEBRAS . E XTERIOR ALGEBRAS. SYMMETRIC ALGEBRAS ................................................... 1 . Definition of an algebra ............................ $ 1 . Algebras ............................................. 2 . Subalgebras . Ideals . Quotient algebras ................ 3 . Diagrams expressing associativity and commutativity .... 4 . Products of algebras ................................ 6 . Inverse and direct limits of algebras 7 . Bases of an algebra . Multiplication table ............... 5 . Restriction and extension of scalars ................... .................. 427 428 428 429 431 432 433 434 436 tj 2 . Examples of algebras ................................... 438 1 . Endomorphism algebras ............................ 438 2 . Matrix elements ................................... 438 3 . Quadratic algebras ................................. 439 4 . Cayley algebras ................................... 441 5 . Construction of Cayley algebras . Quaternions .......... 443 6 . Algebra ofa magma, a monoid, a group ............... 446 448 450 452 454 455 457 457 459 460 7 . Free algebras ...................................... 8 . Definition of an algebra by generators and relations .... 9 . Polynomial algebras ................................ 10 . Total algebra of a monoid ........................... 11 . Formal power series over a commutative ring .......... 9 3 . Graded algebras ....................................... 1 . Graded algebras ................................... 2 . Graded subalgebras, graded ideals of a graded algebra . . 3 . Direct limits of graded algebras ...................... tj 4 . Tensor products of algebras ............................. 2 . Universal characterization of tensor products of algebras 3 . Modules and multimodules over tensor products of algebras 4 . Tensor product of algebras over a field ................ 5 . Tensor product of an infinite family of algebras ........ 6 . Commutation lemmas .............................. 7 . Tensor product of graded algebras relative to commutation factors ......................................... 8 . Tensor product of graded algebras of the same types .... 9 . Anticommutative algebras and alternating algebras ..... 1 . Tensor product of a finite family of algebras ........... 460 460 463 465 468 470 472 474 480 482 xvi xvii CONTENTS ................................ $ 5 . Tensor algebra . Tensors 484 484 485 489 490 491 492 497 497 498 ........... 1 . Definition of the tensor algebra of a module 2 . Functorial properties of the tensor algebra 4 . Direct limit of tensor algebras ............. ....................... ........................ 3 . Extension of the ring of scalars 5 . Tensor algebra of a direct sum . Tensor algebra of a free 6 . Tensors and tensor notation module . Tensor algebra of a graded module .......... ......................... .................................... 5 6 . Symmetric algebras ...................... 1 . Symmetric algebra of a module 2 . Functorial properties of the symmetric algebra ......... 3 . n-th symmetric power of a module and symmetric multi- 4 . Extension of the ring of scalars 5 . Direct limit of symmetric algebras 6 . Symmetric algebra of a direct sum . Symmetric algebra of a free module . Symmetric algebra of a graded module . . linear mappings ................................. 500 ....................... .................... 502 503 503 CONTENTS ...................................... 5 7 . Exterior algebras . .......... 1 Definition of the exterior algebra of a module ........... 2 . Functorial properties of the exterior algebra 3 . Anticommutativity of the exterior algebra 5 . Extension of the ring of scalars 6 . Direct limits of exterior algebras 8 . Exterior algebra of a free module 9 . Criteria for linear independence ............. 4 . n-th exterior power of a module and alternating multi - linear mappings ................................. ....................... ..................... 7 . Exterior algebra of a direct sum . Exterior algebra of a gradedmodule ................................... ..................... ..................... 507 507 508 510 51 1 513 514 .......................................... $ 8 . Determinants .................... 1 . Determinants of an endomorphism 3 . Determinant of a square matrix ...................... 6 . Expansions of a determinant ........................ 7 . Application to linear equations 9 . The unimodular group SL(n, A) 2 . Characterization of automorphisms of a finite-dimensional ...................................... free module ......................... ................................. 4 . Calculation of a determinant 5 . Minors of a matrix ....................... ......................... ...................... 8 . Case of a commutative field 515 517 519 522 522 523 524 526 528 530 534 536 537 10 . The A[X]-module associatcd with an A-module endo- 11 . Charactcristic polynoniial of an endomorphism ......... morphism ...................................... 538 540 $9 . Norms and traces ...................................... 2 . Properties of norms and traces relative to a module ..... 1 . Norms and traces relative to a module ................ 3 . Norm and trace in an algebra ........................ 4 . Properties of norms and traces in an algebra ........... 5 . Discriminant of an algebra .......................... 5 10 . Derivations ........................................... 1 . Commutation factors ............................... 2 . General definition of derivations 3 . Examples of derivations ............................. 4 . Composition of derivations .......................... 5 . Derivations of an algebra A into an A-module .......... ..................... 6 . Derivations of an algebra ........................... 7 . Functorial properties ............................... 8 . Relations between derivations and algebra homomor - phisms ......................................... 9 . Extension of derivations 10 . Universal problem for derivations : non-commutative case 11 . Universal problem for derivations: commutative case .... 12 . Functorial properties of K-differentials ................ ............................ 5 11 . Cogebras. products of multilinear forms. inner products and duality ............................................ . 1 Cogebras .......................................... 2 . Coassociativity. cocommutativity. counit .............. 3 . Properties of graded cogebras of type N ............... 5 . The graded dualsT(M)*gr. S(M)*g' and A (M)*gr ..... 6 . Inner products: case of algebras 7 . Inner products: case of cogebras ...................... 8 . Inner products : case of bigebras ...................... 4 . Bigebras and skew-bigebras ......................... ..................... 9 . InnerproductsbetweenT(M) andT(M*). S(M) and S(M*). A (M) and A (M*) ............................. 10 . Explicit form of inner products in the case of a finitely generated free module ........................... 11 . Isomorphisms between A" (M) and An-' (M*) for an n- dimensional free module M ........................ 12 . Application to the subspace associated with a p.vector .... 13 . Pure $.vectors . Grassmannians ....................... 541 541 542 543 545 549 550 550 551 553 554 557 559 560 561 562 567 568 570 574 574 578 584 585 587 594 597 600 603 605 607 608 609 xviii xix CONTENTS .................... Appendix. Alternative algebras. Octonions 6 1 1 1. Alternative algebras 61 1 2. Alternative Cayley algebras 61 3 ................................ ......................... 3. Octonions ......................................... 615 Exercisesfor$2 ........................................... 618 Exercisesfor $3 ........................................... 626 Exercisesfor$5 ........................................... 627 Exercisesfor$8 ........................................... 637 ........................................... Exercises for $ 1 6 18 ........................................... Exercises for $ 4 626 ......................................... ........................................... Exercises for $ 6 632 Exercises for 3 7 633 ........................................... .......................................... .......................................... .................................. Exercises for $ 9 644 Exercises for $ 10 645 Exercises for $ 11 647 Exercises for the Appendix 654 655 INDEXOFNOTATION ........................................... 669 INDEXOF TERMINOLOGY ......................................... 677 HISTORICAL NOTE ON CHAPTERS I1 AND I11 ........................ INTRODUCTION Agebra is essentially concerned with calculating, that is, performing, on ele- ments of a set, “algebraic operations”, the most well-known example of which is provided by the “four rules” of elementary arithmetic. This is not the place to retrace the slow process of progressive abstraction by which the notion of algebraic operation, at first restricted to natural numbers and measurable magnitudes, little by little enlarged its domain, whilst the notion of “number” received a parallel generalization, until, passing beyond the latter, it came to be applied to elements with no “numerical” character, for example permutations of a set (see the Historical Note to Chapter I). It is no doubt the possibility of these successive extensions, in which theform of the calculations remained the same, whereas the nature of the mathematical entities subjected to these calculations varied considerably, which was responsible for the gradual isolation of the guiding principle of modern mathematics, namely that mathematical entities in themselves are of little importance; what matters are their relations (see Book I). It is certain in any case that Algebra attained this level of abstraction well before the other branches of Mathematics and it has long been accustomed to being considered as the study of algebraic operations, independent of the mathematical entities to which they can be applied. Deprived of any specific character, the common notion underlying the usual algebraic operations is very simple: performing an algebraic operation on two elements a, b of the same set E, means associating with the ordered pair (a, b) a well-defined third element c of the set E. In other words, there is nothing more in this notion than that of function: to be given an algebraic operation is to be given a function defined on E x E and taking its values in E; the only peculi - arity lies in the fact that the set of definition of the function is the product of two sets identical with the set where the function takes its values; to such a func- tion we give the name law of composition. Alongside these “internal laws”, mathematicians have been led (chiefly xxi INTRODUCTION INTRODUCTION under the influence of Geometry) to consider another type of “law of composi- tion”; these are the “laws of action”, where, besides the set E (which remains as it were in the foreground), an auxiliary set Q occurs, whose elements are called operators: the law this time associating with an ordered pair (a, a) con- sisting of an operator a E Q and an element a E E a second element b of E. For example a homothety of given centre on the Euclidean space E associates with a real number k (the “homothety ratio”, which is here the operator) and a point A of E another point A’ of E : this is a law of action on E. In conformity with the general definitions (Set Theory, IV, 4 1, no. 4), being given on a set E one or several laws of composition or laws of action defines a structure on E; for the structures defined in this way we reserve precisely the name algebraic structures and it is the study of these which constitutes Algebra. There are several species (Set Theory, IV, 5 1, no. 4) of algebraic structures, characterized on the one hand by the laws of composition or laws of action which define them and on the other by the axioms to which these laws are sub- jected. Of course, these axioms have not been chosen arbitrarily, but are just the properties of most of the laws which occur in applications, such as associ - ativity, commutativity, etc. Chapter I is essentially devoted to the exposition of these axioms and the general consequences which follow from them; also there is a more detailed study of the two most important species of algebraic structure, that of a group (in which only one law of composition occurs) and that of a ring (with two laws of composition), of which ajeld structure is a special case. In Chapter I are also found the definitions of groups with operators and rings with operators, where, besides the laws of composition, there occur one or several laws of action. The most important groups with operators are modules, of which vector spaces are a particular example, which play an important role both in Classical Geometry and in Modern Analysis. The study of module structures derives its origin from that of linear equations, whence its name Linear Algebra; the results on this subject are to be found in Chapter 11. Similarly, the rings with operators which occur most often are called algebras (or hypercomplex systems). In Chapters I11 and IV a detailed study is made of two particular types of algebras: exterior algebras, which with the theory of determinants which is derived from them is a valuable tool of Linear Algebra; and polynomial algebras, which are fundamental to the theory of algebraic equations. In Chapter V there is an exposition of the general theory of commutative jelds and their classification. The origin of this theory is the study of algebraic equations in one unknown; the questions which gave birth to this today have little more than a historical interest, but the theory of commutative fields re- mains fundamental to Algebra, being basic to the theory of algebraic numbers on the one hand and Algebraic Geometry on the other. As the set of natural numbers has two laws of composition, addition and multiplication, Classical Arithmetic (or Number Theory), which is the study of natural numbers, is subordinate to Algebra. However, related to the algebraic structure defined by these two laws is the structure defined by the order relation “a divides b” ; and the distinguishing aspect of Classical Arithmetic is precisely the study of the relations between these two associated structures. This is not the only example where an order structure is thus associated with an algebraic structure by a “divisibility” relation: this relation plays just as important a role in polynomial rings. A general study will therefore be made of this in Chapter VI; this study will be applied in Chapter VII to determining the structure of modules over certain particularly simple rings and in particular to the theory of “elementary divisors”. Chapters VIII and IX are devoted to more particular theories, but which have many applications in Analysis: on the one hand, the theory of semisimple modules and rings, closely related to that of linear representations ofgroups; on the other, the theory of quadratic forms and Hermitian forms, with the study of the groups associated with them. Finally, the elementary geometries (affine, projective, euclidean, etc.) are studied in Chapters 11, VI and IX from a purely algebraic point of view: here there is little more than a new language for expressing re - sults of Algebra already obtained elsewhere, but it is a language which is particularly well adapted to the later developments of Algebraic Geometry and Differential Geometry, to which this chapter serves as an introduction. xxii xxiii CHAPTER I Algebraic Structures 3 1. LAWS OF COMPOSITION; ASSOCIATIVITY; COMMUTATIVITY 1. LAWS OF COMPOSITION DEFINITION 1. Let E be a set. A mapping f ofE x E into E is called a law of composition on E. The value f (x, y) off for an orderedpair (x, y) E E x E is called the composition of x and y under this law. A set with a law of composition is called a magma. The composition of x and y is usually denoted by writing x and y in a definite order and separating them by a characteristic symbol of the law in question (a symbol which it may be agreed to omit). Among the symbols most often used are + and . , the usual convection being to omit the latter if desired; with these symbols the composition of x and y is written respectively as x + y and x. y or xy. A law denoted by the symbol + is usually called addition (the composition x + y being called the sum of x and y) and we say that it is written additively; a law denoted by the symbol . is usually called multiplication (the composition x .y = xy being called the product of x and y) and we say that it is written multi - plicatively. In the general arguments of paragraphs 1 to 3 of this chapter we shall generally use the symbols T and I to denote arbitrary laws of composition. By an abuse of language, a mapping of a subset of E x E into E is sometimes called a law of composition not everywhere dejined on E. Examples. (1) The mappings (X, Y) H X v Y and (X, Y) H X n y are laws of composition on the set of subsets of a set E. (2) On the set N of natural numbers, addition, multiplication and ex- ponentiation are laws of composition (the compositions of x E N and y E N under these laws being denoted respectively by x + y, xy or x.y and xy) (Set Theoy, 111, 3 3, no. 4). (3) Let E be a set; the mapping (X, Y) H X o Y is a law of composition on the set of subsets of E x E (Set Theoy, 11, 3 3, no. 3, Definition 6); the mapping (f, g) H f 0 g is a law of composition on the set of mappings of E into E (Set Theory, 11, 3 5, no. 2). (4) Let E be a lattice (Set Theory, 111,s 1, no. 11) ; if sup(x, y) denotes the least upper bound of the set {x, y}, the mapping (x, y) H sup(x, y) is a law of 1 I ALGEBRAIC STRUCTURES composition on E. Similarly for the greatest lower bound inf(x, y). Example 1 above is a particular case of this with q(E) ordered by inclusion. (5) Let (El)isI be a family of magmas. Let Ti denote the law of composi - tion on El. The mapping is a law of composition on the product E = I-I E,, called the product of the laws T(. The set E with this law is called the product magma of the magmas El. In particular, if all the magmas El are equal to the same magma M, we ob- tain the magma of mappings of I from M. ((xi), (yd) ((xi Ti%)) (€1 Let (x, y) H x T y be a law of composition on a set E. Given any two subsets X, Y of E, X T Y (provided this notation does not lead to confusion?) will de - note the set of elements x T y in E, where x E X, y E Y (in other words, the image of X x Y under the mapping (x, y) t+ x T y). If a E E we usually write a T Y instead of {a} T Y and X T a instead of X T {a}. The mapping (X, Y) H X T Y is a law of composition on the set of subsets of E. DEFINITION 2. Let E be a ma<qma and T denote its law of composition. The law of composition (x, y) H y T x on E is callcd the opposite ofthe above. The set E with this law is called the opposite mapa of E. Let E and E’ be two magmas; we shall denote their laws by the same symbol T. Conforming with the general definitions (Set Theory, IV, 5 1, no. 5), a bijec- tive mapping f of E onto E‘ such that for every ordered pair (x, y) E E x E is called an isomorphism ofE onto E‘. E and E‘ are said to be isomorphic if there exists an isomorphism of E onto E’. DEFINITION 3. A mapping f of E into E‘ such that relation (1) holds for every ordered pair (x, y) E E x E is called a homomorphism, or morphism, of E into E‘; ifE = E’, f is called an endomorphism of E. The identity mapping of a magma E is a homomorphism, the composition of two homomorphism is a homomorphism. (1) f (x T Y) =f (2) Tf (Y) More generally: t The following is an example where this principle would lead to confusion and should therefore not be used. Suppose that the law of composition in question is the law (A, B) H A u B between subsets of a set E; a law of composition is derived between subsets of 9,?(E), F(U, 23) being the set of A u B with A E U, B E 23; but F(%, ‘23) should not be denoted by U U 23, as this notation already has a different meaning (the union of U and 23 considered as subsets of P(E)). (a, ‘23) * FW, 23) COMPOSITION OF AN ORDERED SEQUENCE OF ELEMENTS 8 1.2 For a mapping f of E into E’ to be an isomorphism, it is necessary and sufficient that it be a bijective homomorphism and f is then an isomorphism of E’ onto E. 2. COMPOSITION OF AN ORDERED SEQUENCE OF ELEMENTS -1 Recall that a family of elements of a set E is a mapping t H x, of a set I (called an indexing set) into E; a family (xJLE1 is calledjnite if the indexing set is jinite. A finite family (x,)~~~ of elements of E whose indexing set I is totally ordered is called an ordered sequence of elements of E. In particular, every finite sequence where H is a finite subset of the set N of natural numbers, can be considered as an ordered sequence if H is given the order relation induced by the relation m < n between natural numbers. are called similar if there exists an ordered set isomorphism + of I onto K such that ybcl, = xi for all i E I. is similar to a suitable finite sequence. For there exists an increasing bijection of A onto an interval [0, n] of N. DEFINITION 4. Let (x,),.~ be an ordered sequence of elements in a magma E whose indexing set A is non-empty. The composition (under the law T) ofthe ordered sequence (xJcrsA, denoted by T x,, is the element of E dejined by induction on the number of elenzents in A as follows: Two ordered sequences (xJLEI and (yk)ko Every ordered sequence aEA (1) ifA = {p} then T xu = xo; (2) if A has p > 1 elements, p is the least element of A and A’ = A - {p}, then It follows immediately (by induction on the number of elements in the in- dexing sets) that the compositions of two similar ordered sequences are equal; in particular, the composition of any ordered sequence is equal to the composi - tion of a finite sequence. If A = {A, p, v} has three elements (A < p < v) the composition T x, is xA T (x, T xv). asA xu = XD T (LA xu). a(EA Remark. Note that there is a certain arbitrariness about the definition of the composition of an ordered sequence; the induction we introduced proceeds ‘‘from right to left”. If we proceeded “from left to right”, the composition of the above ordered sequence (xA, xu, x,) would be (xA T xu) T x,. As a matter of notation, the composition of an ordered sequence is written a&A for a law denoted by I ; for a law written additively it is usually denoted by xu and called the sum of the ordered sequence (x,),,~ (the xu being called the tern of the sum) ; for a law written multiplicatively it is usually 3 2 ALGEBRAIC STRUCTURES I 5 1.3 We prove the theorem by induction on the cardinal n of A. Let p be the cardinal of1 and h its least element; let J = 1 - {h}. If n = 1, as the Bi are non- empty, of necessityp = 1 and the theorem is obvious. Otherwise, assuming the theorem holds for an indexing set with at most n - 1 elements, we distinguish two cases: (a) B, has a single element p. Let C = ig B,. The left-hand side of (3) is ASSOCIATIVE LAWS equal, by definition, to xp T ; the right-hand side is equal, by defini- tion, to denoted by n x, and called the product of the ordered sequence (x,) (the xu being called the factors of the product).? asA When there is no possible confusion over the indexing set (nor over its ordering) it is often dispensed with in the notation for the composition of an ordered sequence and we then write, for example for a law written additively, 2 x, instead of ,;A x,; similarly for the other notations. For a law denoted by T the composition of a sequence (xi) with indexing set a non-empty interval (p, 9) ofN is denoted by T xi or T xi; similarly for laws denoted by other symbols. Let E and F be two magmas whose laws of composition are denoted by T and f a homomorphism of E into F. For every ordered sequence (x,),~~ of elements of E 4 P9iS4 i=p 3. ASSOCIATIVE LAWS DEFINITION 5. A law of composition (x, y) b-+ x T y on a set E is called associative ;f, for all elements x, y, z in E, (X T y) T z = x T (y T 2). A magma whose law is associative is called an associative magma. The opposite law of an associative law is associative. Examples. (1) Addition and multiplication of natural numbers are associative laws ofcomposition on N (Set Theory, 111,s 3, no. 3, Corollary to Proposition 5) (2) The laws cited in Examples (I), (3) and (4) of no. 1 are associative. THEOREM 1 (Associativity theorem). Let E be an associatiue magma whose law is denoted by T. Let A be a totally ordered non-empty Jinite set, which is the union of an ordered sequence of non-empty subsets (BJiE1 such that the relations u E Bi, E B,, i < j imply u < p; let (X,),~ A be an ordered sequence of elements in E with A as index- ingset. Then T X, = T ( T x,) UEA iEI aEBi (3) The use of this terminology and the notation a€A n x, must be avoided if there is any risk of confusion with the product of the sets x, defined in the theory of sets (Set Theoy, 11,s 5, no. 3). However, if the x, are cardinals and addition (resp. multipli - cation) is the cardinal sum (resp. the cardinal product), the cardinal denoted by 2 x, (resp. ,FA x,) in the above notation is the cardinal sum (resp. cardinal pro- asA duct) of the family (x,),~~ (Set 111, 8 3, no. 3). the result follows from the fact that the theorem is assumed true for C and (b) Otherwise, let p be the least element of A (and hence of B,J; let A’ = A - {p} and let B; = A’ n Bi for i E I; then U; = Bi for i E J. The set A‘ has n - 1 elements and the conditions of the theorem hold for A’ and its sub- sets B;; hence by hypothesis: (Bi) i E 3’ Forming the composition of xp with each side, we have on the left-hand side side, by definition, T x, and on the right-hand side, using the associativity, UEA which is equal, by Definition 3, to the right-hand side of formula (3). For an associative law denoted by T the composition (~t),~r~, 41 is also denoted (since no confusion can arise) by T xi of a sequence pst9q Xp T * . . T xq. A particular case of Theorem 1 is the formula Xg T Xi T *. * T X, = (Xg T Xi T . . * T X,-i) T X,. Consider an ordered sequence of n terms all of whose terms are equal to the Same element x E E. The composition of this sequence is denoted by ? x for a law denoted by T, ?. x for a law denoted by 1. For a law written multiplica- tively the composition is denoted by xn and called the n-th power of x. For a law written additively the composition is usually denoted by nx. The associativity 4 5 PERMUTABLE ELEMENTS. COMMUTATIVE LAWS 5 1.5 ALGEBRAIC STRUCTURES I theorem applied to an ordered sequence all of whose terms are equal gives the equation nl +nz+. . . +np T In particular, ifp = 2, X = ("; x) T (;" X) T ..-T ("4 x). and if n, = n2 = . . . = np = m, If X is a subset of E, we sometimes denote, in conformity with the above nota - tion, by f x the set XI T x, T . T xp, where XI = x, = ... 5 x P- - x; it is thus the set of all compositions x1 T x2 T - * - T X, with x1 E x, x2 E x, . . . , xp E x. It is important not to confuse this set with the set of f x, where x runs through X. 4. STABLE SUBSETS. INDUCED LAWS DEFINITION 6. A subset A ofa set E is called stable under the law ofcomposition T on E ;f the composition of two elements of A belongs to A. The mapping (x, y) * x T y of A x A into A is then called the law induced on A by the law T. The set A with the law induced by T is called a submagma of E. In other words, for A to be stable under a law T it is necessary and sufficient that A T A c A. A stable subset of E and the corresponding submagma are often identified. The intersection of a family of stable subsets of E is stable; in particular there exists a smallest stable subset A of E containing a given subset X; it is said to be generated by X and X is called a generating system of A or a generating sef of A. The corresponding submagma is also said to be generated by X. PROPOSITION 1. Let E and F be two magmas and f a homomorphism of E into F. (i) The image under f ofa stable subset of E is a stable subset of F. (ii) The inverse image under f of a stable subset F is a stable subset of E. liii) Let X be a subset of E. The image under f of the stable subset ofE generated by X is the-itable subset of F generated by f (X). (iv) If e is a second homomo@hh of E from F the set of elements x of E such that \, a" f (x) = g(x) is a stable subset ofE. 6 Assertions (i), (ii) and (iv) are obvious; we prove (iii). Let be the stable subset of E generated by X and f (X) the stable subset of F generated by f (X). By (i) f(x) c f (x) and by (ii) x c >'(J(X)), whence f (x) cf(X). PROPOSITION 2. Let E be an associative magma and X a subset of E. Let X' be the set of x1 T x2 T . . . T xn, where n 3 1 and where x, E x for 1 < i 6 n. The stable subset generated by X is equal to X'. - It follows immediately by induction on n that the composition of an ordered sequence of n terms belonging to X belongs to the stable subset generated by X; it is therefore sufficient to verify that X is stable. Now if u and v are elements of Xtheyareoftheformu =x~Tx~T...T~~-~,~ = x~Tx,,+~T...Tx,,+~ withx,EXforO 6 i 6 n +p;then(Theoreml)uTv =X,,TX,T~..T~,,+, belongs to X. Exumples. (1) In the set N of natural numbers the stable subset under addi- tion generated by {I} is the set of integers 2 1 ; under multiplication the set { 1) is stable. (2) Given a law T on a set E, for a subset {h} consisting of a single element to be stable under the law T it is necessary and sufficient that h T h = h; h is then said to be idempotent. For example, every element of a lattice is idem- potent for each of the laws sup and inf. (3) For an associative law T on a set E the stahle subset generated by a set (u} consisting of a single element is the set of elements ? a, where n runs through the set of integers > 0. 5. PERMUTABLE ELEMENTS. COMMUTATIVE LAWS DEFINITION 7. Let E be a magma whose law is denoted by T. Two elements x and y of E are said to commute (or to be permutable) ;fy T x = x T y. DEFINITION 8. A law of composition on a set E is called commutative $any two elements ofE commute under this law. A magma whose law of composition is commutative is called a commutative magma. A commutative law is equal to its opposite. Examples. (1) Addition and multiplication of natural numbers are commu - tative laws on N (Set Theory, 111, 5 3, no. 3, Corollary to Proposition 5). (2) On a lattice the laws sup and inf are commutative; so, in particular, are the laws U and A between subsets of a set E. (3) Let E be a set of cardinal > 1. The law (f, g) ~fo g between map - pings of E into E is not commutative as is seen by takingfand g to be dis- tinct constant mappings, but the identity mapping is permutable with every mapping. 7 I ALGEBRAIC STRUCTURES (4) Let (x, y) I-> x T y be a commutative law on E; the law (x, Y) x T Y between subsets of E is commutative. DEFINITION 9. Let E be a magma and X a subset of E. The set of elements of E which commute with each of the elements of X is called the centralizer of X. Let X and Y be two subsets of E and X' and Y' their respective centralizers. PERMUTABLE ELEMENTS. COMMUTATIVE LAWS 8 1.5 -~ If X c Y, then Y' c X. Let (X,),,, be a family of subsets of E and for all i E I let Xl be the centralizer .. of X,. The centralizer of Xi is Xi. Let X be a subset of E and X the centralizer of X. The centralizer X" of X is called the bicentralizer of X. Then X c X". The centralizer X" of X" is equal to X. For X is contained in its bicentralizer X and the relation X c X" implies X" c X'. PROPOSITION 3. Let E be an associative magma whose law is denoted by T. Ifan ele- ment x of E commutes with each of the elements y and z of E, it commutes with y T 2. For XT (yT 2) = (xT~) T Z = (YT X) T Z =yT(xTz) =yT(zTx) =(yTz)Tx. COROLLARY . Let E bP an associative magma. The centrnlizer of any subset of E is a stable subset of E. DEFINITION 10. The centralizer of a magma E is called the centre of E. An element of the centre ofE is called a central element of E. If E is an associative magma its centre is a stable subset by the Corollary to Proposition 3 and the law induced on its centre is commutative. PROPOSITION 4. Let E be an associative magma, X and Y two subsets of E. Ifcvery element ofX commutes with every element ofY every element of che stable subset generated by X commutes with every element of the stable subset generated by Y. Let X' and X be the centralizer and bicentralizer of X. They are stable sub - sets of E. Now X c X and Y c X and hence X" (resp. X') contains the stable subset of E generated by X (resp. Y). As every element of X commutes with every element of X', the proposition follows. COROLLARY 1. Ifx and y are permutable under the associative law T so are T x and y for all integers m > 0 and n > 0; in particular T x and -? x are permutable for all x and all integers m > 0 and n > 0. m m COROLLARY 2. If all pails (1/ elemcnts oj a subsri X arejlermutable undcr an associative law T, the law induced by T on the stable subset generated by X is associative and commutative. THEOREM 2 (Commutativity theorem). Let T be an associative law of composition on E; let A be a non-emptyjinite family of elements of E which are pairwise per - mutable; let B and C be two totally ordered sets with A as underlying set. Then T X, = T x,. UEB asC Since the theorem is true if A has a single element, we argue by induction on the numberp of elements in A. Letp be an integer > 1 and suppose the theorem is true when Card A < p. We prove it for Card A = p. It may be assumed that A is the interval (0,p - 1) in N; the composition of the ordered sequence (xJaEA defined by the natural order relation on A is T XI. Let A be given another total ordering and let h be the least element of A under this ordering and A' the set of other elements of A (totally ordered by the induced ordering). Suppose first 0 <: h < p - 1 and let P = (0, I, . . . , h - 1) and Q = {h + 1, . . . , fi - l}; the theorem being assumed true for A', applying the associativity theorem, we obtain (since A' = P u Q) P-1 t=o whence, composing xh with both sides and repeatedly applying the commuta- tivity and associativity of T : which proves the theorem for this case. If h = 0 or h = p - 1, the same result follows, but more simply, the terms arising from P or the terms arising from Q not appearing in the formulae. Under a commutative associative law on a set E the composition of ajinite family (xu),,, of elements of E is by definition the common value of the com- position of all the ordered sequences obtained by totally ordering A in all possible Ways. This composition will still be denoted by T xu under a law denoted by T ; similarly for other notations. THEOREM 3. Let T be an associative law on E and (xJa A a non -emptyfinite family of ehet& of E which are pairwise permutable. If A is a union of non-empty subsets (%,I which are pairwise disjoint, then USA asA T Xu= IEI T(T xu). 8 9 ALGEBRAIC STRUCTURES I 3 1.6 PROPOSITION 5. Let E and F be magmas whose laws are denoted by T and let f and g be homomorphisms ofE into F. Let f T g be the mapping x H f (x) T g(x) ofE into F. IfF is associative and commutative, f T g is a homomorphism. QUOTIENT LAWS For all elements x and y of E: (fT g)(x T Y) =f(X T Y) T T Y) =f (X) Tf (Y) T dX) T g(Y) =f (X) T g(x) Tf (Y) T g(Y) = ((f T g)(X)) T ((f T g)(Y))* 6. QUOTIENT LAWS DEFINITION 1 1. Let E be a set. A law ofcomposition T and an equivalence relation R on E are said to be compatible ifthe relations x = x’ (mod R) and y = y’ (mod R) (for x, XI, y, y’ E E) imply x T y = x’ T y’ (mod R) ; the law of composition on the quotient set E/R which maps the equivalence classes of x and y to the equivalence class of x T y is called the quotient law of the law T with respect to R. The set E/R with the quotient law is called the quotient magma of E with respect to R. To say that an equivalence relation R on E is compatible with the internal law of compositionf: E x E + E on E means that the mapping f is compatible (in the sense of Set Theory, 11, 4 6, no. 5) with the product equivalence relation R x R on E x E and the equivalence relation R on E. (Set Theory, 11, 3 6, no. 8). This also means that the graph of R is a submagma of E x E. If the law T is associative (resp. commutative) so is the quotient law (more briefly we say that associativity, or commutativity, is preserved when passing to the quotient). The canonical mapping from the magma E to the magma E/R is a homo- morphism. For a mapping g of E/R into a magma F to be a homomorphism it is neces- sary and sufficient that the composition of g with the canonical mapping of E onto E/R be a homomorphism. PROPOSITION 6. Let E and F be two magmas and f a homomorphism of E into F. Let R{x,y{ denote the relation f(x) =f(y) between elements x, y of E. Then R is an equivalence relation on E Compatible with the law on E and the mapping ofE/R onto f (E) derived from f bypassing to the quotient is an isomorphism ofthe quotient magma E/R onto the submagma f (E) of F. ~OPOSITION 7. Let E be a magma and R an equivalence relation on E compatible with the law on E. For an equivalence relation S on E/R to be compatible with the quotient law it is mcessary and suicient that S be ofthe form T/R where T is an equivalence relation on E implied by R and compatible with the law on E. The canonical mapping ofE/T onto (E/R)/(T/R) (Set Theory, 11, 4 6, no. 7) is then a magma isomorphism. ~OPOSITION 8. Let E be a magma, A a stable subset ofE and R an equivalence rela- fion On E compatible with the law on E. The saturation B ofA with respect to R (Set The two following propositions are immediate from the definitions : This follows from Theorem 2 if A and I are totally ordered so that the B, satisfy the conditions of Theorem 1. We single out two important special cases of this theorem: 1. If (x,~)(~, D) A is a finite family of permutable elements of an associative magma whose indexing set is the product of two non-empty finite sets A, B (a “double family”), then (7) as follows from Theorem 3 by considering A x B as the union of the sets {a} x B on the one hand and of the sets A x {p} on the other. For example, if B has n elements and for each CL E A all the xu,, have the same value xa, then (8) aEA T (; xa) = ? (JAxa). If B has two elements, we obtain the following results: let (ya)asA be two non-empty families of elements of E. If the xu and the yo are pairwise permutable, then (9) Because of formula (7) the composition of a double sequence (xlI) whose indexing set is the product of two intervals (p, q) and (7, s) in N is often de- noted for a commutative associative law written additively by j=r I=P and similarly for laws denoted by other symbols. 2. Let n be an integer >O and let A be the set of ordered pairs of integers (i,j) such that 0 < i < n, 0 < j < n and i < j; the composition of a family (xtf)(t, (under a commutative associative law) is also denoted by OQtcjGn T xtf (or simply T xt,, if no confusion arises) ; Theorem 3 here gives the formulae i<f There are analogous formulae to (7) for a family whose indexing set is the product of more than two sets and analogous formulae to (1 0) for a family whose indexing set is the set s, of strictly increasing sequences (ik) of# integers such that 0 6 ik 6 n (p < n + 1) : in the latter case the composition of the family (XiIra.. .tJ(iL ... .{&ESP is denoted by k 10 11 I ALGEBRAIC STRUCTURES Theory, 11, 5 6, no. 5) is a stable subset. The equivalence relations RA and RB induced by R on A and B respectively are compatible with the induced laws and the mapping derived from the canonical injection of A into B by passing to the quotients is a magma isomorphism of AIRA onto B/RB. Let T denote the law on E. If x and y are two elements of B there exist two elements x' and y' of A such that x = x' (mod R) and y = y' (mod R) ; then x T y x' T y' (mod R) and x' T y' E A, whence x T y E B. Thus B is a stable subset of E and the other assertions are obvious. Let M be a magma and ((u,, v,)),~~ a family of elements of M x M. Con- sider all the equivalence relations S on M which are compatible with the law on M and such that uu = v, (mod S) for all a E I. The intersection of the graphs of these relations is the graph of an equivalence relation R which is compatible with the law on M and such that u, = v, (mod R). Hence R is thejinest (Set Theory, 111,s 1, nos. 3 and 7) equivalence relation with these two properties. It is called the equivalence relation compatible with the law on M generated by the PRoposInoN 9. Preserving the above notation, let f be a homomorphism of M into a magma such that f (ua) = f (va) for all a E I. Then f is compatible with R. Let T be the equivalence relation associated withf. Then u, = v, (mod T) for all a E 1 and T is compatible with the law on M, hence T is coarser than R; this proves the proposition. ku, 4. 3 2.1 IDENTITY ELEMENT Examples. (1) In the set N of natural numbers 0 is an identity element under addition and 1 is an identity element under multiplication. Each of these two laws gives N a commutative monoid structure (Set Theory, 111, 5 3, no. 3). (2) In the set of subsets of a set E, 0 is an identity element under the law u and E under the law n. More generally, in a lattice the least element, if it exists, is identity element under the law sup; conversely, if there exists an identity element under this law it is the least element of the set. Similarly for the greatest element and the law inf. (3) The set N has no identity element under the law (x, y) H xy. Under the law (X, Y) H X 0 Y between subsets of E x E the diagonal A is the identity element. Under the law (f, g) ~fo g between mappings of E into E the identity mapping of E onto E is the identity element. (4) Let E be a magma and R an equivalence relation on E compatible with the law on E (5 1, no. 6). If e is an identity element of E the canonical image of e in E/R is an identity element of the magma E/R. tj 2. IDENTITY ELEMENT; CANCELLABLE ELEMENTS; INVERTIBLE ELEMENTS 1. IDENTITY ELEMENT DEFINITION 1. Under a law of composition T on a set E an element e of E is called an identity element g, for all x E E, e T x = x T e = x. There exists at most one identity element under a given law T, for if e and e' are identity elements then e = e T e' = e'. An identity element is permutable with every element: it is a central element. DEFINITION 2. A magma with an identity element is called a unital magma. If E, E' are unital magmas, a homomorphism of the magma E into the magma E' which maps the identity element of E to the identity element of E' is called a unital homomorphism (or morphism) of E into E'. An associative unital magma is called a monoid. If E, E' are monoids, a unital morphism of E into E' is called a monoid homomorphism or a monoid morphism of E into E'. The identity element of a unital magma is a unital homomorphism; the composition of two unital homomorphisms is also one. For a mapping to be a unital magma isomorphism it is necessary and sufficient that it be a bijective unital homomorphism and the inverse mapping is then a unital homomorphism. Let E and E' be unital magmas and e' the identity element of E'; the constant mapping of E into E' mapping E to e' is a unital homomorphism, called a trivial homomorphism. The product of a family of unital magmas (resp. monoids) is a unital magma (resp. monoid). Every quotient magma of a unital magma (resp. monoid) is a unital magma (resp. monoid). Let E be a unital magma and e its identity element. A submagma A of E such that e E A is called a unital submagma of E. Clearly e is the identity element of the magma A. Every intersection of unital submagmas of E is a unital submagma of E. If X is a subset of E then there exists a smallest unital submagma of E containing X; it is called the unital submapa of E generated by X; it is equal to {e} if X is empty. If E is a monoid, a unital submapa of E is called a sub- monoid of E. If F is a magma without identity element a submagma of F may possess an identity element. For example, if F is associative and h is an idempotent element of F (I, 9 1, no. 4), the set of h T x T h, where x runs through F, is a submagma of F with h as identity element. If E is a magma with identity element e, a submagma A of E such that c 4 A may still possess an identity element. DEFINITION 3. Let E be a unital magma. The identity element of E is called the com- Podion of the emply family of elements of E. 12 13 I ALGEBRAIC STRUCTURES If (xa),€@ is the empty family of elements of E its composition e is also de- noted by x,. For example, we write T x,=e PdiCP 0 when p < q (p, q E N). Similarly we write T x = e for arbitrary x. With these deJiniti0n.s Theorems 1 and 3 of 3 1 remain true if the hypothesis that the sets A and B, are non-empty is suppressed. Similarly the formulae T x = (? x) (? x) and rnn T x = 7 (F x) are then true for m > 0, n Let E be a unital magma whose law is denoted by T and e its identity ele- ment. The support of a family (xJIGI of elements of E is the set of indices i E I such that x, # e. Let (xJIG1 be a family of elements of E with Jinite support. We shall define the composition T x, in the two following cases: m+n 0. 1EI (a) the set I is totally ordered; (b) E is associative and the xi are pairwise permutable. In these two cases let S be the support of the family (x,). IfJ is a finite subset of I containing S, then T x, = T x,, as is seen by induction on the number of elements in J, applying Theorem 1 of 4 1 in case (a) and Theorem 3 of 3 1 in case (b). Let T xi denote the common value of the compositions 1EJ T xi for all finite subsets of I containing S. When I is the interval (p, -+( of N, we also write T x,. LEJ LES IEI m ;=P With these definitions and notation, Theorems 1 and 3 of 4 1 and the remarks following Theorem 3 extend to families with finite support. The identity element of a law written additively is usually denoted by 0 and called zero or the null element (or sometimes the origin). Under a law written multiplicatively it is usually denoted by 1 and called the unit element (or unit). 9 2.3 INVERTIBLE ELEMENTS PROPOSITION 1. If the law T is associative, then for all x E E and y E E YXTY = YX YY3 'XTU = 'Y 'X' For all z E E: YXTy(') = (' Y) = (Y '1 = yX(yY(')) sXTy(z) = Z T (X T y) = (z T x) T y = sy(Sx(z)) In other words, the mapping x - yx is a homomorphism from the magma E to the set EE of mappings of E into itself with the law (f, g) ~fo g; the mapping x H S, is a homomorphism of E into the set EE with the opposite law. If E is a monoid, these homomorphisms are unital. 2. CANCELLABLE ELEMENTS DEFINITION 4. Given a law ofcomposition T on a set E, the mapping x ++ a T x (resp. x H x T u) of E into itself is called lefit translation (resp. right translation) by an element a E E. On passing to the opposite law, left translations become right translations and conversely. Let ya, (or y(a), &(a)) denote the left and right translations by a E E; then y,(x) = a T x, &,(x) = X T a. DEFINITION 5. An element a 0fa magma E is called left (resp. right) cancellable (or regular) iflefit (resp. right) translation by a is injective. A lft and right cancellable element is called a cancellable (or regular) element. In other words, for a to be cancellable under the law T, it is necessary and sufficient that each of the relations a T x = a T y, x T a = y T a imply x = y (it is said that "a can be cancelled " from each of these equalities). If there exists an identity element e under the law T, it is cancellable under this law: the translations ye and 6, are then the identity mapping of E onto itself. Exumples. (1) Every natural number is cancellable under addition; every natural number # 0 is cancellable under multiplication. (2) In a lattice there can be no cancellable element under the law sup other than the identity element (least element) if it exists; similarly for inf. In particular, in the set of subsets of a set E, 0 is the only cancellable element under the law u and E the only cancellable element under the law n. PROPOSITION 2. The set .f cancellable (resp. lejit Cancellable, resp. right cancellable) elements of an associative magma is a submagma. If y, and yy are injective so is yxTY = yx 0 yy (Proposition 1). Similarly for kc,,. 3. INVERTIBLE ELEMENTS DEFINITION 6. Let E be a unital magma, T its law ofcomposition, e its identity element and x and x' two elements of E. x' is called a left inverse (resp. right inverse, resp. inverse) ofx ifx' T x = e (resp. x T x' = e, resp. x' T x = x T x' = e). An element x ofE is called left invertible (resp. right invertible, resp. invertible) if it ha a lgt inverse (resp. right inverse, resp. inverse). A monoid all of whose elements are invertible is called a group. 14 15 I ALGEBRAIC STRUCTURES Sjmmetric and symmetrizable are sometimes used instead of inverse and in- vertible. When the law on E is written additively, we generally say negative instead of inverse. Examples. (1) An identity element is its own inverse. (2) In the set of mappings of E into E an elementfis left invertible (resp. right invertible) iff is a surjection (resp. injection). The left inverses (resp. right inverses) are then the retractions (resp. sections) associated with f (Set Theory, 11,s 3, no. 8, Definition 11). Forfto be invertible it is necessary and sufficient thatfbe a bijection. Its unique inverse is then the inverse bijection off. Let E and F be two unital magmas and f a unital homomorphism of E into F. Ifx’ is the inverse of x in E,f(x’) is the inverse off (x) in F. Hence, if x is an invertible element of E,f(x) is an invertible element of F. In particular, if R is an equivalence relation compatible with the law on a unital magma E, the canonical image of an invertible element of E is invertible in E/R. PROPOSITION 3. Let E be a monoid and x an element of E. (resp. left) translation by x be surjective. In that case x has a unique inverse, which is also its unique lej? (resp. right) inverse. (i) For x to be 14t (resp. right) invertible it is necessary and suficient that the right (ii) For x to be invertible it is necessary and suflcient that it be lgt and right invertible. If x’ is a left inverse of x then (Proposition 1) SX 0 6, = S,.,, = 6, = Id* and SX is surjective. Conversely, if Sx is surjective, there exists an element x’ of E such that S,(x’) = e and x’ is the left inverse of x. The other assertions of (i) follow similarly. If x’ (resp. x”) is a left (resp. right) inverse of x, then x’ = x‘ T e = x’ T (x T x”) = (x’ T x) T X” = e T x” = x”, whence (ii). Remark. Let E be a monoid and x an element of E. If x is left invertible it is left cancellable; for, if X‘ is a left inverse of x, then YX’ YX = Yx‘ TX = Ye = and yx is injective. In particular, if x is left invertible, the left and right transla - tions by x are bijective. Conversely, suppose that yx is bijective; there exists x’ E E such that xx’ = y,(x‘) = e; then yx(x’x) = (xx‘) x = x = y,(e) and hence x’x = e, so that x is invertible. We see similarly that if Sx is bijective, x is invertible. 5 2.4 MONOID OF FRACTIONS OF A COMMUTATIVE MONOID PROPOSITION 4. Let E be a monoid and x and y invertible elements of E with inverses X’ and y’ respectively. Then y’ T XI is the inverse of x T y. This follows from the relation (y’ T x’) T (x T y) = y’ T (x‘ T x) T y = y‘ T y = e and the analogous calculations for (x T y) T (y‘ T x’). COROLLARY 1. Let E be a monoid; if each of the elements x, of an ordered sequence ofelements ofE has an inverse xi, the composition T x, has inverse T xi, where A’ is the totally ordered set derived from A by replacing the order on A by the opposite order. aEA asA’ This corollary follows from Proposition 4 by induction on the number of elements in A. integer n b 0. In particular, if x and x’ are inverses, T n x and T n x‘ are inverses for every COROLLARY 2. In a monoid the set of invertible elements is stable. PROPOSITION 5. If in a monoid x and x’ are inverses and x commutes with y, so does x’. From x T y = y T x, we deduce x‘ T (x T y) T x’ = x’ T (y T x) T x’ and hence (x’ T x) T (y T x’) = (x’ T y) T (x T x’), that is y T x’ = x’ T y. COROLLARY 1. Let E be a monoid, X a subset of E and X’ the centralizer of X. The inverse of every invertible element of X’ belongs to X’. COROLLARY 2. In a monoid theinverse of a central invertible element is a central element. 4. MONOID OF FRACTIONS OF A COMMUTATIVE MONOID In this no., e will denote the identity element of a monoid and x* the inverse of an invertible element x of E. Let E be a commutative monoid, S a subset of E and S‘ the submonoid of E generated by S. Lemma 1. In E x S‘ the relation Rk, y{ defined by: LI there exist a, b E E and p, q, s E S’ such that x = (a,p), y = (6, q) and U~S = bps” an equivalence relation compatible with the law on the product monoid E x S’. It is immediate that R is reflexive and symmetric. Let x = (a, p), y = (6, q) and z = (c, r) be elements of E x S’ such that Rlx, y{ and Rly, z{ hold. Then there exist two elements s and t of S’ such that aqs = bps, brt = cqt, whence it follows that ar(stq) = bpsrt = cp(stq) 16 17 ALGEBRAIC STRUCTURES I MONOID OF FRACTIONS OF A COMMUTATIVE MONOID § 2.4 and hence Rlx, 21 holds, for stq belongs to S’. The relation R is therefore transitive. Further, let x = (a, p), y = (b, q), x’ = (a’, p‘), and y‘ = (b’, q’) be elements ofE x S’ such that Rlx, y{ and Rlx’, y’{ hold. There exist s and s’ in S’ such that aqs = bps, a’q’s’ = bys’, whence it follows that (aa’) (qq’) (ss’) = (bb‘) (pp’) (ss’) and hence Rlxx’, yy‘i for SS’ E S’. The equivalence relation R is therefore compatible with the law of composition on E x S’. The quotient magma (E x S’)/R is a commutative monoid. DEFINITION 7. Let E be a commutative monoid, S a subset ofE and S’ the submonoid ofE generated by S. The quotient monoid (E x S’)/R, where the equivalence relation R is as described in Lemma 1, is denoted by E, and is called the monoid of.fiactionst of E associated with S (or with denominators in S). For a E E and p E S‘ the class of (a, p) modulo R is in general denoted by a/p and called the fraction with numerator a and denominator p. Then by definition (alp) . (a’/$) = aa‘/pp‘. The fractions a/p and a‘/p’ are equal if and only if there exists s in S’ with spa’ = sp’a; if so, there exist (r and a’ in S’ with aa = a’o‘ and pa = #‘a’. In particular, a/p = sa/sp for a E E and s, p in S’. The identity ele- ment of E, is the fraction e/e. Let ale = €(a) for all a E E. The above shows that E is a homomorphism of E into ES, called canonical. For all p E S’, (p/e) . (e/p) = e/e and hence e/p is the inverse of ~(p) = p/e; every element of E(S’) is therefore invertible. Then alp = (a/e).(e/p), whence for a E E andp E S‘, the monoid E, is therefore generated by e(E) u E(S)*. PROPOSITION 6. The notation is that of DeJinition 7 and E denotes the canonical homo- morphism ofE into E,. (i) Let a and b be in E; in order that €(a) = ~(b), it is necessary and suicient that there exist s E S’ with sa = sb. (ii) For E to be injective it is necessary andsuicient that every element ofS be cancellable. (iii) For E to be bijective it is necessary and suicient that every element of S be in- vertible. Assertion (i) is clear and shows that E is injective if and only if every element of S’ is cancellable; but since the set of cancellable elements of E is a sub- monoid of E (no. 2, Proposition 2), it amounts to the same to say that every element of S is cancellable. (1) a/@ = +).(P)* t It is also called monoid of dzfferences if the law on E is written additively. 18 If E is bijective every element of S is invertible, for E(S) is composed of in- vertible elements of E,. Conversely, suppose every element of S is invertible; then every element of S‘ is invertible (no. 3, Corollary 2 to Proposition 4) and hence cancellable. Then E is injective by (ii) and a/@ = =(a .p*) by (l), hence E is surjective. THEOREM 1. Let E be a commutative monoid, S a subset of E, E, the monoid of frac- tions associated with S and E: E + E, the canonical homomorphism. Further let f be a homomorphism of E into a monoid F (not necessarily commutative) such that every element off (S) is invertible in F. There exists one and only one homomorphism f of E, into F such that f = f 0 E. Iffis a homomorphism of E, into F such that f = fo E, then for a E E and p E S’, whence the uniqueness of$ show that g is a homomorphism of E x S‘ into F. First of all, Let g be the mapping of E x S‘ into F defined by g(a, p) = f(a) . f (p)*. We Let (a, p) and (a’, p’) be two elements of E x S‘; as a’ and p commute in E, f(a’) and f (p) commute in F, whencef(a’)f(p)* =f(P)*f (a’) by no. 3, Pro- position 5. Moreoverf(pp’)* =f(p’p)* = (f(p’) f @))* = f(p)*f(p’)* by no. 3, Proposition 4, whence We show that g is compatible with the equivalence relation R on E x S’: if (a, p) and (a’, p’) are congruent mod. R, there exists s E S’ with spa‘ = sup‘, whence f (s)f(p)f(a‘) = f(s) f (a)f(p’). As f (s) is invertible, it follows that f(@) f (a’) =f(a)f(p’) and then by left multiplication by f(p)* and right multiplication by f (p’ ) * Hence there exists a homomorphismfof E, into F such thatf(a/p) = g(a, p), whencef(c(a)) = f(a/e) = f(a) f (e)* = f(a). Hencefo E = f. COROLLARY. Let E and F be two commutative monoids, S and T subsets ofE and F respectively, f a homomorphism of E into F such that f (S) c T and E: E --f E,, F -+ F, the canonical homomorphism. There exists one and only one homomorphism g: E, 4 F, such that g o E = q of. The homomorphism vertible element of FT. -q 0 f of E into FT maps every element of S to an in- 19 I ALGEBRAIC STRUCTURES Remarks. (1) Theorem 1 can also be expressed by saying that (Es, E) is the solution of the universal mapping problem for E, relative to monoids, monoid homomorphisms and homomorphisms of E into monoids which map the ele- ments of S to invertible elements (Set Theory, IV, 4 3, no. 1). It follows (loc. cit.) that every other solution of this problem is isomorphic in a unique way to (2) For the existence of a solution to the above universal mapping problem it is unnecessary to assume that the monoid E is commutative, as follows from Set Theory, IV, 4 3, no. 2 (cf. Exercise 17). (Es, 4. We mention two important special cases of monoids of fractions. (a) Let E = E,. As the monoid E is generated by the set E(E) u E(E)*, which is composed of invertible elements, every element of E is invertible (no. 3, Corollary 2 to Proposition 4). In other words, E is a commutative goup. Moreover, by Theorem 1 every homomorphismfof E into a group G can be uniquely factorized in the form f =fa E, wheref: E -+ G is a homo - morphism. E is called the group offructions ofE (or group ofdz$rences ofE in the case of additive notation). (b) Let @ = E,, where C consists of the cancellable elements of E. By Pro- position 6, (ii), the canonical homomorphism of E into CD is injective; it will be profitable to identify E with its image in @. Hence E is a submonoid of @, every cancellable element of E has an inverse in @ and every element of @ is of the form alp = a .p* with a E E and p E C; then alp = a'lp' if and only if up' = pa'. It is easily seen that the invertible elements of @ are the fractions alp, where a and p are cancellable, and p/a is the inverse of alp. Now let S be a set of cancellable elements of E and S' be the submonoid of E generated by S. If alp and a'lp' are two elements of E,, then alp = a'lp' if and only if ap' = pa' (for sap' = spa' implies up' = pa' for all s E S') . E, may there- fore be identified with the submonoid of @ generated by E u S*. If every element of E is cancellable, then @ = E and E is a submonoid of the commutative group 0. Conversely, if E is isomorphic to a submonoid of a group, every element of E is cancellable. APPLICATIONS : I. RATIONAL INTEGERS 4 2.5 5. APPLICATIONS: I. RATIONAL INTEGERS Consider the commutative monoid N of natural numbers with law of composi - tion addition; all the elements of N are cancellable under this law (Set Theory, 111,s 5, no. 2, Corollary 3). The group of differences of N is denoted by Z; its elements are called the rational integers; its law is called addition ofrational integers and also denoted by + . The canonical homomorphism from N to Z is injective and we shall identify each element of N with its image in Z. The elements of Z are by definition the equivalence classes determined in N x N by the relation between (m,, nl) and (m,, n,) which is written m, + n, = m2 + n,; an element m ofN is identified with the class consisting of the elements (m + n, n), where 11 E N; it admits as negative in Z the class of elements (n, m + n). Every element (p, q) ofN x N may be written in the form (m + n, n) ifp 2 q or in the form (n, m + n) ifp 6 q; it follows that Z is the union ofN and the set of negatives of the elements of N. The identity element 0 is the only element of N whose negative belongs to N. For every natural number m, - m denotes the negative rational integer of m and -N denotes the set of elements -m for m E N. Then Z =Nu (-N) and Nn (-N) = (0). form EN, m = -m if and only if m = 0. Let m and n be two natural numbers; (a) if m 2 n, then m + (-n) = p, where p is the element of N such that (b) if m f n, then m + ( -n) = -p, where p is the element of N such that m = n +p; m+p=n; (c) (-m) + (-n) = -(m + n). Properties (b) and (c) follow from no. 3, Proposition 4; as Z = N u (-N), addition in N and properties (a), (b) and (c) describe completely addition in Z. More generally -x is used to denote the negative of an arbitrary rational integer x; the composition x + (-y) is abbreviated to x - y (cf. no. 8). The order relation 6 between natural numbers is characterized by the following property: m 6 n if and only if there exists an integerp E N such that m + p = n (Set Theory, 111,s 3, no. 6, Proposition 13 and Ej 5, no. 2, Proposition 2). The relation9 - x E N between rational integers x and y is a total order rela- tion on Z which extends the order relation < on N. For, for all XEZ, x-x=O~N;ify-x~Nandz-y~N, then t - x = (Z - y) + (y - X) EN, for N is stable under addition; ify - x E N and x - y E N, then y - x = 0, for 0 is the only element of N whose negative belongs to N; for arbitrary rational integers x and y, y - x E N or x - y E N, for Z = N u ( -N) ; finally, if x and y are natural numbers, then y - x E N if and only if there exists p E N such that * + p = y. This order relation is also denoted by < . Henceforth when Z is considered as an ordered set, it will always be, unless otherwise mentioned, with the ordering that has just been defined, the natural numbers are identified with the integers >, 0; they are also called positive in- )egem; the integers < 0, negatives of the positive integers, are called lzegative integers; the integers > 0 (resp. < 0) are called strictlypositive (resp. strictly mgatiVe) ; the set of integers > 0 is sometimes denoted by N*. Let x, y and t be three rational integers; then x < y if and only if 20 21 I ALGEBRAIC STRUCTURES + z < y + z. For x - y = (x + t) - (y + 2). This property is expressed by saying that the order relation on Z is invariant under translation. 6. APPLICATIONS: II. MULTIPLICATION OF RATIONAL INTEGERS Lemma 2. Let E be a monoid and x an element of E. (i) There exists a unique homomorphism f of N into E with f (1) = x and f(n) = ? xfor all n EN. (ii) If x is invertible, there exists a unique homomorphism g of Z into E such that \I - g(1) = x and g coincides with f on N. Writing f (n) = f x for all n E N, the formulae 0 m+R T x = e and (? x) T (?x) = T x (no. I) express the fact that f is a homomorphism of N into E and obviously f (1) = x. Iff’ is a homomorphism of N into E such that f’( 1) = x, then f = f ’, by 4 1, no. 4, Proposition 1, (iv). Suppose now that x is invertible. By no. 3, Corollary 2 to Proposition 4, f(n) = 7 x is invertible for every integer n 2 0. By construction, Z is the group of differences of N and hence (no. 4, Theorem 1) f extends uniquely to a homo - morphism g of Z into E. If g’ is a homomorphism of Z into E with g’( 1) = x, the restriction f‘ of g’ to N is a homomorphism of N into E withf‘( 1) = x. Hence f’ = f, whence g’ = g. We shall apply Lemma 2 to the case where the monoid E is Z; for every integer m E Z there therefore exists an endomorphism fm of Z characterized by fm( 1) = m. If m is in N, the mapping n H mn of N into N is an endomorphism of the magma N (Set Theory, 111, 3 3, no. 3, Corollary to Proposition 5) ; hence fm(n) = mn for all m, n in N. Multiplication on N can therefore be extended to multiplication on Z by the formula mn = f,(n) for m, n in Z. We shall establish the formulae: xy = yx (2) (3) (XY)Z = X(P> (4) (5) (6) (7) (8) x(y + 2) = xy + xz (x + y). = xz + yz 0.x = x.0 = 0 1.x = x.1 = x (-l).x = x.(-I) = --x for x,y, z in Z. (*In other words, Z is a commutative ring.,) The formulae x(y + t) = xy + xz and x.0 = 0 express the fact thatf, is an endomorphism NOTATION 4 2.8 of the additive monoid Z andf,(l) = x may be written x. 1 = x. The endo- morphismf’ 0 f, of Z maps 1 to xy and hence is equal to fry, whence (3). NOW f,(-y) = -f,(y), that is x(-y) = -xy; similarly, the endomorphism y t-+ -xy of Z maps 1 to -x, whence (-x) .y = -xy and therefore (-x)(-y) = -(x(-y)) = -(-xy) = xy. Form, n in N, mn = nm (Set Theory, HI,§ 3, no. 3, Corollary to Proposition 5), whence(-m).n =n(-m)and(-m)(-n) = (-n)(-m);asZ =Nu (-N), xy = yx for x,y in Z; and this formula means that (5) follows from (4) and completes the proof of formula (6) to (8). 7. APPLICATIONS: III. GENERALIZED POWERS Let E be a monoid with identity element e and law of composition denoted by T. If x is invertible in E, let g, be the homomorphism of Z into E mapping 1 to x. Let gx(n) = T x for all n E Z; by Lemma 2 this notation is compatible for n E N with the notation introduced earlier. Then n 0 Tx=e 1 Tx=x m for x invertible in E and m, n in Z. Further, ify = T x, the mapping n +% g,(mn) of Z into E is a homomorphism mapping 1 to y, whence g,(mn) = gy(n), that is AS - 1 is the negative of 1 in Z, -1 T x is the inverse of x = x in E. If we write n = -m in (9), it is seen that -m T x is the inverse of m T x. 8. NOTATION (a) As a general rule the law of a commutative monoid is written additively. It is then a convention that -x denotes the negative of x. The notation * + (-y) is abbreviated to x - y and similarly xi-y-2, x-y- 2, x-y+z-t, etc .... represent respectively x +Y + (-21, x + (-Y) + (-z), x + (-y) + z + (-t), etc.. . . 23 22 ALGEBRAIC STRUCTURES I ACTIONS 4 3.1 For n E Z the notation '4 x is replaced by nx. Formulae (9) to (12) then become (m + n).x = m.x + n.x 0.x = 0 1.x = x m.(n.x) = (mn).x where m and n belong to N or even to Z if x admits a negative. Also in the latter caSe the relation (- 1) .x = -x holds. We also note the formula (17) n.(x + y) = n.x + n.y. (b) Let E be a monoid written multiplicatively. For n E Z the notation ? x is replaced by xn. We have the relations xm+n = p.Xn xo = 1 x1 = x (p)n = Xm" and also (xy)" = x"y" if x and y commute. 1 When x has an inverse, this is precisely x-l. The notation - is also used in- stead of x-l. Finally, when the monoid E is commutative, - or x/y is also used for xy-'. X X Y $3. ACTIONS 1. ACTIONS DEFINITION 1. Let Q and E be two sets. A mapping of Q into the set EE of mappings ofE into itself is called an action of Q on E. Let a ++fa be an action of Q on E. The mapping (a, x) -fa(x) (resp. (x, a) -fa(x)) is called the law oflejit (resp. right) action of Q on Et associated with the given action of Q on E. Given a mapping g of Q x E (resp. E x Q) into E, there exists one and only one action a H fa of Q on E such that the associated law of left (resp. right) action is g (Set Theory, 11, 3 5, no. 2, Proposition 3). In this chapter we shall say, for the sake of abbreviation, "law of action " t Or sometimes the external law of composition on E with R as operating set. 24 instead of "law of left action ". 'lhe elementf,(x) of E (for a E i1 and x E E) is Sometimes called the transform of x under a or the composition of a and x. It is often denoted by left (resp. right) multiplicative notation a.x (resp. x.a), the dot may be omitted; the composition of a and x is then called the product of a and x (resp. x and a). The exponential notation xa is also used. In the arguments ofthe following paragraphs we shall generally use the notation a x. The ele- ments of Q are often called operators. Exarnfdes. (1) Let E be an associative magma written multiplicatively. The mapping which associates with a strictly positive integer n the mapping x H x" of E into itself is an action of N* on E. If E is a group, the mapping which associates with a rational integer a the mapping x H xa of E into E is an action of Z on E. (2) Let E be a magma with law denoted by T. The mapping which associates with x E E the mapping A H x T A of the set of subsets of E into itself is an action of E on T(E). (3) Let E be a set. The identity mapping of EE is an action of EE on E, called the canonical acfion. The corresponding law of action is the mapping (f, x) ~f(x) of EE x E into E. be a family ofsets. For all i E I, letf;: Rl-t EE be an action of Rt on E. Let R be the sum of the Rt (Set Theory, 11,s 4, no. 8). The map- ingfof R onto EE, extending the 5, is an action of R on E. This allows us to reduce the study of a family of actions to that of a single action. (5) Given an action of R on E with law denoted by I, a subset 8 of R and a subset X of E, 8 I X denotes the set of a I x with a E 3 and x E X; when 8 consists of a single element a, we generally write a I X instead of {a} I X. The mapping which associates with a E R the mapping X - a I X is an action of R on T(E), which is said to be derived from the given action by ex- tension to the set of subsets. (6) Let a H fa be an action of R on E. Let g be a mapping of R' into R. Then the mapping p H fgto, is an action of R' on E. (7) Letf: E x E -+ E be a law of composition on a set E. The mapping y: x ++ yx (resp. 6: x H 6,) (8 2, no. 2) which associates with the element x E E left (resp. right) translation by x is an action of E on itself; it is called the Zeft (resp. right) action of E on itself derived from the given law. When f is commutative, these two actions coincide. The law of left (resp. right) action associated with y isf (resp. the opposite law tof). The law of right (resp. left) action associated with 6 is f (resp. the opposite law tof). (4) Let Let R, E, F be sets, a H fa an action of Q on E and a H ga an action of Q on F. An Q-morphism ofE into F, or mapping ofE into F compatible with the action of Q, is a mapping h of E into F such that 25 I ALGEBRAIC STRUCTURES for all a E Q and x E E. The composition of two Q-morphisms is an Q-mor- phism. Let $2, Z, E, F be sets, a -fa an action of Q on E, p H go an action of Z on F and C+ a mapping of Q into 0'. A C+-morphism of E into F is a mapping h of E into F such that for all a E Q and x E E. gmca,(h(x)) = h(fa(X)) 2. SUBSETS STABLE UNDER AN ACTION. INDUCED ACTION DEFINITION 2. A subset A ofa set E is called stable under an action a H fa of Q on E $fa@) c A for all a E Q. An element x of E is called invariant under an element a of The intersection of a family of stable subsets of E under a given action is stable. There therefore exists a smallest stable subset of E containing a given subset X of E; it is said to be generated by X; it consists of the elements ( fal 0 fa, o . . - o fan) (x), where x E X, n > 0, ui E Q for all i. Remark. Let E be a magma with law denoted by T. It should be noted that a subset A of E which is stable under the left action on E on itself is not necessarily stable under the right action of E on itself; a subset A of E stable under the left (resp. right) action of E on itself is stable under the law on E but the converse is not in general true. More precisely, A is stable under the law on E if and only if A T A c A whereas A is stable under the left (resp. right) action on E on itself if and only if E T A c A (resp. A T E c A). Q $fa(%.) = x. Example. Take the magma E to be the set N with multiplication. The set {I} is stable under the internal law of N, but the stable subset under the action of N on itself generated by {I} is the whole of N. DEFINITION 3. Let a H fa be an action of Q on E and A a stable subset of E. The mapping which associates with an element a E Q the restriction of fa to A (considered as a mapping of A into itself) is an action of Q on A said to be induced by thegiven action. 3. QUOTIENT ACTION DEFINITION 4. Let u H fa be an action ofa set Q on a set E. An equivalence relation R on E is said to be compatible with the given action ;f, for all elements x and y of E such that x =_ y (mod R) and all a E Q, fa(x) = fa(y) (mod R). The mapping which associates with an element a E Q the mapping of E/R into itself derived from fa bypassing to the quotients is an action of Q on E/R called the quotient of the action of Q on E. Let E be a magma and R an equivalence relation on E. R is said to be le$t (resp. right) compatible with the law on E if it is compatible with the left (resp. 26 DISTRIBUTIVITY 5 3.4 right) action of E on itself derived from the law on E. For R to be compatible with the law on E it is necessary and sufficient that it be left and right compatible with the law on E. We leave to the reader the statement and proof of the analogues of Proposi - tions 6, 7 and 8 of 5 1, no. 6. 4. DISTRIBUTIVITY DEFINITION 5. Let El,. . ., En and F be sets and u a mapping of E, x . x E, into F. Let i E (1, n). Suppose E, and F are given the structures of magmas. u is said to be distributive relative to the index variable i ;f the partial mapping XI CJ u(a1,. . ., a,-,> xi, a,+,, . . ., a,) is a homomorphism of E, into F for alljxed a, in E, andj # i. the equations If T denotes the internal laws on E, and F, the distributivity of u is given by (1) u(a1, *. ., at-,, xi T 4, at+,, an) = u(a1,. . ., at-,, xi, at+,, . . .y an) T u(a1,. . ., at-,, x:, at+,, . . ., a,) fori= 1,2 ,..., n,a,~E, ,..., ~,~,EE,~,,~,EE,,~~EE,,~,,,EE,+, ,..., a, E En. Example. Let E be a monoid (resp. group) written multiplicatively. The mapping (n, x) I+ x" of N x E (resp. Z x E) into E is distributive with respect to the first variable by the equation xm+" = xmxn (with addition as law on N). If E is commutative, this mapping is distributive with respect to the second variable by the equation (xy)" = x"f. PROPOSITION 1. Let El, E,, . . . , En and F be commutative monoids written additively and let u be a mapping of El x . + . x E, into F, which is distributive with respect to all the variables. For i = 1, 2, . . . , n, let L, be a non-emptyjnite set and (xi, JhoL, a Yamily of elements of E,. Let y, = xzt xi, ),for i = 1, 2, . . . , n. Then the sum being taken over all sequences a = (u,, . . . , u,) belonging to L, x . . . x L,. We argue by induction on n, the case n = 1 following from formula (2) of $ 1, no. 2. From the same reference (3) 27 ALGEBRAIC STRUCTURES I DISTRIBUTIVITY OF ONE INTERNAL LAW WITH RESPECT TO ANOTHER 3 3.5 for y, = 2 xn, an and the mapping t * u(y,, . . . , y,- ,, z) of En into F is a magma homomorphism. By the induction hypothesis applied to the distribu - tive mappings (tl,. . . , z,-,) H ~(zl, . . . , zn-1, x,,,,) from El x . * to F, a. E Ln x (4) ~(y1,. . .,Yn-1, xn.a,) = a1.. . 2 . ,a,- 1 u(xl.al,. . .? xn-l.an-l, xn.a,), the sum being taken over the sequences (a1, . . . , an-,) belonging to M = L1 x . . . x L,-l. NOW L, x . . . x L, = M x L,; writing - ta1, ... .u, - u(xl,al, * . *> xn,an), we have (5) by formula (7) of 3 1, no. 5. (2) follows immediately from (3), (4) and (5). Remark. If u(al, . . ., ai.-l, 0, a,+l,. . ., a,) = 0 for i = 1, 2,. . ., n and a, E E, (j # i), then formula (2) remains true for families (xi,A)AaLI ofjnite support. A special case of Definition 5 is that where u is the law of action associated with the action of a set C2 on a magma E. If u is distributive with respect to the second variable, it is also said that the action of !2 on the magma E is distribu- tive. In other words: DEFINITION 6. An action a ++fa ofa set !2 on a magma E is said to be distributive ;f, for all a E Q, the mapping fa is an endomorphism Ofthe magma E. If T denotes the law of the magma E and I the law of action associated with the action of Q on E, the distributivity of the latter is then expressed by the formula (6) distributive) with respect to the law T. (xh) a 1 (x T y) = (a I x) T (a I y) (for a E !2 and x, y E E). By an abuse of language, it is also said that the law I is distributive (or right Formula (2) of 3 1, no. 2 shows that then, for every ordered sequence of elements of E and every a E Q, (7) If an action a -.fa is distributive and an equivalence relation R on E is compatible with the law ofcomposition of E and the action a -fa, the quotient action on E/R is distributive. When the law on E is written multiplicatively, we often use the exponential notation xa for a law of action which is distributive with respect to this multipli- cation, so that distributivity is expressed by the identity (xy)" = xaf. If the law 28 on E is written additively, we often use left (resp. right) multiplicative notation q.x (resp. %.a) for a law of action which is distributive with respect to this addition, the distributivity being expressed by the identity a(x + y) = ax + ay (resp. (x + y)a = xa + y~). We may also consider the case where C2 has an internal law, denoted by 7, and the law of action is distributive with respect to the first variable, which means that (8) for all a, p E Q and x E E. Then, by formula (2) of 3 1, no. 2 (a T p) I x = (a I x) T (p I x) (9) for every ordered sequence of elements of Q and all x E E. 5. DISTRIBUTIVITY OF ONE INTERNAL LAW WITH RESPECT TO ANOTHER DEFINITION 7. Let T and I be two internal laws on a set E. The law I is said to be distributive with respect to the law T if x I (y T 2) = (x I y) T (x I Z) (X T y) I z = (x I 2) T (y I Z) (10) (11) for all x, y, z in E. Note that (10) and (11) are equivalent if the law I is commutative. In general, one of the laws is written additively and the other multiplicatively; if multiplication is distributive with respect to addition, then: x.(y + z) = x.y + x.2 (x +y).z = X.Z +y.z Examples. (1) In the set g(E) of subsets of a set E, each of the internal laws n and u is distributive with respect to itself and the other. This follows from formulae of the form A n (B u C) = (A n B) u (A n C) A u (B n C) = (A u B) n (A u C). (2) In Z (and more generally, in any totally ordered set) each of laws sup (3) In Z (*and more generally in any ring*) multiplication is distributive and inf is distributive with respect to the other and with respect to itself. with respect to addition. (4) In N addition and multiplication are distributive with respect to the laws sup and inf. 29 ALGEBRAIC STRUCTURES I GROUPS WITH OPERATORS 8 4.2 $4. GROUPS AND GROUPS WITH OPERATORS 1. GROUPS Recall the following definition (4 2, no. 3, Definition 6). DEFINITION 1. A set with an associative law of composition, possessing an identity ele- ment and under which every element is invertible, is called a group. In other words, a group is a monoid (5 2, no. 1, Definition 1) in which every element is invertible. A law of composition on a set which determines a group structure on it is called a group law. If G and H are two groups, a magma homomorphism of G into H is also called a group homomorphism. Such a homo- morphismfmaps identity element to identity element; for, let e (resp. e') be the identity element of G (resp. H) ; writing the group laws of G and H multipli - catively, e.e = e, whencef(e) .f(e) =f(e) and, multiplying byf(e) -',f(e) =el. Hencefis unital. It then follows from no. 3 of 3 2 thatf(x-I) = f(x) -I for all x E G. Example. In any monoid E the set of invertible elements with the structure induced by that on E is a group. In particular, the set of bijective mappings of a set F onto itself (or set of permutations of F) is a group under the law (f, g) H f o g, called the symmetric group of the set F and denoted by eF. In this paragraph, unless otherwise indicated, the law of composition of a group will always be written multiplicatively and e will denote the identity ele- ment of such a group law. A group G is calledjnite if the underlying set of G is finite; otherwise it is called injnite; the cardinal of a group is called the order of the group. If a law of composition on G determines a group structure on G, so does the opposite law. The mapping of a group G onto itself which associates with each x E G the inverse of x is an isomorphism of G onto the opposite group (3 2, no. 3, Proposition 4). Following our general conventions (Set Theory, 11, 9 3, no. 1)) we shall denote by A-1 the image of a subset A of G under the mapping x H x-l. But it is important to note that, in spite of the analogy of notation, A-l is definitely not the inverse element of A under the law of composition (X, Y) H XY between subsets of G (recall that XY is the set of xy with x E X, y E Y) : the identity element under this law is {e} and the only in- vertible elements of Cp(G) under this law are the sets A consisting of a single element (such an A, moreover, certainly has inverse A-l). The identity (AB) -l = B-'A-l holds for A c G, B c G. A is called a symmetric subset of G if A = A-l. For all A c G, A U A-l, A n A-l and AA-l are sym- metric. 2. GROUPS WITH OPERATORS DEFINITION 2. Let SZ be a set. A group G together with an action of R on G which is distributive with respect to the group law, is called a group with operators in SZ. In what follows xa will denote the composition of CL E SZ and x E G. Distri- butivity is then expressed by the identity (xy)" = xaya. In a group with operators G, each operator defines an endomorphism of the underlying group structure; these endomorphisms will sometimes be called the homotheties of the group with operators G. A group with operators G is called commutative (or Abelian) if its group law is commutative. In what follows a group G will be identified with the group with operators in 0 obtained by giving G the unique action of ia on G. This allows us to consider groups as special cases of groups with operators and to apply to them the defini - tions and results relating to the latter which we shall state. Example. In a commutative group G, written multiplicatively, (xy)" = x"y" for all n E Z (9 2, no. 8, equation (1)) ; the action n H (x H x") of Z on G therefore defines, together with the group law, the structure of a group with operators on G. DEFINITION 3. Let G and G' be groups with operators in R. A homomorphism of groups with operators of G into G' is a homomorphism of the group G into the grout G' such that for all u E R and all x E G. group G which is permutable with all the homotheties of G. Axa) = (f(4)" An endomorphism of the group with operators G is an endomorphism of the As two homotheties of a group with operators G are not necessarily per- mutable, a homothety of G is not in general an endomorphism of the group with operators G. 2 The identity mapping of a group with operators is a homomorphism of groups with operators ; the composition of two homomorphisms of groups with operators is also one. For a mapping to be an isomorphism of groups with operators, it is necessary and sufficient that it be a bijective homomorphism of groups with operators and the inverse mapping is then an isomorphism of groups with operators. 30 31 1 I ALGEBRAIC STRUCTURES More generally, let G (resp. G) be a group with operators in R (resp. Q'). Let (p be a mapping of R into R'. A +-homomorphism of G into G' is a homo - morphism of the group G into the group G' such that f(x") = (f for all a 6 R and all x E G. In the rest of this paragraph we shall be given a set R. Unless otherwise mentioned the groups with operators considered will admit R as set ofoperators. SUBGROUPS 5 4.3 xx-l = e is an element of H. For every element x of H, x-l = ex-l belongs to H; hence the relations x E H, y E H imply x(y-') -l = xy E H. Clearly (a) implies (d). We show that (d) implies (a) : the canonical injection ofH into G is a group homomorphism; hence e E H and the relation x E H implies x - E H (no. 1). 3. SUBGROUPS DEFINITION 4. Let G be a group with operators. A stable subgroup ofG is a subset H of G with the following properties : (i) e EH; (ii) x, y E H implies xy E H; (iii) x E H implies x-l E H; (iv) x E H and a E R imply xu E H. If H is a stable subgroup of G, the structure induced on H by the structure of a group with operators on G is the structure of a group with operators and the canonical injection of H into G is a homomorphism of groups with operators. (no. 2), which is a subset of G satisfying conditions (i), (ii), (iii) of Definition 4, is called a sub- group of G. When we speak of a subgroup of a group of operators we shall always mean a subgroup of the underlying group of G. A subgroup of a group with operators G is not necessarily a stable subgroup of G. Let G be a group. A stable subgroup of G with the action of Example (1). Let I: be a species of structure (Set Theory, IV, 9 1, no. 4) and S a structure of species C on a set E (loc. cit.). The set of automorphisms of S is a subgroup of 6*. PROPOSITION 1. Let G be a group with operators and H a subset of G which is stable under the homotheties of G. The following conditions are equivalent: (a) H is a stable subgroup of G. (b) H is non-empty and the relations x E H, y E H imply xy E H and x-l E H. (c) H is non-empty and the relations x E H, y E H imply xy-l E H. (d) H is stable under the law on G and the law of comflosition induced on H by the law of composition on G is a group law. Clearly (a) implies (b). We show that (b) implies (a). It suffices to show that H contains the identity element of G. As the subset H is non-empty, let x E H. Then x-1 E H and e = xx-l E H. Clearly (b) implies (c). We show that (c) implies (b). First of all since H is non-empty it contains an element x. Hence 32 Remarks. (1) Similarly it can be shown that condition (b) is equivalent to the condition (c') H # 0 and the relations x E H and y E H imply y-lx E H. (2) For every subgroup H of G there are the following relations H.H = H and H-I = H. (1) F0rH.H c HandH-' C Hby(b).AscEH,H.H 3 e.H = Handtaking inverses transforms the inclusion H-' c H into H c H-l, whence formulae (1). IfH is a stable subgroup of G and K is a stable subgroup of H, clearly K is a stable subgroup of G. The set {e} is the smallest stable subgroup of G. The intersection of a family of stable subgroups of G is a stable subgroup. There is therefore a smallest stable subgroup H of G containing a given subset X of G; it is called the stable subgroup generated by X and X is called a generating system (or generating set) of H. PRoPosInoN 2. Let X be a non-empty subset ofa group with operators G and fi the stable subset under the action of R on G generated by X. The stable subgroup generated by X is the stable subset under the law on G generated by the set Y = 2 u 2-l. The latter subset Z is the set of compositions of finite sequences all of whose terms are elements of 2 or inverse of elements of 2: the inverse of such a com- position is a composition of the same form (5 2, no. 3, Corollary 1 to Propo- sition 5) and Z is stable under the action of Q, as is seen by applying 5 3, no. 4, Proposition 1 to the homotheties of G, hence (Proposition 1) Z is a stable subgroup of G. Conversely, every stable subgroup containing X obviously contains Y and hence Z. COROLLARY 1. Let G be a group with operators and X a subset of G which is stable under the action of R. The subgroup generated by X and the stable suvbgroup generated by x coincide. COROLLARY 2. Let G be a group and X a subset of G consisting ofpairwire permut - able elements. The subgroup of G generated by X is commutative. The set Y = X u X-l consists of pairwise permutable elements (3 2, no. 3, Proposition 5) and the law induced on the stable subset generated by Y is commutative (5 1, no. 5, Corollary 2). 33 I ALGEBRAIC STRUCTURES QUOTIENT GROUPS 5 4.4 relation R is then equivalent to x-'y E H. We show that H is a stable subgroup of G. For every operator a, the relation x = e implies xu = ea = e, hence HZ c H and H is stable under the action of R. It suffices to establish (Propo - sition 1) that x E H and y E H imply x-'y E H, that is x = e and y E e imply x z y, which is a consequence of the transitivity of R. Conversely, let H be a stable subgroup of G; the relation x-'y~H is reflexive since x-lx = e E H; it is symmetric since x-ly E H implies y-1x = (x-'y)-'~H; it is transitive, for x-'y~H and y-lz~H imply x-1~ = (x-'y) (y-lz) E H; it is left compatible with the law of composition on G, for x-'y = (zx)-'(zy) for all z E G; finally, for every operator GI, the rela- tion y E xH implies y" E xaHa c xaH and hence the equivalence relation x-ly E H is compatible with the action of R on G. Let G be a group and H a subgroup of G; the relation x-'y E H (resp. yx-1 E H) is also written in the equivalent form y E xH (resp. y E Hx). Every subgroup H of G thus defines two equivalent relations on G, namely y E xH and y E Hx: the equivalent classes under these relations are respectively the sets xH, which are called 14t cosets ofH (or modulo H), and the sets Hx, which are called right cosets ofH (or modulo H). By saturating a subset A c G with respect to these relations (Set Theory, 11, 3 6, no. 4), we obtain respectively the sets AH and HA. The mapping x H x-I transforms left cosets modulo H into right cosets modulo H and conversely. The cardinal of the set of left cosets (mod. H) is called the index of the sub- group H with respect to G and is denoted by (G:H); it is also equal to the cardinal of the set of right cosets. If a subgroup K of G contains H, it is a union of left (or right) cosets of H. Since a left coset of K is obtained from K by left translation, the set of left cosets of H contained in a left coset of K has cardinal independent of the latter. Hence (Set Theory, 111, 5 5, no. 8, Proposition 9) : PROPOSITION 4. Let H and K be two subgroups Ofa group G such that H c K. Then COROLLARY. If G is ajnite group of order g and H is a subgroup of G of order h, then (2) (G:H) = (G:K)(K:H). (3) h.(G:H) = g (in particular, the order and index of H are divisors of the order of G) . Theorem 1 allows us to determine the equivalence relations compatible with the laws on a group with operators G: ifR is such a relation, it is both left and right Compatible with the group law on G and with the action of R. Hence, if H is the class of e (mod. R), H is a stable subgroup such that the relations Y E XH and y E Hx are equivalent (since both are equivalent to R); hence *H = Hx for all x E G. Conversely, if this is so, one or other of the equivalent If G is a group with operators, the stable subgroup generated by a subset of G consisting of pairwise permutable elements is not necessarily commuta - tive. COROLLARY 3. Let f: G + G' be a homomorphism of groups with operators and X a subset of G. The image under f of the stable subgroup of G generated by X is the stable subgroup of G' generated by f (X) . Let X =f(X). Then X' = f (2) and XI-' =f(X-'). Hence f(A u 2-1) = s' u Xt-1. The corollary then follows from 5 1, no. 4, Proposition 1. Example (2). Let G be a group and x an element of G. The subgroup gener - ated by {x} (called more simply the subgroup generated by x) is the set of x", n E Z. The stable subset (under the law on G) generated by {x} is the set of x" where n E N*. These two sets are in general distinct. Thus, in the additive group Z, the subgroup generated by an element x is the set x.Z of xn, n E Z, and the stable subset generated by x is the set of xn, n E N*. These two sets are always distinct if x f. 0. The union of a right directed family of stable subgroups of G is obviously a stable subgroup. It follows that, if P is a subset of G and H a stable subgroup of G not meeting P, the set of stable subgroups of G containing H and not meeting P, ordered by inclusion, is inductive (Set Theory, 111, 3 2, no. 4). Applying Zorn's Lemma (Set Theory, 111, 5 2, no. 4), we obtain the following result: PROPOSITION 3. Let G be a group with operators, P a subset of G and H a stable sub - group G not meeting P. The set oj'stable subgroups of G containing H and not meeting P has a maximal element. 4. QUOTIENT GROUPS THEOREM 1. Let R be an equivalence relation on a group with operators G; if R is 14t (resp. right) compatible (5 3, no. 3) with the group law on G and compatible with the action of Q, the equivalence class 0fe is a stable subgroup H of G and the relation R is equivalent to x-'y E H (resp. yx-' E H). Conversely, ifH is a stable subgroup of G, the relation x-'y E H (resp. yx-l E H) is an equivalence relation which is lejl (resp. right) compatible with thegroup law on G and compatible with the action OfR and under which H is the equivalence class of e. We restrict our attention to the case where the relation R is left compatible with the law on G (the case of a right compatible relation follows by replacing the law on G by the opposite law). The relation y = x (mod. R) is equivalent to x-1y e (mod. R), for y _= x implies x-'y = x- 'x = e and conversely x-1~ e implies y = x(x-'y) = x. If H denotes the equivalence class of e, the 34 35 I ALGEBRAIC STRUCTURES relations y E xH, y E Hx is compatible with the group law, since it is both left and right compatible with this law (5 3, no. 4) and is compatible with the action of R. Since the equation xH = Hx is equivalent to xHx-l = H, we make the following definition: DEFINITION 5. Let G be a group with operators. A stable subgroup H ofG is called a normal (or invariant) stable subgroup ofG ifxHx-' = H for all x E G. If R = m, a normal stable subgroup of G is called a normal (or invariant) subgroup of G. In a commutative group every subgroup is normal. To verify that a stable subgroup H is normal, it suffices to show that xHx-1 c H for all x E G; for if so then x-'Hx c H for all x E G, that is H c xHx-1, and hence H = xHx-l. Let H be a normal stable subgroup of G and R the equivalence relation y E xH defined by H; on the quotient set G/R, the internal law, the quotient by R of the law of the group G, is associative; the class of e is the identity element of this quotient law; the classes of two inverse elements in G are inverses under the quotient law and the action of R, the quotient by R of the action of Q on G, is distributive with respect to the internal law on G/R (5 3, no. 5). Hence, summarizing the results obtained: THEOREM 2. Let G be a group with operators. For an equivalence relation R on G to be compatible with the group law and the action of R, it is necessary and suficient that it be of the form x-'y E H, where H is a normal stable subgroup of G (the relation x-ly E H being moreover equivalent to yx-' E H for such a subgroup). The law of composition on G/R the quotient of that on G and the action of R on G/R the quotient of that of R on G by such a relation R give G/R the structure ofa group with operators, called the quotient structure, and the canonical mapping of the passage to the quotient is a homomorphism of groups with operators. DEFINITION 6. The quotient of a group with operators G by the equivalence relation dtjined by a normal subgroup H of G, with the quotient structure, is called the quotient group with operators of G by H and is denoted by G/H. The canonical mapping G -+ G/H is called a canonical homomorphism Let G be a group and H a normal subgroup of G. The quotient G/H, with its group structure, is called the quotient group of G by H. For a mapping from G/H to a group with operators to be a homomorphism of groups with operators, it is necessary and sufficient that its composition with the canonical mapping of G onto G/H be one: this justifies the name "quotient group " (Set Theory, IV, 5 2, no. 6). The equivalence relation defined by a normal stable subgroup of G is denoted by x = y (mod. H) or x = y(H). DECOMPOSITION OF A HOMOMORPHISM 5 4.5 PROPOSITION 5. Let f: G + G' be a homomorphism of groups with operators and H and H' normal stable subgroups of G and G' respectively such that f (H) c H'. The mapping f is compatible with the equivalence relations dejned by H and H'. Let x: G -+ G/H and x': G' -+ G'/H' be the canonical homomorphism. The mapping J: G/H --f G'/H' derived from f by passing to the quotients is a homomorphism. If x = y (mod. H), then x-ly E H, whence f (4 - XY) = f b- ')f (Y) = f (x- 'Y) Ef (H) = H' and hence f (x) = f (y) (mod. HI). The second assertion follows from the univer - sal property of quotient laws (5 1, no. 6). Remarks. (1) If A is any subset of a group G and H is a normal subgroup of G, then AH = HA; this set is obtained by saturating A with respect to the relation x sz y (mod. H). (2) If H is a normal subgroup of G of finite index, the quotient group G/H is a finite group of order (G : H) . Note that if H is a normal subgroup of a group G and K is a normal subgroup Let G be a group with operators. The intersection of every family of normal stable subgroups of G is a normal stable subgroup. Hence, for every subset X of G, there exists a smallest normal stable subgroup containing X, called the normal stable subgroup generated by X. In a group with operators G, the stable subgroups G and {e} are normal. DEFINITION 7. A group with operators G is called simple ifG # {e} and there exists no normal stable subgroup of G other than G and {e}. 5. DECOMPOSITION OF A HOMOMORPHISM ~OPOSITION 6. Let G be a group with operators and G' a magma with an action by Q, wdtm exponentially. Let f: G -+ G' be a homomorphism of the magna G into the magma G' such that, for all tc E R and all x E G, f (x") =f(x)%. Then f (G) is a stable subset ofG' under the law on G' and the action of R; the set f (G) with the induced laws is a group with operators and the mapping x H f (x) of G into f (G) is a homo- morphism of groups with operators. 2 of H, K is not necessarily a normal subgroup of G (I, 3 5, Exercise 10). By virtue of 3 1, no. 4, Proposition 1, f (G) is a stable subset of G' under the internal law on G'. For every element XEG and for every operator a, f(~)~ =f(x") E f (G) and therefore f (G) is stable under the action of R on G. Writing the internal law of G' multiplicatively, (.mf(Y))f(Z> = f (XYIf(4 = f ((2Y)Z) = f (4YZ)) = f (X)f(Y4 = fW(f(Y)f (z>) for all elements x, y, z in G; therefore the induced law onf(G) is associative. 36 37 I ALGEBRAIC STRUCTURES Let e be the identity element of G. Its imagef(e) is an identity element of f (G) (9 2, no. 1). Every element off (G) is invertible inJ'(G) (6 2, no. 3). Therefore the law induced on f (G) by the internal law on G' is a group law. For all elements x and y in G and every operator a, (f (x)f(y))" = (f (XY))" =f =f =f (4f (Y") = (.f(x))*(f (Y))" which shows that the action of l2 is distributive with respect to the group law on f (G). Therefore f (G) with the induced laws is a group with operators and clearly the mapping x - f (x) is a homomorphism of groups with operators. DEFINITION 8. Let f: G -+ G' be a homomorphism of groups with operators. The inverse image of the identity element of G' is called the kernel off. The kernel off is often denoted by Ker( f) and the imagef(G) offis some- times denoted by Im (f) . THEOREM 3. Letf: G -+ G' be a homomorphism of groups with operators. (a) Ker( f) is a normal stable subgroup of G; (b) Im( f) is a stable subgroup ofG'; (c) the mapping f is compatible with the equiualence relation dejined on G by Ker( f) ; (d) the mapping3 G/Ker( f) + Im( f) derived from f by passing to the quotient is an isomorphism Ofgroups with operators; (e) f = t of0 x, where t is the canonical injection of Im( f) into G' and x is the canonical homomorphism of G onto G/Ker( f) . Assertion (b) follows from Proposition 6. The equivalence relation f (x) = f(y) on G is compatible with the group with operators structure on G. By Theorem 2 (no. 4), it is therefore of the form y E xH, where H is a normal stable subgroup of G and H is the class of the identity element, whence H = Ker(f). Assertions (a), (c) and (d) then follow. Assertion (e) is obvious (Set Theory, 11, 3 6, no. 5). SUBGROUPS OF A QUOTIENT GROUP 4 4.6 6. SUBGROUPS OF A QUOTIENT GROUP PROPOSITION 7. Let G and H be two groups with operators, f a homomorphism of G into H and N the kernel off. (a) Let H' be a stable subgroup of H. The inverse image G' = f (H') is a stable subgroup ofG and G' is normal in G ifH' is normal in H. Further, N is a normal sub- group of G'. Iff is surjective, then H' = f (G') and f dtjines an isomorphism of G'/N onto H' on passing to the quotient. (b) Let G' be a stable subgroup ofG. The image H' = f (G') is a stable subgroup ofH and f (H') = G'N = NG'. Inpartieular, f (H') = G' ifandonly ifN c G'. Iff is surjective and G' is normal in G, then H' is normal in H. (a) Let x and y be in G' and a E 0; then f (x) E H' and f (y) E H', whence -1 -1 -1 f (xy-') =f(x)f(y)-' E H', that is xy-l E G'; hence G' is a subgroup of G. Nowf(x") =f(x)" E H', whence x" E G' and therefore G' is stable. Suppose H' is normal in H and let x E G', y E G; then f (x) E H' and f (YXY - l) = f (Y)f (4s (!I) - ' E H' whence yxy-' E G; hence G' is normal in G. For all n E N, f(n) = e E H', whence N c G' ; as N is normal in G, it is normal in G'. Finally, iff is surjective, f( f (A)) = A for every subset A of H, whence H' = f (G') ; the restriction of f to G' is a homomorphism f' of G' onto H' of kernel N, hence f' defines on passing to the quotient an isomorphism of G'/N onto H'. (b) Let a and b be in H' and a in R; there exist x, y in G' with a = f (x) and b = f (y), whence ab-l = f (xy-') E H', hence H' is a subgroup of H which is stable, for aa = f (xu) E H'. Let x E G; then x E f (HI) if and only if f (x) E H' = f (G'), that is if and only if there exists y in G with f (x) =f(y); the relation f (x) = f (y) is equivalent to the existence of n EN with x = yn; finally, XE f (H') is equivalent to XE G'N = NG'. Clearly the relation G' = G'N is equivalent to G' 3 N. Suppose finally that f is surjective and G' is normal in G; let a E H' and b E H; there exist x E G' and y E G with a = f (x) and b = f (y), whence bab-l = f(yxy-') E f (G') = H'. Hence H' is normal in H. COROLLARY 1. Suppose that f is surjective. Let 6 (resp. Q') be the set of stable (resp. normal stable) subgroups of G containing N and 4 (resp. 4') the set of stable (resp. normal stable) subgroups of H, these sets being ordered by inclusion. The mapping G' *> f (G') is an ordered set isomorphism @: Q + 4; the inverse isomorphism 'l!: 4 +- 6 is the mapping H' ++ j'(Hf). Further CD and Y? induce isomorphisms 0': (5' -+ 4' and Y?': 4' -+ 6'. COROLLARY 2. Let f: G -+ H be a homomorphism of groups with operators, N the kernel off, G' a stable subgroup ofG and L a normal stable subgroup of G'. Then LN, L. (G' n N) and f (L) are normal stable subgroups ofGN, G' and f (G') respectively and the three quotient groups with operators GN/LN, G'/L. (G' n N) and f (G')/f(L) are isomorphic. Let H = f (G') and f' denote the homomorphism of G' onto H' which coincides with f on G; the kernel off' is G n N and f'(L) =f(L); by Proposition 7,f'(L) is a normal stable subgroup of H' and -1 -1 -1 f' (f'(L)) = L. (G' n N) is a normal stable subgroup of G'. Let A be the canonical homomorphism of H' onto H/f'(L) =f(G')/f(L) : as A of' is surjective with kernel 38 39 I ALGEBRAIC STRUCTURES THE JORDAN-HdLDER TIIEOREM 4 4.7 it defines an isomorphism of G'/L. (G' n N) ontof(G')/f(L). By Proposition 7, (b), f (H) = G'N; iff" is the homomorphism of G'N onto H' which co- incides with f on G'N, the homomorphism A 0 f" of G'N onto f (G')/f(L) is ' sur- jective with kernel f (f (L)) = LN; this proves that LN is a normal stable sub- group of G'N and that h 0 f" defines an isomorphism of G'N/LN onto COROLLARY 3. Let f: G -+ H be a homomorphism of groups with operators, N its kernel, X a subset of G such that f (X) generates H and Y a subset ofN which generates N. Then X u Y generates N. Let G be the stable subgroup of G generated by X u Y. As Y c G', N c G'. -1 -1 f (G') If(L). -1 As f(X) c f(G'), f(G') = H, whence G' = f (H) = G. Remark. In the notation of Proposition 7, the fact that the inverse image of a subgroup of H is a subgroup of G follows from the following more general fact. IfA and B are subsets ofH and f is surjective, then jl(A-l) = >'(A)-l. Obviously jl(A). ?'(B) c ?'(A.B) ; on the other hand, if z E fiA.B), there exists a E A and b E B such that f (2) = ab; as f is surjective, there exists x E G such that f(x) = a; writing y = x-lz, f(y) = a-'f(z) = b and t = xy, whence z E f (A). f (B). The relation x E f (A-l) is equivalent to f (x) E A- ', hence to f (x-l) E A, that is to x-l E jl(A) and finally to x E f (A) -l. PROPOSITION 8. Let G be a group with operators and A and B two stable subgroups of G. Suppose that the relations a E A and b E B imply aba-I E B *(in other words, A normalizes B)*. Then AB = BA is a stable subgroup of G, A n B is a normal stable subgroup ofA and B is a normal stable subgroup of AB. The canonical injection of A into AB dejnes on passing to the quotient an isomorphism of A/(A n B) onto AB/B. -1 -1 -1 -1 The formulae (ab) (a'b') = aa'(a'- lba' . b') (ab)-l = a-l(ab-la-l) (ab)" = aubu for a, a' E A, b, b' E B and every operator a on G, show that AB is a stable subgroup of G. Let a E A and x E A n B; then axa-l E B by the hypotheses made on A and B and clearly axa-' belongs to A, hence A n B is normal in A. Let a E A and b, b' be in B; the formula (ab)b'(ab) -' = a(bb'b-l)a-l shows that B is normal in AB. Let I$ be the restriction to A of the canonical homomorphism ofAB onto AB/B; thcn +(a) = aB and hence the kernel of 4 is equal to A n B. Clearly + is surjcctive and hcncc defines an isomorphism of A/(A n B) onto AB/B. THEOREM 4. Let G be a group wilh operators and N a normal stable subgroup of G. (a) The mapping G' H G'/N is a bijection of the set ofstable subgroups ofG con- taining N onto the set of stable subgroups of G/N. (b) Let G' be a stable subgroup ofG containing N. For G'/N fa be normal in G/N, it is necessary and su@cient that G' be normal in G and the groups G/G' and (G/N)/(G'/N) are then isomorphic. (c) Let G' be a stable subgroup of G. Then G'N is a stable subgroup of G and N is normal in G". Further G' n N is normal in G' and the groups G'/(G' n N) and GN/N are isomorphic. Let f denote the canonical homomorphism of G onto G/N. For all x E G, f(x) E xN; therefore, f (G') = G'/N for every subgroup G' of G containing N. f is surjective, assertion (a) follows from Corollary 1 to Proposition 7; similarly for the equivalence "G' normal " o "G'/N normal ". Suppose that G' is a normal stable subgroup of G containing N. By no. 4, Proposition 5 applied to Id,, there exists a homomorphism u of G/N into G/G' defined by u(xN) = xG' for all x E G. It is immediate that u is surjective with kernel G'/N whence the desired isomorphism of (G/N)/(G'/N) onto G/G'. Finally, (c) follows immediately from Proposition 8. 7. THE JORDAN -HOLDER THEOREM DEFINITION 9. A composition series of a group with operators G is a finite sequence (Gt)oGt<n of stable subgroups of G, with Go = G and G, = {e} and such that Gi + is a normal subgroup of Gi for 0 < i < n - 1. The quotients GJG, + I are called the quotients ofthe series. A composition series C' is said 10 be finer than a composition series C ifC is a series taken -from C'. ,,, are respectively conlposition series of two groups with operators G and H, they are said to be equivalent ifm = n and there exists a permutation 4 of the interval (0, n - 1) of N, such that the groups with operators Gi/G,+ and Ha&-&,(t) + are isomorphic for all i. If (Gt) ,, and (H,) Note that in general a series taken from a composition series (G,) is not a composition series, since forj > i + 1, G, is not in general a normal sub- group of G,. 5 (Schreier). Given two composition series XI, C, of a group with operators GY there exist two equivalent conlposition series Xi, Eh, finer respectively than XI and Let XI = (&)o<t<n and C., = (K,)o<,cp be the two given composition =ries with respectively n + 1 and p + 1 terms; we shall see that the com- Position series Xi can be formed by inserting p - 1 subgroups Hi,, 41 2 z,. 40 I ALGEBRAIC STRUCTURES (1 < j < p - 1) between Hi and Hi+, for 0 < i < n - 1 and the series 2; by inserting n - 1 subgroups KJ, (1 < i < n - 1) between K, and K,+ for 0 < j < p - 1 ; thus two series of pn + 1 stable subgroups of G will be obtained ; by choosing suitably the inserted stable subgroups, we shall show that these series are equivalent composition series. To this end note that Hi n K, is a stable subgroup of Hi and of K, and hence (Theorem 4) Hi+, . (H, n Kj) is a stable subgroup of Hi containing Hi+, and K,+,. (Hi n K,) is a stable subgroup of K, containing K,+,. If we write H;,, = Hi+, . (H, n Kj) and K;,, = K,+ (H, n Kj), Hi,,+ , is a stable sub- group of Hi,, (0 < j < p - 1) and K;,*,., is a stable subgroup of K;,( (0 < i < n - 1). Moreover Hi,o = Hi, Hisp = Hi+,, K;,o = K, and K;,p = K,+l. To show the theorem, it suffices to show that Hi,,+l (resp. K;, , + ,) is a normal stable subgroup of HI., (resp. K;, i) and that the quotient groups Hi, ,/Hi, ,+ , and K;, JK;, , + , are isomorphic (0 < i < n - 1, 0 < j < p - 1). This follows from the following lemma by taking H = Hi, H’ = Hi+,, K = Kj, K’ = K,+1. Lemma 1 (Zassenhaus). Let H and K be two stable subgroups of a group with operators G and H’ and K’ normal stable subgroups of H and K respectively; then H’. (H n K’) is a normal stable subgroup of H’. (H n K), K’. (K n H’) is a normal stable subgrouf of K’. (K n H) and the quotient groups with operators (H’. (H n K))/(H’. (H n K’)) and (K’. (K n H))/(K’. (K n H’)) are iso- morphic. By Theorem 4, H’ n K = H’ n (H n K) is a normal stable subgroup of H n K; similarly K’ n H is a normal stable subgroup of K n H; hence (no. 6, Corollary 2) (H’ n K) (K’ n H) is a normal stable subgroup of H n K. By Theorem 4 applied to the group H, H’ . (H’ n K) . (K’ n H) = H’ . (H n K) (H’. (H n K))/(H’. (H n K)) (H n K)/((H‘ n K). (K’ n H)). is a normal stable subgroup of H’. (H n K) and the quotient group is isomorphic to In the last quotient, H and H on the one hand and K and K‘ on the other appear symmetrically; permuting them the stated result is obtained. DEFINITION 10. A Jordan -Holder series of a group with operators G is a strictly decreasing decomposition series I: such that there exists no strictly decreasing decomposition series distinct from C andjner than C. PROPOSITION 9. For a decomposition series of G to be a Jordan -Holder series it is necessary and suficient that all the quotients ofthe series be simple. 3 4.7 THE JORDAN -HOLDER THEOREM A decomposition series is strictly decreasing if and only if none of its succes - sive quotients is reduced to the identity element. If a strictly decreasing com- position series C is not a Jordan-Holder series, there exists a strictly decreasing composition series C‘ which is finer than C and distinct from C. There are there - fore two consecutive terms G,, G, ,. , of C which are not consecutive in C’; let H be the first term which follows G, in C’; H is a normal stable subgroup of G,, containing G,+, and distinct from the latter; hence H/Gitl is a normal stable subgroup of Gi/G,+,, distinct from the latter and from the identity element; therefore Gi/G, ,. , is not simple. Conversely, if Z is a strictly decreasing com- position series one of whose quotients G,/G, ,. , is not simple, this quotient con- tains a normal stable subgroup other than itself and {e}, whose inverse image in Gi is a normal stable subgroup H of G,, distinct from G, and G, ,. , (Theorem 4) ; it suffices to insert H between G, and Gi+, to obtain a strictly decreasing composition series distinct from C and finer than C. THEOREM 6 (Jordan-Holder). Two Jordan -Holder series of a group with operators are equivalent. I Let C,, C, be two Jordan -Holder series of a group with operators G; by applying Theorem 5 two equivalent composition series C;, Ck are obtained which are respectively finer than C, and X2; since the latter are Jordan - Holder series, C; is identical with Cl or is derived from it by repeating certain terms; the series of quotients of C; is derived from that for 2, by inserting a number of terms isomorphic to the group {e}; since C, is strictly decreasing, the series of quotients of Z1 is derived from that of 2; by suppressing in the latter all the terms isomorphic to {e}. Similarly for C, and 2;. As the series of quotients of differ (up to isomorphism) only in the order of the terms, the same is true of El and C2; the theorem is proved. ‘COROLLARY . Let G be a group with operators in which there exists a Jordan -Holder series. If C is any strictly decreasing composition series of G, there exists a Jordan- Holder seriesjner than C. Let I;, be a Jordan-Holder series of G; by Theorem 5, there exist two equivalent composition series, C’ and Ch respectively finer than C and C,; the argument of Theorem 6 shows that, by suppressing from 2’ the repetitions, a sequence C“ is obtained which is equivalent to C, and hence a Jordan-Holder series, since all its quotients are simple (Proposition 9). As I: is strictly decreas- hi?, z” is finer than C, whence the corollary. and Remark. A group with operators does not always possess a Jordan -Holder series; an example is given by the additive group Z of rational integers: the sequence (2”. Z), a is a strictly decreasing infinite sequence of (normal) subgroups of Z; for all p, the first p terms of this sequence form with the 42 43 I ALGEBRAIC STRUCTURES group {0} a strictly decreasing composition series; if there existed a Jordan - Holder series for Z, it would have at least p f 1 terms, by the Corollary to Theorem 6; absurd, since p is arbitrary. On the other hand, there exists a Jordan -Holder series in every Jinite group with operators G: if G # {e}, among the normal stable subgroups of G distinct from G, let HI be a maximal subgroup; similarly define H,,, by induction as a maximal element in the set of normal subgroups of H, dis- tinct from H,, when H, # {e}; the sequence of orders of the H, is strictly decreasing, hence there exists n such that H, = {e} and the sequence con- sisting of G and the H, (I < i < n) is by its formation a Jordan -Holder series. DEFINITION 1 1. Let G be a group with operators; the length of G is the upper bound of the integers n such that there exists a strictly decreasing composition series ofG (Gi) 9i ,,. If G admits a Jordan -Holder series, the length of G is the number of successive quotients of this series, as follows from the Corollary to Theorem 6. If G does not admit a Jordan -Holder series, its length is infinite; by Proposition 9, for every strictly decreasing series of G there exists a strictly finer strictly decreasing series. The group consisting of the identity element is the only group with operators of length zero. A group with operators is simple if and only if it is of length 1. Let G be a group with operators, H a normal stable subgroup of G, K the quotient G/H and x: G -+ K the canonical homomorphism. Let C' = (HJost<n be a decomposition series of H and C" = (K,)oGj<p be a composition series of K. Writing GI = kl(Ki) for 0 < i < p and Gi = Hi-p for p < i < n + $, a composition series 2 = (Gi)oci~n+P of G is obtained. The sequence of quo- tients of X is obtained by juxtaposing the sequence of quotients of 2;" and the sequence of quotients of X'. If 2;' and X"are Jordan -Holder series,X is a Jordan - Holder series of G, by Proposition 9. If H or K admits composition series of arbitrary length, so does G. We have proved : PROPOSITION 10. Let G be a group with operators and H a normal stable subgroup gfc. The length of G is the sum of the lengths OfH and G/H. COROLLAR Y. Let G be a group with operators and (GJ0 <, <,, a composition series of G. The length of G is the sum of the lengths ofthe Gi/Gl+l, 0 < i < n - 1. If G and G' are isomorphic groups with operators and G admits a Jordan - Holder series, so does G' and the Jordan -Holder series of G and G' are equiva - /I lent. However, non -isomorphic groups can have equivalent Jordan -Holder series; such is true for 2/42 and (Z/2Z) x (Z/2Z), cf. no. 10. 5 4.8 PRODVCTS AND FIBRE PRODUCTS 8. PRODUCTS AND FIBRE PRODUCTS Let (Gi)raI be a family of groups with operators. Let G be the product monoid of the Gi. Consider the action of 0 on G defined by ((Xi)ieI)' = (xT)isI (a E Q, x, E Gt). With this structure G is a group with operators. For all i E I, the projection mapping pr,: G -+ G, is a homomorphism of groups with operators. DEFINITION 12. The group with operators G = n G, defined above is the product projection homomorphisms. A particular case of the product of gsoups with operators is the group GE consisting of the mappings of a set E into a group with operators G, the laws being defined by : group with operators of the G,. The mappings {€I pr,: G -+ G, are called the Let (bl: H -+ Gi), be a family of homomorphisms of groups with operators. The mapping h c--f (+,(h))iGl of H into n G, is a homomorphism of groups 161 with operators. It is the only homomorphism @: H -+ n G, satisfvinp J -0 1€I * pr, 0 @ = +i for all i. This justifies the name "product group with operators " (Set Theory, IV, $2, no. 4). Let (+{: H, -+ Gi), be a family of homomorphisms of groups with operators. G, is a homomor- The mapping 5 +i: (hi)ieI (+t(hi))ieI of n Hi into phism of groups with operators. i€I ~OPOSITION 11. Let (&: H, -+ GJieI be a family of homomorphisms Ofgroups with OPemtors and let CD = n +i. Then: iEI (a) Ker (CD) = n Ker (+,) ; in particular, if all the +, are injective, CD is injective. (b) Im(@) = n 1EI Im(+i); in particular, if all the +, are surjective, CD is sur- jectiue. 1EI This is immediate. In particular, let (GJtsr be a family of groups with operators and, for all i, 44 45 I ALGEBRAIC STRUCTURES let H, be a stable (resp. normal stable) subgroup of G,. The product n IEI Hi is a stable (resp. normal stable) subgroup of ,€I n G, and the canonical mapping of n Gi onto n (GJH,) defines when passing to the quotient an isomorphism of ,€I 161 $ 4.9 RESTRICTED SUMS 9. RESTRICTED SUMS Let (G,),,, be a family of groups with operators and, for i E I, let H, be a stable subgroup of G,. The subset of I-I G, consisting of the (xt),,I such that the set of i E I with x, $ H, isfinite is a stable subgroup of n G, equal to n G, if I except for a finite number, H, is a normal stable subgroup of G,, the restricted sum is a normal stable subgroup of the product. When, for all i, the subgroup H, is reduced to the identity element of GI, the direct sum of the G, with respect to the H, is called simply the restricted sum of the G, and is sometimes denoted by n G,. For all i, E I, the mapping Lto: G,, -+ n G,defined by Lto(x) = operators called the canonical injection. G, is identified with the stable subgroup Im(t,). The subgroups G, are normal. For i # j, the elements of G, and G, commute and G, n G, = {e}. The group n G, is generated by the set u G,. PROPOSITION 12. Let (&: G, --+ K)! I be a family of homomorphisms of groups with operators such that, for all i E I and j E I with i # j, x E G,, y E G,, the elements &(x) and +,(y) of K commute; there exists one and only one homomorphism of groups with okerators CD of G, into K such that 4, = CD 0 1, for all i E I. For euery element ,€I is finite. It is called the restricted sum of the G, with respect {€I to the H,. When, iEI for all i where !€I xi, = x and x, = e, if i # i,, is an injective !€I homomorphism ofgroups with ,€I iE1 LEI If @ and @ are solutions to the problem, they coincide on ,; G, and hence on n G,, whence the uniqueness of @. We now show the existence of CD: for every element x = of n G,, let @(x) = n +i(x,) (3 1, no. 5). Clearly @ 0 ti = +* for all i and @ commutes with homotheties; the formula @(.Y) = CD(x)CD(y) follows from tj 1, no. 5, formula (9). DEFINITION 14. Let G be a group with operators and (HJtE1 a family of stable sub - .POUPS of G. G is called the internal restricted sum (or restricted sum) of the family ofsubgroups (HI) ifevery element of H, is permutable with every element ofH, for j # i and the unique homomorphism of a Hi into G whose restriction to each of the H, is the canonical injection is an isomorphism. When I is finite, we also say, by an abuse of language, internal direct product (Or direct product, or product) instead of internal restricted sum. Every stable If1 ,€I (€1 onto n (Gi/H,). For example, let J be a subset of I. The subgroup GJ of 1EI n GL consisting of the (xJiE1 such that x, = e, for i 4 J is a normal stable sub- group. The mapping tJ which associates with x = (x,),,~ the element y = (y,) , ,I such that yi = el for i $ J and y, = x, for i E J, is an isomorphism of n G, onto G,. The mapping pr,- , defines when passing to the quotient an jEJ isomorphism OJ of the quotient group G/GJ onto iE n I-.J G,. The composition pr, 0 tJ is the identity mapping of n G,. G/GJ is often identified with , €1 jfJ n Gi because of OJ and n G, with Gj because of iJ. 1~1-4 teJ If J1 and J2 are disjoint subsets of I, it follows from the definitions that every element of GJ, commutes with every element of GJ2. DEFINITION 13. Let G be a group with operators and (H,) , I a family of normal stable subgroups of G. Let p,: G --+ G/Hi be the canonical homomorphism. G is called the internal product (or product) of the family of quotient groups (G/H,) ;f the homomor - phism g H (pt(g))lEI is an isomorphism of G onto n LEI G/H,. Let G and H be groups with operators and let + and 9 be two homomor- phisms of G into H. The set of elements x in G such that +(x) = +(x) is a stable subgroup of G, called the coincidence group of + and 3. In particular, let : G, -+ H and +, : G, -+ H be homomorphisms of groups with operators; the coincidence group of the homomorphisms +, 0 pr, and +, o pr2 of GI x G2 into H is called thejbreproduct of G, and G, over H relative to and +2. It is denoted by G, xH G, when there is no ambiguity about +1 and +2 and the restrictions p1 and p, of prl and pr2 to G1 xH G, are also called projection homomorphisms. Then +1 opl = +, op2. The elements of G1 xH G2 are the ordered pairs (gl, 9,) E G1 x G, such that +l(gl) = +,(g,). IfJ is a homo- morphism of a group with operators K into G, (i = 1,2) and 0 fl = +, of,, there exist one and only one homomorphism f of K into G, x G, such that J;=fi,offori= 1,2. 46 47 I ALGEBRAIC STRUCTURES subgroup H of G for which there exists a stable subgroup H’ of G such that G is the direct product of H and H‘ is called a direct factor of G. PROPOSITION 13. Let G be a group with operators and a family ofstable sub- groups of G such that every element of H, is permutable with every element of H, for j # i. For G to be the restricted sum of the family of subgroups (Hi)isI, it is necessary and s@cient that every element x of G be expressible uniquely in the form y,, where (yJlEI is a family withjinite support of elements ofG with y, E H, for all i. Obvious. PROPOSITION 14. Let G be a group with operators and (Hi)ieI ajnite family of stable subgroups of G. For G to be the restricted sum of the family of subgroups (Hi), it is necessary and su@cient that each Hi be normal and that G be the product of the quotient groups (G/H‘), where Hi is the subgroup generated by the H, for j # i. The condition is obviously necessary. Conversely, suppose G is the product of the Ki = G/Hi and let G be identified with the product of the K,. Then H, is identified with a subgroup of Ki, so that, for i #j, every element of Hi is permutable with every element of H,; on the other hand, Hi is identified with the product of the K, for j # i, hence Hi = K, for all i and G is the direct product of the Hi. PROPOSITION 15. Let G be a group with operators and (HJ1 a sequence ofnormal stable subgroups of G such that (H,H,,. . .H,) n Hi+, = {e} for 1 < i < n - 1, the set H,H,. . . H, is a normal stable subgroup of G, the restricted sum of the H,. By induction on n, this is immediately reduced to proving the proposition for n = 2. We show first that, if x E H, and y E H,, x and y are permutable; for xyx-ly-l = (xyx-l)y-l = x(yx-’y-’) and hence (sinceH,andH,arenormal) xyx-ly-l E H, n H,, that is xyx-ly-l = e, by the hypothesis. Moreover HIH, is a subset of G which is stable under the homotheties of G. It follows (by no. 3, Proposition 1) that H,H, is a stable subgroup of G and it is immedi - ately verified that this subgroup is normal. Suppose finally that xy = x’y‘, with x E HI, x’ E H,, y E H,, y’ E H,; then x’-lx = y’y-l, hence x’-l E H, n Hz = (e}, x’ = x and similarly y’ = y; H,H, is thus the direct product of H1 and Ha. When the group considered are commutative, the term direct sum is used instead of direct product. 10. MONOGENOUS GROUPS Let a E Z; since aZ is a subgroup of Z, the relation between elements x, y of Z which st&es “there exists z EZ such that x - y = az” is an equivalence 3 4.10 relation, which we agree, once and for all, to write as x = y (mod. a) or x y(a) and which is called congruence modulo a. Replacing a by -a an equiva - lent relation is obtained, hence it may be supposed that a > 0; for a = 0, x y(0) means x = y, hence a relation distinct from equality is obtained only ifa # 0: we shall therefore suppose in what follows that a > 0 unless other - wise indicated. For a > 0, the quotient of Z by the congruence x f y(a), that is the group ZlaZ, is called the group of rational integers modulo a. PROPOSITION 16. Let a be an intqer > 0. The integers r such that 0 < r < a form a system of representatives of the equivalence relation x F y (mod. a) on Z. If x is an integer 2 0, there exist (Set Theory, 111, fj 5, no. 6) integers q and r such that x = aq + r and 0 < r < a and x = r (mod. a). If x is an integer GO, the integer --x is 20 and by the above there exists an integer r such that 0 < r < a and -x = r (mod. a). Writing r‘ = 0 if r = 0 and r‘ = a - r if Y > 0, then MONOGENOUS GROUPS x = -r = r’ (mod. a) and 0 < rf < a. We now show that if 0 < r < r’ < a, then r f r‘ (mod. a). Now r’ - r < nu for n > 1 and rf - r > nu for n < 0, whence r’ - r 6 aZ. COROLLARY . Let a be an integer > 0. The group ZlaZ of rational integers modulo a is a group of order a. PROPOSITION 17. Let H be a subgroup ofZ. There exists one and only one integer a 2 0 such that H = aZ. If H = {0}, then H = OZ. Suppose that H # (03. The subgroup H has an element x # 0. Then x > 0 or -x > 0, and therefore H has elements >O. Let a be the smallest element >O in H. The subgroup aZ generated by a is contained in H; we show that H c aZ. Let y E H. By Proposition 16, there exists an integer r such that y = r (mod. a) and 0 < r < a. A fortiori y = r (mod. H), whence YEH. But this is only possible if r = 0 and therefore YEa. The integer a is unique: if H = {0}, then necessarily a = 0, and if H # {0}, the integer a is the order of Z/H. DEPINITION 15. A group is called monogenous ;fit admits a system ofgenerators con- sisting Of a single element. Ajinite monogenous group is called cyclic. Every monogenous group is commutative (no. 3, Corollary 2 to Proposition 2). Every quotient group of a monogenous group is monogenous (no. 3, Corollary 3 to Proposition 2). The additive group Z is monogenous: it is generated by (1). For every positive integer a, the group Z/aZ is monogenous, for it is a quotient of Z. 48 49 ’ I ALGEBRAIC STRUCTURES PROPOSITION 18. A jinite monogenous group of order a is isomorphic to Z1.Z. An injnite monogenous group is isomorphic to Z. Let G be a monogenous group (written multiplicatively) and x a generator ofG. The identity xmxn = xm+" (Cj 1, no. 3, formula (1)) shows that the mapping n ++ ~n is a homomorphism of Z into G. Its image is a subgroup of G containing x and hence it is G. By no. 5, Theorem 3, the group G is isomorphic to the quotient of Z by a subgroup, which is necessarily of the form aZ (Proposition 17). If a > 0, the group G is finite of order a and if a = 0, the group G is iso- morphic to Z. PROPOSITION 19. Let a be an integer > 0. Let H be a subgroup of ZlaZ, b the order of H and c its index in ZlaZ. Then a = bc, H = cZ1.Z and H is isomorphic to 2/62. Conversely, let b and c be two integers >O such that a = bc. Then aZ c cZ and cZlaZ is a subgroup ofZ/aZ, of order b and index c. a = bc by no. 4, Corollary to Proposition 4. By no. 7, Theorem 4, H is of the form H'/aZ, where H' is a subgroup of Z and Z/H' is isomorphic to (Z/aZ)/H and hence of order c. By Proposition 17 and the Corollary to Proposition 16, H' = CZ and hence H is monogenous. Finally, H is isomorphic to Z/bZ by Proposition 18. Conversely, if a = bc, then aZ c CZ for a E GZ: the quotient group (Z/aZ)/(cZ/aZ) is isomorphic to Z/cZ (no. 7, Theorem 4) and hence of order c (no. 4, Corollary to Proposition 4) and index 6 (no. 4, Corollary to Proposition 4). COROLLARY . Every subgroup of a monogenous group is monogenous. Let a and b be two integers #O. The relation b E aZ is also written: b is a multiple of a, and also a divides b or a is a divisor of b. DEFINITION 16. An integerp > 0 is calledprime ifp # 1 and it admits no divisor > 1 other than p. PROPOSITION 20. An integer p > 0 is prime if and only if the group ZlpZ is a simple group. COROLLARY . Every commutative simple group is cyclic of prime order. Let G be such a group. Then G # {e}; let a # e be an element of G. The sub- group generated by a is normal since G is commutative, it is not reduced to {e} and hence is equal to G. Therefore G is monogenous and hence isomorphic to a group of the form Z/pZ with p > 0, since Z is not simple, and p is necessarily prime. Remark. A finite group G of prime order is necessarily cyclic. G admits no subgroup other than G and {e} and hence it is generated by every element This follows from Proposition 19. # e. 4 4.10 Lemma 2. Let a be an integer >O. By associating with every composition series (Hi) bib , of the group ZIaZ the sequence (s,) bi ,1, Lelhcre si is the order of Hi - JHi, a one-to-one correspondence is obtained between the composition scrics of Z/aZ and the Jinite sequences (si) of integers > 0 such that a = s1 . . . s,. The composition series (HI) Qi , is a Jordan -Holder series if and only ;f the si are prime. If (Hi)osicn is a composition series of ZlaZ, it follows, by induction on n, MONOGENOUS GROUPS n from no. 4, Proposition 4, that a = n (Hi-l:Hi). Conversely, let (si) 9i , be a scquence of integers > 0 such that a = sl. . . s,. If (HJOsisn is a composition series of Z/aZ such that (HI-l:Hi) = si for 1 < i < n, then necessarily ((ZlaZ) : Hi) = IT sj as is seen by induction on i, whence HI = (0 s)Z/aZ (Proposition 19), which shows the injectivity ofthe mapping in question. We now show its surjectivity. Writing H, = (h s)Z/aZ for 0 < i < n, a composition series ofZ/aZ is obtained such that (Hi - : Hi) = s1 (Proposition 19). The second assertion follows from Proposition 20 and no. 7, Proposition 9. i=l 1cjs i j= 1 Let 5)3 denote the set of prime numbers. THEOREM 7 (decomposition into prime factors). Lei a be a strictlypositive integer. There exists one and only one family (vp(a)),,, of integers >O such that the set of @ E with .,(a) # 0 isjnite and (5) As the group Z/aZ is finite, it admits a Jordan-Holder series. Lemma 2 then implies that a is a product of prime integers, whence the existence of the family (v,,(a)) ; further, for every family (vp(a))pE Lp satisfying the conditions of Theorem 7, the integer up(.) is, for all p E '$3, equal to the number of factors of a Jordan- Holder series of Z/aZ isomorphic to Z/pZ (Lemma 2). The uniqueness of the family (up(.)) therefore follows from the Jordan-Holder theorem (no. 7, Theorem 6). COROLLARY. Let a and b be two integers > 0. Then v,(ab) = up(.) + v,(b). For a to divide b, it is necessary and suicient that up(.) < v,(b) for every prime number p. In any group G, if the (monogenous) subgroup generated by an element x E G is of finite order d, x is called an element of order d; the number d is there - fore the least integer > 0 such that xd = e; if the subgroup generated by x is in- finite, xis said to be of infinite order. These definitions, together with Proposition 4 50 51 MONOID OPERATING ON A SET 5 5.1 In the rest of this paragraph, we shall consider, unless otherwise stated, only left M-sets which we shall call simply M-sets. Their law of action will be denoted by left multiplication. Let E be a set. Let G be a group operating on E. For all a in G, the element of EE defined by a is a permutation of E (4 2, no. 3, Example 2). Being given an operation of G on E therefore amounts to being given a homomorphism of G into 6,. In conformity with 3 3, no. 3, we make the following definition: DEFINITION 2. Let M be a monoid and E and E' M-sets. A maptjing f of E inco E' such that, for all x E E and all a E M, f (a.x) = a. f (x) is called an M-set homo - morphism (or M-morphism, or mapping cGmpatible with the operations of 16\ I ALGEBRAIC STRUCTURES (no. 4), imply in particular that in a finite group G the order of every element of G is a divisor of the order of G. PROPOSITION 21. In ajnite group G of order n, xn = e for all x E G. Ifp is the order of x, then n = pq, with q an integer, and hence x" = (xp)q = e. $5. GROUPS OPERATING ON A SET 1. MONOID OPERATING ON A SET DEFINITION 1. Let M be a monoid, with law written mult$licatively and identity element denoted by e, and E a set. An action a H fa of M on E is called a left (resp. right) operation ofM on E iffe = Id, and fao =fa 0 fo (resk.fao = fo 0 fa) for all a, EM. In other words, a left (resp. right) operation of a monoid M on a set E is a monoid homomorphism of M into the monoid EE (resp. the opposite monoid of EE) with composition of mappings. If the law of action corresponding to the action of M is denoted by left (resp. right) multiplication, the fact that this action is a left (resp. right) operation may be expressed by the formulae (1) e.x = x; a. (p.x) = (ap) .x (resp. x.e = x; (..a) .p = x. (ap) for a, p E M and x E E. for a, (3 E M and x E E). Under these conditions, it is also said that M operates on E on the ldt (resp. right) and that the corresponding laws of action are laws of le$ (resp. right) operation of the monoid M on E. Let M be a monoid; a set E with a left (resp. right) operation of M on E is called a left (resp. right) M-set. The monoid M is said to operate on the left (resp. right) faithfully if the mapping a -fa of M into EE is injective. Examples. (1) Let E be a set; the canonical action of EE on E (3 3, no. 1, Example 3) is a left operation. (2) Let M be a monoid. The left (resp. right) action of M on itself derived from the law on M (3 3, no. 3, Example 7) is a left (resp. right) operation of M on itself. When considering this operation, we say that M operates on itself by leJt (resp. right) translation. Let E be a left (resp. right) M-set and Mo the opposite monoid to M. Under the same action, the monoid Mo operates on E on the right (resp. left). The Mo-set obtained is called opposite to the M-set E. The definitions and results relating to left M -sets carry over to right Mo-sets when passing to the opposite structures. LVL) . The identity mapping of an M-set is an M-morphism. The composition of two M -morphisms is an M-morphism. For a mapping of one M-set into another to be an isomorphism, it is necessary and sufficient that it be a bijective M-morphism and the inverse mapping is then an M-morphism. Let (E,),,, be a family of M-sets and let E be the product set of E,. The monoid M operates on E by a. (x,),,, = (a.xJteI and E, with this action, is an M-set; let E' be an M-set ; a mapping f of E' into E is an hl-morphism if and only if pr, ofis an M-niorphism of E' into E, for all i E I. Let E be an M-set and F a stable subsct of E under the action of M; with the induced law, F is an h4-set and the canonical injection F + E is an M-mor- phism. Let E be an M-set and R an equivalence relation on E compatible with the action of M; the quotient E/R with the quotient action is an M-set and the canonical mapping E -+ E/R is an M-morphism. Let +: M --f M' be a monoid homomorphism, E an M-set and E' an M'-set. A mappingfof E into E' such that, for all x E E and a E M, f (a.4 = 444 .f(4 is called a +morphism of E into E' (cf. 5 3, no. 1). Extension ofa law of operation. Given (for example) three sets F,, F,, F,, permuta- f'onsfi,fz,f3 of F,, F,, F3 respectively and an echelon F on the base sets F,, F2, F3 (Set Theory, IV, fj 1, no. l), we can define, proceeding step by step on the construction of the echelon F, a permutation of F called the canonical extension OffI,f2,f3 to F (Set Theory, IV, fj 1, no. 2); we shall denote it by ~+~(f,,f~,f,). Then let G be a group and h, a homomorphism of G into the symmetric group of F, (i = 1,2, 3), in other words an operation of G on F,. The mapping X xF = +F(hl(~), h2(x), hg(x)) is a homomorphism of G into 6,, in other words an operation of G on F, called the extension of h,, h,, h3 to F. Let P be a subset of F such that, for all x E G, xF(P) = P; let xp be the restriction of xF to 52 53 I ALGEBRAIC STRUCTURES p: then the mapping x H x is an operation of G on P, also called the extension INNER AUTOMORPHISMS $ 5.3 of hl, h,, h3 to PI For example, let K and L be two echelons on F1, F,, F,; take F to be the set of subsets of K x L and P to be the set of mappings of K into L, identified with their graphs. If w E P and x E G, xp(w) is the mapping k H xL(w(xil(k))) of^ into L. 2. STABILIZER, FIXER DEFINITION 3. Let M be a monoid operating on a set E and A and B subsets of E. The set of a E M such that UA c B (resp. aA = B) is called the transporter (resp. strict transporter) ofA to B. The transporter (resp. strict transporter) of A to A is called the stabilizer (rest. strict stabilizer) of A. The set of a E M such that aa = a for all a E A is called the fixer of A. An element a of M is said to stabilize (resp. stabilize strictly, resp. fix) a sub- set A of E if a belongs to the stabilizer (resp. strict stabilizer, resp. fixer) of A. A subset P of M is said to stabilize (resp. stabilize strictly, resp. fix) a subset A of E if all the elements of P stabilize (resp. strictly stabilize, resp. fix) A. The fixer of A is contained in the strict stabilizer of A which itself is contained in the stabilizer of A. PROPOSITION 1. Let M be a monoid operating on a set E and A a subset of E. (a) The stabiliter, strict stabilizer andjxer ofA are submonoids of M. (b) Let a be an invertible element of M; ;fa belongs to the strict stabilizer (resp. fixer) of A, so does a-l. Let e be the identity element of M; then ea = a for every element a E A and therefore e belongs to the fixer of A. Let a and p be elements of E which stabilize A. Then (aP)A = a(PA) c aA c A and therefore the stabilizer of A is a submonoid of M. Similarly for the strict stabilizer and fixer of A, whence (a). If aA = A, then A = a-l(aA) = a-lA. If for all a E A, aa = a, then a = a-l(cta) = a-la, whence (b). COROLLARY . Let G be a group operating on a set E and A be a subset of E. The strict stabilizer S and thejxer F of A are subgroups of G and F is a normal subgroup of S. The first assertion follows from the proposition and F is the kernel of the homomorphism of S into 6, associated with the operation of S on A. A group G operates faithfully on a set E if and only if the fixer of E consists of the identity element of G. The fixer of E is the kernel of the given homomor - phism of G into QE; this homomorphism is injective if and only if its kernel consists of the identity element (3 4, no. 5, Theorem 3). Let M be a monoid, E an M-set and a an element of E. The fixer, strict stabilizer and stabilizer of (a} are equal; this monoid is called equally the fixer or stabilizer of a. The fixer of a subset A of E is the intersection of the fixers of 54 the elements of A. a is called an invariant element of E if the fixer of a is the monoid M. M is said to operate trivially on E if every element of E is invariant. PROPOSITION 2. Let G be a group operating on a set E and, for all x E E, let S, be the stabilizer of x. For all a E G, Sax = &,a- l. If SES,, then as~-~(ax) = asx = ax, whence US,^-^ c Sax. As x = a-'(ax), a-'S,,a c S,, whence Sax c d3,a-l. It is seen similarly that, if A and B are two subsets of E and T is the trans- porter (resp. strict transporter) of A to B, then the transporter (resp. strict transporter) of aA to aB is equal to aTa-l. 3. INNER AUTOMORPHISMS Let G be a group. The set Aut(G) of automorphisms of the group G is a sub- group of 6G (§ 4, no. 1, Example 2). PROPOSITION 3. Let G be a group. For every element x of G, the mapping Int(x):y H xyx-l of G into itself is an automorphisrn of G. The mapping Int: x H Int(x) of G into Aut(G) is a group homomorphism, whose kernel is the centre ofG and whose image is a normal subgroup ofAut(G). If x, y and z are elements of G, then (xyx-') (xzx-l) = xyzx-l and hence Int(x) is an endomorphism of G. For x and y elements of G, Int(x) 0 Int(y) = Int(xy): for all z E G, x(yzy-l)x-l = (xy)z(xy) -l. On the other hand, Int(e) is the identity mapping of G. The mapping Int is therefore a monoid homomorphism from G to the monoid End(G) of endomorphisms of the group G. As the ele- ments of G are invertible, the mapping Int takes its values in the set Aut(G) of invertible elements of End(G) (3 2, no. 3). Now xyx-l = y if and only if x and y commute and hence Int(x) is the identity mapping of G if and only if x is a central element. Finally, let a be an automorphism of G and let x E G; then (2) For y E G, Int(a(x)) = a 0 Int(x) 0 a-l. a(x) .y.a(x)-' = a(x).a(a-'(y)) .a(.)-' = a(~.a-'(y).x-l). Hence a.Int(G).a-l c Int(G). DEFINITION 4. Let G be a group and x E G. The automorphism y ++ xyx- is called the inner automorphism ofG dejned by x and is denoted by Int x. For x, y E G, we also write xu = y-lxy = (Inty-l)(x). A subgroup of G is normal if and only if it is stable under all inner auto- morphisms of G (§ 4, no. 4, Definition 5). A subgroup of G is called charac- 55 I ALGEBRAIC STRUCTURES ORBITS 4 5.4 terhtic if it stable under all automorphisms of G. The centre of a group G is a characteristic subgroup (formula (2)). The centre of a group G is not necessarily stable under all endomorphisms of G (Exercise 22). In particular, the centre of a group with operators is not necessarily a stable subgroup. PROPOSITION 4. Let G be a group, H a characteristic (resp. normal) subgroup Cf G and K a characteristic subgroup of H. Then K is a characteristic (resp. normal) subgroup of G. The restriction to H of an automorphism (resp. inner automorphism) of G is an automorphism of H and therefore leaves K invariant. Let G be a group, A c G and b E G. b is said to normalize A if bAb-l = A; b is said to centralize A if, for all a E A, bab-l = a. Let A and B be subsets of G; B is said to normalize (resp. centralize) A if every element of B normalizes (resp. centralizes) A. The set of g E G which normalize (resp. centralize) A is called the normalizer (resp. centralizer) of A (cf. 5 1, no. 5, Definition 9); it is often denoted by NG(A) or simply N(A) (resp. CG(A) or C(A)). It is a subgroup of G. When A is a subgroup of G, NG(A) may be characterized as the largest subgroup of G which contains A and in which A is normal. Remarks. (1) The normalizer (resp. centralizer) of A is the strict stabilizer (resp. fixer) of A when G operates on itself by inner automorphisms. In par- ticular the centralizer is a normal subgroup of the normalizer. (2) The set of elements b E G such that bAb-' c A is a submonoid of G. Even when A is a subgroup of G, this set is not necessarily a subgroup of G (Exercise 27). 2 4. ORBITS DEFINITION 5. Let G be a group, E a G-set and x E G. An element y E E is conjugate to x under the operation ofG if there exists an element a E G such that y = orx. The set of conjugate elements of x is called the orbit of x in E. The relation "y is conjugate to x" is an equivalence relation. For x = ex; if y = m, then x = u-ly; ify = ax and z = py, then z = pax. The orbits are the equivalence classes under this relation. A subset X of E is stable if and only if it is saturated with respect to the rela- tion of conjugation. The mapping a H ccx of G into E is sometimes called the orbital mapping defined by X. It is a G-morphism of G (with the operation of G on itself by left translation) into E. The image G. x of G under this mapping is the orbit of X. G is said to operate freely on E if, for all x E E, the orbital mapping defined by x is injective or also if the mapping (g, x) H (gx, x) of G x E into E x E is injective. Exuin@!es. (1) Let G be a group and consider the operation of G on itself by inner automorphisms. Two elements of G which are conjugatc under this operation are called conjugate undcr inner automorphisms or simply conjugate. The orbits are called conjugacy classes. Similarly, two subsets H and H' of G are called conjugate if there exists an element OLE G such that H' = a.H.cr-', that is if they are conjugate under the extension to P(G) of the operation of G on itself by inner automorphisms. (2) *In the space R", the orbit of a point x under the action of the ortho - gonal group O(n, R) is the Euclidean sphere of radius IJxJJ.* The stabilizers of two conjugate elements of E are conjugate subgroups of G (no, 2, Proposition 2). The quotient set of E under the relation of conjugation is the set of orbits of E; it is sometimes denoted by E/G or G\E. (Sometimes the notation E/G is reserved for the case where E is a right G-set and the notation G\E for the case where E is a left G-set.) Let G be a group operating on a set E on the right. Let H be a normal sub- group of G. The group G operates on E/H on the right, the corresponding law of right action being (xH, g) - xHg = xgH; under this operation, H operates trivially, whence a right operation of G/H on E/H. Let + be the canonical mapping of E/H onto E/G; the inverse images under + of the points of E/G are the orbits of G (or of G/H) in E/H. Hence + defines when passing to the quotient a bijection, called canonical, of (E/H)/G = (E/H)/(G/H) onto E/G. Let G (resp. H) be a group operating on a set E on the left (resp. right). Suppose that the actions of G and H on E commute, that is (g.x).h =g.(x.h) forgEG,xEEandhEH. The action of H on E is also a left operation of the opposite group Ho to H. It then follows from 4 4, no. 9, Proposition 12 that the mapping which asso- ciates with the element (g, h) E G x Ho the mapping x H g . x . h of E into itself is a left operation of G x Ho on E. The orbit of an element x E E under this operation is the set GxH. The set of these orbits is denoted by G\E/H. On the other hand, the operation of G (resp. H) is compatible with the relation of conjugation with respect to the operation of H (resp. G) and the set of orbits G\(E/H) (resp. (G\E)/H) is identified with G\E/H: in the diagram 56 57 I ALGEBRAIC STRUCTURES (where a, p, y, 6, E denote the canonical mappings of taking quotients), yoa=iJop=~. Let G be a group and H a subgroup of G. Consider the right operation of H on G by right translation (no. 1, Example 2). The set of orbits G/H is the set of 14t cosets modulo H; note that G operates on G/H on the lejt by the law (9, xH) H gxH (cf. no. 5). Similarly, the set of right cosets modulo H is the set H\G of orbits of the left operation of H on G by left translation. If K is a subgroup of G containing H and I' is a left (resp. right) coset modulo H, then rK (resp. KI') is a left (resp. right) coset modulo K. The mapping r H I'K (resp. I' H KI') is called the canonical mapping of G/H into G/K (resp. of H\G into K\G). It is surjective. Let G be a group and H and K two subgroups of G. Let H operate on G on the left by Ieft translation and K on the right by right translation; these two operations commute, which allows us to consider the set H\G/K. The elements ofH\G/K are called the double cosets of G modulo H and K. When K = H, we simply say double cosets modulo H. For the canonical mapping of G/H onto H\G/H to be a bijection, it is necessary and sufficient that H be a normal sub - group of G. 5. HOMOGENEOUS SETS DEFINITION 6. Let G be a group. An operation of G on a set E is called transitive if there exists an element x E E whose orbit is E. A G-set E is called homogeneous ifthe operation ofG on E is transitive. It is also said that G operates transitively on E; or that E is a homogeneous set under G. It amounts to the same to say that E is non-empty and that, for all elements x and y of E, there exists an element a E G such that GC. x = y. Example. If E is a G-set, each orbit of E, with the induced operation, is a homogeneous set under G. Let G be a group and H a subgroup of G. Consider the set G/H of left cosets modulo H. The group G operates on G/H on the lejt by (9, xH) H gxH. Let N be the normalizer of H. The group N operates on G/H on the right by (xH, n) ++ xHn = xnH. This operation induces on H the trivial operation and hence, on passing to the quotient, N/H operates on G/H on the right. Let + : (N/H)O + 6G,H be the homomorphism corresponding to this operation. PROPOSITION 5. With the above notation, G/H is a homogeneous G-set. The mapping $ induces an isomorphkm of (N/H)O onto the group of automorphisms of the G-set G/H. The orbit in G/H of the element i = H is G/H, whence the first assertion. We now prove the second. If n E N defines by right translation the identity mapping on G/H, then i.n = i, that is H.n = H, whence n E H. Therefore HOMOGENEOUS SETS 5 5.5 58 N/H operates faithfully on G/H on the right and + is injective. The left operation of G and right operation of N/H on G/H commute and hence the operators of N/H define G-morphisms of G/H into itself, which are necessarily G-automorphisms since they are bijective. Therefore + takes its values in the group 0 of G-automorphisms of G/H. We show that the image of + is @. Let f~ @. By transporting the structure, the stabilizer off (6) in G is equal to the stabilizer of e' and hence to H. Let n E G be such that f (i) = ni. The stabilizer of ni in G is nHn-l (no. 2, Proposition Z), whence nHn-l = H and n E N. For every element xH of G/H, f (xH) = f (x.i) = x. f (i) = xnH = xHn and f co- incides with the mapping defined by n. Remarks. (1) Let G be a group, H a subgroup of G and 4: G ---f B,,, the homomorphism corresponding to the operation of G on G/H. The kernel of 4 is the intersection of the conjugates of H (no. 2, Proposition 2). It is also the largest normal subgroup contained in H (no. 3). In particular, G oper - ates faithfully on G/H if and only if the intersection of the conjugates of H reduces to {e}. (2) Let G be a group and H and K subgroups such that H is a normal subgroup of K. Then K/H operates on the G-set G/H on the right and the canonical mapping of G/H onto G/K defines on passing to the quotient a G-set isomorphism (G/H)/(K/H) -+ G/K (cf. no. 4). PROPOSITION 6. Let G be a group, E a homogeneous G-set, a E E, H the stabilizer Ofa and K a subgroup of G contained in H. There exists one and only one G-morphism f of G/K into E such that f (e. K) = a. IfK = H, f is an isomorphism. Iff is a solution, thenf(x.K) = x.a for all x in G, whence the uniqueness; we show the existence. The orbital mapping defined by a is compatible with the equivalence relation y E xK on G. For, if y = xk, k E K, then y.a = xk.a = x.a. A mapping f is thus derived of G/K into H which satisfies f (x. K) = x. a for all x in G. This mapping is a G-morphism and f (K) = a. This mapping is surjective for its image is a non-empty stable subset of E. Suppose now that K = Handletusshowthatfisinjective.Iff(x.H) = f(y.H),thenx.a = y.a, whencex-ly.a =aandx-ly~H,whencex.H =y.H. THEOREM 1. Let G be a group. (a) Every homogeneous G-set is isomorphic to a homogeneous G-set of the form G/H Where H is a subgroup of G. (b) Let H and H' be two subgroups of G. The G-sets G/H and G/H' are isomorphic if and only if H and H' are conjugate. As a homogeneous G-set is non-empty, assertion (a) follows from Proposition 6. We show (b). Let3 G/H -+ G/H' be a G-set isomorphism. The subgroup H is the stabilizer of H and hence, by transport of structure, the stabilizer of an 59 ALGEBRAIC STRUCTURES I element of G/H'. The subgroups H and H' are therefore conjugate (no. 2, Proposition 2). If H' = aHa-l, H' is the stabilizer of the element a. H of G/H (no. 2, Proposition 2) and hence G/H' is isomorphic to G/H (Proposition 6). Examples. (1) Let E be a non -empty set. The group GE operates transitively on E. If x and y are two elements of E, the mapping T: E -+ E such that T(X) = y, ~(y) = x and ~(z) = z for z # x, y, is a permutation of E. Let a E E. The stabilizer of a is identified with GF, where F = E - (a}. The homogeneous G -set E is thus isomorphic to QE/QF. (2) Let E be a set of n elements and (pi)iEI a finite family of integers > 0 such that Ffi, = n. Let X be the set of partitions (F,),,, of E such that Card(F,) = p, for all i. The group QE operates transitively on X. The stabilizer H of an element (F,),,, of X is canonically isomorphic to n Q,, and is hence of order nfi,!. Applying Theorem 1 and 4 4, no. 4, Corollary to Proposition 4, a new proof is obtained of the fact that (€1 is1 n! Card(X) = -. ,€I nP,! In particular, take I = { 1, 2, . . . , r}, E = { 1, 2, . . . , n}, for 1 < i < z. Let S be the set of ~€6~ such that TI^^ is increasing for 1 < i < r. If (Gl,. . ., G,) E X there exists one and only one T E S which maps (Fl,. . ., F,) to (Gl,. . ., G,). In other words, each left coset of 6~ modulo H meets S in one and only one point. (3) *Let n be an integer 2 1. The orthogonal group O(n, R) operates transitively on the unit sphere S,-l in R". The stabilizer of the point (0,. . ., 0, 1) is identified with the orthogonal group O(n - 1, R). The homogeneous O(n, R)-set S,-l is thus isomorphic to O(n, R)/O(n - 1, R).* Fi={p,+...+P,..,+ l,...,k1 +...+pi} 6. HOMOGENEOUS PRINCIPAL SETS DEFINITION 7. Let G be a group. An operation of G on a set E is called simply transitive if there exists an element x OfE such that the orbital mapping dejined by x is a bijection. A set E, together with a simply transitive lejt action OfG on E, is callcd a left homogeneous principal G-set (or left homogeneous principal set under G). It amounts to the same to say that G operates freely and transitively on E, or also that there exists an element x E E such that the orbital mapping defined by x is an isomorphism of the G-set G (where G operates by left translation) onto E; or also that the two following conditions are satisfied: (i) E is non-empty. (ii) for all elements x and y of E, there exists one and only one element tc E G such that ox = y. 5 5.7 PERMUTATION GROUPS OF A FINITE SET Condition (ii) is also equivalent to the following condition: (iii) the mapping (a, x) H (ax, x) is a bijection of G x E onto E x E. We leave to the reader the task of defining right homogeneous principal G- sets. 60 Examples. (1) Let G operate on itself by left (resp. right) translation. Thus a left (resp. right) homogeneous principal G-set structure is defined on G, which is sometimes denoted by G, (resp. Gd). (2) Let E be a homogeneous set under a commutative group G. If G operates faithfully on E, the latter is a homogeneous principal G-set. (3) Let E and F be two isomorphic sets with structures of the same species and let Isom(E, F) be the set of isomorphisms of E onto F (with the given structures). The group Aut(E) of automorphisms of E (with the given structure) operates on Isom(E, F) on the right by the law (a,f) --fo o and Isom(E, F) is a right homogeneous principal Aut(E)-set. Similarly, the group Aut(F) operates on Isom(E, F) on the left by the law (o,f) H o of and Isom(E, F) is a left homogeneous principal Aut(F)-set. (4) *A homogeneous principal set under the additive group of a vector space is called an ajh space (cf. 11, 5 9, no. l).* The group of automorphisms of the homogeneous principal G-set G, (Example 1) is the group of right translations of G which is identified with Go (no, 5, Proposition 5). Let E be a homogeneous principal G-set and a an element of E. The orbital mapping a, defined by a is an isomorphism of the G-set G, onto E. By transporting the structure an isomorphism +, is derived of Go onto Aut(E). It should be noted that in general depends on a; more 7 precisely, for a E G, +aa = +, 0 Int GO(a) = +a 0 Int(a-l). (3) For, writing for the translation x H xa on G, and whence +aa(X) = w, 0 6, 0 6, 0 6,10 W,l = w, 0 6a-1xa 0 all = qa(a-lxa). 7. PERMUTATION GROUPS OF A FINITE SET IfE is a finite set with n elements, the symmetric group G, (3 4, no. 1) is a finite group of order n!. When E is the interval (1, n) of the set N of natural numbers, the corresponding symmetric group is denoted by 6,; the symmetric group of any set with n elements is isomorphic to G,. 61 PERMUTATION GROUPS OF A FINITE SET 3 5.7 I ALGEBRAIC STRUCTURES DEFINITION 8. Let E be afinite set, < E 6, a permutation of E and the subgroup of 6, generated by <. < is called a cycle ;f, under the operation oft on E, there exists one and only one orbit which is not reduced to a single element. This orbit is called the support of <. Let 1: be a cycle. The support supp(<) of < is the set of x E E such that <(x) # x. The order of a cycle is equal to the cardinal of its support. The subgroup 5 generated by < operates transitively and faithfully on supp(<). As < is com- mutative, supp(<) is a principal set under (no. 6, Example 2) and hence Card(supp(<)) = Card(t). Lemma 1. Let be a family of cycles whose supports are pairwise disjoint. Then the <, are pairwise permutable. Let (I = n <, and Ti be the subgroup generated by (I. Then ~(x) = <,(.)for x E S,, i E I, and (I(.) = x for x # ,g S,. The mapping i H S, is a bijection of1 onto the set of %orbits not consisting ofa single element. Let < and <' be two cycles whose supports are disjoint. If ,€I x $ SUPP(C) " SUPP(Y), then <r(x) = r<(x) = x. If x belongs to the support of <, then r(x) = x and <(x) belongs to the support of <, whence <r(x) = r<(x) = <(x). Similarly when x belongs to the support of <', then <'<(x) = <r(x) = r(x). Hence <r = <'<. Therefore the c, are pairwise permutable and, for i E I and x E S,, Q(X) = t(x) E S,. The mappings (I and <, coincide on S,, hence S, is stable under (I and the subgroup of GS, generated by the restriction of Q to S, operates transitively on S,; therefore S, is a Ti-orbit. As the S, are non-empty and are pairwise disjoint, the mapping i H S, is injective. As $J S, is the set of x such that Q(X) # x, every Ti-orbit not consisting of a single element is one of the S,. PROPOSITION 7. Let E be ajnite set and Q apermutation of E. There exists one and only onejnite set C of cycles satis$jng the two following conditions: (a) the supports of the elements of C are pairwise disjoints; (b) (I = 1: (the elements of C being pairwise permutable by Lemma 1). Let Ti be the subgroup generated by (I and let S be the set of Ti-orbits not con - sisting of a single element. For s E s, write Cs(x) = O(X) if x E s and <,(x) = x if x $ s. For all s E S, '5, is a cycle whose support is s and Q = a c,, as is seen by applying the two sides to any element of E. The uniqueness of C follows from Lemma 1. DEFINITION 9. A cycle of order 2 is called a transposition. position with support {x, y}. whose support is {(~(x), (~(y)}. Hence: Let x and y be two distinct elements of E. Let -rxSy denote the unique trans - For every permutation c of E the permutation c. T~, y. c-l is a transposition (4) ~.Tx.Y'~-l = Ta(x),a(y). Transpositions thus form a conjugacy class in the group 6,. PROPOSITION 8. Let E be ajinite set. The group 6, is generated by the transpositions. For every permutation (I let Fa be the set of x E E such that (I(.) = x. We show by descending induction on p that every permutation Q such that Card(F,) = p is a product of transpositions. Ifp 2 Card(E), the permutation Q is the identity mapping of E; it is the product of the empty family of trans- positions. Ifp < Card(E), suppose that the property is proved for every per- mutation (I' such that Card(F,.) > p. NOW E - Fa # @; let x E E - Fa and y E (I(.). Then y # x and y E E - Fa. Let (I' = T~, y. Q. The set Fa, con- tains Fa and x and hence Card(F,.) > Card(F,) =p. By the induction hypothesis (I' is a product of transpositions and hence (I = T~, y. Q' is a product of transpositions. PROPOSITION 9. Let n be an integer 20. The group 0, is generated by the transpo - By virtue of Proposition 8, it suffices to show that every transposition T~,~, 1 < p < q < n, belongs to the subgroup H generated by the T,,,+~, 1 < i < n - 1. We show this by induction on q - p. For q - p = 1, it is obvious. If q - p > 1, then (formula (4)) T~,~ = T~-~.~T~,~-~T~-~,~. By the induction hypothesis T~, - E H and therefore T~, 'I E H. If Q E 6,, every ordered pair (i,j) of elements of (1, n) such that i < j and o(i) > ~(j) is called an inversion of Q. Let V(Q) denote the number of inversions of (I. Let P be the additive group of mappings from Z" to Z. For J E P and Q E a,, let of be the element of P defined by sitions (Ti,i+1)1,t<n-l. (5) of(z1,. . .> 2,) =f (Za<l), . * *> za(n))* The action of 6, on P thus defined is an operation; for all 0, 7 E 6, and feP,ef= fand (~(.f)) (21, - . *> zn) = ~f(~(1). . . .J +nJ =f(zra(i), . . . , zra(,,)) = ((Tc)f)(Z1,..*, zn). Formula (5) shows that (I( -f) = - offor Q E 6, and f E P. 62 63 I ALGEBRAIC STRUCTURES I Let p be the element of P defined by (6) P(z1, . . .> 2,) = n i<l (Zj - Zi). Lemma 2. p # O and up = (- l)"(') for 5 E 6,. ~(1.2,. . . , n) = n (j - i) # 0 and hence p # 0. On the other hand, if i<l 0 E 6,, then Gp(z1, . . ., 2,) = ~(zo<l,, . . -9 zocnl) = Q (zo(j) - zoci))* Let C be the set of ordered pairs (i, j) such that 1 6 i < n, 1 < j < ti, i < j. A permutation 0 is defined on C by setting O(i,j) = (cr(i), u(j)) if (i,j) is not an inversion, 0(i,j) = (~(j), u(i)) if (i,j) is an inversion. This implies up = ( - 1)""'p. THEOREM 2. Let E be ajnite set. There exists one and only one homomorphism E from 8, to the multiplicative group { - 1, + 1) such that E(T) = - 1 for every transposition 7. The uniqueness follows from Proposition 8. We show the existence. By transporting the structure, it may be assumed that E = (1, n). Using the above notation, let E(G) = (-I)"(@. Then (Lemma 2) a(a'p) = a(~(o')p) = E(Q')(QP) = ~(o')~(o)p. On the other hand, o(a'p) = (5u')p = .(..')p. Asp # -p, it follows that ~(55') = E(U)E(~') and thus E is a homomorphism. We now show that, for every transposition T, E(T) = - 1. V(T,,-~,,) = 1, whence E(T,-~,~) = - 1. As every transposition T is conjugate to .c~-~,~ and the group {- 1, + l} is commutative, E(T) = E(T,,-~,,) = - 1. DEFINITION 10. In the notation of Theorem 2, the number E(O) (also denoted E,,) is called the signature of the permutation 5. The kernel ofthe homomorphism E is called the alternating group of E. o is called euen (resp. odd) if ~(5) = 1 (resp. E(Q) = -1). The alternating group of E is denoted by U,. It is a normal subgroup of 6,. When E = (1, n), it is simply denoted by U,. When the cardinal n of E is 2 2, it is a subgroup of index 2 and hence of order n!/2. It can be shown that, for n = 3 or n 2 5, the group U, is a simple group (cf. Exercise 16). EXTENSIONS 3 6.1 Example. If 5 is a cycle of order d, then E(b) = (-1)d-1. The number of inversions of the permutation (1,2,3 ,..., d)++(d,1,2 ,..., d- 1) is equal to d - 1. 5 6. EXTENSIONS, SOLVABLE GROUPS, NILPOTENT GROUPS Throughout this paragraph, the group laws are, unless expressly mentioned otherwise, written multiplicatively. 1. EXTENSIONS DEFINITION 1. Let F and G be two groups. An extension of G by F is a triple 8 = (E, i, p), where E is a group, i an injective homomorphism of F into E and p a surjective homomorphism of E onto G such that Im(i) = Ker(p). A homomorphism s: G -+ E (resp. r: E + F) such that p 0 s = Id, (resp. r 0 z = Id,) is called a section (resp. retraction) of the extension 8. An extension 8 = (E, i, p) of G by F is often denoted by the diagram 8: F -!+ E -% G, in which i and p are sometimes omitted if no confusion can arise. It is sometimes said simply that the group E is an extension of G by F. For a group E to be an extension of G by F, it is necessary and sufficient that it contain a normal subgroup F' isomorphic to F such that the quotient group E/F' is isomorphic to G. An extension Q: F -!+ E -% G is called central if the image i (F) is contained in the centre of E; this is only possible if F is commutative. Let Q: FA E 4 G and b': F G be two extensions of G by F. A morphism of Q into 8' is a homomorphism u: E -+ E' such that p' 0 u = p and u 0 i = i', or, in other words, such that the following diagram is commutative: E' 64 65 I ALGEBRAIC STRUCTURES PROPO~ITION 1. Let b: F A E -% G and 8‘: F 1; E‘ G be extensions ofG by F. Ifu: E -+ E‘ is a morphism of B into b‘, u is an isomorphism of E onto E‘ and u- is a morphism of&’ into b. Let x E E be such that ~(x) = e. Thenp(x) = p’(u(x)) = e, whence x E i(F). Let y E F be such that x = i(y); then i’(y) = u(i(y)) = e. As i’ is injective, y = e and x = e. Therefore u is injective. By virtue of 5 4, no. 6, Corollary 1 to Proposition 7, u is surjective since u(i(F)) = i’(F). The last assertion is immedi - ate. In other words, the extensions 8 and 8’ are isomorphic if and only if there exists a morphism of d into 8‘. Let F and G be two groups and write E, = F x G; let i: F -+ E, be the canonical injection and p: E, -+ G be the canonical projection. Every exten - sion of G by F isomorphic to the extension 8,: F A E, -% G is called a trivial extemion. PROPOSITION 2. Let 8: F E 5 G be an extension of G by F. The following con- ditions are equivalent: (i) d is a trivial extension; (ii) d has a retraction r; (iii) b has a section s such that s(G) is contained in the centralizer ofi(F). Clearly (i) implies (ii) and (iii). If (ii) holds, the mapping (r, p) : E -+ F x G is a morphism of b into do, whence (i). If (iii) holds, the homomorphism of F x G into E corresponding to (i, s) (5 4, no. 9, Proposition 12) is a mor- phism of do into 8, whence (i). It may be that an extension 8: F -+ E --f G is not trivial and yet the group E is isomorphic to F x G (Exercise 6). DEFINITION 2. Let F and G be two groups and T a homomorphism of G into the automorphism group of F. Write T(g) (f) = “f for g E G and f E F. The set F x G with the law of composition (1) ((f, g), (.I-‘, g’)) ++ (f, s) -5 (f‘, g’) = (f.”f’, is?’) is called the external semi -direct product of G by F relative to T. The external semi-direct product of G by F relative to T is denoted by F x,G. PROPOSITION 3. The external semi -direct product F x, G is a group. The mappings i:F-+F x,G dejined by i(f) = (f, e),p: F x.G-+G defined byp(f, g) = g, and s: G + F x, G defined by s(g) = (e, g) are group homomorphism. The tZiple (F x G, i, p) is an extension ofG by F and s is a section ofthe extension. 66 EXTENSIONS 5 6.1 We have : Now g( f ’ . g’f”) = ”f’ .g”f”, which shows that the law of composition defined by (1) is associative. The element (e, e) is the identity under this law. The ele- ment (f, g) admits as inverse (9-Y-I) g-I). Hence the law of composition on F x G is a group law. The other assertions are immediate. Using the notation of Proposition 3, B, will denote the extension FAF X,G~:G. Let 8’: F -$ E’ -% G be an extension of G by F and s’: G -+ E’ a section of 8’. We define an operation T of G on the group F by: (2) i‘(T(g,J)) = s’(g)i‘(f)s’(g) -I = WS’(d)(i’(f))- PRoposx-rioN 4. With the above notation, there exists one and only one isomorphism u of 8% onto 8‘ such that u 0 s = s‘. (5 g) = (f, e) ., (e, g) = i (f) .7 s(g). Therefore, if u is a solution to the problem, of necessity u(f, g) = i‘( f) .s’(g), whence the uniqueness of u. We prove the existence. We write u(f, g) = i’( f) .s’(g). Then u(f, 9) .U(f’, g’) = i‘(f)s’(g)i’(f’)s’(g’) = i’(f )(s’(‘f)i’(f’)s‘(p.) - ‘)s’(g)s’(g’) = i’(f)i‘(T(&Y)) .s’(g)s’(g’) = i’(f..(g,f‘)) .s‘(gg’) = u((f,,d *7 (f’, g’)). Therefore, u is a homomorphism of F x ‘I G into E’. Obviously u o i = if, @’ 0 u = p and u o s = s’. Remark. The definition of the operation r by formula (2) depends on the extension 8’ and the section s’. When F is commutative, the oDeration t does not depend on s’. For Int(s’(g)) I i’(F) depends thenbnly 0; the coset ~~ . of s’(g) mod. i’(F). More generally, let B: F -j E + G be an extension of G by a commuta - tive group F (it is not assumed that 0 admits a section). The group E operates on F by inner automorphisms, this image is trivial on the image of F and hence defines an operation of G on F. If 8 admits a section, this operation is that defined by formula (2). COROLLARY . Let G be a group and H and K two subgroups ofG such that H is 67 I ALGEBRAIC STRUCTURES normal, H n K = {e} and H.K = G. Let -T be the operatioti of K on H by inner automorphisms of G. The mapping (h, k) H hk is an isomorphism of H x K onto G. Under the hypotheses of this corollary, G is said to be the semi-direct product of K by H. Examples. (1) Let G be a group and E a homogeneous principal G -set; let I? denote the automorphism group of G. Let A be the set of permuta- tionsf of E with the following property: There exists y E r such that f is a y-morphism of E into E (that is, f(gb) = y(g)f(b) for b E E and g E G). The above formula f(gb) = y(g)f(b) shows that iff€ A there exists a unique y E I? such thatfis a y-morphism, we shall denote it by p(f). LetJf' be in A, y = p(f), y' = p(f'). Then, for all b E E and all g E G, which proves thatf' of E A and that p(f' of) = p(f')p(f). On the other hand,f(y-l(g)f-'(b)) = gb, whencef-l(gb) = y-'(g)f-'(b) and f -'€A. Thus A is a subgroup of eE and p is a homomorphism of A into r. The ker- nel ofp is the set Aut,(E) of automorphisms of the G-set E. of Go onto Aut,(E) such that +,(x) (ga) = gxa for all g, x in G. On the other hand, for y E r, let s,(y) be the permutation of E defined by s,(y)(ga) = y(g)a for all g E G; it is immediately verified that s, is a homomorphism of r into A such that p o s, = Id,. Thus Go 2 A -% r is an extension of r by Go and s, is a section of this extension. This extension and this section define an operation of r on Go, sa(I') acting on +,(Go) by inner automorphisms; we write this operation exponentially. We show that this operation is the natural operation (8 3, no. 1, Example 3) : for x, g in G and y E I', (f' of)(gb) = f'(y(g)f(b)) = y'(y(g))f'(f(b)) We fix a E E. We have defined in 5 5, no. 6 an isomorphism (+,(Yx))(ga) = (Sa(Y) O +,(XI 0 s,(y)-')(ga) = (SdY) 0 +a(4)(Y-1(da) = s,(y)(y-l(g)xa) = gywa = +a(y(.))ga 4 6.2 COMMUTATORS (x, y) is used to denote the commutator of x and y. Then obviously (Y, 4 = For x and y to commute it is necessary and sufficient that (x, y) = e. More generally, On the other hand we write XY = !I)- xy = y-'q = x(x,y) = (y, x-')x. (3) As the mapping x H xu is the inner automorphism Int(y-'), (~,y)~ = (xz,~~) for all x, y, z E G. For x, y, z E G, we prove the following relations: (4) (x, yz) = (x, 4. (x, Y)" = (x, 4. (2, (Y, 4) ' (x, Y) (4 bis) (.!I, 4 = (x, 4". (Y, 4 = (x, 4. ((x, 4, Y) ' (Y, 4 (5) by, (Y, 4). (Y", (2, Y)). (ZX, (x, Y)) = e (6) (x, Y.1. (2, "9) ' (Y, 4 = e (6 bis) (.!I, 2). (YZ, ' (zx, Y) = p. = (x, z)(z, (x, Y) - 'k Y) Now (.,yz) = x-'z-'y-'xyz = (x, z)z-'x-'y-1xyz = (x, z)(x,y)' by (3), which proves (4). Formula (4 bis) follows similarly. On the other hand, (XU, (y, 4) = (4 - YZ, Y)(X")(!/, 2) = y - 1x - 'yz - 'y - 'zyy - 'xyy - 1z - 'yz = (yzy-'xy) -1(zxz-1yz). (XY, (y, 2)) = u-lu. (zX, (x, Y)) = w- lu, Then writing u = yzy-'xy, u = zxz-'xy and w = xyx-lzx, we obtain By cyclically permuting x, y, z, we deduce (y", (z, x)) = u-lw and which immediately imply (5). Finally, (6) follows by multiplying together the two sides in the three formulae obtained by cyclically permuting x, y, z in the formula (x,yz) = ~-'z-~y-'xyz = (yzx)-'(xyz), and similarly for (6 bis). If A and B are two subgroups of G, (A, B) denotes the subgroup generated by the commutators (a, b) with a E A and b E B.t Then (A, B) = {e} if and only if A centralizes B. (A, B) c A if and only if B normalizes A. If A and B are normal (rap. characteristic), so is (A, B). Of composition to subsets. - t We here reject the notational convention made in Q 1, no 1 of extending a law whence yx= y(x). Proposition 4 then shows that A is isomorphic to the semidirect product of r = Aut(G) by Go under the natural operation of Aut(G) on GO. Note that the isomorphism which we have constructed depends in general on the choice of the element a E E. (2) *Let A be a commutative ring. The upper triangular group T(n, A) is the semi -direct product of the diagonal subgroup D(n, A) by the upper strict triangular group Tl(n, A).* 2. COMMUTATORS DEFINITION 3. Let G be a group and x and y two elements of G. The element X- 1y- lxy of G is called the commutator of x and y. 68 69 ~ I ALGEBRAIC STRUCTURES PROPOSITION 5. Let A, B, C be three subgroups of G. (i) The subgroup A normalizes the subgroup (A, B). (ii) If the subgroup (B, C) normalizes A, the subgroup (A, (B, C)) is generated by (iii) If A, B and C are normal, then the elements (a, (b, c)) with a E A, b E B and c E C. (A, (B, C)) = (C, (B, A)) * (B, (C, -4)). By (4 bis), for a, a' E A and b E B, (a, b)"' = (aa', b).(a', b)-l, whence (i). Suppose now that (B, C) normalizes A. For a E A, b E B, c E C and XEG, (4) implies and ((6, c), a) E A since (B, C) normalizes A, whence by induction on p the fact that (a, fi 1=1 (b,, ci)), fo; b, E B, ci E C, belongs to the subgroup generated by the elements of the form (a, (6, c)). If finally A, B and C are normal, so are the sub- groups (A, (B, C)), (C, (B, A)) and (B, (C, A)). It therefore suffices by (ii) to show that for all a E A, b E B and c E C. Now by (5), writing ab-' = u (a, (b, 4.4 = (a, 4. (x, ((b, 4, a))(a, (b, 4) (b, '1) (c, (B, (c> A)) (a, (6, 4) = ((ub), (b, c)) = (c", (u, b))-'.(bC, (c, u))-', whence (iii). DEFINITION 4. Let G be a group. The subgroup generated by the commutators ofele- ments of G is called the derived group of G. The derived group of G is thus the subgroup (G, G). It is also denoted by D(G). By an abuse of language, it is sometimes called the commutator group of G although it is in general distinct from the set of commutators of elements of G (Exercise 16). D(G) = (e} if and only if G is commutative. PROPOSITION 6. Let f: G -+ G' be a group homomorphism. Then f (D(G)) c D(G'). Iff is surjective, the homomorphim of D(G) into D(G') the restriction off is surjective. The image under f of a commutator of elements of G is a commutator of elements of G'. Iff is surjective, the image under f of the set of commutators of G is the set of commutators of G'. The proposition thus follows from $4, no. 3, Corollary 3 to Proposition 2. COROLLARY 1. The derived group ofa group G is a characteristic subgroup of G. In particular it is a normal subgroup of G. 70 $ 6.3 COROLLARY 2. Let G be a group. The quotient group G/D(G) is commutative. Let x: G -+ G/D(G) be the canonical homomorphism. Every homomorphism f of G into a Commutative group G' can be expressed uniquely in the form f = fo x, where 3 G/D(G) .j G' is a homomo?phism. Now x(D(G)) = {e}. As x is surjective, it follows that D(G/D(G)) = (e}, whence the first assertion. The second follows from $ 4, no. 4, Proposition 5. COROLLARY 3. Let H be a subgroup of G. The following conditions are equivalent: LOWER CENTRAL SERIES, NILPOTENT GROUPS (i) H 3 D(G); (ii) H is a normal subgroup and G/H is commutative. (ii) (i) by Corollary 2 and (i) 3 (ii) by $4, no. 7, Theorem 4, since every subgroup of a commutative group is normal. COROLLARY 4. Let G be a group and X a subset of G which generates G. The group D(G) is the normal subgroup of G generated by the commutators of elements of X. Let H be the normal subgroup of G generated by the commutators of ele- ments of X and 4: G -+ G/H the canonical homomorphism. The set +(X) generates G/H. The elements of +(X) are pairwise permutable and hence H is commutative (5 4, no. 3, Corollary 2 to Proposition 2). Hence (Corollary 3) H contains D(G). On the other hand, obviously H c D(G). Remarks. (1) Corollary 2 can also be expressed by saying that G/D(G), together with x, is a solution of the universal mapping problem for G, relative to commutative groups and homomorphisms from G to commuta - tive groups. (2) Under the hypotheses of Corollary 4, the subgroup generated by the commutators of elements of X is contained in D(G) but is not in general equal to D(G) (cf. Exercise 15e). Examfiles. (1) If G is a non-commutative simple group, then D(G) = G. Therefore every homomorphism of G into a commutative group is trivial. (2) The derived group of the symmetric group 6, is the alternating group U,. For 9L, is generated by the products of two transpositions; if T = T~, and T' = T~,,~, are two transpositions, let o be a permutation such that o(x') = x and o(y') = y. Then 7' = o-l~o and TT' = 7-l~' = T-~O-~TS is a commutator. Hence U, c D(6,). As G,/%, is commutative, 3, =I D(6,) (Corollary 3). 3. LOW ER CENTRAL SERIES, NILPOTENT GROUPS Let G be a group, H a subgroup of G and K a normal subgroup of G. The image of H in G/K is contained in the centre of G/K if and only if (G, H) c K. DE FINITION 5. Let G be a group. The lower central series of G is the sequence (c"(G))n,l ofsubgroups ofG defined inductively by: C1(G) = G, C"+'(G) = (G, C"(G)). 71 I ALGEBRAIC STRUCTURES Let5 G + G' be a group homomorphism. It is seen, by induction on n, that f (C"(G)) cn(G) and that, if f is surjective, f (C"(G)) = Cn(G). In particular, for all n 2 1, C2(G) is a characteristic (and hence normal) sub - group of G. For all n > 1, C"(G)/C"+l(G) is contained in the centre of Let (Gl, Gz, . . .) be a decreasing sequence of normal subgroups of G such that (1) GI = G; (2) for all i, G,/G,+, is contained in the centre of G/G,+,. Then C'(G) c G,, as is seen by induction on i. G/c~+ yG). Now (7) (Cm(G), Cn(G)) c Cm+"(G). For, ifthis relation is denoted by (F,,,"), it follows from (Fm,n), by no. 2, Prop- osition 5, that (Crn(G), C"+l(G)) = (G, (Crn(G), C"(G))).(C"(G), (G, Crn(G))) c Cm+"+l(G) .(Cmcl(G), Cn(G)). Hence ((Frn, n) and (F, + I. n)) * (Fm, n + 1)- As (Fm, 1) and (F1, n) are obvious (F,,,, ,,) follows by induction. DEFINITION 6. A group G is called nilpotent if there exists an integer n such that C"+ l(G) = {e}. The least integer n such that C"+ '(G) = {e} is called the nilpotency class of a nilpotent group G. If R E N, a group of nilpotency class n is called a nilpotent group of class n. It is sometimes said that the nilpotency class of a group G is finite if G is nil- potent. Examples. (1) A group is nilpotent of class 0 (resp. < 1) if and only if it consists of the identity element (resp. is commutative). (2) *For every commutative ring A and every integer n 2 1, the upper strict triangular group T,(n, A) is nilpotent of class <n - 1 (and exactly of class n - 1 if A # {O}).* (3) Let G be a nilpotent group of class n. Every subgroup (resp. every quotient group) of G is nilpotent of class <n. For, if H is a subgroup of G, then Cn(H) c C"(G). If G' is a quotient group of G and x: G -+ G' is the canonical homomorphism, then C"(G') = z(C"(G)). (4) A finite product of nilpotent groups is nilpotellt. PROPOSITION 7. Let G be a group and n an integer. The following conditiom are equivalent : (a) G is nilpotent of class <n. (b) There exists a series of subgroups of G: G = G1 1 G2 2.. . 2 Gn+l = (8) such that (G, G") c Gk+lfor all k E (1, n). 3 6.3 (c) There exists a subgroup A gG contained in the centre of G such that G/A is nil- (a) 3 (b) : it suffices to take Gk = C"(G). (b) 3 (a): by induction on k, Ck(G) c G". (a) * (c) : it suffices to take A = C"(G). (c) + (a) : let x: G + G/A be the canonical homomorphism; then LOWER CENTRAL SERIES, NILPOTENT GROUPS potent of class <n - 1. n(Cn(G)) = C"(G/A) = {e} and hence Cn(G) c A, whence C"+l(G) = {e}. More briefly: a group is nilpotent of class <n if it can be obtained from the group {e} by n successive central extensions. COROLLARY . A central extension of a nilpotent grouk (by a necessarily commutative group) is nilpotent. PROPOSITION 8. Let G be a nilpotent group of class 6 n and let H be a subgroup of G. There exists a sequence of subgroups G = H1 3 H2 2. *a 2 H"+l = H, such that Hk+' is normal in Hk and Hk/Hk+ is commutatiue for all k < n. Proposition 7 (b) for all k; Gk is normal in G. Write: Choose a sequence (G") of subgroups of G satisfying the conditions of H" = H.Gk. It is necessary to verify that Hk+l is normalized by Hk = H.Gk; as it is normalized by H, it suffices to verify that it is by G". Now, if s E Gk and h E H, shs-l = shs-lh-l.hE(G, Gk).H and (G, Gk) .H is contained in Gk+l.H = Hk+l; hence s.HJC+l.s-l = H+kl, which shows that Hk+l is normal in H". Finally, the canonical homomorphism Gk/Gk + +- Hk/Hk + is obviously surjective; as the first group is commutative, so is the second. COROLLARY 1. Let G be a nilpotent group and H a subgroup of G. If H is distinct from G, the normalizer N,(H) ofH in G is distinct from H. Let k be the largest index such that H" # H. The group Hk normalizes H ad is distinct from H. COROLLARY 2. Let G be a nilpotent group and H a subgroup .f G. If H is distinct ftom G, there exists a normal subgroup N ofG, containing H, distinctfiom G and such that G/N is commutative. kt k be the least index such that Hk # G. The group Hk satisfies the re- 'lui.ed conditions. 72 73 ALGEBRAIC STRUCTURES I 5 6.4 n EN, D"(G) c C2"(G), as is seen by induction on n using formula (7) of no. 3. Let fi G --f G' be a group homomorphism. It is seen, by induction on n, that f (D"(G)) c D"(G) and that, iff is surjective, f (D"(G)) = D"(G'). In particular, for all n E N, D"(G) is a characteristic (and therefore normal) sub- group of G. For all n E N, the group Dn(G)/D"+ '(G) is a commutative normal (but not in general central) subgroup of G/D"+l(G). Let (Go, GI,. . .) be a decreasing sequence of subgroups of G such that: (1) Go = G; (2) for all i, Gi+, is normal in G, and Gl/Gl+, is commutative. Then D'(G) c Gi for all i, as is seen by induction on i. DEFINITION 8. A group G is called solvable ;f there exists an integer n such that Dn(G) = {e}. IfG is a solvable group, the least integer n such that Dn(G) = {e} is called the solvability class of G. A solvable group of solvability class n is called a solvable group of class n. A group is sometimes said to be of finite solvability class if it is solvable. DERIVED SERIES, SOLVABLE GROUPS I Examfles. (1) A group is solvable of class 0 (resp. < 1) if and only if it is reduced to {e} (resp. is commutative). (2) Every nilpotent group of class <2, - 1 is solvable of class <n; this follows from the relation Dn(G) c C2"(G) proved above. (3) Let G be a solvable group of class <n. Every subgroup (resp. quo- tient group) of G is solvable of class <n (proof analogous to that of no. 3, Example 3). (4) If G is a solvable group of class p and F is a solvable group of class q, every extension E of G by F is a solvable group of class <p + q. For, let x: E + G be the projection; then x(Dp(E)) c D*(G) = {e} and therefore Dp(E) c F; it follows that Dp+q(E) = Dq(DP(E)) c Dq(F) = {e}. (5) The symmetric group 6, is solvable if and only if n < 5 (cf. 8 5, Exercises 10 and 16). (6) *If A is a commutative ring, the upper triangular group T(n, A) is soIvable but not in general nilpotent., COROLLARY 3. Let G be a nilpotent group and H a subgroup of G. IfG = H . (G, G), then G = H. Every subgroup N of G which contains H and such that G/N is commutative contains H. (G, G). Corollary 3 thus follows from Corollary 2. Corollary 3 can also be formulated thus: let X be a subset of G. For X to generate G, it is necessary and sufficient that the image of X in G/D(G) generate G/D(G). COROLLARY 4. Let f: G' --f G be a grou$ homomorphism. Suppose that (a) G is nilpotent. (b) The homomorphism fl: G'/(G', G') -+ G/(G, G), derived from f by passing Then f is surjective. This follows from Corollary 3 applied to the subgroup H = f (G'). to the quotients, is surjective. PROPOSITION 9. Let G be a nilpotent group of class < n and let N be a normal subgroup of G. There exists a series of subgroups N=N1 IN~~...IN,+~= (4 such that (G, N") c Nk+l for k = 1, . . . , n. If (G") satisfies condition (b) of Proposition 7, then take Nk = Gk n N. COROLLARY 1. Let G be a nilpotent group, Z the centre of G and N a normal subgroup of G. IfN # {e}, then N n Z # {e}. Let k be the largest index such that N" # {e}. The group Nk is contained in N. On the other hand, (G, N") c Nktl = {e}; hence Nk is contained in the centre Z of G. COROLLARY 2. Let f be a homomorphism from a nilpotent group G to a group G'. If the restriction off to the centre of G is injective, f is injective. This is Corollary 1 applied to Ker( f). 4. DERIVED SERIES, SOLVABLE GROUPS DEFINITION 7. Let G be a group. The derived series ofG is the series (D"(C)),,N dejined inductively by : DO(G) = G; D*+l(G) = D(D"(G)) for n EN. Then D*(G) = C1(G) = G, D1(G) = C2(G) = D(G) = (G, G). For all ~OPOSITION 10. Let G be a group and n an integer. The following conditions are e pivalent : (i) G is solvable of class < n. (ii) There exists a series of normal subgroups of G G = GO 3 G1 I...I Gn = (e} such that the groups Gk/Gk + are commutative. (s) There exists a series ofsubgroups ofG G = GO 3 G1 1...3 G" = e such that, for all k, Gk + 1 is a normal subgroup of Gk and Gk/Gk + is commutative. 74 75 I ALGEBRAIC STRUCTURES (iv) There exists a normal commutative subgroup A ofG such that G/A is solvable ofclass <n - 1. For (i) (ii) it suffices to take Gk equal to Dk(G). (ii) * (iii) trivially. (iii) + (i) for Dk(G) is necessarily contained in Gk. The equivalence of (ii) and (iv) is immediate by induction on n. More briefly: a group is solvable of class <n if it can be obtained by successive extensions of n commutative groups. COROLLARY . Let G be ajnite group and G=GOr>G1~...~G"={e} n .Tordan-Holder series of G For G to be solvable, it is necessary and su$cicnt that the &'-GROUPS 3 6.5 E - EG is a disjoint union of orbits not reduced to a point. The cardinal of such an orbit is a power of! distinct from Po = 1 and hence a multiple COROLLARY. Let G be ap-group. If G is not reduced to e, its centre is not reduced to e. Let G operate on itself by inner automorphisms. The set of fixed points is the ofp. centre Z of G. By Proposition 11, Card(Z) = Card(G) E 0 (mod.@), whence Card(Z) # 1 and Z # {e}. THEOREM 1. Let G be a p-group andp' its order. There exists a sequence of subgroups of G {el G = G1 2 G2 >...I G'+1 = such that (G, Gk) c Gk+l, 1 < k < r, and Gk/Gk+', 1 < k < r, is cyclic of The theorem is true for G = {e}. We prove it by induction on Card(G). Let Z be the centre of G, x # e an element of Z (Corollary to Proposition 11) and p*, s # 0, the order of x. Then xPs-l is an element of order p and therefore Z contains a subgroup G' which is cyclic of orderp. By the induction hypothesis, the group G' = G/G' has a series of subgroups (G'") with the required properties. Let x: G -+ G' be the canonical homomorphism. The sequence of subgroups of G defined by Gk = x-l(G''), 1 < k < r, Grfl = Cel is a solu- order p. k - - .. . quotients Gk/Gk+ be cyclic ofprime order. If the quotients of a composition series of G are cyclic and hence commuta- tive, G is solvable by Proposition 10. Conversely, if G is solvable, the group Gk/Gk+ is, for all k, solvable and simple (3 4, no. 7, Proposition 9). Now, every solvable simple group H is cyclic of prime order. For D(H) is a normal sub- group ofH; D(H) = His impossible for in that case Dk(H) = H for all k; then D(H) = {e) and H is commutative. The corollary then follows from 3 4, no. 10, , I ._ Corollary to Proposition 20. 5. p-GROUPS In this number and the following, the letter p denotes a prime number (3 4, no 10, Proposition 16). DEFINITION 9. Ajnite group whose order is a power of p is called a p-group. Let G be a p-group of order p'. Every divisor of pr is a power of p (4 4, no. 10, Corollary to Theorem 7). Therefore every subgroup and every quotient group of G is a p-group (9 4, no. 4, Corollary to Proposition 4) ; the cardinal of every homogeneous space of G is a power ofp (4 5, no. 5, Theorem 1). An extension of a p-group by a p-group is a p-group. *Examples. (1) A commutative p-group is isomorphic to a product of cyclic groups Z/p"Z (cf. Exercise 19 and also VII, 0 4, no. 7, Proposition 7). (2) Let k be a finite field of characteristic p. The strict triangular group Tl(n, k) is a p-group. (3) The quaternionic group (2 1, +i, +j, +k} is a 2-group (cf. Exercise 4)** PROPOSITION 1 1, Let E be ajnite set and G a @-group operating on E. Let EG denote the set of x E E such that gx = x for all g E G (the fixed points). Then Card(EG) = Card(E) (mod.!). <I tion for- GL/Gk+' is isomorphic to G'k/G'k+r for I < k < r (3 4, no. 7, Theorem 4). COROLLARY . Every p-group is nilpotent. This follows from no. 3, Proposition 7. ~OPOSITION 12. Let G be a p-group and H a subgroup ofG distinct from G. Then: (a) The nornzalizer NG(H) ofH in G is distinctfrom G. (b) There exists a normal subgroup N of G ofindex p in G, which contains H. Assertion (a) follows from no. 3, Corollary 1 to Proposition 8. We prove (b). BY no. 3, Corollary 2 to Proposition 8, there exists a normal subgroup N' of G containing H, distinct from G and such that G/N' is commutative. Let N be a maximal subgroup distinct from G containing N'. Then N is normal (no. 2, corollary 3 to Proposition 6) and G/N is a simple commutative p-group and cyclic of order p (5 4, no. 10, Corollary to Proposition 20). ~ROLLARY. Let G be a p-group. Every subgroup of G of index p is normal. 76 77 I ALGEBRAIC STRUCTURES 6. syLoW SUBGROUPS DEPINI~ON 10. Let G be ajinite group. A Sylow p-subgroup of G is any subgroup P of G satisfying the two following conditions: (a) P is up-group. (b) (G: P) is not a multiple of p. If the order of G is written in the form p'm, where rn is not a multiple of p, conditions (a) and (b) are equivalent to Card(P) = p'. Examples. (1) In the group 8, let < be a cycle of order p. The subgroup generated by < is a Sylow p-subgroup of 8, for p does not divide (p - 1) !. (2) *Let k be a finite field of characteristic p and let n be a positive in- teger. The strict triangular group Tl(n, k) is a Sylowp -subgroup of the group GW, 4.* THEOREM 2. Every jinite group contains a Sylow p-subgroup. Lemma. Let n = p'm, where m is an integer which is not a multiple of p. Then The proof depends on the following lemma. Let S be a group of orderp' (for example Z/p'Z) and T a set with m elements. Write X = S x T and let E be the set of subsets of X with p' elements. Then Card(X) = n, whence Card(E) = (Set Theory, III,§5, no. 8, Corollary 1 to Proposition 11). Let S operate on X by s.(x, y) = (sx, y)(s, x E S, y E T) and consider the canonical extension of this operation to E. In the notation of no. 5, Proposition 11, the set ES is the set of orbits of X, that is the set of subsets Y c X ofthe form S x {t), t E T, whence Card(ES) = m. By no. 5, Proposition 11 , (i7) = Card(E) = Card(ES) = m f 0 (mod.p), which proves the lemma. We now prove the theorem. Let G be a finite group and n its order; we write n = p'm, where m is not a multiple ofp. Let E be the set of subsets of G with p' elements. Then Card(E) = (i7); whence, by virtue of the lemma, Card(E) $ 0 (mod. p). Consider the extension to E of the operation of G on itself by left translation. There exists X E E whose orbit has non -zero cardinal m0d.p. If Hx denotes the stabilizer of X, then 3 6.6 (G: H,) $ 0 (mod. p) , which means that p' divides Card(H,). But Hx consists of the s E G such that SX = X; if x E X, then H, c X.x-l, whence Card(Hx) 6 Card(X) = p'. Hence Card(Hx) = p'. COROLLARY. If the order of G is divisible by p, the group G contains an element of order p. By virtue of Theorem 2, this is reduced to the case where G is a p-group # {e}; if x E G is different from e, the cyclic group generated by x is then of order p" with n 3 1 and it therefore contains a subgroup of order p. Remark. For every prime number q dividing Card(G), let P, be a Sylow q-sub- group of G. Then the subgroup H of G generated by the P, is of order a multiple ofCard(P,) for each q and of order a divisor of Card(G), hence it is equal to G. THEOREM 3. Let G be ajnite group. (a) The Sylow p-subgroups of G are conjugate to one another. Tfieir number is congruent to 1 m0d.p. (b) Every subgroup of G which is a p-group is contained in a Sylow p-subgroup. Let P be a Sylow p-subgroup of G (Theorem 2) and let H be a p-subgroup of G. Let E = G/P and consider the operation of H on G/P. As Card(E) $ 0 mod.p, Proposition 11 of no. 5, shows that there exists x E G/P such that hx = x for all h EH. If g is a representative of x in G, this means that H c gPg-l, whence assertion (b). If H is a Sylow p-subgroup, then Card(H) = Card(P) = Card(gPg-'), whence H = gPg-', which proves the first assertion of (a). We now prove the second assertion of (a). Let Y be the set of Sylow p-sub- groups of G and let P operate on Y by inner automorphisms. The element p E Y is a fixed point under this operation, we show that it is the only one. Let QE Y be a fixed point; Qis a Sylow subgroup of G normalized by P and hence p is contained in the normalizer N of Q. The groups P and Qare Sylowp-sub- groups of N; hence there exists n EN such that P = nQn-l = Q. By no. 5, Proposition 1 1, Card(Y) = Card(YP) = 1 (mod. p). COROLLARY 1. Let P be a Sylow p-subgroup of G, let N be its normaliter in G and let M be a subgroup ofG containing N. The norrnalizer ofM in G is equal to M. Let s E G be such that sMs-l = M. The subgroup sPs-l of M is a Sylow &subgroup of M. There thus exists tEM such that sPs-l = tPt-l; then t-'s E N, whence s E tN c M. COROLLARY 2. Let$ GI -+ G, be a homomorphism ofjinite groups. For every Sylow *~bP'ou@ PI of GI there exists a Sylow fl-subgroup P, of G, such that f (Pl) c P,. SYLOW SUBGROUPS This follows from Theorem 3 (b) applied to the subgroupf(P,) of G,. 79 78 I ALGEBRAIC STRUCTURES COROLLARY 3. (a) Let H be a subgroup of G. For every Sylow p-subgroup P of H there exists a Sylow p-subgroup Q of G such that P = Q n H. (b) Conversely, ;S Q is a Sylow p-subgroup of G and H is normal in G, the group Q n H is a Sylow p-subgroup of H. (a) The p-group P is contained in a Sylow p-subgroup Q of G and Q n H is a p-subgroup of H containing P and is hence equal to P. (b) Let P' be a Sylow p-subgroup of H. There exists an element g E G such that gP'g-l c Q. As H is normal, P = gP'g-' is contained in €I and hence in Qn H. As Qn H is a p-subgroup of H and P is a Sylow p-subgroup of H, P=QnH. COROLLARY 4. Let N be a normal subgroup of G. The image in G/N ofa Sylow p-sub- group of G is a Sylow p-subgroup of G/N and every Sylow fl-subgrolrp of GIN is ob- tained in this way. Let G' = G/N and P' be the image in G' of a Sylow p-subgroup P of G. The group G operates transitively on G'/P' and hence G'/P' is equipotent to G/S, where S is a subgroup of G containing P. Thcrefore (G':P') divides (G: P), is thus not a multiple ofp and the p-group P' is a Sylow p-subgroup of G'. Let Q' be another Sylow p-subgroup of G'; then Q' = g'P'g'-l for some g' E G'; if g E G is a representative of g', the group Q' is the image of Q = gPg- l. 7. FINITE NILPOTENT GROUPS THEOREM 4. Let G be ajnite group. The following conditions are equivalent: (a) G is nilpotent. (b) G is a product of p-groups. (c) For every prime number p there exists a normal Sylow p -subgroup of G. (b) * (a) (no. 5, Corollary to Theorem 1). Suppose (a) holds and let P be a Sylow p-subgroup of G. If N is the norma - lizer of P in G, Corollary l to Theorem 3 shows that N is its own normalizer. By $6, no. 3, Corollary to Proposition 8, this shows that N = G. Hence Suppose (c) holds and let I be the set of prime numbers dividing Card(G). For all p E I, let P, be a normal Sylow p-subgroup of G. For all p # q, P, n P, is reduced to e for it is both ap-group and a q-group, hence P, and P, centralize one another ($4, no. 9, Proposition 15). Let 4 be the canonical homo- morphism (§ 4, no. 9, Proposition 12) ofn PSI P, into G. The homomor- phism 4 is surjective by the Remark of no. 6. As Card(n ,PI P,) = Card(G), it (4 * (C)' $ 7.1 Remarks. (1) Let G be a finite group and p a prime number. By no. 6, Theorem 3 (a) and no. 6, Theorem 2, the following conditions are equiva - lent: FREE MAGMAS (i) there exists a normal Sylow p-subgroup of G; (ii) every Sylow p-subgroup of G is normal; (iii) there exists only one Sylow p-subgroup of G. (2) Let G be a nilpotent finite group. Let I be the set of prime divisors of Card(G). By Theorem 4 and Remark 1, G = n G,, where G, is the unique (3) Applied to commutative groups, Theorem 4 gives the decomposition, of commutative finite groups as a product of primary components, which will be studied from another point of view in Chapter VII. Example. The group 6, is of order 6. It contains a normal Sylow 3-subgroup of order 3: the group ?13. It contains three Sylow 2-subgroups of order 2: the groups {e, T}, where T is a transposition. Thr group 6, is thus not nilpotent. Sylow !-group of G. ,€I $7. FREE MONOIDS, FREE GROUPS In thisparagraph X will denote a set. Unless otherwise mentioned, the identity element ofa monoid will be denoted by e. 1. FREE MAGMAS Asequence ofsets M,(X) is defined by induction on the integer n 3 1 as follows: writing M,(X) = X, for n 2 2, M,(X) is the set the sum of the sets Mp(X) x M,,-,(X) for 1 < p < n - 1. The set the sum of the family (M,,(X)),,31 is denoted by M(X); each of the sets M,(X) is identified with its canonical image in M(X). For every element w of M(X) there exists a unique integer n such that w E M,(X) ; it is called the length of w and denoted by 1 (w). The set X consists of the elements in M(X) of length 1. Let w and w' be in M(X); write p = l(w) and q = l(w'). The image of (w, w') under the canonical injection of M,(X) x M,(X) into the sum set Mpfq(X) is called the composition of w and w' and is denoted by ww' or w. w'. Then I(w.w') = l(w) + l(w') and every element of M(X) of length 32 can be written uniquely in the form w'w'' with w', w" in M(X). The set M(X) with the law of composition (w, w') &+ w. w' is called thefree magma constructed on X ($ 1, no. 1, Definition 1). POSITION 1. Let M be a magma. Every mapping f ofX into M may be extended in a unique way to a morphism ofM(X) into M. follows that + is bijective. By induction on n 2, mappings fn: Mn(X) -+ M are defined as follows: let 80 81 I ALGEBRAIC STRUCTURES fi =fi for n > 2, the mappingf, is defined byf,(w.w') =fp(w) ..f,-,(w') for p= 1,2 ,..., n- land(w,w')inM,(X) x Mn-,(X).Letgbethemapping ofM(X) into M which inducesf, on M,(X) for every integer n > 1. Clearly g is the unique morphism of M(X) into hf which extends$ Let be a mapping of X into a set Y. By Proposition 1, there exists one and only one homomorphism of M(X) into M(Y) which coincides with u on X. It will be denoted by M(u). If u is a mapping of Y into a set Z, the homomorphism M(u) M(u) of M(X) into M(Z) coincides with u o u on X, whence M(u) 0 M(u) = M(u o u). PROPOSITION 2. Let u: X 3 Y be a mapping. Ifu is injective (resp. suiective, bijec- tive), so is M (u) . Suppose u is injective. When X is empty, M(X) is empty, hence M(u) is in- jective. If X is non-empty, there exists a mapping u of Y into X such that u o u is the identity mapping of X (Set Theory, 11, 3 3, no. 8, Proposition 8) ; the mapping M(v) 0 M(u) = M(u 0 u) is the identity mapping of M(X) and hence FREE MONOIDS 3 7.2 length 0, namely the empty sequence e. X will be identified with the set of words of length 1. and w' = (x;)lGis<n be two words. The composition of w and w' is the word u = (Y~)~ Gk<<m+n defined by (1) Yk={:f-m form+I fk<m+n. In other words, the sequence wn is obtained by first writing the elements of the sequence w and then those of w'. The composition of w and w' is generally de- noted by ww' or w. w'; it is sometimes said that it is obtained byjiixtuposition of w and w'. Then by construction l(w.w') = l(w) + l(w'). The relation we = ew = w is immediately established for everv word w. Let Let w = for1 <k<m M(uj is-injeitive. When u is surjective, there exists a mapping w of Y into X such that u 0 w is the identity mapping of Y (Set Theory, 11, 3 3, no. 8, Proposition 8). Then M(u) o M(w) = M(u o w) is the identity mapping of M(Y) and hence M(u) is .. surjective. Finally, if u is bijective, it is injective and surjective and hence M(u) has the same properties. Let S be a subset of X. By Proposition 2 the injection of S into X can be ex- tended to an isomorphism of M(S) onto a submagma M(S) of M(X). The magmas M(S) and M'(S) are identified by means of this isomorphism. Then M(S) is the submagma of M(X) generated by S. Let X be a set and (u,, u~),~~ be a family of ordered pairs of elements of M(X). Let R be the equivalence relation on M(X) compatible with the law of M(X) and generated by the (u,, v,) (3 1, no. 6). The magma M(X)/R is called the magma dejined by X and the relators (u,, u,),,~. Let h be the canonical morphism of M(X) onto M(X)/R. Then M(X)/R is generated by h(X). Let N be a magma and (nx)xcx a family of elements of N. Let k be the morphism from M(X) to N such that k(x) = n, for all x E X (Proposition 1). If k(ua) = k(v,) for all a€ I, there exists one and only one morphism f: M(X)/R -+ N such thatf(h(x)) = n, for all x E X (3 1, no. 6, Proposition 9). 2. FREE MONOIDS Any finite sequence w = of elements of X indexed by an interval (1, n) of N (possibly empty) is called a word constructed on X. The integer n is called the length of the word w and denoted by 1 (w). There is a unique word of 82 w = (xt)lSiGm, w' = (~;)IG~G, and w" = (~:)1~k., be three words; clearly the words ~(w'w") and (ww')~'' arc both equal to the word (~l)l~lGm+n+p defined by if1 fl<m xl -m-n ifm + n + 1 < 1 f nz + n + p. (2) The above shows that the set of words constructed on X with the law of composition (w, w') H w. w' is a monoid with identity element e. It is denoted by Mo(X) and called the free monoid constructed on X. It follows immediately from the definition of product of words that every word w = (xi) <,, is equal to the product PRoPosrTroN 3. Let M be a monod. Every mapping f of x into M extends uniquely lo a homomorphism OfMo(X) into M. Let g be a homomorphism of Mo(X) into M extending f. If w = (xJ1 <, <,, n xi. A word may therefore be written in the form xl. . .xn. 1=1 n is a word, then w = n xi in the monoid Mo(X), whence 1=1 n n in the monoid M (8 1, no. 2, formula (2)). This proves the uniqueness ofg. n Let h(w) = nf(xf) for every word w = (xJIGcGn. The associativity imply h(ww') = h(w)h(w'). By convention the empty product h(e) is the identity element of M and h(x) = f(x) for x E X. Hence h is a homomorphism OfMo(x) into M extending$ Let u: X -+ Y be a mapping. By Proposition 3, there exists one and only one theorem (3 1, no. 1=1 3, Theorem 1) and the definition of product in Mo(X) 83 I ALGEBRAIC STRUCTURES homomorphism of Mo(X) into Mo(Y) which coincides with u on X; it is denotedbyMo(u). Itmapsaword (x,),,,,,to theword (u(xi))lGt,n.Asin the case ofmagmas (no. l), the equation Mo(v 0 u) = Mo(v) 0 Mo(u) is established for every mapping u: Y + Z and it can be shown that Mo(u) is injective (resp. surjective, bijective) if u is. For every subset S of X, Mo(S) is identified with the submonoid of Mo(X) generated by S. Let X be a set and (u,, v,),~~ be a family of ordered pairs of elements of Mo(X). Let R be the equivalence relation on Mo(X) compatible with the law on Mo(X) and generated by the (u,, v,) (9 1, no. 6). The monoid Mo(X)/R is called the monoid dejned by X and the relators (u,, u,),,~. Let h be the canonical morphism of Mo(X) onto Mo(X)/R. Then Mo(X) is generated by h(X). Let N be a monoid and (nJXEx a family of elements of N. Let k be the morphism of Mo(X) into N such that k(x) = n, for all x E X (Proposition 3). If k(u,) = k(u,) for all a E I, there exists one and only one magma morphisrn fi Mo(X)/R --f N such that f (h(x)) = n, for all x E X (5 1, no. 6, Proposition 9) ; as k is unital, f is a monoid morphism. 3. AMALGAMATED SUM OF MONOIDS Let denote a family of monoids and e, the identity element ofMi. We are given a monoid A and a family of homomorphisms hi: A + M, (for i E I). The set S the sum of the family (MJte1 has elements the ordered pairs (i, x) with i E I and x E M,. For every triple a = (i, x, x’) with i E I, x, x’ in Mi, write u, = (i, xx’) and u, = (i, x) . (i, x’); for every triple A = (i,j, a) in I x I x A, write PA = (i, h,(a)) and qh = (j, h,(a)) ; for all i E I, write E, = (i, e,). The monoid M defined by S and the relators (u,, u,), (PA, qA) and (c,, e) is called the sum ofthe family amalgamated by A. Let + denote the canonical homo- morphism of Mo(S) onto M and write +,(x) = +(i, x) for (i, x) E s. It is said that +, is the canonical mapping of M, into M. For all a E A, the element +(i, h,(a)) is independent of i and denoted by h(a).t The universal property ofmonoids defined by generators and relators (no. 2) implies the following result : PROPOSITION 4. (a) For all i E I, the mapping 4, is a homomorphism of M, into M and +i 0 h, = hfor all i E I. Further, M is generated by ig +,(Mi). (b) Let M be a monoid and f’: M, + M‘ (for i E I) homomorphism such that 5 0 h, is independent of i E I. There exists one and only one homomoqhism fi M -+ M’ such that5 = f o +,for all i E I. In what follows we shall make the following hypothesis : (A) For all i E I, there exists a subset Pi of M, containing el such that the mapping (u,)) H hi(a) .p of A x P, into Mi is bijective. t When I is empty, M = {e} and h(a) = e for all a E A. 3 7.3 AMALGAMATED SUM OF MONOIDS It implies that the homomorphisms h, are injective. Let x E M; every finite sequence c= (a:il ,..., in;pl ,..., p,,) with aEA, i,EI and paEPi, for 1 6 c( < n, satisfying (3) is caIled a decomposition of x. The integer n 0 is called the length of the decom- position 0 and is denoted by I(.) ; the sequence (e) is a decomposition of length 0 of the identity element of M. The decomposition cs is called reduced if i, # i, + for 1 6 c( < nandp, # et,for 1 < a < n. PROPOSITION 5. Under hypothesis (A) every element x of M admits a unique reduced decomposition 0. Every decorr$osition 0‘ # 0 of x satisJies ~(cs’) > I(.). (A) Uniqueness of a reduced decomposition : Let C denote the set of sequences 0 = (a; i,, . . . , in; pl, . . . , p,,) with n 2 0, aEA, i,EI and Pa~Pi, -{cia} for 1 < c( 6 n, such that i, # ia+l for 1 < M < n. Let 0 denote the mapping of C into M defined by (4) A reduced decomposition of x E M is an element 0 of C such that ~(cs) = x. (e; i,, . . ., in;pl,. . . , pn) with i # i, when n > 0. Let For all isI; let 2, be the subset of I; consisting of the sequences 0 = (e;il,...,in;Pi,...,Pn) be in C, and 5 in M,; let 5 = h,(a) .p with a E A and p E P,, and ifp = e, . . .> in;Pi,. . .,Pn) (5) (a; i, i1,. . .> in;P,Pi,. * .,Pn) ifp # 8,. It is immediate that Y, is a bijection of M, x Ci onto C.. fined by Let i E I and x E Mi; as Y, is bijective, a mappingJ;, of 2 into itself is de- .A,x(yi(C, 0)) = Yt(x5, 0) (E€Mi, o~Ct). (6) Further, for a E A, fa denotes the mapping of C into itself defined by Clearlyf,, is the identity mapping ofC and&, ,.,, = J;, , of;, ,., for x, x‘ in M, Then Proposition 4 may be applied to the case where M’ is the monoid of mappings of Y, into itself with law of composition (Jf’) H f 0 f‘ and where =fa for a E A and i E I. 84 85 I ALGEBRAIC STRUCTURES fi is the homomorphism x ~f,,~ of Mi into M‘; then there exists a homo- morphismfof M into M‘ such thatf,,, =f(&(x)) for i E I and x E Mi. Let = = (a;i,,...,in;Pl,...,Pn) = = (faofil.Pl O.. *O.L”,P”)(e) be in C. Formulae (5) to (7) imply by induction on n the relation =f(h(a)+t1(~1). * *+in(Pn)) (el, that is Q = f(@(~)) (e). This proves that @ is injective. (B) Existence of a decomposition: Let D be the set of elements of M admitting a decomposition. Then e E D and M is generated by !-I+i(Mi) and hence by h(A) u!JI+i(Pi). Then D.+[(P,) c D for all i E I; to prove that D = M, it thus suffices to prove the relation D .h(A) c D. This follows from the following more precise lemma: Lemma 1. Let i,, . . . , i, be in I and pa in Pi, for 1 < a < n. For all a E A there exists a’ E A and a sequence (ph) , a , with p; E Pta such that , +[,(A) * * .+tn(Pn)h(a) = h(a’)+ti(Pi). . *+i,(pb)* h(a) = +in(h,n(a)) and there exists a, E A and ph E Pin with pn.hi,(a) = ht,(an) .Ph* It follows that &,(p,)h(a) = h(a,)+i,,(pb), whence +ii(pl). * .+t,-i(Pn-l)+t,(Pn)h(a) = +[,(PI) * . .+t,-i(pn-l)h(un)+tn(p~); the lemma follows from this by induction on n. AMALGAMATED SUM OF MONOIDS 5 7.3 (C) End of the proof: Let x E M and let n be the minimum of the lengths of decompositions of X. We shall prove that every decomposition 0 of x of length n is reduced. This will establish the existence of a reduced decomposition of x; the uniqueness of the reduced decomposition then implies 1 (a’) > 1(0) for every decomposition 0’ # 0 of x. The case n = 0 being trivial, suppose n > 0. Let o = (a; il, . . ., i,;pl, . . . ,pn) be a decomposition of x of length n. If there existed an integer a with 1 < a 6 n and pa = eta, the sequence (a; * ., ia-1, ;a+,, * . .> in;kl, *. *,Pa-l,Pa+l,. . .>Pn) would be a decomposition of x of length n - 1, which is excluded. Suppose that there exists an integer a with 1 6 a < n and i, = i,+, and let Pda + 1 = hi,(a’) with a’ E A and ph E Pi=; by Lemma 1 there exists elements a” E A, and the sequence .. (aa”; i,, . . ., ia-1, Za, za+z,. . .> in;Pi, *. .,PL-1,Ph,Pa+z, * * *,PJ is a decomposition of x of length n - 1, which is a contradiction. COROLLARY . Under hypothesis (A) the homomorphisms +[ and h are injective. For First h is injective: if h(a) = h(a‘), then (u) and (a’) are two reduced de- compositions of the same element of M, whence a = a’. Let iE1; then h(A) = +;(h$(A)) c +;(Mi) ; the uniqueness of reduced decompositions implies We have thus proved that 0 is reduced. i #j in I, +(Md n +,(MI) = h(A). h(A) n +[(Mi - hi(A)) = @, whence +[(Mi - h,(A)) = +[(Mi) - h(A). The injectivity of the homomorphisms +, and the relation +t(Mt) n +,(M,) h(A) for i # j are then consequences of the following fact: for i, j in I, x in M, - h,(A) and y in M, - h,(A), the relation +,(x) = +,(y) implies i =j and x = y. Let x = hi(a) .p and y = h,(b) .q with a, b in A, p in Pi - {e,} and y in P, - {e,}. Then +[(x) = h(a)+,(p) and +,(y) = h(b)+,(q) and hence (a; i;p) and (6;j; q) are two reduced decompositions of the same element of M. It follows that 2 =J, a = b andp = q, whence x = h,(a)p = h,(b)q = y. When hypothesis (A) is fulfilled, we shall identify each monoid M, with a submonoid of M by means of +[; similarly, we shall identify A with a sub- monoid of M by h, Then M is generated by [; M, and Mi n M, = A for i # j. Every element of M can be written uniquely in the form a.np, with a €A, PI E Pil- {e}, . . .,fin E P,n - {e} and i, # i,,, for 1 < a < n. Finally, if M‘ a monoid and (A: Mi + M’) (for i E I) a family of homomorphisms whose restrictions to A are the same homomorphism of A into M‘, there exists one and only one homomorphismf: M -+ M’ inducing& on M, for all i E I. .. n a=l 87 86 I ALGEBRAIC STRUCTURES Hypothesis (A) is satisfied in two important cases: (a) A = {e}. In this case, there is a family (Mi)tEI of monoids and M is called the monoidal sum of this family. Each Mi is identified with a submonoid of M and M is generated by gI Mi; further, Mi n M, = {e} for i # j. Every clement of M may be written uniquely in the form xl. . . x, with x1 E Mi, - {e}, . . . , x, E Min - {e} and i, # ia+l for 1 < a < n. Finally, for every family of homomorphisms (A: Mi --f MI), there exists a unique homomorphism f: M --f M’ whose restriction to M, is& for all i E I. (b) There is a family of groups (G,),,, containing as subgroup the same group A and hi is the injection of A into Gi. The sum of the family (Gi),EI amalgamated by A is then a group G: the monoid G is generated by FI +i(Gi) and every element of !-I +{(Gi) admits an inverse in G (cf. 4 2, no. 3, Corollary 1 to Proposition 4); it is denoted by *A Gi or G1 *A G, when I = (1, 2). When A consists of the identity element, it is also said that G is the free product of the family (Gi)iEI of groups and it is denoted by * Gi (or G1 * G, if 1 = {1,2)).t FREE GROUPS 9 7.5 4. APPLICATION TO FREE MONOIDS Lemma 2. Let M be the monoidal sum of the family (MX)XEX deJined by M, = N for all x E X and let +, denote the canonical homomorphism of M, into M. The mapping x H +,(1) ofX into M extends to an isomorphism h of Mo(X) onto M. Let h be the homomorphism of Mo(X) into M characterized by h(x) = +,(l). For every integer n 2 0, +,(n) = +x(l) - - h ( x )” and as M is generated by ,L? +x(N), it is also generated by h(X). Hence h is surjective. Moreover, for all x in X, the mapping nHxn is a homomorphism of N = M, into Mo(X) ; there thus exists (no. 3, Proposition 4) a homomorphism h’ of M into Mo(X) such that h’(+,(n)) = xn for x E X and n E N; in particular, h’(h(x)) = x for x E X and hence h’ o h is the identity homomorphism of Mo(X). Therefore h is injective. It has thus been proved that h is bijective. PROPOSITION 6. Let w be an element of Mo(X). (a) There exist an integer n 2 0, elements x, of X and integers m(a) > 0 (for t Note that G1 * G, is not the “product” of G1 and Gz in the sense of Set Theory, IV, 5 2, no. 4 (nor in the sense of the “theory of categories”; in the context of this theory, G1 * Gz is the “sum” of G1 and Gz). n 1 6 a < n) such that x, # xaCl for 1 < a < n and w = n x:(~). The sequence (x,, m(a)) < (b) Let p be a positive integer, x; in X and m’(p) in N for 1 < p < p such that w= rI ~’“’(0). Then p 2 n. If p = n, then x; = xp and m’(p) = m(p) for a=l , is determined uniquely by these conditions. P p=1 1<p<p. -1 In the notation of Lemma 2, h (+,(n)) = xn for x E X and n E N. Proposition 6 then follows from no. 3, Proposition 5. 5. FREE GROUPS Let G, = Z for all x E X. The free product of the family (GJXEx is called the free group constructed on X and is denoted by F(X). Let +x denote the canonical homomorphism of G, = Z into F(X). By no. 3, Corollary to Proposition 5, the mapping x H +,(1) of X into F(X) is injective; we shall identify X with its image in F(X) under this mapping. Then X generates F(X) and e $ X. PROPOSITION 7. Let g be an element of the free group F(X). There exist an integer n 2 0 and a sequence (x,, m(a;)l determined uniquely by the relations x, E X, x, # x,+lfor 1 < a < n, m(a) E Z, m(a) # 0 for 1 < a < n, andg = xZ(~). Applying no. 3, Proposition 5, we obtain the following result: n ,=1 The free group F(X) enjoys the following universal property: PROPOSITION 8. Let G be a group and f a mapping ofX into G. There exists one and only one homomorphismfof F(X) into G which extendsf. The uniqueness offfollows from the fact that the group F(X) is generated by x. For all x in X, letf, be the homomorphism n ~f(x), of Z into G. By no. 3, Proposition 4, there exists a homomorphism f of F(X) into G such that f(x”) =f,(n) for x E X and n E Z; in particular, f(x) = f,( 1) = f (x) for all x EX and hencefextendsf. Let u : X --f Y be a mapping. By Proposition 8 there exists one and only one homomorphism of F(X) into F(Y) which coincides with u on X; it is denoted by F(u). As in the case of magmas (no. 1) the formula F(u 0 U) = F(v) 0 F(u) is established for every mapping u: Y +- Z and it is shown that F(u) is injective (resp. surjective, bijective) if u is. For every subset S of X, F(S) will be identified With the subgroup of F(X) generated by S. Let I be a set. In certain cases it is of interest not to identify i in I with its 89 88 I ALGEBRAIC STRUCTURES canonical image +i(l) in the free group F(1); the latter will be denoted by T, (or Ti, Xi, . . . as the case may be) and called the indeterminate of index i. The free group F(1) is then denoted by F((Ti)isI) or F(T,, . , ., T,) if 9 7.6 Let I be a set and r = (rj)joJ a family of elements of the free group F(1). Let N(r) be the normal subgroup of F(1) generated by the rj for j E J. Let F(1, r) = F(I)/N(r) and denote the class ofi modulo N(r). The ordered pair (T, r) with 7 = (Ti)iEI is a presentation of the group F(1, r) ; if G is a group and (t, r) is a presentation of G with t = (ti)iEI, there exists a unique isomorphism u of F(1, r) onto G such that u(T~) = ti for all i E I. The group F(1, r) is said to be defined by the generators Ti and the relators rj, or by an abuse of language that it is dejined by the generators T, and the relations rj(7) = e. When I = (1, n) and J = (1, m), it is said that F(1, r) is defined by the presentation PRESENTATIONS OF A GROUP (71, * . .> ‘n; r1, * . ., rm)- Ifr, = u,-’v, with uj and vj in F(I), this presentation is equally denoted by the symbol (T,, . . ., r,; u1 = vl,. . ., u, = v,). Examples. (1) The group defined by the presentation (T; -r* = e} is cyclic of order q. I = (1, 2,. . ., n}. Let G be a group and t = (ti)iEI a family of elements of G. By Proposition 8 there exists a homomorphism ft of F((Tt)tsI) into G characterized by &(Ti) = ti for all i E I. The image of an element w of F((TJiEI) under ft will be denoted by w(t) or w(t,,. . ., t,) if I = (1, 2,. . ., n}; w(t) is said to result from the substitution Ti ++ ti in W. In particular, if we take G = F((T,),EI) and (ti) = (T,) = T, fT is the identity homomorphism of G, whence w(T) = w; for I = {1,2,. . ., n}, then w(Tl, . . ., T,) = w. Let G and G’ be two groups, u a homomorphism of G into G’ and t = (tl, . . . , t,) a finite sequence of elements in G. Let t’ = (u(tl), . . . , u(t,)) ; the homomorphism u 0 ft of F(Tl, . . ., T,) into G’ maps Ti to .(ti) for 1 < i < n and hence is equal toft,; for w in F(T,, . . . , T,), then (8) u(w(t1,. . ., tn)) = ~(~(tl), . . .> U(tn))* Let w be given in F(T1,. . ., T,) and elements vl, . . ., v, in the free group F(T;, . . . , TL). The substitution Ti H vt defines an element w’ = w(vl, . . . , v,) of F(T;, . . . , TL). Let G be a group, tl, . . . , t, elements of G and u the homo - morphism ofF(T;, . . . , TL) into G characterizcd by u(T;) = tj for 1 < j < m. Then u(v,) = v,(tl,. . ., tm) and u(w’) = w(tl, , . ., t,); formula (8) thus implies (9) W’(tl,. . ., t,) = W(Ul(t,,. . ., tm), . . ., V,(tl,. . ., t,)). This justifies the “functional notation” w(t,, . . ., t,). The reader is left to ex- tend formulae (8) and (9) to the case of arbitrary indexing sets. 6. PRESENTATIONS OF A GROUP Let G be a group and t = (ti)iEI a family of elements of G. Letft be the unique homomorphism of the free group F(1) into G which maps i to t,. The image of ft is the subgroup generated by the elements ti of G. The elements of the kernel of ft are called the relators of the family t. t is called generating (resp. free, basic) if ft is surjective (resp. injective, bijective). Let G be a group. Apresentation of G is an ordered pair (t, r) consisting of a generating family t = (ti)ieI and a family r = (r,) jEJ of relators such that the kernel Nt offt is generated by the elements grjg-l for g E F(1) and j E J. It amounts to the same to say that Nt is the normal subgroup of F(1) generated by the rj forj E J (in other words, the smallest normal subgroup of F(1) con- taining the elements rj (j E J), cf. § 4, no. 4). By an abuse of language the generators ti and the relations rj(t) = e are said to constitute apresentation of the group G. 90 (2) The group defined by the presentation (x, y; xy = yx) is isomorphic to z x z. PROPOSITION 9. Let G be a group, t = (ti), I a generatingfamily ofG andr = (rj) a family of relators oft. The following conditions are equivalent: (a) The ordered pair (t, r) is a presentation of G. (b) Let G‘ be a group and t‘ = (ti)iEI a family of elements of G‘. qrj(t’) = e for all j E J, there exists a homomorphism u ofG into G’ such that .(ti) = t; for all i E 1. (c) Let G be a group and i = (I,), a generating family ofG such that rj(f) = e for all j E J. Every homomorphism of G into G which maps 2, to ti for all i E I is an iso- morphism. Letfdenote the homomorphism of F(1) into G which maps i to ti for all i E I and N the kernel of$ (a) be a family of elements of a group G’ with r,(t‘) = e for all j E J. Let f’ be the hnomorphism of F(1) into G‘ characterized by f ‘(i) = ti for all i E I. By hypothesis f ‘(rj) = e for all j E J and, as N is generated by the elements Fjg-’ forj E J and g E F(I), f ‘(N) = {e}. As the homomorphism$ F(1) --f G 1s surjective with kernel N, there exists a homomorphism u: G --f G’ such that f’ = u of. Then u(t,) = u(f(i)) =f‘(i) = tl. (c): Suppose condition (b) holds. Let t = (tJiEI be a generating family ofa group G such that rj(t) = e for allj E J and let v be a homomorphism Ofc into G such that .(ti) = ti for all i E I. As the family (t,)aEI generates G, the homomorphism v is surjective. By property (b) there exists a homomorphism u: G 3 such that .(ti) = 2, for all i E I. Then u(v(ta)) = t, for all i E I and (b) : Suppose that (t, r) is a presentation of G and let t’ = (b) 91 ~ 3 7.7 ALGEBRAIC STRUCTURES FREE COMMUTATIVE GROUPS AND MONOIDS I hence u o v is the identity on c, which proves that v is iriective. Hence v is an isomorphism and condition (c) holds. (c) + (a) : Suppose condition (c) holds. Let ti be the canonical image of i in F(I, r) and t' = (ti)iEI; then rj(t') = e for allj E J. As r,(t) = e for all j E J, there exists one and only one homomorphism v of F(1, r) into G such that v(iJ = ti for all i E I. By (c), u is an isomorphism of F(1, r) onto G which trans - forms the presentation (t', r) of F(1, r) into a presentation (t, r) of G. thatg(6,) = f(x)@r all x E X. IfM is written additively, theng(a) = 2 a(.) .f(x) Let g be a homomorphism of Ncx' (resp. Z'x)) into M such that g(6,) = f(x) for all a in N(x) (resp. Z(x)). nex for all x E X. For all a in N(x' (resp. ZX)), it follows from (12) that 7. FREE COMMUTATIVE GROUPS AND MONOIDS whence the uniqueness of R. 93 I ALGEBRAIC STRUCTURES and therefore H is the set of elements of Z'x) of support contained in S. Hence - forth Z(S) will be identified with H by means of$ Formula (1 5) shows that the restriction of ZU) to N(x) induces a homomor - phism of N(x) into NCy). Then N('O") = N(") o N(") for every mapping v: y --f Z; further, N(") is injective (resp. surjective, bijective) if u is. If S is a subset of X, "S) = Z(S) n "XI. 3 7.9 RELATIONS BETWEEN THE VARIOUS FREE OBJECTS When M is commutative, U" can be defined for every family u of elements of M and formulae (15) to (20) hold without restriction. 9. RELATIONS BETWEEN THE VARIOUS FREE OBJECTS As the free monoid Mo(X) is a magma, Proposition 1, of no. I shows the exis- tence of a homomorphism A: M(X) -+ Mo(X) whose restriction to X is the identity. Similarly, as the free group F(X) is a monoid the identity mapping of X extends to a homomorphism p: Mo(X) -+ F(X) (no. 2, Proposition 3). By no. 4, Proposition 6 and no. 5, Proposition 7, p is injective. Similarly Proposition 10 of no. 7 and Proposition 8 of no. 5 show the existence of homo - morphisms v: Mo(X) -+ N") and TC: F(X) -+ Z") characterized by v(x) = 8, and x(x) = sx for all x E X. If I is the injection of N(x) into Z(x), the two homo - morphisms 10 v and TC Q p of Mo(X) into Z(x) coincide on X, whence L a v = x 0 p. The situation may be summarized by the follo~ing commutative diagram; M(X) A Mo(X) 4 x T1 F(X) ---+ Z(x). The homomorphisms A, p, v and x will be called canonical. length of the word h(w) is equal to that of w. Moreover Let w be in M(X); it is immediately shown by induction on I (w) that the Remark. Let M be the multiplicative monoid of strictly positive integers and let 5Q be the set of prime numbers (3 4, no. 10, Definition 15). By Proposi - tion 10 there exists a homomorphism u of N(V) into M characterized by ~(6,) = p for every prime numberp. Then u(u) = n pa(p) for a in "9) and Theorem 7 of 5 4, no. 10 shows that u is an isomorphism of N(@) onto H. PE?, 8. EXPONENTIAL NOTATION Let M be a monoid, written multiplicatively, and u = (u,),,~ a family of elements of M, commuting in pairs. Let a be in N(x); the elements u;',) and ,;(,) of M commute for x, y in X and there exists a finite subset S of X such that uzcx) = 1 for x in X - S. We may therefore write: ua = n (16) ,EXUX Let M be the submonoid of M generated by the family (uJXEx; it is com- mutative (3 1, no. 5, Corollary 2 to Proposition 4). There thus exists (no, 7, Proposition 10) a unique homomorphismfofN(X) into M' such thatf(8,) = u, for all x E X and f(a) = ua for all a in N(x). We deduce the following for - mulae (17) n"+P = u~.uD (18) uo = 1 (19) u6x = u, for a, @ in NCX) and x in X. Let v = (u,),~~ be another family of elements of M; suppose that u,u, = v,u, and u,u, = up, for x, y in X. Then there exists (3 1, no. 5, Corollary 2 to Proposition 4) a commutative submonoid L of M such that u, E L and v, E L for all x E X. The mapping a H u" . u" of into L is then a homomorphism (5 1, no. 5, Proposition 5) mapping 8, to u,.u,. Thus we have the formula (20) u".v" = (u.v)", where u . v is the family (u, . v,) , X' 94 for xl,. . ., x, in X, whence Iv(xl.. .xn)l = n by (13) and (14). In other words, Iv(u)I = l(u) (U E Mo(X)). (22) PROPOSITION 1 1. The canonical homomorphism v af Mo (X) into N(x) is surjective. Let w = xl. . .x, and w' = xi. . .xA be two elements of Mo(X); in order that V(W) = v(w'), it is necessary and sujicient that rn = n and that there exist apermutation Q E 6, with x; = xu,,, for 1 < i < n. The image of v is a submonoid I of N(x) containing the elements 8, (for x EX). Formula (12) (no. 7) shows that N(x) is generated by the family (ax),,x, where I = N(x). Therefore v is surjective. Ifm = n and x; = for 1 < i < n, then "r. by formula (21) and the commutativity theorem (5 1, no. 5, Theorem 2). 95 I ALGEBRAIC STRUCTURES Conversely, suppose that V(W) and v(w') are equal to the same element a of "XI; by formula (22), n = Id] = m. For all x E X, let I, (resp. I:) be the set of integers i such that 1 < i < n and xi = x (resp. xi = x). Hence (I,) and (I;),,~ are partitions of the interval (1, n) of N; further, the formula = 2 8,, shows that a(.) is the cardinal of I,; similarly the formula Q = 2 s,,, shows that a(.) is the cardinal of I:. There thus exists a permutation of (1, n) such that ~(1:) = I, for all x E X, that is xi = x,(~) for i = 1,. . . , n. Remark. Let S be a subset of X. Recall that we have identified M(S) with a submagma of M(X), Mo(S) with a submonoid of Mo(X) and N"' with a submonoid of NCx). Then f=1 1=1 (23) M(S) = A-'(Mo(S)). Clearly h(M(S)) c Mo(S). Let w E A-I(Mo(S)); we show by induction on l(w) thatwEM(S).Itisobviousifl(w) = l.Ifl(w) > 1,wemaywritew = wlw2 with w,, w2 E M(X), 1 (wl) < 1 (w), 1 (wz) < l(w). Then h(w,)A(w,) E Mo(S), hence h(wl) E Mo(S) and A(w,) E Mo(S), whence w1 E M(S) and w2 E M(S) by the induction hypothesis and finally w E M(S). (24) Mo(S) = v-l(N(')). This follows immediately from formula (2 1). Further, NCs) is the set of elements of N(x) whose support is contained in S; if (Sl),,, is a family of subsets of X of intersection S, then N@) = PI N@i) and formulae (23) and (24) imply Also RINGS fj 8.1 In what follows, (x, y) - x + y denotes addition and (x, y) H xy multi - plication; 0 denotes the identity element for addition and 1 that for multi - plication. Finally, -x denotes the negative of x under addition. The axioms of a ring are therefore expressed by the following identities : (1) x + (y + z) = (x + y) + z (associativity of addition) (zero) (negative) (associativity of multiplication) (unit element) x+y=y+x (commutativity of addition) 0 + x = x + 0 = x x + (-x) = (-x) + x = 0 x.1 = 1.x = x (2) (3) (4) (5) (6) x(y4 = (XY)Z $8. RINGS 1. RINGS DEFINITION 1. A ring is a set A with two laws of composition called respectively addition and multiplication, satisfying the following axiom: (AN I) Under addition A is a Commutative group. (AN 11) Multiplication is associative and possesses an identity element. (AN 111) Multiplication is distributive with respect to addition. The ring A is said to be commutative if its multiplication is commutative. 96 (x +y).z = xz +yz x.(y + 2) = xy + xz } (distributivity) (7) (8) Finally the ring A is commutative if xy = yx for x, y in A. With addition alone A is a commutative group called the additive group of A. For all x E A, we define the left homothety y, and right homothety 6, by y,(y) = xy, 6,(y) = yx. By formulae (7) and (8), y, and 6, are endomor- phisms of the additive group of A and thus map zero to zero and negative to negative. Therefore x.0 = 0.x = 0 (9) (10) (1 1) (-X)(--Y) = XY. x.(-y) = (-x).y = -xy; it follows that ( -x) (-y) = - (( - x) .y) = - ( - xy), whence Formulae (10) and (1 1) constitute the sign rule. It follows that and (-l)(-1) = 1. -x = (-1)x = x(-1) From (1 1) it follows by induction on n that xn ifn is even = { -xn if n is odd. When we speak of cancellable elements, invertible elements, permutable elements, central elements, centralizer or centre of a ring A, all these notions will refer to the multiplication on A. If x, y E A and y is invertible, the element xy -' of A is also denoted by x/y when A is commutative. The set of invertible elements of A is stable under multiplication. Under the law induced by multiplication it is a Soup called the multiplicative group of A, sometimes denoted by A*. kt X, y be in A. x is said to be a lejl (resp. right) multiple of y if there exists Y' E A such that x = y'y (resp. x = yy') ; it is also said that y is a right (resp. 97 I ALGEBRAIC STRUCTURES left) divisor of x. When A is commutative, there is no need to distinguish between "left" and "right". In conformity with the above terminology, every element y E A would be considered as a right and left divisor of 0; but, by an abuse of language, in general the term "right (resp. left) divisor of0" is reserved for elements y such that there exists x # 0 in A satisfying the relation xy = 0 (resp. yx = 0). In other words, the right (resp. left) divisors of zero are the right (resp. left) non- cancellable elements. Let x E A. xis called nilpotent if there exists an integer n > 0 with X" = 0. The element 1 - x is then invertible, with inverse equal to 1 + x + x2 +...+ xn-1. As A is a commutative group under addition, the element nx for n E Z and x E A has been defined (3 2, no. 8). As y, and 6, are endomorphisms of the additive group A, yx(ny) = ny,(y) and su(nx) = nsu(x), whence x. (ny) = (nx) .y = n. (xy). In particular, nx = (n. 1)x. A set A with addition and multiplication satisfying the axioms of a ring with the exception of that assuring the existence of the identity element under multiplication, is called a pseudo -ring. 2. CONSEQUENCES OF DISTRIBUTIVITY Distributivity of multiplication with respect to addition allows us to apply Proposition 1 of 3 3, no. 4, which gives where the sum extends over all sequences (a1,. . . , a,,) belonging to L, x - - . x L, and for i = 1,. . . , n the family (xi, A)AELf of elements of the ring A is of finite support. PROPOSITION 1. Let A be a commutative ring and (x~)~~~ afinite family ofelements $A. For every family ofpositive integers P = ( PA) L, let I P I = sL (3&. Then (14) We apply formula (13) with Li = L and = xa for 1 < i < n. Then the sum extending over all sequences a = (al, . . . , a,,) E L". 3 8.2 Let a be in L"; for all A E L let Ui denote the set of integers i such that 1 < i < n and at = A and let @(a) = (Ut)AEL. It is immediate that is a bijection of L" onto the set of partitions of {I, 2, . . . , n} indexed by L. For all p E NL such that I PI = n, let L; denote the set of a E Ln such that Card U; = PA for all A E L. It follows that the family (Li) Dl =" is a partition of L" and that CONSEQUENCES OF DISTRIBUTIVI'IT and formula (14) thus follows from (15). COROLLARY 1 (binomial formula). Let x and y be two elements of a comniutahe ring A. Then: Formula (14) applied to L = (1,2}, x1 = x and x2 = y gives the sum extending over ordered pairs of positive integers p, q with p + q = n. The binomial formula follows immediately from this (Set Theory, 111, 4 5, COROLLARY 2. Let A be a commutative ring, X a set, u = (uJ, two families of elements of A. Let u + v denote the family (ux + u,),,~. For all A EN") we write A! = n A(x) !. Then for all a E N(X), in the notation of§ 7, no. 8, no. 8). and v = (vx) XEX 98 99 ALGEBRAIC STRUCTURES I For x E X, EXAMPLES OF RINGS 3 8.3 3. EXAMPLES OF RINGS I. Zero ring. Let A be a ring. For 0 = 1 in A, it is necessary and sufficient that A consist of a single element. The condition is obviously sufficient. On the other hand, if 0 = 1, then, for all x E A, x = x. 1 = x.0 = 0. Such a ring is called a zero ring. 11. Ring qf rational integers. With the addition defined in 3 2, no. 5, and the multiplication defined in 3 2, no. 6, Z is a commutative ring. The notation 0, 1, -x is in accordance with the notation introduced earlier. *III. Ring of real-valued functions. Let I be an interval in the set R of real numbers and let A be the set of continuous functions defined on I with real values. The sum f + g and pr0ductj.g of two functions f and g are defined by (f+ g)(t) =f (4 + dt), (fi!)(t) =f (t)g(t) (tE 1). A commutative ring is obtained whose unit element is the constant l., *IV. Convolution pseudo -ring. Let E be the set of real-valued continuous functions on R, which are zero outside a bounded interval. The sum of two functions is defined as in 111, but the product is now defined by by Corollary 1. Taking the product of these equations for x E X and using (1 3), we obtain the corollary. PROPOSITION 2. Let A be a ring, xl, . . . , x, elements ofA and I = (1, 2, . . . , n}. For H c I, we write xH = ,C x,. Then IEH In particular, if A is commutative, (-l)nn!x,x,. . .x, = HCI C (--1)card~(x~~)n. Let C be the set of mappings of I into (0, 1). If each H c I is mapped to its characteristic function, a bijection is obtained of ?$(I) onto C. The right hand side of (16) is thus equal to: where (1) Suppose that (il,. . ., in) is not a permutation of I. There exists aj E I distinct from il, . . ., in. Let C' be the set of a E C such that a(j) = 0. For all a E C', let a* be the sum of a and the characteristic function of {j}. Then a*(1) +a + a*(n) = a(1) +. . . + a(.) + 1 and hence - - C ((-l)a(l)+...+(n) + (-l)a(l)+...+a(n)+l )a(il) . . .a(in) = 0. asC' (2) Suppose that there exists ts E 6, such that il = ~(l), . . .,in = ts(n). Then a(il). . .a(in) = 0 unless a only takes the value 1. Thus cil,..*, = (- 1)". 100 ("convolution product "). Thus a commutative pseudo-ring is obtained which is not a ring (cf. Integration, VIII, 34).* V. OFposite ring Ofa ring A. Let A be a ring. The set A with the same addition as A and the multiplication (x, y) H yx is often denoted by AD. It is a ring (called the opposite ring of A) with the same zero and same unit as A and which coincides with A if and only if A is commutative. VI. Endomorphism ring of a commutative group. Let G be a commutative group written additively. Let E denote the set of endomorphisms of G. Given f and g in E, the mappings f + g and fg of G into G are defined by By $1, no. 5, Proposition 5, f + g is an endomorphism of G and so obviously also is fg =fog. By 5 4, no. 8, E is a (commutative) group under addition. Multiplication is obviously associative and has identity element Id,. Also for f, g and h in E, we write + =f.(g + h); for all x E G, forf is an endomorphism of G; hence 4 = fi + fh and dearly (f + g)(x) =f (4 + ,.(XI, (fa (4 =f (g(x)) (2 E G). +(XI =f(k + h)(x)) =S(g(x) + hb)) =f(g(x)) +f (h(x)) (g + h)f= g_f+ hf. =" 101 ALGEBRAIC STRUCTURES I IDEALS 3 8.6 Therefore E is a (not in general commutative) ring called the endomorphism ring of G. VII. Pseudo -ring of zero square. A pseudo-ring A is said to be of zero square if XY = 0 for all x, y E A. Let G be a commutative group. If the set G is given the addition of the group G and multiplication (x, y) H 0, a pseudo-ring of zero square is obtained. It is only a ring if G = (01, in which case it is the zero ring. 4. RING HOMOMORPHISMS DEFINITION 2. Let A and B be two rings. A morphism, or homomorphism, of A into B is any mapping f ofA into B satisjying the relations: for all x, y in A. The composition of two ring homomorphisms is a ring homomorphism. Let A and B be two rings and f a mapping of A into B; for f to be an isomorphism, it is necessary and sufficient that it be a bijective homomorphism; in that case, f is a homomorphism of B into A. A homomorphism of a ring A into itself is called an endomorphism of A. Let f: A -+ B be a ring homomorphism. The mapping f is a homomorphism of the additive group of A into the additive group ofB; in particular, f (0) = 0 and f (-x) = - f (x) for all x E A. The image under f of an invertible element of A is an invertible element of B and f induces a homomorphism of the multiplicative group of A into the multiplicative group of B. -1 Examples. (1) Let A be a ring. It is immediately seen that the mapping n H n. 1 of Z into A is the unique homomorphism of Z into A. In particular, the identity mapping of Z is the unique endomorphism of the ring Z. In particular, take A to be the endomorphism ring of the additive group Z (no. 3, Example VI). The mapping n H n. 1 of Z into A is an isomorphism of Z onto A by the very construction of multiplication in Z (5 2, no. 6). (2) Let a be an invertible element of a ring A. The mapping x H axa-' is an endomorphism of A for a(x + y)a-l = axa-l + aya-l, a(xy)a-' = (axa-')(aya-l). It is bijective, for the relation x' = nxa-l is equivalent to x = a-lx'a. It is therefore an automorphism of the ring A, called the inner automorphism asso- ciated with a. 5. SUBRINGS DEFINITION 3. Let A be n ring. A subring of A is any subset U of A which is a subgrouk ofA under addition, which is stable under mu1ti;blication and which contains the unit ofA. The above conditions may be written as follows OEB, B+BcB, -B cB, B.BcB, ~EB. If B is a subring of A, it is given the addition and multiplication induced by those on A, which make it into a ring. The canonical injection of B into A is a ring homomorphism. Examples. (1) Every subgroup of the additive group Z which contains 1 is equal to Z. Thus Z is the only subring of Z. a family of subrings of A; it is immediate that n A, is a subring of A. In particular, the intersection of the subrings of A containing a subset X of A is a subring called the subring of A generated by X. (3) Let X be a subset of a ring A. The centralizer of X in A is a subring of A. In particular, the centre of A is a subring of A. (4) Let G be a commutative group with operators; let R denote the set of operators and a -fa the action of R on G. Let E be the endomorphism ring of the group without operators G and F the set of endomorphisms of the group with operators G. By definition, F consists of the endomorphisms 4 of G such that +.fa =fa.+ for all a E R. Therefore F is a subring of the ring E. F is called the endomorphism ring of the group with operators G (cf, 11, 3 1, no. 2). Let F, be the subring of E generated by the fa. Then F is the centralizer of F, in E. (2) Let A be a ring and (AJL LEI 6. IDEALS DEFINITION 4. Let A be a ring. A subset a ofA is called a left (resp. right) ideal if it a subgroup of the additive group of A and the relations a E A, x E a imply ax E: a (resp. xu E a). a is called a two-sided ideal ofA ifit is both a l~t ideal and a right ideal of A. The definition of a left ideal may be expressed by the relations OE~, afaca, A.aca the relation -a c a following from the formula ( - 1) . x = -x and A. a c a. For all x E A, let yx be the mapping a- xu of A into A; the action x ++ yx g;ves the additive group A+ of A the structure of a group with operators with A as set of operators. The left ideals of A are just the subgroups of A+ which are stable under this action. The left ideals in the ring A are just the right ideals of the opposite ring A'. In a commutative ring the three species of ideals are the same; they are simply aUed ideals. 102 103 ALGEBRAIC STRUCTURES I 3 8.7 let (ah)AeL be a family with finite support in A. Then ahxh E a’ for all A E L, whence c ahxh. E a’ ; hence a c a’. Hence a is the left ideal of A generated by the x,. The argument for b is analogous. PROPOSITION 4. Let A be a ring and (aA)hEL a family of left ideals of A. The left ideal generated by ,VL ah consists ofthe sums zLyA where (yh)AeL is a family withjnite sup- port such that y, E a, for all Let a be the set of sums 2 Y, with Y,, E a, for all h E L. The formulae QUOTIENT RINGS AsL E: L. ICT. Examfiles. (1) Let A be a ring. The set A is a two-sided ideal of A; so is the set consisting of 0, which is called the zero ideal and sometimcs denoted by 0 or (0) instead of (0). (2) For every element a of A, the set A.a of left multiples of a is a left ideal; similarly the set a.A is a right ideal. When a is in the centre of A, A.a = a.A; this ideal is called the principal ideal generated by a and is denoted by (a). (a) = A if and only if a is invertible. (3) Let M be a subset of A. The set of elements x E A such that xy = 0 for all y E M is a left ideal of A called the left annihilator of M. The right annihilator of M is defined similarly. (4) Every intersection of left (resp. right, two-sided) of A is a left (resp. right, two-sided) ideal. Given a subset X of A, there thus exists a smallest left (resp. right, two-sided) ideal containing X; it is called the left (resp. right, two-sided) ideal generated by X. Let a be a left ideal of A, The conditions 1 $ a, a # A are obviously equiva - lent. DEFINITION 5. Let A be a ring. By an abuse of language, a left ideal a is said to be maximal ifit is a maximal element of the set of left ideals distinct from A. In other words, a is maximal if a # A and the only left ideals of A containing a are a and A. THEOREM 1 (Krull). Let A be a ring and a a ldt ideal of A distinct from A. There exists a maximal ideal m of A containing a. Consider A as operating on the additive group A+ of A by left multiplication. Then the left ideals of A are the stable subgroups of A+. The theorem thus follows from Ej 4, no. 3, Proposition 3 applied to the subset P = (1) of A+. PROPOSITION 3. Let A be a ring, (x,)~~~ a family of elements of A and a (resp. 6) the set of sums zL ahxh where (a,) , is a family withjnite support of elements ofA (resp. 2 a x b where (a,JAoL, (bh)hsL are families withjnite support of elements ofA). Then a (resp. b) is the left (resp. two-sided) ideal ofA generated by the elements xA. AEL A The formulae (18) 0 = ZL0.Xh 2 aAXA + c a~xA = , c (ah + a~)xx a. , c ~AXA = c (aah)~~ (19) AEL (20) prove that a is a left ideal. Let a’ be a left ideal such that x, E a’ for all A E L and 104 .._I 2 xh + zLy, = zL (%A + y~) and a. zL X, = zL ax, shows that a is a left ideal of A. Let AEL and x €a,; write y, = x and yu = 0 for p # A; then h€L Y- x = A+L y,, whence x E a and finally a, c a. If a left ideal a’ contains a, for all 1 E L, it obviously contains a and hence a is generated by ,vL a,. The ideal a generated by ,YL a, is called the sum ofthe left ideals a, and is denoted by zL a, (cf. 11, 3 1, no. 7). In particular, the sum a1 + a2 of the two left ideals consists of the sums a, + a2 where a, E a, and a2 E a2. 7. QUOTIENT RINGS Let A be a ring. If a is a two-sided ideal of A, two elements x and y of A are said to be congruent modulo a, written x = y (mod. a) or x = y(a), if x - y E a. This is an equivalence relation on A. The relations x = y(a) and x’ = ~‘(a) imply x + x‘ = y + y’(a), xx’ = xy’(a) for a is a left ideal and xy’ = yy’(a) for a is a right ideal, whence xx’ = yy’(a). Conversely, if R is an equivalence relation on A compatible with addition and multiplication, the set a of x = 0 mod. R is a two-sided ideal and x = y mod. R is equivalent to x = y mod. a. Let A be a ring and a a two-sided ideal of A. A/a denotes the quotient set of A by the equivalence relation x = y(a), with addition and multiplication the quotients ofthose on A (5 1, no. 6, Definition 1 1). We show that A/a is a ring: (a) Under addition, A/a is the quotient commutative group of the additive group ofA by the subgroup a. (b) Under multiplication, A/a is a monoid (4 2, no. 1). (c) Let E,, q, < be in A/a and let x: A -+ A/a be the canonical mapping; we choose elements x, y, tinA such that x(x) = E,, ~(y) = q and n(z) = <. Then E(? + C) = ..(x)x(y + z) = x(x(y + z)) = x(xy + XZ) = x(x)x(y) + x(x)44 = + E< and the relation (< + q)< = << + q< is established similarly. 105 I ALGEBRAIC STRUCTURES DEFINITION 6. Let A be a ring and a a two-sided ideal of A. The quotient ring of A by a, denoted by A/a, is the quotient set of A by the equivalence relation x = y(a), with addition and multiplication the quotients of those on A. The ring A/{O} is isomorphic to A and A/A is a zero ring. (a) The canonical mapping x of A onto A/a is a ring homomorphism. (b) Let B be a ring and f a homomorphism of A into B. Iff (a) = {0}, there exists By construction, x(x + y) = X(X) + x(y) and x(xy) = x(x)x(y) for x, y THEOREM 2. Let A be a ring and a a two-sided ideal of A. one and only one homomorphism f of Ala into B such that f = f o x. $8.9 (c) If 6' is a two-sided ideal of A', the composite mapping of the canonical morphism A' -+ A'/b' and$ A + A' dejines, when passing to the quotient, an injective morphismf ofA/b into A'/b'. Iff is surjective,fis an isomorphism ofA/b onto A'/b'. (d) Suppose f is surjective. Let CD be the set of subrings (resp. l~t ideals, right ideals, two-sided ideals) of A containing a. Let CD' be the set of subrings (resp. le$t ideals, right bijections of@ onto CD' and CD' onto 0. MULTIPLICATION OF IDEALS ideals, two-sided ideals) of A'. The mappings B + f (B) and B' ++ f -1 (B') are inverse (a) and (b) are obvious, except the last assertion of (a) which follows from no. 7, Theorem 3. The composite morphism g: A -+ A' -+ A'/b' considered in (c) has kernel b and henceJis an injective morphism of A/b into A'/b' (3 8, no. 7, Theorem 3). Iff is surjective, g is surjective and hencefis surjective. mapping of CD' into CD. Clearly the mapping y1: B H f (B) is a mapping of CD into CP'. Then 0 o y1 = Id,, q 0 8 = Id,,, whence (d). Remark. In the above notation, 0 and y1 are ordered set isomorphisms (CD and @' being ordered by inclusion). COROLLARY . Let A be a ring and a a two-sided ideal of A. b/a, where b is a lejit (resp. right, two -sided) ideal of A containing a. whnpassing to the quotient an isomorphism ofA/b onto (A/a)/(b/a). A/a. Suppose f is surjective. By the above, the mapping 0: B'H f -1 (B') is a (a) Every left (resp. right, two -sided) ideal nfA/a can be written uniquely in the form (b) If b is two-sided, the composite homomorphism A -+ A/a + (A/a)/(b/a) deJines It suffices to apply Proposition 5 to the canonical morphism of A onto in A; also x(1) is the unit E of A/a, whence (a). Let A+ be the additive group of A and B + that of B ; as f is a homomorphism of A+ into B+, zero on the subgroup a of A+, there exists ($ 4, no. 4, Proposition 5) one and only one homomorphismfof A+/a into B+ such that f =fo x. Let c, q be in A/a; choose x, y in A with x(x) = 5 and x(y) = 3; then Eq = ~~(xy), whence f(Ed = f(+Y)) = f (XY) = f (4 .f (Y) = fcn fh) andf(c) = f (x(1)) = f (I), hencefis a ring homomorphism. THEOREM 3. Let A and B be rings and f a homomorphism of A into B. (a) The kernel a off is a two-sided ideal of B. (b) The image B' = f (B) off is a subring of B. (c) Let x: A -+ A/a and i: B' -+ B be the canonical morphisms. There exists one and only one morphismfof A/a into B' such that f = i of0 x andfis an isomorphism. As f is a morphism of the additive group of A into that of B, a is a subgroup of A. If x E a and a E A, then f (ax) = f (a) f (x) = 0, hence ax E a and similarly xu E a; hence a is a two-sided ideal of A. Assertion (b) is obvious. As f is zero on a, there exists a morphismJof A/a into B' such that f = i of. x (Theorem 2). The uniqueness of f and the fact that f is an isomorphism follow from Set Theory, 11, $ 6, no. 4. 8. SUBRINGS AND IDEALS IN A QUOTIENT RING PROPOSITION 5. Let A and A' be two rings, f a homomorphism ofA into A' and a the kernel off. (a) Let B' be a subring of A'. Then B = f (B) is a subring of A containing a. Iff is surjective, then f (B) = B' and f I B defines when passing to the quotient an iso- morphism of B/a onto B'. (b) Let b' be a left (resp. right, two -sided) ideal ofA'. Then b = jl(b') is a left (resp. right, two-sided) ideal of A containing a. 106 -1 9. MULTIPLICATION OF IDEALS Let A be a ring and a and b two-sided ideals of A. The set of elements of the form x,y, + . . . + x,y, with n > 0, xi E a and yi E b for 1 < i < n, is obviously a two-sided ideal of A, which is denoted by ab and called the product of the two- sided ideals a and 6. Under this multiplication, the set of two-sided ideals of A is a monoid with unit element the two-sided ideal A. If a, b, c are two-sided ideals ofA, then a(b + c) = ab + ac, (b + c)a = ba + ca. IfA is commutative, multiplication of ideals is commutative. ab C aA c a and ab c Ab c b, hence ab c a n b. PROPOSITION 6. Let a, b,, . . ., 6, be two-sided ideals of A. IfA = a + 6, for all i,hzA = a + b,b ,... b,, = a + (6, n b,n.-.n b,,). 107 PRODUCT OF RINGS 5 8.10 Suppose that a, is a two-sided ideal for all i E I and let& denote the canonical mapping of A, onto A,/ai. Then the mapping$ (xJtEI H (&(xi))iEI of n ,€I At onto n (Ai/ai) is a ring homomorphism of kernel n a, and hence defines ,€I ,€I I ALGEBRAIC STRUCTURES By (21) it suffices to prove that A = a + b,b, . . . 6,. By induction, it suffices to consider the case where n = 2. By hypothesis, there exist a, a’ E a, b1 E b,, b, E b2 such that 1 = a + 6, = a’ + b,. Then 1 = a’ + (a + b,)b, = (a’ + ab,) + b,b, E a + b1b2, whence A = a + bib,. PROPOSITION 7. Let b,, . . . , 6, be two-sided ideals ofA such that b, + bj = A for i # j. Then b, n b, n. . . n b, = 2 bo(l)bo(2). . . bo(n). In particular, ;f A is commutative, b, n 6, n. . . n 6, = b,b,. . . 6, (cf. Exercise 2). Suppose first n = 2. There exist a, E b,, a, E bz such that a, + a, = 1. If x E bl n b,, then Hence 0.56, x = .(al + az) = xu, + xu, E b,b, + b,b,. b, n 6, = bib, + 6261. Suppose now the equation of the proposition is established for all integers c c br(l)br(2). . . b,(,) c 6, n 6, n. . . n 6,. ’~€6~ 10. PRODUCT OF RINGS Let (AJ,,, be a family of rings. Let A be the product set fl ,€I A,. On A addition and multiplication are defined by the formulae (22) (xi) + (Yi) = (xi + ~i), (xi)(.Yi) = (XiYi). It is immediately verified that A is a ring called the product of the rings A,, with zero the element 0 = (Oi),.51, where 0, is the zero of A,, and unit 1 = (li),€I, where 1 , is the unit of A,. If the A, are commutative, so is A. If C, is the centre of A,, the centre of A is For all i E I, the projection pr, of A onto A, is a ring homomorphism. If B is a ring andfi: B --f A, is a family of homomorphisms, there exists a unique homomorphism $ B --f A such that ft = pr, 0 f for all i E I; it is given by For all i E I, let ai be a left ideal of Ai. Then a = n 1EI a, is a left ideal of A. There is an analogous result for right ideals, two -sided ideals and subrings. C,. f (b) = (A(b))iSI* when passing to the quotient an isomorphism of - C,! (Ai/ai)* Let (Ih)h.5L be a partition of I. The canonical bijection of n A, onto (,g A,) is a ring isomorphism, under which these two rings are identified. Let J c I. Let e, denote the element (xi)iEI ofA defined by xi = 1, for i E J, X, = 0, for i E I - J. Then e, is a central idempotent (3 1, no. 4) of A. The following formulae follow immediately : ,€I e, = 1; eJnK e, = = 0; eJeK eJuK = e, + e, for J c I, K c I; for J c I, K c I, J n K = 0; if (Jh) is a finite partition of I. 7 e,). = 1 Let A - n -4,. Let -qJ be the canonical projection of A onto A,. For x = (xi),€ J E A,, let E,(x) be the element (yi)iEI of A defined by y, = xi for i E J, y, = 0, for i E I - J. Then -qJ is a ring homomorphism of A onto A,, E, is an injective homomorphism of the additive group of A, onto A and in the diagram A, A % AI-, the kernel a, of qr- J is equal to the image of Ej. Then E,(xx’) = E,(x)E,(x’) for all X, X’ €A,; but E, is not in general a ring homomorphism for ~~(1) = e,. Clearly a, = e,A = Ae,. - iSJ Let qt) = e, and a, = a{,) = e,A = Ae, for all i E I. Then e; = e,, eie, = e,e, = 0 for i # j. If I is finite, then ,z ei = 1, the additive group A is the direct sum Of the two-sided ideals a, and if x E A its component in a, is xei. The following Proposition is immediately deduced. POSITION 8. Suppose I isJinite. If b is a left or right ideal of A, b is the direct sum ofthe b n a,. 108 109 ALGEBRAIC STRUCTURES I 3 8.1 1 DIRECT DECOMPOSITION OF A RING 11. DIRECT DECOMPOSITION OF A RING Let A be a ring and (6,) , I a family of two-sided ideals of A. We shall call the homomorphism where +, is the canonical homomorphism of A onto A/b,, the canonical homo - morphism of A into n ,€I (Alb,). PROPOSITION 9. Let A be a ring and (bl, . . . , b,) two-sided ideals n 0s A such that 6, + 6, = A for i # j. The canonical homomorphism of A into n i=l (A/b,) is surjective ofkerneltfil bt = o& bo(l)bo~2). . .ba(n)- Clearly the kernel is ,nl 6,. To prove the surjectivity, it is necessary to show that, for every family (xi)l <,<, of elements of A, there exists x E A such that x z xi (6,) for all 1 < i < n. We prove this assertion by induction on the cardinal n of I, the case n 6 1 being trivial. By the induction hypothesis, there exists y E A such that y z xi (6,) for 1 < i < n - 1. We look for an x of the form y + z with z E A. Of necessity z = 0 (6,) for i < n, that is z E b = ,=1 n b,, and on the other hand z = x, - y (6,). Now b, + b = A by no. 9, Proposition 6, whence the existence of z. Finally, the second expression of the kernel follows from no. 9, Proposition 7. DEFINITION 7. Let A be a ring. A jnite family (b,),,I of two-sided ideals of A such that the canonical homomorphism ofA into a (A/b,) is an isomorphism is called a direct decomposition of A. PROPOSITION 10. Let A be a ring, A' its centre and ( b,), I afinite family of two-sided ideals of A. The following conditions are equivalent: x ++ (+i(x))iEI, n a-1 (a) the family (b,)iEI is a direct decomposition of A; (b) there exists a family (ei)i,I of idempotents of A' such that e,e, = 0 for i # j, 1 = e, and b, = A(l - e,) for i E I; (c) 6, + 6, = A for i # j and GI b, = (0); (d) b, + b, = A for i # j and (e) there exists a direct decomposition (bi),EI of A' such that 6, = Abi for i E I. (a) 3 (b). If condition (a) holds, Amay be identified with the ring a (A/b,) ie1 6, = {O}for every total order on I; and 6, with the kernel of pr,. The existence of the e, with the properties in (b) then follows from no. 10. (b) * (d). Suppose that the e, exist with the properties in (b). For i #j, 1 - e, E b,, el = e,(l - ej) E bj, hence 1 E bi + b, and A = 6, + 6,. On the other hand, if I is given a total ordering and (xJtSI is a family of elements ofA, then, since the ei are central, hence 6, = (0). (d) 3 (c). This follows from no. 9, Proposition 7. (c) + (a). This follows from Proposition 9. Thus conditions (a), (b), (c) and (d) are equivalent. Suppose they hold. By (b) + (a), the family of 6; = A'( 1 - e,) is a direct decomposition of A'. Then 6, = A(l - e,) = Ab: for all i E I. Hence condition (e) holds. Finally, suppose condition (e) holds. By (a) => (b), there exists a family (el)1eI of idempotents of A' such that e,e, = 0 for i #j, 1 = 2 e, and (b) holds. hark. Let A be a ring. Let (a,), I be a finite family of subgroups of the addi - tive group A+ of A such that A+ is the direct sum of the a,. Suppose a,a, c ai for i E I and a,a, = (0) for i # j. Then a, is for all i E I a two-sided ideal of A. With addition and multiplication induced by those on A, a, is a ring with unit element the component of 1 E A in a,. If 6, = 2 aj, clearly the 6, satisfy con- which is said to be dejned by Example: Ideals and quotient rings of Z An ideal of Z iy an additive subgroup of Z and hence of the form n . Z with > 0; conversely, for every integer n 0, the set n .Z is an ideal, the principal Ideal (n). Thus every ideal of Z is principal and is represented uniquely in the form nZ with n 2 0. The ideal (1) is equal to Z, the ideal (0) consists of 0 and the ideals distinct from Z and (0) are therefore of the form nZ with n > 1. If m > 1 and n 1, mZ 2 nZ if and only if n E m.Z, that is m divides n. There - fore, for the ideal nZ to be maximal, it is necessary and sufficient that there exist no integer m > 1 distinct from n and dividing n; in other words, the maximal tdeals OfZ are the ideals of the form pZ where p is a prime number (4 4, no. 10, Definition 16). Let m and n be two integers 2 1. The ideal mZ + nZ is principal, whence there is an integer d 1 characterized by dZ = mZ + nZ; for every integer b( = A'( 1 - e,) for i E I. Then b, = Abi = A( 1 - e,) for i E I, hence ,€I condition dition (c) of Proposition 10 and hence (b,),,, ,#I is a direct decomposition of A, 110 111 ALGEBRAIC STRUCTURES I r 2 1, the relation “r divides d” is equivalent to rZ 2 dZ and hence to C‘rZ 3 mZ and rZ 2 nZ’, that is to “r divides m and n”. It is thus seen that the COmOn divisors of m and n are the divisors of d and that d is the greatest of the divisors 1 common to m and n; d is called the greatest common divisor (abbrevi - ated to g.c.d.) of m and n. AS dZ = mZ + nZ, there exist two integers x and y such that d = mx + ny. m and n are said to be relativelyprime if their g.c.d. is equal to 1. It amounts to the same to assume that there exist integers x and y with mx + ny = 1. The intersection of the ideals mZ and nZ is non-zero for it contains mn and hence is of the form rZ with r 2 1. Arguing as above, it is seen that the multiples ofr are the common multiples of m and n and that r is the least of the integers 2 1 which are common multiples of m and n; it is called the least common mult$le (1.c.m.) of m and n. r The product of the ideals mZ and nZ is the set of 1=1 2 mx,nyl = mn for xl, . . . , yT E Z and hence is equal to mnZ. For every integer n 2 1, the quotient ring Z/nZ is called the ring of integers modulo n; it has n elements, which are the classes modulo n of the integers 0, 1,2,. . ., n - 1. For n = 1, we obtain the zero ring. PROPOSITION 1 1. Let n,, . . . , nd be integers 2 1 which are relatively prime in pairs and n = n, . . . n,. The canonical homomorphism of Z into the product ring n i=1 Z/nlZ is surjective of kernel nZ and dejines a ring isomorphism of Z/nZ onto n 1=1 Z/n,Z. r r Let a, = n,Z for i = 1,. . ., r. By hypothesis, a, + a, = Z for i #j. The proposition then follows from Proposition 9. The above results, as also those concerning decomposition into prime factors, will be generalized in Chapter VII, Q 1, which is devoted to the study of princi- pal ideal domains and in Commutative Algebra, Chapter VII, 3 3, which is de- voted to the study of factorial domains. 12. RINGS OF FRACTIONS THEOREM 4. Let A be a commutative ring and S a subset of A. Let A, be the monoid of fractions of A (provided only with multiplication) with denominators in S (Q 2, no. 4). Let E: A --f A, be the canonical morphism. There exists on A, one and only one addition satishing the following conditions: (a) As, with this addition and its multiplication, is a commutative ring; (b) E is a ring homomorphism. Suppose an addition has been found for A, satisfying conditions (a) and (b) . 3 8.12 Let x, y E A,. Let S’ be the stable multiplicative submonoid of A generated by s. There exist a, b E A and p, q E S‘ such that x = alp, y = b/q. Then RINGS OF FRACTIONS x = E(aq)E(Pq)-l, Y = E(bP)E(Pd-l, whence (23) x + Y = (44 f E(bP))E(PQ)-l = E(aq + bP)E(Pd -l = (aq + bP)/Pq. This proves the uniqueness of the addition. We now deJine an addition on A, by setting x + y = (aq + bp)/pq. It is necessary to show that this definition does not depend on the choice of a, b, p, q. Now, if a’, b’ E A,p’, q‘ E S’ are such that x = a’/p’, y = b‘/q’, there exist s and t in S‘ such that ap‘s = a’ps, bq’t = b’qt, whence (aq + bP)(P’q‘)(st) = (a’q’ + b’p’)(Pq)W (aq + bP)IPq = (a’q‘ + b‘p’)/P’q’* and hence It is easily verified that addition in A, is associative and commutative, that O/l is identity element for addition, that (-a)/p is the negative of a/# and that x(y + z) = xy + xz for all x, y, z E A,. If a, 6 E A, then E(U + 6) = (a + b)/l = all + b/l = E(a) + E(b) and hence E is a ring homomorphism. DEFINITION 8. The ring dejined in Theorem 4 is called the ring of fractions associated with S, or with denominators in S, and is denoted by A[S-l]. The zero of A[S-l] is 011, the unit of A[S-l] is 111. We shall return to the properties of A[S-l] in Commutative Algebra, Chapter IfS is the set of cancellable elements of A, the ring A[S-l] is called the total ring of fractions of A. A is then identified with a subring of A[S - ‘3 by means of the mapping E, which is then injective (I, 3 2, no. 4, Proposition 6). 11, 0 2. THEOREM 5. Let A be a commutative ring, S a subset $A, B a ring and f a homomor- PhLm of A into B such that every element off (S) is invertible. There exists one and only oWfofA[S-l] into B such that f =Yo E. We know (3 2, no, 4, Theorem 1) that there exists one and only one mor- phism$ of the multiplicative monoid A[S - ‘1 into the multiplicative monoid such that f =Jo E. Let a, b E A, p, q E S’ (stable multiplicative submonoid 113 112 FIELDS 9 9.1 THEOREM 1. Let A be a ring. The following conditions are equiualetit : (a) A is afield; (b) A is not reduced to 0 and the only le$t ideals of A are 0 and A. Suppose that A is a field. Then A is not reduced to 0. Let a be a left ideal of A distinct from 0. There exists a non-zero a belonging to a. For a11 x E A, x = (xa-l)a E a; hence a = A. Suppose that A satisfies condition (b). Let x # 0 be in A. We need to prove that x is invertible. The left ideal Ax contains x and hence is not zero, whence Ax = A. There thus exists x’ E A such that x‘x = 1. Then x’ # 0 since 1 # 0. Applying the above result to x’, it is seen that there exists X” E A such that x”x’ = 1. Then X” = x”. 1 = X”X’X = 1 .x = x, hence xx’ = 1. Hence x’ is the inverse of x. ALGEBRAIC STRUCTURES I $9. FIELDS 1. FIELDS DEFINITION 1. A ring K is called afield if it does not consist only of 0 and every non-zero element of K is invertible. The set of non-zero elements of the field K, with multiplication, is a group, which is precisely the multiplicative group K* of the ring K (5 8, no. 1). The opposite ring of a field is a field. A field is called commutative if its multiplication is commutative; such a field is idcntified with its opposite. Anon -commutative field is sometimes called a skewjcld. Examples. *(1) We shall define in no. 4 the field of rational numbers; In General Topology the field of real numbers will be defined (General Topology, IV, Q 1, no. 3), also the field of complex numbers (General Topology, VIII, Q 1, no. 1) and the field of quaternions (General Topology, VIII, 5 1, no. 4). These fields are commutative with the exception of the field of quaternions., (2) The ring 2/22 is obviously a field. Let K be a field. Every subring L of K which is a field is called a subjeld of K and K is then called an extensionfiefield of L; it amounts to the same to say that L is a subring of K and that x-l E L for every non -zero element x of L. If (LJtE1 is a family of subfields of K, then tQl L, is a subfield of K; for every subset X of K there thus exists a smallest subfield of I( containing X; it is said to be generated by X. PROPOSITION 1. Let K be ajeld. For euery subset X of K, the centralizer (9 8, no. 5, Example 3) X’ of X is a subfield of K. We know (loc. cit.) that X is a subring of K. On the other hand, if x # 0 is permutable with z E X, so is x-l (I, 3 2, no. 3, Proposition 5) and hence X’ contains the inverse of every non -zero element of X. COROLLARY . The centre of ajield K is a (commutative) subjeld of K. Remark. In Theorem 1, left ideals may be replaced by right ideals. In Chapter VIII, 9 5, no. 2, we shall study non-zero rings A which have no two-sided ideal distinct from 0 and A; such rings (called quasi-simple) are not neces - sarily fields *(for example, the ring M,(Q) of square matrices of order 2 with rational cocficients is quasi -simple but is not a field),. COROLLARY 1. Let A be a ring and a a two-sided ideal of A. For the ring A/a to be a Jield, it is necessary and su@cient that a be a maximal le$t ideal of A. The left ideals of A/a are of the form b/a where b is a left ideal of A containing a (3 8, no. 8, Corollary to Proposition 5). To say that A/a # 0 means that a # A. Under this hypothesis, A/a is a field if and only if the only left ideals of A containing a are a and A (Theorem l), whence the corollary. COROLLARY 2. Let A be a commutative ring which is not 0. There exists a homo- moqhism of A onto a commutatiuefield. By Krull’s Theorem (3 8, no. 6, Theorem l), there exists in A a maximal ideal a. Then A/a is a field (Corollary 1). COROLLARY 3. Let a be an integer 0. For the ring Z/aZ to be ajield, it is necessary and sujicient that a be prime. This follows from Corollary 1 and 4 8, no. 11. For fi prime the field ZlpZ is denoted by F,. THEOREM 2. Let K be afield and A a non-zero ring. Iff is a homomorphism of K into A, then the subring f (K) of A is afield and f dejines an isomorphism ofK onto f (K). Let a be the kernel off. Then 1 6 a for f (1) = 1 # 0 in A and, as a is a left ideal of K, a = (0) by Theorem 1. Thereforefis injective and hence an iso- morphism of K onto the subring f (K) of A; the latter ring is therefore a field. 115 114 3 9.4 THE FIELD OF RATIONAL NUMBERS The implications (c) => (b) * (a) are obvious. If A/p is an integral domain, let f be the canonical injection of A/?' into its field of fractions and g the canonical homomorphism of A onto A/p ; then p is the kernel off 0 g, whence the implication (a) 3 (c). DEFINITION 3. In a commutative ring A, an ideal p satisjying the conditions of Proposi - tion 4 is called a prime ideal. Examples. (1) Let A be a commutative ring. If m is a maximal ideal of A, m is prime; the ring Aim is a field (no. 1, Corollary 1 to Theorem 1). (2) If A is an integral domain, the ideal (0) of A is prime (but not maximal in general, as the example of the ring Z proves). 4. THE FIELD OF RATIONAL NUMBERS DEFINITION 4. Thejeld of fractions of the ring Z of rational integcrs is called thejield of rational numbers and is denoted by Q. The elements of Qare called rational numbers. Every rational number is thus of the form a/b where a and b are rational integers with b # 0 (and we may even take b > 0 as the relation a/b = (-a)/(-b) proves). Q+ is used to denote the set of rational numbers of thc form a/b with a E N and b E N*. We have the relations: Q+ + Q+ = Q+ Q+.Q+ = Q+ Q+ n (-Q+) = (0) Q+ U (-Q+) = Q (5) The first two follow from the formulae a/b + a'/b' = (ab' + a'b)/bb', (alb)(a'/b') = aa'/bb', 0 E Q+, 1 E Q+ and the fact that N is stable under addition and multiplication and N* under multiplication. Obviously 0 E Q+, whence 0 E (-Q+); let x be in Q+ n (-Q+); then there exist positive integers a, b, a', b' with b # 0, 6' # 0 and x = alb = -a'/b'; then ab' + ba' = 0, whence ab' = 0 (for ab' > 0 and bar > 0) and therefore a = 0 since b' # 0; in other words, x = 0. Finally, obviously N c Z and N c Q+. Conversely, if x belongs to Z n Q+, it is a rational integer; there exist two rational integers a and b with a 0, b > 0 and x = a/b, whence a = bx; if *$N, then -x > 0, whence -a = 6(-x) > 0 and consequently a < 0 Given two rational numbers x and y, we write x < y if y - x E Q+. It is easily deduced from formulae (I), (3) and (4) that x < y is a total ordering on (1) (2) (3) (4) Q+ n Z = N. to the hypothesis. ALGEBRAIC STRUCTURES I 2. INTEGRAL DOMAINS DEFINITION 2. A ring A is called an integral domain (or a domain Ofintegrity) fit is commutative, non -zero, and the product of two non-zero elements of A is non-zero. The ring Z of rational integers is an integral domain; it is commutative and non-zero; the product of two integers > 0 is non-zero; every non -zero integer is ofthe form a or -a with a > 0 and (-a)b = -ab, (-a)(-b) = ab for a > 0, b > 0, whence our assertion. Every commutative field is an integral domain. A subring of an integral domain is an integral domain. In particular, a subring of a commutative field is an integral domain. We shall show that conversely every integral domain A is isomorphic to a subring of a commutative field. Recall (3 8, no. 12) that A has been identified with a subring of its total ring of fractions. Our assertion then follows from the following proposition : PROPOSITION 2. If A is an integral domain, the total ring of fractions K of A is a commutative ring. The ring K is commutative. It is non-zero since A # (0). As A is an integ - ral domain, every non -zero clement of A is cancellable and K consists of the fractions a/b with b # 0. Now a/b # 0 implies a # 0 and the fraction bla is then inverse of alb. The total ring of fractions of an integral domain is called itsfield of fractions. Such a ring is identified with its image in its field of fractions. PROPOSITION 3. Let B be a non-zero ring and A a commutative subring of B such that every non-zero element ofA is invertible in B. (a) A is an integral domain. (b) Let A' be thejeld of fractions A. The canonical injection of A into B can be (c) The elements off (A') are the xy-l where x E A, y E A, y # 0. Assertion (a) is obvious. The canonical injection of A into B extends uniquely to a homomorphismfof A' into B (3 8, no. 12, Theorem 5). Assertion (b) then follows from no. 1, Theorem 2. The elements of A' are the fractions x/y with x E A, y E A, y # 0 andf(x/y) = xy-l, whence (c). extended uniquely to an isomorphism f of A' onto a subjeld of B. 3. PRIME IDEALS PROPOSITION 4. Let A be a commutative ring and p an ideal of A. The following conditions are equivalent : (a) the ring A/p is an integral domain; (b) A # p and, ifx~A - p andy €A - p, thenxy EA - p. (c) p is the kernel of a homomorphism of A into ajeld. 116 117 ALGEBRAIC STRUCTURES I 4 10.2 INVERSE LIMITS OF ACTIONS inverse limit magma of /he magmas E,. It enjoys the following universal pro- perty : (a) For all c1 E I, the canonical mappingf, of E into E, is a magma homo - morphism of E into E,.f, =fap ofD for a < p. (b) Suppose a magma F is given and homomorphisms u, : F -+ E, such that u, = fa, 0 up for a < p. There exists one and only one homomorphism u: F -+ E such that u, =fa 0 u for all c( E I (namely x ++ ~(x) = (U,(X)),~~). If the magmas E, are associative (resp. commutative), so is E. Suppose that each magma E, admits an identity elemcnt e, and that the homomorphisms yap are unital. Then e = (ea)asI belongs to E for e, =fao(ep) for a 6 p and it is an identity element of the magma E; with the above notation, the homo- morphismsf, are unital and if the u, are unital then u is unital. Further, an element x = (x,),, I of E is invertible if and only if each of the x, is invertible in the corresponding magma E, and x-l = (x,'),,~; this follows from the formulafaO(xp1) =fao(xo)-l = xi1 for c1 From thesc remarks it can be deduced that if the magmas E, are monoids (resp. groups) and thcf,, are monoid homomorphisms, then the magma E is a monoid (resp. a group). In this case we speak of an inverse system of monoids (resp. groups). The univcrsal property goes over immediately to this case. It is left to the reader to define an inverse system of rings (Ea,fao) and to verify that E = lim E, is a subring of the product ring n E,, called the inverse limit ring Ofthe rings E,; it can be verified that the universal property extends to this case. Let Q = (Ea,fap) and B' = (Ed,f&) be two inverse systems of magmas (resp. monoids, groups, rings) relative to the same indexing set. A homo - morphism of (3 into Q' is an inverse system (u,),~~ of mappings u,: E, + EL such that each u, is a homomorphism. Under these conditions, the mapping = lirn u, of lirn E, into lirn EL is a homomorphism (cf. Set Theory ,111, 3 7, no. 2). p. t aeI c t- t Q, from (5) that this relation induces the usual order rclation on Z. Finally, from (1) it follows that the relations .x < y and x' < y' imply x + x' < y + y' and from (2) that the relation x < y implies xz < yz for all z 2 0 and xz 2 yz -.-~ for Let x be a rational number. x is said to be positive if x >/ 0, strictly positive if x > 0, negative if x < 0 and strictly negative if x < 0.7 The set of positive rational numbers is Q+ and that of negative numbers is -Q+. If Q* denotes the set of nnn-zero rational numbers, the set Q*, of strictly positive numbers is equal to < o '(if. VI, 4 2, no. I). ____- -.~ Q* n Q+ and -Q*+ is the set of strictly negative numbers. The sets Q*+ and (1, - 11 are subgroups of the multiplicative group Q*. Every rational number x # 0 can be expressed in one and only one way in the form 1. y, (- 1). y, where y > 0; hence the multiplicative group Q* is the product of the subgroups Q*+ and { 1, - I}, the coniponent of x in Q*+ is called the absolute value of x and is denoted by 1x1 ; the component of x in { - 1, 1) (equal to 1 if x > 0, to - 1 if x < 0) is called the sign of x and denoted by sgn x. Usually these two functions are extended to the whole of Q by setting 101 = 0 and sgn 0 = 0. 5 10. INVERSE AND DIRECT LIMITS Throughout this paragraph, I will denote a non-empty preordered set, a < p the preordering on I. The notion of inverse (resp. direct) system of sets relative to the indexing set I is defined in Set Theory, 111, 3 7, no. 1 (resp. Set Theory, 111, 3 7, no. 5, under the hypothesis that I is right directed). 1. INVERSE SYSTEMS OF MAGMAS DEFINITION 1. An inverse system of magmas relative to the indexing set I is an inverse system of sets (Ea,fao) relative to I, each E, having a magma structure and each fap being a magma homomorphism. Let (Ea,fao) be an inverse system of magmas whose laws are written multi- plicatively. The inverse limit set E = lim E, is a subset of the product magma n E, consisting of the families (x,),~~ such that x, =fa0(x0) for a < p. If ,€I (x,) and (y,) belong to E, then for a < p, x, =fap(xp) andy, =fas(yo), hence x,y, = fap(xp)faB(yp) = fap(xpyp) ; hence E is a submagma of E,. E will be given the law induced by that on E,; the magma obtained is called the t- t We avoid the current terminology where positive means > 0. 118 2. INVERSE LIMITS OF ACTIONS Suppose there are given two inverse systems of sets (Q,, +,,J and (Ea,fap) rela- tive to the same indexing set I. Suppose there is given for all a E I an action of Q, on E, such that fao(w~x~) = +ao(Wo) Aao(xp) (1) for a p, xp E ED, wp E Qp. Then the family of actions considered is said to be an inverse system of actions. Let Q = lim Qu and E = lim E,; if x = (x,),~~ belongs to E and w = (w,),EI belongs t to Q, then w.x t = (W,.X,),~~ belongs 119 2 I ALGEBRAIC STRUCTURES to E by (1). Thus an action of Q on E is defined called the inverse limit ofthe 5 10.3 yv =f.,,(y,), x: =fv,(xh), y: =fvD(yh). By the definition of direct limit, there exists y in I with y > a, y 2 p, x., = x!,, y., = y:. Then DIRECT SYSTEMS OF MAGMAS fa(xaYa) =f,(fva(xa~cO) = fy(x.,~,) = f,(xCy:) =fy(fy~(xh~h)) = fB (‘byb). Suppose the magmas E, are associative. Let x, y, z be in E. There exist a E I and elements x,, y,, z, in E, such that x =fa(%,), Y =faha), =fa(za)* Then xy = fa(xaya), whence (xy)z =fa( (xaya)za) ; similarly x(y.1 = fa (xa(YaZa) 1, whence (xy)z = x(yz) for (xaya)za = xa(yaza). The case of commutative magmas is treated analogously. Suppose finally that each magma E, has an identity element ear and that foa(e,) = e, for a 6 p. For a, p in I, there exists y in I with y tc and y 2 p, whence fa(ea) =fy(fva(ea)) =fv(ey) =fv(fv~(e~)) =f~(e~) and there thus exists an element e in E such that fa(e,) = e for all a E I. Let x be in E; choose u E I and x, E E, such that x =fa(xa). Then ex =fa(ea) .fa(xa) =fa(ea.~a) =fa(xa) = x and similarly x.e = x, hence e is the identity element of E. The magma E is called the direct limit ofthe magmas E,. PROPOSITION 2. Let (E,, fDa) be a direct system of magmas and let E be its direct limits f,: E, + E the canonical homomorphisms. Suppose a magma F and homomorphisms u,: E, -+ F are given such that u, = up 0 fpa for a 6 p. There exists one and only one homomorphism u: E --f F such that ua = u 0 fa for all a E I. If the magmas E, and F and the homomorphisms fBa and u, are unital, the homomorphism u is unital. We know (Set Theory, 111, 3 7, no. 6, Proposition 6) that there exists one and only one mapping u: E --f F such that u, = u 0 fa for all a E I. We verify that is a homomorphism: let x, y be in E, a in I and x,, ya in E, such that x = fa(xa) and y = fa(ya). Then xy = fa(xaya), whence u(~.Y) = U(fa(xaya)) = ua(xaYa) = Ua(xa)Ua(Ya) = u(fa(Ua) )u(fa(Ya) 1 = U(~)U(Y) - we consider now the unital case and let e, denote the identity element of E,, e that of E and e‘ that of F. Choose a E I, then e = fa(ea), whence for U, is unital. Hence u is unital. ~(e) = U(fa(ea)) = ua(ea) = e’ actions ofthe Q, on the E,. The above applies especially in the case where the Qa are monoids and each action of Q, on E, is an operation. Then the inverse limit of these operations is an operation of the monoid on E. with operators and to verify that this limit is a group with operators. It is left to the reader to define the inverse limit of an inverse system of groups 3. DIRECT SYSTEMS OF MAGMAS In this no. and the following we shall assume that I is right directed. DEFINITION 2. A direct system of magmas relative to the indexing set I is a direct system of sets (E,, fpa) relative to I, each E, having a magma structure and each fBe being a magma homomorphism. Let (Ea,fpa) be a direct system of magmas. E will denote the direct limit set lim E, andf, the canonical mapping of E, into E. Recall that 3 (2) fp ofoa =fa for a 6 P, (3) By (2), also (4) fa(Ea) ~f,(Ep) for a 6 P* If x,,y, E E, are such that fa(xa) = fa(ya), there exists a p 2 a such that PROPOSITION 1. There exists on E one and only one magma structure for which the mappings fa: E, --f E are homomorphisms. If the magmas E, are associative (resp. commutative), so is E. Ifthe magmas E, and the homomorphism f,, are unital, so are the magma E and the homomorphisms fa. fpa(xu) =fpa(ya). The magmas E, will be written multiplicatively. Let x, y be in E. There exist a in I and x,, ya in E, such that x = fa(xa) and y = f,(y,). If there exists a magma structure on E for whichf, is a homo - morphism, then x.y = fa(xaya), whence the uniqueness of this magma structure. To prove the existence, we must prove that for a, fi in I, x,,y, in E, and xb, yb in E,, the relations (5) fa(xa) =fp(xb), fa(Ya) =f~(~b) imply fa(xay,) =fo(xbyb). For y 2 a and y 2 B, we set xy =fya(xa), 120 121 I ALGEBRAIC STRUCTURES By analogy with the notion of direct system of magmas, that of a direct system of monoids or groups can be formulated. Proposition 1 shows that the magma E which is the limit of a direct system of monoids (Ea,foa)a,oeI is a monoid. We show that E is a group if the E, are groups: let x E E, a E I and x, E E, be such that x = f,(xa) ; the element y =fa(.; ') of E is the inverse of x (9 2, no. 3). The universal property of Proposition 2 goes over immediately in the case of a direct system of monoids or groups. The reader is left to define a direct system of rings. Let (Aa,fDa) be such a direct system; let A = lim A, andf,: A, + A the canonical homomorphisms. There exists (Proposition 2) on A one and only one addition and multiplica- tion characterized by x + y = fa(xa + y,), xy = fa(xaya) for a E I, x,, y, in A, and x = f,(x,), y =f,(y,). Under addition A is a commutative group and multiplication is associative and unital. Finally, for x, y, z in A, choose a in I and x,, ya and z, in A, with --+ x =f(Xa), Y =fa(Ya) and z =fa(Za)* Then (x + Y) *z =fa(xa + ycJfa(za) =fa((xa + YcJza) =fa(xaza + ~cxza) =fa(xaza) + fa(YaZa) = xz + YZ and the relation x(y + z) = xy + xz is proved analogously. In other words, A has the structure of a ring, characterized by thc fact thatf, is a ring homo- morphism for all a E I. The ring A is called the direct limit of the rings A,. Proposition 2 extends immediately to the case of rings. PROPOSITION 3. (a) Ifthe A, are non-zero, A is non-zero. (b) Ifthe A, are integral domains, A is an integral domain. (c) Ifthe A, arefields, A is ajeld. Let 0,, 1 , be the zero and unit of A, and 0, 1 the zero and unit of A. There exists a E I such thatf,(O,) = O,fa(la) = 1. If 0 = 1, there exists 2 a such thatf,,(O,) =fDa(la), that is 0, = 1,. This proves (a). Suppose that A, are integral domains. Then A is commutative and non-zero by (a). Let x, y be elements of A such that xy = 0. There exists a E I and x,, y, E A, such that x = fa(xa), y = f,(y,). Then f,(x,y,) = xy = 0 =f,(O,). Hence there exists p 2 a such thatfDa(xaya) =f,,(O,). As A, is an integral domain, it follows that fOa(xa) = 0, or foa(ya) = O,, hence x = 0 or y = 0. This proves (b). Suppose that the A, are fields. Then A # (0) by (a). Let x be a non-zero element of A. There exist a E I and x, E A, such that x = f,(x,). Then xu # 0 andf,(x; ') is the inverse of x in A. This proves (c). Let Q = (E,,f,,) and Q' = (Eh,fia) be two direct systems of magmas 122 3 10.4 (resp. monoids, groups, rings). A homomorphism of Q into &' is a direct system (u,),~~ of mappings u,: E, + Ek such that each u, is a homomorphism. Under these conditions, the mapping u = lirn u, from E = lirn E, to DIRECT LIMIT OF ACTIONS E' = lirn Ek is a homomorphism (cf. Set Theory, --+ 111, 3 7, no. 6). --f --+ 4. DIRECT LIMIT OF ACTIONS Suppose that there are given two direct systems of sets (R,, and (Em&,) relative to the same indexing set I and for each a E I an action of Q, on E,. Suppose that f,a(Wa.xa) = +,a(wa) .fpa(xa) for a < p, w, E R, and xu E E,. Then the family of actions under consideration is said to be a direct system of actions. It is easily verified as in Proposition 2 that there exists an action h of Q = lim SZ, on E = lim E, which is described as follows: let w E !2 and x EE; choose UE I and W,E R,, x, E E, such that o = +a(wa) and x = fa(xa) (+,: Ra + R and fa: E, + E denote the canonical mappings); then w.x =f,(o,.x,). The action of C2 on E is called the direct limit of the actions of the Q, on the E,. If the R, are monoids and each action of R, on E, is an operation, the direct limit action is an operation. It is left to the reader to define the direct limit of a direct system of groups with operators and to verify that this limit is a group with operators. .--, --f 123 EXERCISES EXERCISES 31 1. Let E be a set, A C E x E and (x,y) HXT~, (x,y) EA, an internal law of Composition not everywhere defined on E. Given two subsets X and Y of E, let X T Y denote the set of elements of E of the form x T y with x E X, y E Y and (x, y) E A. A law of composition is thus defined which is everywhere defined on !Q(E). If (Xa)aeA and are any two families of subsets of E, then (,lk xa) T LV! YD) = (a. J2A x B (xa T YO)' 2. Let T be a law of composition not everywhere defined on a set E. Let E' be the subset of p(E) consisting of the sets {x} where x runs through E and the empty subset @ of E; show that E' is a stable subset of T(E) under the law (X, Y) X T Y (Exercise 1) ; deduce that, if E denotes the set obtained by adjoining (Set Theory, 11, 9 4, no. 8) to E an element w, the law T can be ex- tended to E x E, so that the law T is identical with the law induced on E by this extended law. 3. Let T be a law of composition not everywhere defined on E. (a) For the law (X, Y) H X T Y (Exercise 1) between subsets of E to be associative, it is necessary and sufficient that, for all x E E, y E E, z E E, if one of the two sides of the formula (x T y) T z = x T (y T z) is defined, so is the other and is equal to it (use Exercise 1 to show that the condition is sufficient). (b) If this condition holds, show that Theorem 1 of no. 3 can be generalized as follows: if one of the two sides of formula (3) is defined, the other is defined and is equal to it. 4. (a) Given a set E, let @ be the set of mappings into E of any subset of E; iff, g, h are three elements of @, show that if the composition (YO g) o h is defined so isfo (g o h), but that the converse is not true; if these two composi - tions are defined they are equal. 124 (b) Let 5 be a family of non-empty subsets of E such that no two have an element in common and let Y be the subset of consisting of the bijective mappings of a set of 5 onto a set of 5. Show that, under the law induced by the law f 0 g on Y, the condition of Exercise 3(a) holds. 5. Show that the only triples (m, n,p) of natural numbers #O such that (rnf)P = mnP are: (1, n,p), n andp being arbitrary; (m, n, l), m and n arbitrary and (m, 2,2) where m is arbitrary. 6. Let T be a law of composition on a set E; let A be the subset of E con- sisting of the elements x such that x T (y T z) = (x T y) T z for ally E E, z E E; show that A is a stable subset of E and that the law induced on A by T is associative. 7. If T is an associative law on E and a and b two elements of E, the sets {a} T E, E T {b}, {u} T E T {b}, E T {u} T E are stable subsets of E under the law T. 8. Let T be an associative law on E and u an element of E; for all x E E, y E E, we write x I y = x T a T y: show that the law I is associative. 9. On a set E the mappings (x, y) ++ x and (x, y) - y are opposite associative laws of composition. 10. Let X and Y be any two subsets of a set E and let X T Y = X U Y if XnY= @,XTY =EifXnY# @;showthatthelawofcomposition thus defined on T(E) is associative and commutative. 11. Let T be an associative law on E and A and B two subsets of E which are stable under this law; show that, if B T A c A T B, A T B is a stable subset of E. 12. The only distinct natural numbers #O which are permutable under the law (x, y) xu are 2 and 4. 13. Show that under the law (X, Y) H X 0 Y between subsets of E x E the centre is the set consisting of o and the diagonal. 14. Show that under the law of composition f 0 g between mappings of E into E the centre consists of the identity mapping. 15. A law T on a set E is called idempotent if all the elements of E are idempo- tent (no. 4) under this law, that is if x T x = x for all x E E. Show that, if a law T on E is associative, commutative and idempotent, the relation x T y = y is an order relation on E; if we write x < y, any two elements x, y of E admit a least upper bound (under this order relation) equal to x T y. Obtain the con- verse. 125 I ALGEBRAIC STRUCTURES 16. Let T be an associative and commutative law of composition on a set E. Let n be an integer > 0. The mapping x t-f 7 x of E into E is a morphism. EXERCISES §2 1. Let T be a law of composition on a set E. Denoting by F the sum set (Set Theory, 11, 5 4, no. 8) of E and a set {e} consisting of a single element and identifying E and {e} with the corresponding subsets of F, show that it is possible in one and only one way to define on F a law of composition T’ which induces on E the law T and under which e is identity element; if T is associa- tive, the law ’i: is associative. (F is said to be derived from E “by adjoining an identity element”.) 2. Let T be a law everywhere defined on E. (n) For T to be associative, it is necessary and sufficient that every left translation yx be permutable with every right translation Su in the set of mappings of E into E (with the lawfo g). (b) Suppose that E has an identity element. For T to be associative and commutative, it is necessary and sufficient that the mapping (x, y) c-f x T y be a morphism of the magma E x E into the magma E. 3. In the set F of mappings of E into E, for the relationf o g = f o h to imply g = h, it is necessary and sufficient that f be injective; for the relation g 0 f = h of to imply g = h, it is necessary and sufficient that f be surjective; forfto be cancellable (under the law a), it is necessary and sufficient thatfbe bijective. 4. For there to exist on a set E a law of composition such that every permuta - tion of E is an isomorphism of E onto itself under this law, it is necessary and sufficient that E have 0, 1 or 3 elements. 5. For 2 < n < 5, determine on a set E with n elements all the laws every - where defined admitting an identity element and under which all the elements of E are cancellable; for n = 5 show that there exist non-associative laws satis- fying these conditions. (N.B. Exercises 6 to 17 inclusive refer to associative laws written mult$li- catively; e denotes the identity when it exists, ya and Sa the translations by u E E; for every subset X of E we write ya(X) = ax, Sa(X) = Xu.) 6. For an associative law on ajnite set, every cancellable element is inver - tible (use no. 3, Proposition 3). 7. Given an associative law on a set E and an element x E E, let A be the set of xR for n E N* ; if there is an identity element, let B be the set of X“ for n E N; if further x is invertible, let C be the set of xa for a E Z. Show that, if A (resp. B, C) is infinite, it is isomorphic (with the law induced by the given law on E) to N* (resp. N, Z) with addition. 8. In the notation of Exercise 7, suppose A is finite; show that A contains one and only one idempotent (5 1, no. 4) (observe that if xp and xq are idempotents, then xp = xpq, xq = xpq, hence xf, = xq) ; if h = xp, the set of xn for n 2 p is a stable subset D of E such that under the law induced on D, D is a group. 9. Under a multiplicative law on a set E let a be a left cancellable element of E. (a) If there exists an element u such that au = a, show that ux = x for all x E E; in particular, if xu = x for all x E 15, u is identity element. (6) If there exist u E E such that au = a and b E E such that ab = u, show that ba = u (form aba) ; in particular, if there exists an identity element e and an element b such that ab = e, b is the inverse of a. 10. Ifa and b are two elements of E such that ba is left cancellable, show that a is left cancellable. Deduce that under an associative and commutative law on E the set S of non-cancellable elements of E is such that ES c S (and in particular is stable). 7 11. E is called a l$t semi-group if every element x of E is left cancellable. (n) If u is an idempotent (4 1, no. 4), then ux = x for all x E E (use Exercise 9(a)) ; u is identity element for the law induced on Eu. (b) If u and v are two distinct idempotents of E, then Eu n Eu = 0 and the sets Eu and Ev (with the laws induced by that on E) are isomorphic. (c) Let R be the complement of the union of the sets Eu, where u runs through the set of idempotents of E. Show that ER c R; it follows that R is a stable subset of E. If R is non-empty, then aR # R for all a E R (use Exercise 9(n) to prove that, in the contrary case, R would contain an idempotent) ; in particular, R is then infinite. R is called the residue of the left semi -group E. (d) If R is non-empty and there exists at least one idempotent u in E (that is E # R), for all XEREU, xEu # Eu; in particular, no element of REu is invertible in Eu and REu is an infinite set (use Exercise 8). (e) If E has an identity element e, e is the only idempotent in E and R is empty (note that E = Ee). v) If there exists a right cancellable element a in E (in particular if the given law on E is commutative), either E has an identity element or E = R (note that if there exists an idempotent u, then xu = xua for all x E E). For example, if E is the set of integers > 1, with addition, E is a commutative semi- group such that E = R. (g) If there exists a E E such that Sa is surjective, either E has an identity element or E = R (examine separately the case where a E R and the case where a E Eu for an idempotent u). 126 127 I ALGEBRAIC STRUCTURES 7 12. Under a multiplicative law on E, let a E E be such that ya is surjective. (a) Show that, if there exists u such that ua = a, then ux = x for all x E E. (b) For an element b E E to be such that ba is left cancellable, it is necessary and sufficient that ya be surjective and that b be left cancellable. 7 13. Suppose that, for all x E E, yx is surjective. Show that if, for an element a E E, ya is a bijective mapping, yx is a bijective mapping for all x E E (use Exercise 12 (b)) ; this holds in particular if there exist two elements a, b of E such that ab = b (use Exercise 12(a)) ; E is then a left semi-group whose residue R is empty; moreover, for every idempotent u, every element of Eu is invertible in Eu. 7 14. For an associative law on afinite set E there exist minimal subsets of the form aE (that is minimal elements of the set of subsets of E of this form, EXERCISES __~~ ordered by inclusion). (a1 If M = aE is minimal, then xM = xE = M for all x E M; with the law \--f -~ induced by that on E, M is a left semi-group (Exercise 11) in which every left translation is bijective (cf. Exercise 13). (b) If M = aE and M’ = a’E are minimal and distinct, then M n M‘ = % ; for all b E M, the mapping x’ H bx‘ of M’ into M is a bijective mapping of M’ onto M. Deduce that there exist an idempotent u‘ E M’ such that but = b and an idempotent u EM such that uu‘ = u (take u to be the idempotent such that bu = bl. Show that u‘u = u‘ (consider uu’u) and that every y’ E M‘ such that y’u = y’ belongs to M’u’. (c1 Show that the maDDing x’ H ux‘ of M’u’ into M is an isomorphism of II v \-I --~- Mu‘ onto Mu; deduce that M and M’ are isomorphic left semi-groups. (d) Let M, (1 < i < r) be the distinct minimal subsets of the form aE: deduce from (b) that the idempotents of each left semi-group Mf can be arranged in a sequence (uU) (1 < j < s) so that U,~U~, = uij for all i,j, k. If K is the union of the Mi, show that Eufj c K (note that, for all x E E, xuilE is a minimal subset of the form aE) ; deduce that Euil is the union of the UkjEukj for 1 < k < r (show that (Eu,,) n Mk = ukjEukj) and that Eu,,E = K. Prove finally that every minimal subset of the form Eb is identical with one of the s sets Euij (note, using (a), that Ebui,b = Eb and deduce that Eb c K). 15. Let (xJl <,<,, be a finite sequence of left cancellable elements. (a) Show that the relation xlxz. . .x, = e implies all the relations x,+~. . .x,,~~x~. . .xi = e which are obtained from one another by “cyclic permutatxon” for 1 < i < n. (b) Deduce that if the composition of the sequence (xi) is invertible each of the xf is invertible. 7 16. Let T be an associative law on a set E and E* the set of cancellable elements of E; suppose that E* is non-empty and that every cancellable element is a central element. Let 5 denote the set of subsets X of E with the following property: there exists y E E* such that S,(E) c X. (a) Show that the intersection of two sets of 5 belong to 5. (b) Let @ be the set of functions f defined on a set of 5 taking their values in between elements f and g of CD: “there exists a set X E 5 such that the restric - tions off and g to X are identical”. Show that R is an equivalence relation; let YP = @/R be the quotient set of @ under this relation. (6) Letfand g be two elements of @, A E 5 and B E 5 the sets where f and g are respectively defined; there exists X c B and belonging to 5 such that g(X) c A; if gx is the restriction of g to X, show that the mapping f 0 g, belongs to @ and that its class (mod. R) depends only on the classes off and g (and not on X) ; this class is called the composition of that off and that of g; show that the law of composition thus defined on ‘K is associative and has an identity element. (d) For all a E E, show that the left translation ya belongs to (D; let +a be its class (mod. R). Show that the mapping x H dX is an isomorphism of E onto a submonoid of Y and that, if x E E*, bX is invertible in Y (consider the inverse mapping of y,., show that it belongs to @ and that its class (mod. R) is the in- verse of +,.). Deduce a generalization of Theorem 1 of no. 4. 17. Let E be a monoid and S a stable subset of E with the following proper - ties : (a) For all s E S and all a E E, there exist t E S and b E E such that sb = at. (p) For all a, b E E and all s E S such that sa = sb, there exist t E S such that (a) In E x S let (a, s) N (b, t) denote the relation: “there exist s’ and t‘ in Show that - is an equivalence relation. We write E = E x S/-, denote by the equivalence class of (a, S) (mod -) and by E: E + E the mapping a - ae-l. (b) Show that there exists on E one and only one monoid structure such that E is a unital homomorphism and such that, for all s E S, E(S) is invertible. (c) Show that (E, E) has the universal property described in Theorem 1 (d) Show that E is injective if and only if the elements of S are cancellable. E and such that, for X E 5, f -1 (X) belongs to 5. Let R denote the relation at = bt. S such that tt’ = ss’ and as’ = bt’.” (no. 4). §3 1. Let a .-.fa be an action of s2 on E. Let F be the image of Q in EE and G the stable subset of EE under the law (f, g) ++ f 0 g generated by F and the identity mapping of E. (a) Show-t-hatevery subset of E which is stable under the action of Q is also 128 129 EXERCISES I ALGEBRAIC STRUCTURES stable under the restriction to G of the canonical action of EE on E (no. 1, Example 3). (b) Let X be a subset of E; show that the stable subset of E generated by X is the set off(x) wherefruns through G and x runs through X. 2. Let I be an internal law on E which is doubly distributive with respect to the associative internal law T ; show that, if x I x’ and y I y’ are cancellable under the law T, x I y’ is permutable with y I x’ under the law T (calculate the composition (x T y) I (x‘ T y’) in two different ways). In particular, if the law I has an identity element, two cancellable elements under the law T are permutable under this law; if all the elements of E are cancellable under the law T, this law is commutative. 3. Let T and I be two internal laws on a set E such that I is right distribu - tive with respect to T. (u) If the law T has an identity element e, x I e is idempotent under the law T for all x E E; if further there existsy such that x I y is cancellable under the law T, then x I e = e. (b) If the law I has an identity element u and there exists z E E which is cancellable for both laws I and T, u is cancellable under the law T . 7 4. Let T and I be two internal laws on E, each with an identity element; if the left action of E on itself derived from each of the laws is distributive with respect to the other, every element of E is idempotent under both these laws (if e is the identity element for T, u the identity element for I, prove first that e I e = e, noting that e = e I (u T e)). 5. On a set E suppose three internal laws of composition are given: an addition (not necessarily associative nor commutative), a multiplication (not necessarily associative) and a law denoted by T . Suppose that the multiplica - tion has an identity element e, that the law T is right distributive with respect to multiplication and the law T is left distributive with respect to addition. Show that, if there exist x,y, z such that x T z, y T z and (x + y) T z are cancellable under multiplication, then e + e = e (use Exercise 3(a)). 76. An internal law of composition T on E is said to determine on E a quasi -group structure if, for all x E E, the left and right translations yx and Sx are bijective mappings of E onto itself. A quasi-group is said to be distributive if the law T is doubly distributive with respect to itself. (a) Determine all the distributive quasi-group structures on a set of n elements for 2 < n < 6. (b) *Show that the set Q of rational numbers, with the law of composition (x, y) H $(x + y), is a distributive quasi -group.* (c) In a distributive quasi -group E, every element is idempotent. Deduce that, if E has more than one element, the law T can neither have an identity element nor be associative. (d) The right and left translations of a distributive quasi-group E are auto- morphisms of E. (e) If E is ajinite distributive quasi-group, the structure induced on every stable subset of E is a distributive quasi -group structure. (f) Let E be a distributive quasi-group. If R is an equivalence relation which is left (resp. right) compatible (no. 3) with the law T, the classes mod. R are stable subsets of E. If E is finite, all these classes are obtained one from another by left (resp. right) translation; under the same conditions, if R is compatible with the law T, the quotient structure on E/R is a distributive quasi-group structure. (g) Let E be a distributive quasi-group. The set A, of elements of E which are permutable with a given element a is stable; if E is finite, for all x E A,, A, = A, (note, using (e), that there exists y E A, such that x = a T y) and when x runs through E the sets A, are identical with the equivalence classes with respect to a relation compatible with the law T. (h) If E is a finite distributive quasi-group and the law T is commutative, the number of elements of E is odd (consider the ordered pairs (x, y) of elements of E such that x T y = y T x = a, where a is given). 7. Suppose given on a set E an (associative and commutative) addition under which all the elements of E are invertible and an (associative) multipli - cation which is doubly distributive with respect to addition *(in other words, a ring structure),; write [x, y] = xy - yx; the law (x, y) H [x, y] is doubly distributive with respect to addition. For x and y to be permutable under multiplication, it is necessary and sufficient that [x,y] = 0; and we have the identities CX,YI = - CY, .I; [x, ry, 211 + [y, [z, XI] + [z, [x, yI1 = 0 (the second is known as the “Jacobi identity”). The second identity may also be written [x, CY, 211 - “x,Yl, 21 = “2, x1,yl which expresses the “deviation from associativity’’ of the law [x, y]. 8. With the same hypotheses as in Exercise 7, let x T y = xy + yx; then the law T is commutative and doubly distributive with respect to addition but not in general associative. (a> For all x E E, show that m+n T x = (? X) T (T .). (b) If we write [x, y, Z] = (x T y) T z - x T (y T z) (deviation from associativity of the law T), prove the identities [x, y, 21 + [z, y, XI = 0 131 130 I ALGEBRAIC STRUCTURES EXERCISES [x,y, zl + ry, z,xl + [t,x,yl = 0 [x T Y, u, z1 = [% [(X T Y), .)I] (the notation [x, y] meaning the same as in Exercise 7) [x Ty,u, 21 4- [y T z,u,x] 4- [z T X,u,y] = 0. 7 9. Suppose given on E an (associative and commutative) addition under which all the elements of E are invertible and a multiplication which is non- associative, but commutative and doubly distributive with respect to addition. Suppose further that n E Z, n # 0 and nx = 0 imply x = 0 in E. Show that if, writing [x, y, z] = (xy)z - x(yz), the identity [xy, u, 21 + [yz, u, 21 + [zx, u,yl = 0 holds, then xm+” = xmxn for all x (show, by induction on p, that the identity [xq, y, xp-q] = 0 holds for 1 < q < p). 10. Let E be a commutative monoid whose law is written additively and whose identity element is denoted by 0. Let T be an internal law on E which is distributive with respect to addition and such that 0 T x = x T 0 = 0 for all x E E. Let S be a subset of E which is stable under addition and under each of the external laws derived from T ; let E denote the monoid of differences of E with respect to S under addition and E the canonical homomorphism of E into E. Then there exists on E one and only one law f which is distributive with respect to addition and such that E is a homomorphism for the laws T and T. If the law T is associative (resp. commutative), so is the law T. §4 1. Determine all the group structures on a set of n elements for 2 < n < 6 (cf. Q 2, Exercise 5). Determine the subgroups and quotient groups of these groups and also their Jordon-Holder series. 7 2. (a) An associative law (x, y) H xy on a set E is a group law if there exists e E E such that, for all x E E, ex = x and if, for all x E E, there exists X’ such that x’x = e (show that xx’ = e by considering the composition x’xx’; deduce that e is the identity element). (b) Show that the same is true if, for all x E E, the left translation yx is a mapping of E onto E and if there exists some a E E such that the right translation So is a mapping of E onto E. 3. In a group G everyjnite non-empty stable subset H is a subgroup of G (cf. 9 2, Exercise 6). 4. Let A and B be two subgroups of a group G. (a) Show that the least subgroup containing A and B (that is the subgroup generated by A u B) is identical with the set of compositions of the sequences (x,) Gi ~ 2n + of an (arbitrary) odd number of elements such that x1 E A for i odd and x, E B for i even. (6) For AB to be a subgroup of G (in which case it is the subgroup generated by A v B), it is necessary and sufficient that A and B be permutable, that is that AB = BA. (c) IfA and B are permutable and C is a subgroup containing A, A is permutable with B n C and A(B n C) = C n (AB). 5. If a subgroup of a group G has index 2 it is normal in G. 6. Let (G,) be a family of normal subgroups of a group G such that n G, = {e}; show that G is isomorphic to a subgroup of the product group 7. If G is the direct product of two subgroups A and B and H is a subgroup 8. Let A and A’ be two groups and G a subgroup of A x A‘. We write: of G such that A c H, H is the direct product of A and H n B. N = G n (A x {e}), N’ = G n ({e} x A’), H = prl(G) H’ = pr2(G). (a) Show that N is normal in H and N‘ is normal in H’; define isomorphisms H/N --f G/(N x N’) -+ H’/N‘. (b) Suppose that H = A, H’ = A’, that the groups A and A’ are of finite length and that no quotient of a Jordan-Holder series of A is isomorphic to a quotient of a Jordan -Holder series ofA’. Show that G = A x A’. 9. Let G be a group which does not consist only of the identity element. Suppose that there exists a finite subset S of G such that G is generated by S (resp. by the elements gsg- l, g E G, s E S) . Let 9 be the set of subgroups of G bp. of normal subgroups of G) distinct from G. Show that 9, ordered by inclusion, is inductive. Deduce the existence of a normal subgroup H of G such that G/H is simple. 10. Let H be a normal subgroup of a group G contained in the centre of G. show that if G/H is a monogenous group G is commutative. 11. If all the elements of a group G other than the identity element are of order 2, G is commutative; if G is finite, its order n is a power of 2 (argue by induction on n). 132 133 EXERCISES I ALGEBRAIC STRUCTURES 12. Let G be a group such that, for a fixed integer n > 1, (xy)" = x"y" for all x E G, y E G. If G(") denotes the set of x", where x runs through G, and G(") the set of x E G such that X" = e, show that G(") and G(") are normal subgroups of G; if G is finite, the order of G(") is equal to the index of G("). Show that, for all x,y in G, also xl-'yl-" = (xy)'-" and deduce that xn-lyn = y"x"-l; con- clude from this that the set of elements of G of the form x"("-l) generates a commutative subgroup of G. 13. Let G be a group. If S is a simple group, S is said to occur in G if there exist two subgroups H, H' of G with H a normal subgroup of H' such that H'/H is isomorphic to S. Let in(G) denote the set of isomorphism classes of simple groups occurring in G. (a) Show that in(G) = 0 o G = {e}. (b) If H is a subgroup of G, show that in(H) c in(G); if moreover H is normal, then in(G) = in(H) u in(G/H). (c) Let G, and G, be two groups. Show the equivalence of the two following DroDerties : A (i) in(G,) n in(G,) = 0; (ii) Every subgroup of G, x G, is of the form H, x H, with HI C Gl and .. H, c G,. (Use Exercise 8.) (iii) Card(G,) and Card(G,) are relatively prime., 14. Let G be a group and H a subgroup of G. Any subset T of G which meets every left coset mod. H in one and only one point is called a representa - tive system of the left cosets mod. H (cf. Set Theory, 11, 5 6, no. 2) ; this con- dition is equivalent to the following: *When G1 and G, are finite groups, show that (i) and (ii) are equivalent to: The mapping (x, y) H xy is a bijection of T x H onto G. (a) Let T be such a representative system; for all g E G and all t E T, let x(g, t) E T and y(g, t) E H be such that gt = x(g, t)y(g, t). Let S be a subset of G generating G. Show that the elements y(g, t), g E S, t E T generate (If H' denotes the subgroup of H generated by these elements, show that T . H' is stable under the ys, g E S, hence also under the yp, g E G, and deduce that T.H' = G, whence H' = H.) (b) Suppose that (G:H) is finite. Show that G can be generated by a finite subset if and only if H can. 7 15. Let 5 be a set of stable subgroups of a group with operators G; 5 is said to satisfy the maximal condition (resp. minimal condition) if every subset of 5, ordered by inclusion, has a maximal (resp. minimal) element. Suppose that the set of all stable subgroups of a group G satisfies the minimal condition. (u) Prove that there exists no stable subgroup of G isomorphic to G and distinct from G (argue by reductio ad absurdum by showing that the hypothesis would imply the existence of a strictly decreasing infinite sequence of stable subgroups of G). (b) The minimal elements of the set of normal stable subgroups of G not consisting only of e are called minimal normal stable subgroups. Let 2R be a set of minimal normal stable subgroups of G and S the smallest stable subgroup of G containing all the subgroups belonging to M; show that S is the direct product of ajnite number of minimal normal stable subgroups of G (let (M,) be a sequence of minimal normal stable subgroups of G belonging to 9 and such that M,,, is not contained in the stable subgroup generated by the union of MI, M,, . . ., M,; let S, be the stable subgroup generated by the union of the M, of index n 2 k; show that S,, , = S, after a certain rank and therefore that (M,) is a finite sequence; finally use no. 9, Proposition 15). (c) If G is a group without operators, show that every minimal normal sub- group M of G is the direct product of a finite number of simple groups isomor - phic to one another (let N be a minimal normal subgroup ofM; show that M is the smallest subgroup of G containing all the subgroups aNa-' where a runs through G, and then apply (b) to the group M). 16. If the set of stable subgroups of a group with operators G satisfies the maximal and minimal conditions (Exercise 15), G has a Jordan-Holder series (consider, for a subgroup H of G, a maximal element of the set of normal stable subgroups of H distinct from H). 7 17. Let G be a group with operators; a composition series (GI) of G is called normal if all the G, are normal stable subgroups of G; a normal series Z is called principal if it is strictly decreasing and there exists no normal series distinct from Z, finer than Z and strictly decreasing. (a) If (GI) and (H,) are two normal series of G, show that there exist two equivalent normal series finer respectively than (G,) and (H,) (apply Schreier's Theorem by considering a suitable domain of operators for G). Give a second Proof of this proposition by "inserting " the subgroups Glj = G, n (Gi+,H,) and Hit = H, n (H,+lGi) in the sequences (G,) and (H,) respectively. (6) If G has a principal series, any two principal series of G are equivalent; for every strictly decreasing normal series Z there exists a principal series finer *an z. Deduce that, for G to possess a principal series, it is necessary and sufficient that the set of normal stable subgroups of G satisfy the maximal and (c) If G is a group without operators and possesses a principal series and if *e *t of subgroups of G satisfies the minimal condition, every quotient group 135 conditions. 134 I ALGEBRAIC STRUCTURES Gi/G, + is the direct product of a finite number of simple subgroups isomorphic to one another (use Exercise 15). 7 18. (u) Let G be a group with operators generated by the union of a family (Hi),EI of simple normal stable subgroups of G. Show that there exists a subset J of I such that G is the restricted sum of the family (HJtEJ (apply Zorn’s Lemma to the set of subsets K of I such that the subgroup generated by the union of the Hi for i E K is the restricted sum of this family). (6) Let A be a normal stable subgroup of G. Show that there exists a subset JA of I such that G is the restricted sum of A and the Hi for i E JA. Deduce that A is isomorphic to the restricted sum of a subfamily of the Hi. 7 19. (u) Let G be a group such that every normal subgroup of G distinct from G is contained in a direct factor of G distinct from G. Show that G is the restricted sum of a family of simple subgroups. (Let K be the subgroup of G generated by the union of all the simple subgroups of G. Suppose K # G; if x E G - K, consider a normal subgroup M of G which is maximal amongst those containing K and not containing x. If G = M’ x S, where M’ contains M and S # {e}, show that x $ M’ and therefore M’ = M; on the other hand, if N is a normal subgroup of S, show that M x N = G and conclude that S is simde. whence a contradiction. Conclude with the aid of Exercise 18.) Obtain EXERCISES -I‘ the converse (cf. Exercise 18). Ih\ In order that there exist in a group G a family (N,) of normal simple -. \-/ ~~ subgroups such that the G/N, are simple and N, = {e}, it is necessary and sufficient that, for every normal subgroup N # {e} of G, there exist a normal subgroup N‘ # G such that NN‘ = G. (To see that the condition is necessary, consider an x E N distinct from e and an N, not containing x. To see that the condition is sufficient, for all x # e in G, consider the normal subgroup N generated by x and a normal subgroup N‘ # G such that NN’ = G. Show that if a normal subgroup M of G is maximal amongst the normal subgroups of G containing N’ and not containing x, it is maximal in the set of all normal sub- groups #G and deduce the conclusion.) 20. (u) Let G be a group, the direct product of two subgroups A and B. Let C, be the centre of A and let N be a normal subgroup of G such that N n A = (1). Show that N c CA x B. (6) Let (SJiEI be a finite family of non-commutative simple groups. Show that every normal subgroup of a S, is equal to one of the partial products n S, where J is a subset of I. is J 21. Let G be a group of finite length. Show the equivalence of the following (u) G is a product of simple groups isomorphic to one another. properties: 136 (6) No subgroup of G distinct from (1) and G is stable under all the auto- morphisms of G. 722. Let E be a quasi-group (5 3, Exercise 6) written multiplicatively, possessing an idempotent e and such that (xy) (zt) = (xz) (yt) for all x, y, z, t. Let u(x) denote the element of E such that u(x)e = x and U(X) the element ofE such that eu(x) = x; show that the law of composition (x, y) - u(x)u(y) is a commutative group law on E under which e is identity element, that the mappings x - xe and y - ey are permutable endomorphisms of this group structure and that xy = u(xe)u(ey) (start by establishing the identities e(xy) = (ex)(ey), (xy)e = (xe)(ye), e(xe) = (ex)e; then note that the relations x = y, ex = ey and xe = ye are equivalent). Obtain the converse. 7 23. Consider on a set E an internal law, not everywhere deJined, written multiplicatively and satisfying the following conditions : (1) if one of the compositions (xy) z, x(yz) is defined, SO is the other and they are equal; (2) if x, x’, y are such that xy and x‘y (resp. yx and yx’) are defined and equal, then x = x’; (3) for all x E E, there exist three elements ex, ek and x-l such that e2 = x, xek = x, x-lx = ek; ex is called the left unit of x, ek the right unit of x, x-l the inverse of x (by an abuse of language). (a) Show that the compositions xx-l, x-le,, ekx-l, exex, eke: are defined and that xx-l = ex, x-leX = e2x-l = x-l, exex = ex, eke: = ek. (6) Every idempotent e of E (5 1, no. 4) is a left unit for all the x such that ex is defined and right unit for all the y such that ye is defined. (c) For the composition xy to be defined, it is necessary and sufficient that the right unit of x be the same as the left unit ofy (to see that the condition is sufficient, use the relation ey = yy-l); if xy = z, then x-lz = y, zy-l = x, y-lx-l = z-l, z-lx = y-l, yt-’ = x-l (the compositions appearing on the left-hand sides of these relations being defined). (d) For any two idempotents e, e‘ of E, let Ge, e, denote the set of x E E such that e is left unit and e’ is right unit of x. Show that, if Ge, is non-empty, it is a group under the law induced by that on E. E is called a groupoid if it also satisfies the following condition: (4) for all idempotents e, e’, G,, e, is non-empty. (e) In a groupoid E, if u E Ge, et, show that x - xu is a bijective mapping of %e onto Ge, ec, y H uy a bijective mapping of Ge,, ef onto Ge, e, and x H a- lxu an isomorphism of the group Ge, onto the group Ge,, e,. cf) Show that the law defined in 3 1, Exercise 4(b) determines a groupoid structure on the set Y, if all the sets of the family 8 are equipotent. 24. (a) Given any set E, consider in the set E x E the law of composition, not everywhere defined, written multiplicatively, under which the composition 137 I ALGEBRAIC STRUCTURES of(x, y) and (y', z) is only defined ify' = y and in this case has the value (x, z). Show that E x E, with this law of composition, is a groupoid (Exercise 23). (b) Let (x, y) .= (x', y') (R) be an equivalence relation compatible with the above composition (that is such that (x,y) = (x',y') and (y, z) = (y', z') imply (x, z) (x', z')); suppose further that the relation R satisfies the following condition : for all x, y, z there exists one and only one t E E such that (x,y) Show that under these conditions the quotient structure by R of the groupoid structure on E x E is a group structure (prove first that the quotient law is everywhere defined, since, if i,j, i are three classes such that iy = xi, then j = i; establish finally that, for all x E E, y E E, (x, x) 3 (y, y)). (c) Let G be the group obtained by giving the quotient of E x E by R the above structure. Let a be any element of E; if, for all x E E,fQ(x) denotes the class mod. R of (a, x), show thatfa is a biective mapping of E onto G and that the relation (x, y) s (x', y') is equivalent to the relation (z, t) and there exists a u E E such that (x, y) = (u, z). fa(x) (fa(x')) - = fa(y) t fa(^')) - '. For G to be commutative, it is necessary and sufficient that the relation (x,y) = (x',~') imply (x, 2') = (Y, Y'). 7 25. Let E be a set and f a mapping of Em into E; we write f(X1, xz, . . ., x,) = x1xz. . .x,; (XlX,.. .x,)x,+,.. .X,,-l = Xl(XZX3.. .x,+1)x,+z.. .XZ,-l suppose that f satisfies the following conditions : (1) identically ; (2) For all a,, a,, . . . , a,-,, the mappings x - xa,a,. . . a, - 1 x Hala,. . .a,-1x are bijections of E onto E. (u) Show that identically (~1x2, * *x,)Xm+1* * .~zm-l = x1xz.. .xi(xt+1.. .~i+m)~i+m+l*. *Xzm-1 for every index i such that 1 < i < m - 1 (argue by induction on i, by con- sidering the element ((x1xz. . .x,)x,+,. . .xz,-,)a,a2. . . am-,). (b) For all a,, a,, . . . , am-2, there exists u E E such that for all x x = xala2. . .U,-~U = ua,a,. . .a,-,x. EXERCISES (c) In the set Ek of sequences (ul, uz, . . ., uk) of k elements of E (1 < k < m - 1) consider the equivalence relation Rk which states: for all the quotient set Ek/Rk and G the sum set (Set Theory, 11, $4, no. 8) of El = E, E,, . . ., Em-l. Let aEE,, p EE,; if (ul, uz,. . ., u,) is a sequence of class a, (ul, . . . , u,) a sequence of class p, consider the sequence X1, X2, . . . , X, - k, UIUz. . . UkXlXZ. . . X, - k = UlUz. . . ukxlX2. . . x,- k; kt Ek denote (u1, u2,. . ., u,, u1,. . ., v,) in E*+' if i + j < m, the sequence (211, ' * 'Y 'i+I-mJ ('i+,-?l+l' "u~v~u~'~ .',>> in Ei+f-,+l if i + j m; show that the class of this sequence in Ei+j (resp. Ei+,-, + 1) depends only on the classes a and p; it is denoted by a. p. Show that a group law is thus defined on G, that H = Em - , is a normal subgroup of G and that the quotient group G/H is cyclic of order m - 1 ; prove finally that E is identical with a class (mod. H) which generates G/H and that xlxz. . . x, is just the composition, in the group G, of the sequence (xl, xz, . . . , x,). 7 26. Let G, G' be two groups, f: G --f G' a mapping such that, for two arbitrary elements x, y of G,f(xy) = f(x)f(y) or f(xy) = f(Y)f(x). It is pro- posed to prove thatfis a homomorphism of G into G' or a homomorphism of G into the opposite group G'O (in other words, eitherf(xy) = f(x)f(y) for every ordered pair (x, y) orf(xy) = f(y)f(x) for every ordered pair (x, y)). (a) Show that the set N = >'(e') (e' the identity element of G') is a normal subgroup of G and that f factors into G 5 G/N -% G', where g is injective. Attention may therefore be confined to the case where f is injective, which will be assumed throughout what follows. (6) Show that if xy = yx then f(x)f(y) = f(y)f(x) (consider f(x2y) and f(xzya), expressing them in several ways). (c) Show that iff(xy) =f(x)f(y) thenf(yx) =f(y)f(x). (Show with the aid of (b) that attention may be confined to the case where xy # yx.) (4 Show thatf(xyx) =f(x)f(y)f(x). (Use (a) or (6) according to whether *y = yx or xy # yx; in the second case, considerf(x2y) and f (yxz).) (e) Let A be the set of x E G such thatf(xy) =f(x)f(y) for ally E G and B the set of x E G such that f(xy) = f(y)f(x) for all y E G. Show that A # G, €3 # G and A u B = G is an impossible situation. (By virtue of (b) there Would then exist x, y in A such thatf(xy) = f(x)f(y) # f (y)f(x) and u, v in such thatf(uv) =f(v)f(~) #f(u)f(u). Deduce from this a contradiction bY considering successively the two possibilities xu E A, xu E B; in the first case Co~iderf(rmu) and in the secondf(yxu).) The rest of the proof consists of proving that A u B # G is impossible, by reductio ad absurdum. If A U B # G, there exist a, 6, c in G such that f(ab) = f(a)f(b) # f(b)f(a) and f(ac) = f(c)f(a) # f(a)f(c). 138 I39 I ALGEBRAIC STRUCTURES (f) Show thatf(c)f(a)f(b) = f(b)f(a)f(c). (Consider two cases according to whether bc # cb or bc = cb. In the first case, consider f (bac) ; in the second, use (b) and (c) and considerf(abc),f(bca) andf(bac).) (g) Considerf(abac) and obtain a contradiction. EXERCISES §5 1. In a finite group G, show that the number of conjugates (no. 4) of an element a E G is equal to the index of the normalizer of u and is therefore a divisor of the order of G. 7 2. *If G is a finite group of order n, the number of automorphisms of G is <n10g"/log2 1 (show that there exists a generating set {al, a2, . . . , a,} of G such that a, does not belong to the subgroup generated by a,, u2, . . . , a, - for 2 < i < m; deduce that 2m < n and that the number of automorphisms of G is <nm).* 3. Let I' be the group of automorphisms of a group G and A the group of inner automorphisms of G; show that A is a normal subgroup of r. For an automorphism Q of G to be permutable with all the inner automorphisms of G, it is necessary and sufficient that, for all x E G, x-'o(x) belong to the centre of G; deduce that, if the centre of G is reduced to the identity element, so is the centralizer of A in I?. 7 4, (u) Let G be a non-commutative simple group, r the automorphism group of G and A the group of inner automorphisms of G (isomorphic to G). If s is an automorphism of the group F, show that s(A) = A (using Exercise 3 above and 3 4, no. 9, Proposition 15, note that the intersection A n s(A) can- not consist only of the identity element of r). (6) Show that the only automorphism of r leaving invariant each of the elements of A is the identity element (use the fact that this automorphism leaves invariant a and Dab- for all a E A and Q E r and use Exercise 3). (c) Lets be an automorphism of F, + the isomorphism x H Int(x) of G onto A, + the inverse isomorphism and Q the automorphism + 0 s o + of G; show that the automorphism 5 H Q-'s(~)o of I' is the identity automorphism (use (b), noting that, for all x E G, s(Int(x)) = Int(o(x))). Deduce that every auto- morphism of r is an inner automorphism. 5. Let G be a group. (u) If H is a subgroup of G of finite index n, show that the intersection N of the conjugates of H is of index a divisor of n! (note that G/N is isomorphic to a subgroup of 6,). (b) G is called residuallyfinite, if the intersection of its subgroups of finite index is fe). Show that this is equivalent to saying that G is isomorphic to a subgroup \I of a product of finite groups. 140 (c) Suppose that G can be generated by a finite set. Show that, for every (d) Under the hypothesis of (c), consider an endomorphismfof G which is tion of P, onto itself (observe that it is an injection). Deduce that the kernel of f is contained in every subgroup of G of finite index. In particular, if G is residually finite, f is bijective. 6. Let H be a subgroup of finite index in a group G. Suppose that G is the union of the conjugates of H. Show that H = G. (Reduce it to the case where G is finite by means of Exercise 5. In the latter case, note that: Card( u xHx-') 6 1 + (Card(H) - l)Card(G/H).) integer n, the set P, of subgroups of G of index n is finite. surjective. Show that, for every integer n, the mapping H H f -1 (H) is a bijec- XEG 7. Let p be a prime number. (u) Show that np = n (mod. p) for all n E Z. (Reduce it to the case where n is (b) Let x, y be two elements of a group G. Suppose that positive and argue by induction on n using the binomial formula.) yxy-' = x", with n EZ, and that xp = 1. Show that ypxy-p = x"; deduce that yP-l commutes with x. (c) Let G be a group all of whose elements # 1 are of order p and are conjugate to one another. Show that Card(G) is equal to 1 or 2. (Use (b) to prove that G is commutative.) 8. Let A and B be subgroups of a finite group G, NA and NB the normalizers of A and B, vA and vB the indices of NA and N, in G, rA the number of conju. gates ofA which contain B and rB the number of conjugates of B which contain 7 9. Let G be a group of permutations of a finite set E; for all s E G, let (a) Show that sz x(s) = Nt, where N = Card(G) and t is the number o A; show that vArB = vnrA. X(S) denote the cardinal of the set of fixed points of s. crbits of G in E (evaluate in two ways the number of ordered pairs (s, X) E G x E such that s(x) = x). (b) Suppose that, for all 'a E Ej the stabilizer Ha of a is not reduced to e. Show that, if ~(s) is independent of s for s # e and is equal to an integer k,then (Note that Card(E) 6 Nk.) In the particular case where k = 2, find the possible orders of the groups Ha; &OW that, if t = 3, the order of Ha cannot be > 3 for elements of two of the three orbits unless N has one of the values 12, 24 or 60. k < t < 2k. 141 I ALGEBRAIC STRUCTURES (c) Suppose that G operates transitively on E and let H denote the stabilizer ofan element of E; show that ssG 2 x(s), is equal to N. t,, where tH is the number oforbits of H in E. (For every (s, u), write ~(s, u) = ~(s). Evaluate in two ways x(s, u), where R is the set of ordered pairs (s, u) such that 2 the sum (s. u) E €2 usu-l E H.) 10. (a) Show that the elements 712~34, 713724, 714r23 of the alternating group U4 form with the identity element a commutative subgroup H of 3,. Show that H is normal in U, and in 6, and that U,/H is cyclic of order 3. (6) Show that the centralizer of H in 6, is equal to H. Deduce that the mapping s H (h H shs-') of 6, into Aut(H) defines when passing to the quo- tient an isomorphism of 6,/H onto Aut(H) and that the latter group is iso - morphic to 63. (c) Let K be a subgroup of H of order 2. Show that K is not normal in 91,. 11. Let E be a finite set, < E GE a cycle of support E and r a transposition. Show that GB is generated by 1: and r. 7 12. (a) Show that every permutation Q E U, is a product of cycles of order 3 (which are not in general component cycles of 0). (Prove it for a product of two transpositions and use 5 5, no. 7, Proposition 9.) (b) If a,, . . . , a, are p distinct elements of (I, n), let (alaz. . .a,) denote the cycle of order p whose support is {al, a2, . . . , a,} and which maps a, to a, + for 1 < i < p-1 and up to a,. Show that 3, is generated by the n - 2 cycles (1 2 31, (1 2 41,. . ., (1 2 n). (Use (a).) (c) Deduce that, if n is odd, U, is generated by (1 2 3) and (1 2 . . . n) and, if is even, by (1 2 3) and (2 3 . . . n). (d) Show that, if a normal subgroup of U, contains a cycle of order 3, it is identical with 91, (prove that such a subgroup contains all the cycles (1 2 k), 3 < k < n). 7 13. Let G be a transitive group of permutations of a set X and H the stabilizer of an element x E X. Show the equivalence of the following proper- ties : (a) Every subgroup H' of G containing H is equal to H or G. (b) Every subset Y of X such that, for all g E G, gY is either contained in y or disjoint from Y, is equal to X or consists of a single element. (If H' satisfies (a) and Y = Hx, then gY = Y for all ~EH' and gY n Y = 0 for all g $ H'. Conversely, if Y enjoys the property in (b), the set H' of g E G such that gY = Y is a subgroup of G containing H.) A transitive permutation group G satisfying (a) and (6) and not consisting only of the identity element is called primitive. EXERCISES 14. A group of permutations I' of a set E is called r-ply transitive if, for any two sequences (al, a,, . . . , a,), (61, b,, . . . , 6,) of r distinct elements of E, there mists a permutation Q E I' such that Q(UJ = 6, for 1 < i < r, this property not holding for at least one ordered pair of sequences of r + 1 distinct elements of E. (a) Show that an r-ply transitive group is primitive if r > 1. (b) The order of a permutation group I' operating on n objects which is r-ply transitive is of the form n(n - 1). . .(n - r + l)d, where d is a divisor of (n - r)! (consider a subgroup of the permutations of r leaving invariant r elements and calculate its index). 7 15. Let I' be an r-ply transitive group of permutations of a set E with n elements; for a permutation Q E I' distinct from the identity permutation, let n - s be the number of elements of E invariant under Q. If s > r, show that there exists a permutation r E r such that Q-~TQ'F-~ is distinct from the identity permutation and the number of elements of E it leaves invariant is pn - 2(s - r + 1) (use the decomposition of Q into its component cycles). Ifs = r, show similarly that there exists r E I' such that Q-~TQT-~ is a cycle of order 3. Deduce that, if r 2 3 and I' does not contain the alternating group a,, then s 3 2r - 2 for every permutation in F (use Exercise 10). Conclude finally that, if I' is not identical with 9, or 6,, then r ,< n/3 + 1. 7 16. (u) Show that the alternating group U, is (n - 2)-ply transitive. (6) Show that 91, is a non-commutative simple group for n 2 5. (Use (a), the method of Exercise 15 and Exercise 12(d) to prove that U, is simple if n > 6; examine analogously the cases n = 5 and n = 6.) (c) Show that, for n 2 5, the only normal subgroups of 6, are {e}, 21, and 6,. 17. Let G be a transitive group of permutations of a set X; suppose G is primitive (Exercise 13). Show that every normal subgroup of G distinct from {e} is transitive. 18. Let I' be a group of permutations of a set E. Let A and B be two subsets of E, complements one of the other and stable under I'. Let I'A and rB denote the groups consisting of the restrictions of the permutations in I' to A and B respectively and AA and A, the subgroups of I' which leave invariant respec- tively every element of A and every element of B. Show that AA and A, are nOrmal subgroups of I' and that FA is isomorphic to r/AA and PB to I'/AB; if AAB (resp. ABA) is the group consisting of the restrictions of the permutations in (reSP. AB) to B (resp. A), show that the quotient groups rA/ABA, FB/AAB and r/(AAAB) are isomorphic (use Exercise 8). l9- Let G be a group, H a subgroup of G and G/H the set of left cosets bod. H) in G. Let r: G/H + G be a section associated with the canonical Projection G -+ G/H. Then for all X E G/H, X = r(X) .H. An internal law Of composition is defined on G/H by setting X T Y = r(X)r(Y) . H. 143 1 42 EXERCISES I ALGEBRAIC STRUCTURES (a) Show that X T H = X for all X and that under the law T every left translation is bijective. If G' is the subgroup of G generated by the set of elements r(X) and H' = H n G', the internal law defined analogously on G'/H' by the mapping r determines on this set a structure isomorphic to that de;ermined on G/H by the law T. (6) For the law T to be associative, it is necessary and sufficient that H' be a normal subgroup of G' in which case the structure determined by T is iso- morphic to the structure of the quotient group G/H' (to see that the condition is sufficient, show first, with the aid of 3 4, Exercise 2(a), that if T is associative it determines on G/H a group structure; denoting by K the largest normal sub- group of G' contained in H', show then, using the associativity condition on T, that (r(X T Y)) -lr(X)r(Y) E K for all X, Y; deduce that the mapping X i--f Y (X) , K is an isomorphism of the group G/H (with the law T) onto the quotient group G'/K; conclude that H' = K, noting that H' is a union of Eosets mod. K). (c) Conversely, suppose given on a set E an internal law of composition T such that every left translation is bijective and there exists e E E such that x T e = x for all x E E. Let r be the group of permutations of E generated by the left translations y, and A the subgroup of permutations in r leaving e invariant; show that to every left coset X modulo A there corresponds one and only one element x E E such that y, E X; if we write r(X) = yx, the mapping x H y, is an isomorphism of the set E with the law T onto the set I'/A with the law (X, Y) ++ r(X)i(Y). A. 20. Let G be a simple group and H a subgroup of G of finite index n > 1. (a) Show that G is finite and that its order divides n! (use Exercise 5). (6) Show that G is commutative if n 6 4 (use Exercise 10). 21. Letp(n) be the number of conjugacy classes of the symmetric group 6,. Show that p(n) is equal to the number of families (xl, . . . , x,) of integers 2 0 such that 2 i.xi = n. n 1=1 m *Deduce the identity n=O 2 p(n)T" = 22. Let A = 2/22, B = Q3 and G = A x B. If s is an element of B of order 2, let + be the endomorphism of G whose kernel is equal to B and whose image is {I, s}. Show that the centre of G is not stable under +. 7 23. (a) Let s be an element of order 2 in the group 6, which is a product of k transpositions t,, . . . , t, with mutually disjoint supports. Show that the centralizer C, of s in 6, is isomorphic to 6,- 2k x A, where A admits a normal subgroup of order 2, such that A/B is isomorphic to 6, (observe that every element of C, permutes the fixed points of s and also the supports oft,, . . . , tk). 144. (6) Show that, if n = 5 or n 2 7, every element of order 2 in 6, whose centralizer is isomorphic to that of a transposition is a transposition (compare the Jordan -Holder quotients of these centralizers using Exercise 16). Deduce that every automorphism of 6, permutes the transpositions. (6) If a, 6 are distinct elements of (1, n), let Fa (resp. Fa,) denote the stabi- lizer of a (resp. the fixer of {a, b}) in 6, and (a 6) the transposition T,,,. Let a, 6, c, d be distinct elements of (1, n). Show that, if n 2 5, the group generated by Fa, and F,, is 6, (show that this group contains all the transpositions; for this use the fact that, if x E (1, n) - {a, 6, c. d}, then (a c) = (c x) (a x) (c x)). Show that the group generated by Fa, and Fa, is Fa (use the same equation as in the above case). (d) Show that an automorphism a of 6, such that a(F,) = Fa for all a E (1, n) is the identity (show that a maps every transposition to itself). Show that, if there exists a E (1, n) such that a(Fa) = Fa, a is an inner automorphism (show that there exists an inner automorphism fl such that Poa maps each of the groups F,, 6 E (1, n), to itself). (e) Let a, 6 be two distinct elements of (1, n) and let Cab be the centralizer of (a, 6) in Q ,. Show that, if n 2 5, the derived group of Cab is of index 4 in Cab (use the fact that the derived group of ~5,,-~ is %,-2). Show that Fa, is the unique subgroup of Cab of index 2 which does not contain (a 6) and is not contained in a,. (f) Show that, if n 2 5, an automorphism a of 6, which leaves fixed a transposition (a 6) is inner. (Show by means of (e) that a(Fab) = Fab; show equally that, if c $ {a, 6}, a(F,,) is either of the form Fa,, x $ {a, 6}, or of the form Fbx, x $ {a, 6); in the first case conclude that a(F,) = Fa and apply (d) ; reduce the second case to the first by multiplying a by the inner automorphism of 6, defined by (a b).) (9) Deduce from (6) and (f) that every automorphism of Q, is inner if n = 5orn 2 7. 24. (a) Show that there exists a subgroup H of 66 which is isomorphic to 6, and leaves fixed no element of (1,6) (let e5 operate by inner automorphisms on the set of its subgroups of order 5, which has 6 elements). (6) Show that there exists an automorphism 0 of 6, which maps (1 2) to (1 2) (3 4) (5 6). (Let 66 operate on 6& where H is chosen as above.) (c) Show that the group of inner automorphisms of 66 is of index 2 in the group of all automorphism of 6,. (Let a be an automorphism of e6; show that Q maps a transposition either to a transposition or to a conjugate of (1 2)(3 4)(5 6). Using Exercise 23, show that in the first (resp. second) case a (resp. a 0 a) is an inner automorphism.) 25. Let n be an integer 2 5. Show that the subgroups of 6, of order (n - 1) ! are isomorphic to GnT1 and form a single conjugacy class (resp. two conjugacy classes) if n # 6 (resp. if n = 6). (Let H be such a subgroup and xl, . . . , x, the 145 I ALGEBRAIC STRUCTURES elements of 6JH. Show that the action of 6, on the x, defines an automorphism of 6, and apply Exercises 23 and 24.) 26. (a) Let G be a group operating on two finite sets El and E,. If s E G, let s1 (resp. s,) denote the permutation of El (resp. E,) defined by s and Es (resp. Ei) the set of elements invariant under s1 (resp. s,). Show the equivalence of the following properties : (i) For all s E G, Card(ES,) = Card(Ei). (ii) For all s E G, the orders of the component cycles of s1 (cf. no. 7, Propo - (iii) For all SEG, there exists a bijection fs: El -+ E, such that If these properties hold, the G-sets El and E, are said to be weakly equiva - (b) Give an example of two weakly equivalent G-sets which are not iso- (c) Let HI and H, be two subgroups of a finite group G. Show the equiva - (1) For every conjugacy class C of G, sition 7) are the same (to within a permutation) as those of s,. s, 05 =J* O s1. lent. morphic (take G to be a non-cyclic group of order 4). lence of the following properties : Card(C n H,) = Card(C n H,). (2) The G-sets G/H, and G/H, are weakly equivalent. (d) Show that, if H, and H, satisfy properties (1) and (2) above and H, is normal in G, then H, = H,. (e) Let G = 9, and let H, = {e, (1 2x3 4), (1 3x2 4), (1 4~2 3)) H, = {e, (1 2~3 4), (1 2~5 61, (3 4x5 6)). Show that H1 and H, satisfy properties (1) and (2) and that H, and H, are not conjugate in 91, (nor in G6). 27. Let Y be a subset of a set X and let A be the fixer of Y in the permutation group 6, of X. Let M be the submonoid of GX consisting of the elements s such that sAs-l c A. Show that M is a subgroup of 6, if and only if one of the sets Y and X - Y is finite. 28. Let G be a group, A and B two subgroups of G and + an isomorphism of A onto B. Show that there exists a group G, containing G such that 4 is the restriction of an inner automorphism of G, (use the permutation group of the set G). Show that, if G is finite, G, can be chosen to be finite. EXERCISES 29. Let X be an infinite set. Let G be the subgroup of GX consisting of the permutations CT which enjoy the following property: there exists a finite subset Y, of X such that CTX = x if x E X - Y, and the restriction of Q to Y, is even. 146 (a) Show that G is a non-commutative simple group. Show that every subset (6) Show that every finite group is isomorphic to a subgroup of G (use the of G which generates G is equipotent to X. fact that 6, is isomorphic to a subgroup of §6 1. Let G be a group and H a normal subgroup of G such that G/H is cyclic of finite order n. Let x be an element of G whose image 2 in G/H generates G/H. Let 9 be the automorphism h H xhx-l of H and let y = xn; then y E H. Show that +(y) = y and that cJ~ is the inner automorphism of H defined byy. Let T be the operation of Z on H defined by (m, h) ++ +"(h) and let E = H x I Z be the corresponding semi-direct product. Show that the element (y-1, n) of E generates a central subgroup C: of E and that the quotient E/C, is isomorphic to G. 2. Show that every central extension of 2 is trivial. 3. Define extensions (cf. 5 5, Exercise 10) 2/22 x 2/22 + 91, + 91, 2/22 x 2/22+6,+6,. Show that these extensions are non-trivial, that the first admits a section and that the second does not. 7 4. (a) Let G be a finite group of order mn such that there exists a normal subgroup H of G which is cyclic of order m, the quotient group G/H being cyclic of order n. Show that G is generated by two elements a, b such that am = e, b" = a*, bab-l = as, where r and s are two integers such that r(s - 1) and S" - 1 are multiples of m (take a to be an element generating H, b an element of a coset generating G/H; express the elements bhakb-h as powers ofa and apply this in particular to the cases h = n, k = 1 and h = 1, k = r). (b) *Conversely, let G(m, n, r, s) be the group defined by the presentation (a, 6; am = e, bn = d, bab-l = as), where m and n are two integers 30 and r and s arbitrary integers (cf. 4 7, no. 6); show that, if m, r(s - 1) and S" - 1 are not all zero, G(m, n, r, s) is a fin;te group of order qn, where q is the greatest common divisor of m, Ir (s - 1) I ad IS" - 1 I ; in this group, the subgroup H generated by a is a normal sub- Soup of order q and G/H is a cyclic group of order n (prove that every element OfG(m, n, Y, s) can be written in the form axby, where x and y are two integers such that 0 < x < q - 1, 0 < y < n - 1, and G(m, n, r, s) is isomorphic to 147 I ALGEBRAIC STRUCTURES the group consisting of the ordered pairs (x, y) of integers subjected to the above conditions, with law of composition (x -t X'SY, y + y') ify + y' < n - 1 (X,Y) * (X',Y') = { ( x + x'sy + r,y + y' - n) ify + y' 2 n the first coordinate of the right-hand side being a sum modulo q) . Examine the EXERCISES cases where m = r(s - 1) = sn - 1 = 0. The group G(n, 2, 0, - 1) is called the dihedral group of order 2n and is de- noted by D,; the group G(4,2,2, - 1) is a group of order 8 called the quatern- ionic group and denoted by Q. Show that in Q every subgroup is normal and that the intersection of the subgroups distinct from {e} is a subgroup distinct _-... . . from {e}. Prove that D, is not isomorphic to Q.* 5. Let F be a group whose centre is {e} and A its automorphism group. F is identified with a subgroup of A by means of the homomorphism Int: F -+ A; let I' = A/F. (a) Show that the extension F --f A -+ I? is trivial only if r = {e} (note that the centralizer of F in A is {e}). (b) Suppose that J? = {el. Show that every extension F+E+G of a group G by the group F is trivial. 6. Let I beaset; write F = 2, E = 2 x (2/2Z)'and G = 2/22 x (Z/22)'. (a) Define a non-trivial extension F % E 5 G. (b) Show that, if I is infinite, E is isomorphic to F x G. 7. Let G and A be two groups, A being commutative. Let T be a homo- morphism of G into the automorphism group Aut(A) of A; if g E G, a E A, we write 'JQ = 7(g) (a). A crossed homomorphism of G into A is any mapping +: G -+ A such that +(gg') = +(g) + g+(5'), the group A being written additively. The crossed homomorphisms of G into A form a group under addition denoted by Z(G, A). (u) If u E A, the mapping g H 'Ja - a is denoted by €la. Show that a ++ 8, is a homomorphism 8 of A into Z(G, A). The kernel of 0 is the subgroup of A consisting of the elements invariant under G. The image of 8 is denoted by (b) Let X = A x G be the semi-direct product of G by A. If + is a mapping of G into A, show that g c-f (+(g), g) is a section of X if and only if + belongs to Z(G, A). For the sections corresponding to bl, +2 E Z(G, A) to be conjugate by an element of A, it is necessary and sufficient that bl 3 d2 mod. B(G, A). 148 B(G, A). (c) Let F d E 5 G be an extension of G by a group F whose centre is equal to A. Suppose that, ify E E and x = p(y), then i("f) = y.i(f) .y-1 for allfE A. If + E Z(G, A), let u, be the mapping of E into E given by the formula um(y) = i(+(p(y))) .y. Show that + -um is an isomorphism of Z(G, A) onto the automorphism group Aut(E) of the extension E. This iso- morphism maps B(G, A) to the group of automorphisms Int,(x), where x runs through A. 7 8. With the notation of the above exercise, C(G, A) is used to denote the group of mappings of G into A. G operates on C(G, A) by the formula (g+)(s') = g(+(g-ld))* If Q denotes the corresponding homomorphism of G into Aut(C(G, A)), the semi-direct product C(G, A) x ,, G is denoted by E,. (a) Let E: A -+ C(G, A) be the mapping which associates with each a E A the constant mapping equal to a. Show that E is an injective homomorphism, compatible with the action of G. E 5 G be an extension ofG by A such that i (ga) = xi (a).- 1 if a E A, x E E and g = p(x). Let p be a mapping G + E such that p o p = Id,. For all x E E, let +x be the mapping of G into A such that Show that (b) Let A xp(P(x-')g) = i(+X(d)Pk) for allg E G. = +x + p(x)+y if x, y E E. Deduce that the mapping @: E --f E, defined by W) = (+X,P(X)) is a homomorphism which makes the following diagram commutative: P ALE-G (6) A G-mean on A is a homomorphism m: C(G, A) --f A Satisfying the two following conditions: (cl) m 0 E = Id, (c2) rn(g+) = %(+) ifg E G, + E C(G, A). 1 49 I ALGEBRAIC STRUCTURES Let m be a G-mean on A. Show that, if + is a crossed homomorphism and a = m(+), then + = e-a (cf. Exercise 7). In particular, Z(G, A) = B(G, A). Show that the mapping (+, g) ++ (m(+), g) is a homomorphism of E, into A x ~ G. Deduce that every extension E of G by A satisfying the condition in (b) is isomorphic to A x TG (use the composition E -+ E, +A x G) and hence admits a section. Show that two such sections are transformed one into the other by an inner automorphism of E defined by an element of A (use Exercise 7 and the fact that Z(G, A) = B(G, A)). (d) Suppose that G is finite of order n and that the mapping a ++ na is an automorphism of A. For all c$ E C(G, A), let m(+) denote the unique element of A such that @ Show that m is a G-mean on A. This applies in particular when A is finite of order prime to n. "9. Let F+ E-+G be an extension of finite groups. Suppose no prime number divides both the order of F and the order of G. (a) Suppose F is solvable. Show that there exists a section s: G -+ E and that two such sections are conjugate by an element of F. (Argue by induction on the solvability class of F; where F is commutative, use the preceding Exercise.) (b) Show the existence of a section7 s: G + E without assuming that F is solvable. (Argue by induction on the order of G. If p divides the order of F, choose a Sylow p-subgroup P of F and consider its normalizer N in E. The image of N in G is equal to G, cf. Exercise 25. If N # E, conclude by means of the induction hypothesis; if N = E, use (a) and the induction hypothesis applied to F/P --+ E/P -+ G.) fi 10. Let G be a solvable finite group of order mn, where m and n have no common prime factor. Show that there exists a subgroup H of G of order m and that every subgroup of G whose order divides m is contained in a conjugate of H ("Hall's Theorem ").' (Argue by induction on the order of G. If mn # 1, choose a commutative subgroup A of G which is normal and not equal to {e} and whose order a is a power of a prime number. Apply this induction hypothesis to G/A and to (rn/a, n) or (m, n/a) according to whether a divides m or n. In the second case, use Exercise 9 to pass from G/A to G.) EXERCISES t Here again it can be shown that two such sections are conjugate, cf. FEIT and THOMPSON, Proc. Nat. Acad. Sci., U.S.A., 48, 1962, 968 -970. 150 11. Show that the simple group 91, of order 60 contains no subgroup of 7 12. Let G be a nilpotent group and H the set of elements of G of finite (a) Show that H is a subgroup of G. (b) Show that every finite subset of H generates a finite subgroup. (c) Show that two elements of H of relatively prime orders commute. 13. Let G be a nilpotent group. Show that G is finite (resp. countable) if 14. Let G be a finite group. Suppose that, for all x, y E G, the subgroup of G generated by {x, y} is nilpotent. Show that G is nilpotent (apply the hypothesis to x and y of orders p:' and p?, where p1 and pz are distinct prime numbers). order 15. order. and only if G/D(G) is. 15. Let G be a group and X a subset of G generating G. (a) Write X, = X; for n 2, define X,, by induction on n, as the set of commutators (x, y), .K E X, y E X,-l. Show that X, = {e} if and only if G is nilpotent of class <n. (Argue by induction on n. If X, = {e}, show that the subgroup H of G generated by X, - is contained in the centre of G and apply the induction hypothesis to G/H.) (6) Show that C"(G) is a normal subgroup of G generated by X,. Deduce that the image of X, in C"(G)/C"+l(G) generates the group C"(G)/C"+'(G). (c) Take G = 6, (rn 2 4) and X = {s, t}, where 5 is a transposition and t a cycle of order m. Show that X, does not generate the group C2(G) = 91,. 16. *Let V be a vector space over a commutative field k and 2 W=AV (cf. 111, 5 7, no. 1). On E = V x W a law of composition is defined by the formula (u1, w1) * (uz, w2) = (u1 + u2, w1 + w2 + u1 A 4. (a) Show that this law gives E a group structure, that of a central extension (6) Suppose the characteristic of k is different from 2. Show that the derived POUP D(E) is the set of (0, w), w E W; it is a group isomorphic to W. Show that, ifdim(V) 2 4, there exist elements ofD(E) which are not commutators., of v by w. 17. Let G be a finite commutative group. Show the equivalence of the followhg conditions : (a) G is cyclic. (b) Every Sylow subgroup of G is cyclic. 151 I ALGEBRAIC STRUCTURES (c) For every prime number p, the number of elements x E G such that xp = e is <p. 18. Let n be an integer 21 and let G be a commutative group written additively, such that nx = 0 for all x E G. Let H be a subgroup of G and f a homomorphism of H into Z/nZ. (a) Let x E G and let q be the order of the image of x in G/H; then qx E H. Show that there exists cc E Z/nZ such that qa = f(x). Deduce the existence of an extension off to the subgroup of G generated by H and x. (b) Show that f extends to a homomorphism of G into Z/nZ (use Zorn's Lemma). 19. Let G be a finite commutative group. (a) Show that there exists an element x E G whose order n is a multiple of the orders of the elements of G. (Use the decomposition of G as a product of p-groups.) Show that the subgroup H generated by such an element is a direct factor of G (apply the above exercise to construct a homomorphism G 3 H whose restriction to H is the identity). (6) Show that G is a direct product of cyclic groups. (Argue by induction on Card(G) using (a).) (Cf. VII, 5 4, no. 6.) 20. Let G be a finite commutative group written additively. Let x = ll€Q 2 g and let G, be the subgroup of G consisting of the elements g such that 2g = 0. (u) Show that x = 2 g. (b) Show that x = 0 if Card(G,) # 2. If Card(G,) = 2, show that x is the unique non-zero element of G,. (c) Show that Card(G,) = 2 if and only if the Sylow 2-subgroup of G is cyclic and #O. (d) *Let p be a prime number. Show that PEG2 (p - l)! = -1 (modp). (Apply (6) to the multiplicative group of the field Z/pZ.) EXERCISES Let p and q be two prime numbers such that p > q. Show that every finite group G of order pq is an extension of ZlqZ by (note that every Sylow p-subgroup of G is normal). If further p $ 1 q), show that G is cyclic. Deduce that the centre of a group cannot be of index 69. Show that, ifp = 1 (mod q), there exist a group of order pq whose centre (d) Show that every non-commutative group of order 6 is isomorphic to 22. Let G be a finite group of order n > 1 and let p be the least prime number dividing n. Let P be a Sylow p-subgroup of G and N its normalizer Show that, if P is cyclic, P is contained in the centre of N (show that the of N/P is prime to the order of the automorphism group of P). fT 23. Let o be an automorphism of a group G. (a) Show the equivalence of the following conditions: (i) O(X) = x implies x = e. (ii) The mapping x H x-'o(x) is injective. When these conditions are satisfied, (I is said (by an abuse of language) to be! without jxed point. (b) Suppose G is finite and 0 without fixed point. The mapping x H x-~.(x) is then bijective. If H is a normal subgroup of G stable under O, show that the automorphism of G/H defined by 5 is without fixed point. (c) Under the hypotheses of (b), let fi be a prime number. Show that there exists a Sylow p-subgroup P of G which is stable under 5. (If Po is a Sylow p- subgroup, there exists IJ E G such that .(Po) = yP0y- l; write IJ-, in the form ~-~cr(x) and take P = xP,,x-'.) Show that such a subgroup is unique and contains every p -subgroup of G which is stable under 5. (d) Suppose further that 5 is of order 2. Show that o(g) = g-' for all g E G (write g in the form x-'u(x) with x E G). Deduce that G is commutative and of odd order. 24. Let G be a finite group, H a Sylow subgroup of G and N the normalizer of H. Let X,, X, be two subsets of the centre of H and s E G such that sX,s-l = X,. Show that there exists n EN such that nxn-' = sxs-l for all x E X, (apply the theorem on the conjugacy of Sylow subgroups to the central - izer of XI). Deduce that two central elements of H are conjugate in G if and only if they are so in N. 25. Let +: G -+ G' be a surjective homomorphism of finite groups; let P be a Sylow p-subgroup of G. (a) Let P, be a Sylow p-subgroup of G such that +(P) = +(Pl). Show that there exists an element x of the kernel of + such that P, = xPx-'. (b) Let N (resp. N') be the normalizer in G (resp. G') of P (resp. +(p)). Show that +(N) = N'. In particular, if the order of G' is not divisible by p, then +(P) = {e} and +(N) = G'. 726. Let G be a finite group. G is called supersolvable if there exists a composition series ( Gi) ,, of G consisting of normal subgroups of G such that the quotients GJG, + , are cyclic. (a) Show that every subgroup, every quotient group and every finite Product of supersolvable groups is supersolvable. (b) Show that nilpotent rj supersolvable => solvable and that the converse implications are false. 152 153 I ALGEBRAIC STRUCTURES (c) Suppose G is supersolvable. Show that, if G # {l}, there exists a normal subgroup C of G which is cyclic of prime order. Show that the derived group D(G) ofG is nilpotent. Show that every subgroup of G distinct from G which is maximal is of prime index. (d) A finite group G is bicyclic if there exist two elements a, b of G such that, if c, and C, denote the cyclic subgroups generated by a and b respectively, then G = C,C, = C,C,. Show that every bicyclic group is supersolvable. (Reduce it to proving that there exists in G a normal cyclic subgroup N # (1). Attention may be confined to the case where C, n Cb = (1); let m 2 n> 1 be the orders of a and b respectively; by considering the elements 6- lak, where 0 < k < m - 1, show that there exists an integer s such that as # 1 and b-l& E C,; the set of as with this property is the desired cyclic subgroup N.) 27. Let S be a finite group. S is called a minimal sinlple group if S is simple and non-commutative and if every subgroup of S distinct from S is solvable. (a) Show that the alternating group 21, is minimal simple if and only if n = 3. (b) Let G be a finite group. Show that, if G is not solvable, there exist two subgroups H and K of G with H normal in K such that K/H is a minimal simple group. 7 28. Let G be a finite group and p a prime number. An element s E G is called p-unipotent (resp. p -regular) if its order is a power of p (resp. not divisible by p). (a) Let x E G. Show that there exists a unique ordered pair (u, t) of elements of G satisfying the following conditions: u is p-unipotent, t is p-regular, x = ut = tu. (Consider first the case where G is the cyclic group generated by (b) Let P be a Sylow p-group of G, C its centralizer and E the set ofp- x.1 regular elements of G. Show that Card(E) = Card(E n C) (modp). Deduce that Card(E) $ 0 (modp). (Argue by induction on Card(G) and re- duce it to the case where C = G; then use (a) to show that Card(E) = (G: p).) 7 29. Let G be a finite group of even order and let H be a Sylow 2-subgroup (a) Let s E H. Show that E(S) = - 1 if and only if s generates H. (6) Show that E is surjective if and only if H is cyclic. (c) Suppose that H is cyclic. Show that there exists one and only one normal subgroup D of G such that G is the semi-direct product of H and D. (Argue by induction on the order of H.) Show that the normalizer N of H in G is the direct product of H and N n D. 154 of G. For all s E G, let ~(s) be the signature of the permutation x H sx of G. EXERCISES (d) Show that the order of a non-commutative simple group is either odd7 7 30. Let p be a prime number, r and t integers 2 0, S a cyclic group of and T the subgroup of S of order pt. (a) If S operates on a finite set E, show that or divisible by 4. order p' Card(E) = Card(ET) (modp'+l), where ET denotes the set of elements of E invariant under T. (6) Let m, n 2 0. Show that (Let M be a set with m elements and E the set of subsets of S x M with ptn elements. Define an operation of S on S x M such that there exists a bijection of ET onto the set of subsets of (S/T) x M with n elements; apply (a).) 31. Let G be a finite group and S a Sylow p-subgroup of G. Let r, t be integers 20 such that Card(S) = pr+t. Let E be the set of subgroups of G of order fit. (a) Suppose that S is cyclic. Show that Card@) = 1 (modp'+'). (Let S operate on E by conjugation and use Exercise 30.) Deduce that, if the subgroup ofS oforder$ is not normal in G, there are at least 1 + pr+' Sylowp-subgroups in G. (b) Show that Card(E) = 1 (modp) even if S is not cyclic. (Let G operate by translation on the set F of subsets of G with Pt elements. Show that the elements of E give distinct orbits with (G:S)p' elements and that all the other orbits have a number of elements divisible by p' + l. Apply Exercise 30.) 7 32. Letp be a prime number and G a p-group. Let G* = GPD(G) be the subgroup of G generated by D(G) and the xp, x E G. (a) Show that G* is the intersection of the kernels of the homomorphisms of G into Z/pZ. (6) Show that a subset S of G generates G if and only if the image of S in G/G* generates G/G*. Deduce that, if (G:G*) = p", the integer n is the minimum number ofelements in a subset of G which generates G; in particular, G is cyclic if and only if n < 1. (c) Let u be an automorphism of G of order prime to p and let ii be the corresponding automorphism of G/G*. Show that ii = 1 implies u = 1. 7 33. Let G be a p-group and A its automorphism group. (a) Let (G,) , be a composition series of G such that (Gi : Gi = p for t In fact this case is impossible, as has been shown by FEIT and THOMPSON, Puc. J* Of Math., 13 (1963), 775-1029. 155 F EXERCISES I ALGEBRAIC STRUCTURES 0 < i < n - 1. Let P be the subgroup of A consisting of the automorphisms such that, for all i and all x E G,, u(x)x-l E G,+,. Show that, if an element E p is of order prime top, then u = 1 (argue by induction on n). Deduce that P is a p-group. (b) Conversely, let P be a p-subgroup of A. Show that there exists a com- position series (Gt)og,bn of G which is stable under P and such that (Gi:G,+,) =p for 0 < i < n - 1; show that the G, may be chosen to be normal in G. 34. Letp be a prime number. Show that every group of orderp2 is commuta- 35. Let G be a group such that the derived group D = D(G) is contained in (a) Show that the mapping (x, y) H (x, y) induces a mapping tive. the centre C of G. +: (G/C) x (G/C) -+ D such that, writing these groups additively : +(a + P, Y) = 44% Y) + +(P, Y) +(a, P + Y) = +(a, PI + 44% Y) +(a, P) = -+(P, a) +(a, 4 = 0 for all c(, p, y E G/C. (b) Show that, for every integer n, (yx)" = ynxyx, yp - 1)'2 for x, y E G. Deduce that if n is odd and dn = 1 for all d E D, the mapping x ++ xn induces a homomorphism 8: G/D --f C. 7 36. Let p be a prime number and G a non-commutative group of order (a) Show that, with the notation of the preceding exercise, C = D and that G/D is isomorphic to the product of two groups of order p. (b) Suppose that p is odd and that the homomorphism 8 of the preceding exercise is non-zero. Show that G is the semi-direct product of a group of order p by a cyclic group of order p2. Show that there exist elements x, y generating G such that P3' xp2 = 1, yp = 1, (x, y) = xp, that G is characterized up to isomorphism by this property (and by the fact that it is of order p3) and that such a group G exists. (c) For p = 2, consider the same question as in (6) with the hypothesis 8 # 0 replaced by the hypothesis that G contains a non-central element of order 2. Show that such a group is isomorphic to the dihedral group D, (cf. Exercise 4) and also to a Sylow 2-subgroup of G4. 156 (d) Suppose that p is odd and that the homomorphism 8 of the preceding exercise is zero. Show that there are elements x, y, z generating G such that (x,y) = z; xp = yp = zp = (x, 2) = (y, z) = 1, that G is characterized by this property and that such a group exists. form *Show that G is isomorphic to the multiplicative group of matrices of the with coefficients a, b, c in the field with p elements., (e) Suppose that p = 2 and that no non -central element of G is of order 2 (hence that they are all of order 4). Show that G is generated by elements x, y such that x2 = y2 = (x,y); (x,Y)2 = 1, that G is characterized by this property and is isomorphic to the quaternionic group (cf. Exercise 4). cf) *Is the group of matrices where a, 6, c run through the field with 2 elements, of type (c) or of type (e) ?* 37. (u) Let G be a finite group, H a normal subgroup of G,p a prime number not dividing the order of H and P a Sylow p-subgroup of G. Show that HP is the semi-direct product of P by H. (b) We say that a finite group G has property (ST) if there exists a number- ing pl, . . . , pI of distinct prime numbers dividing the order n = pyl. . .p,"l of G and Sylowp,-subgroups P,, . . ., P, of G such that, for 1 < i < s, the set Gt = PIP,. . . P, is a normal subgroup of G. Show that, if this is so, G, does not depend on the choice of the P,; it is the unique subgroup of G of orderpyl . . .pp'; further, G, is the semi-direct product of Pi by G, - (c) Show that a finite group G has property (ST) if and only if there exists a sylow subgroup P of G which is normal in G and such that G/P has property (d) Show that every subgroup, every quotient group and every central atension of a group with property (ST) has property (ST). (ST). 157 I ALGEBRAIC STRUCTURES 7 38. Let G be a finite group of order n = Pam, where p is a prime number not dividing m, and let X be the set of Sylow p-subgroups of G. Let P E X, let N be the normalizer of P and r = Card(X). (a) Show that r = (G:N). Deduce that r belongs to the set R of positive divisors of m which are congruent to 1 (modp). In particular, if R is {l}, P is EXERCISES normal in G. (6) If a = 1, show that the number of elements of G of order p is r (p - 1). If further N = P, show that the elements of G of order different from p are equal in number to m; if further m is a power of a prime number q, deduce that a Sylow q-subgroup Q of G contains - all the elements of G of order different from p and hence is normal in G. (c) Let H be a group of order 210. Show that H contains a normal subgroup G of order 105 (cf. Exercise 29). Letp = 3 (resp. 5, 7). Show that, if G has no normal Sylow p-subgroup, the group G contains at least 14 (resp. 84,90) elements of order p. Conclude that H and G have property (ST) (cf. the preceding exercise). (d) Show that every group of order n < 100 has property (ST), except possibly if n = 24, 36, 48, 60, 72, 96 (same method as in (c)). (e) Show that the group e4, of order 24, does not have property (ST). Con- struct analogous examples for n = 48,60, 72,96. (For n = 36, see the follow - ing exercise.) 7 39. Let G be a finite group, n = Card(G), p a prime number, X the set of Sylowp-subgroups of G and r = Card(X). Then G operates transitively on X by conjugation; let (b: G --f GX z 6, be the corresponding homomorphism (A z B denotes the relation “A is isomorphic to B”) and K its kernel. (a) Show that K is the intersection of the normalizers of the P E X and that the Sylow p-subgroup of K is the intersection of the P E X. Deduce that if r > 1 the order k of K divides n/rp. (6) Show that if G has no subgroup of index 2, then +(G) c Ux and that if n = kr! (resp. +kr!), then (b(G) = 6x (resp. 21x). (c) Using (a), (b) and the preceding exercise, show the following facts: If n = 12 and a Sylow 3-subgroup of G is not normal, then G z 21,. If n = 24 and G does not have property (ST), then G z 6,. If n = 36, then G has property (ST). (Show that if a Sylow 3-subgroup of G is not normal, then G contains a subgroup K of order 3 such that G/K z U4 and that K is central.) If n = 48 and G does not have property (ST), then G contains a normal sub - group K of order 2 such that G/K = 6,. If n = 60 and G does not have property (ST), then G x 21,. (Show that such a group G has no normal subgroup of order 5, nor of order 2, nor of index 2. Deduce that G is isomorphic to a subgroup of Gj6 and, arguing as in Exercise 25 of 9 5, that G x 2f5.) If n = 72 and a Sylow 3-subgroup of G is not normal, then G contains a If n = 96 and a Sylow 2-subgroup of G is not normal, then G contains a (d) Show that a group G of order < 100 is solvable unless G z a,. normal subgroup K such that G/K is isomorphic either to 6, or to 3,. normal subgroup K such that G/K z e3. 40. Let G,, G, be two groups and G a subgroup of G, x G, such that pr, G = G,, pr, G = G,; identifying G1 and G, with G, x {e,} and {el} x G, respectively, H, = G n G1 and H, = G n G, are normal subgroups of G such that G/H, is isomorphic to G, and G/H, isomorphic to G,. (a) Show that for G to be normal in G, x G,, it is necessary and sufficient that G contain the commutator group D(G1 x G,) = D(G,) x D(G,) (prove that G contains D(G,) by writing xzx-lz-l E G for x E G, and z E G). Deduce that GJH, is commutative. (6) If G is normal in G, x G,, show that a homomorphism is defined of G, x G, onto G1/H, by associating with each ordered pair (s,, s2) the coset of sit;' modulo H,, where t, E G, is such that (t,, s2) E G. Deduce that (G, x G,)/G is isomorphic to G/(H, x H,). 41. Let x, y be two elements of a group G. (a) For there to exist a, b in G such that bay = xa6, it is necessary and (b) For there to exist 2n + 1 elements a,, a,, . . . , a,,, + , in G such that sufficient that xy-l be a commutator. x = a1a2.. .azn+, and y = a2n+la2n. . .a,, it is necessary and sufficient that xy - be a product of n commutators (argue by induction on n). §7 1. Enumerate the elements of length < 4 in M(X) when X consists of a single element. 2. Let X be a set and M one ofthe magmas M(X), Mo(X) or N‘x). Show that every automorphism of M leaves X stable. Deduce an isomorphism of 8, onto the group Aut (M) . Show that the endomorphisms of M correspond bijectively to the mappings of X into M. 3. Let X be a set and Ma the free magma constructed on a set {a} with one element. Let p denote the homomorphism of M(X) into Ma which maps every element of X onto a; on the other hand let A: M(X) -+ Mo(X) be the homo- morphism defined in no. 9. Show that the mapping zu +> (A(w), p(w)) is an isomorphism of M(X) onto the submagma of Mo(X) x Ma consisting of the ordered pairs (u, v) where u and v have the same length. I58 159 EXERCISES I ALGEBRAIC STRUCTURES 7 4. Let X be a set and c an element not belonging to X. We write y = x u {c}. Let M be the magma with underlying set Mo(Y) and law of composition given by (u, u) - cuu. Let L be the submagma of M generated by x and let E be the mapping from Y to Z equal to - 1 on X and to 1 at c. (a) Show that L is the subset of M consisting of the words w = yl. . .Y,, v, E Y, satisfying the two following conditions : ". (i) E(yl) + 3 . . + E(Y,) = - 1 (ii) E(yl) + . . . + E(Y~) 2 0 for 1 < i < p - 1. Show that every element of L - X may be written uniquely in the form cuu with u, v E L. (The elements w satisrying (i) and (ii) form a submagma L' of M containing x. If w = yl.. .y, belongs to L', let k be the least integer such that .(yl) + . . . + ~(y,) = 0; show that y1 = c and that the elements u = y2. . .y, and v = y, + 1. . . y, belong to L' ; then w = cuu, whence by induction on p the relation w EL. Then show that the relations u', u' EL and w = cu'u' imply u' = u, v' = u.) (b) Show that the injection of X into L extends to an isomorphism of M(X) onto L. 5. Let X be a set with one element. If n is an intcger 3 1, let u, denote the (a) For every set Y, show that number of elements in M(X) of length 12. Card(M,(Y)) = u,. Card(Y)". (Use Exercise 3.) (b) Establish the relation m (c) *Letf(T) be the formal power series n=l 2 u,T". Show that J(T) = T +f(T)'. Deduce the formulae 2"-1 f(T) = +(1 - (1 - 4T)19, u, = - 1.3.5.. .(2n - 3) for n 2 2.* n! Obtain the last result using Exercise 11 of Set Theory, 111, 3 5. 7 6. Let X be a set. (a) Let N be a submagma of M(X) and Y = N - (N.N). The injection Y -> N extends to a homomorphism u: M(Y) --f N. Show that u is an iso- morphism. (In other words, every submagma of a free magma is free.) 160 (6) If x E M(X), let M, denote the submagma of M(X) generated by x; by (a), M, is identified with the free magma constructed on x. Show that, if X, y E M(X), then either M, c M, or My c M, or M, n My = ia. (If M, n M, # 0, let z be an element of M, n My of minimum length; show that either z = x or z = y.) 7 7. Let M, be the free magma constructed on a set {x} with one element. (a) For ally E M,, show that there exists a unique endomorphismf, of M, such thatfy(x) = y; it is an isomorphism of M, onto the submagma My generated by y (cf. Exercise 6). (6) Ify, z E M,, we write y 0 z =fy(z). Show that the law of composition (y, 2) H y o z makes M, into a monoid with identity element x which is iso- morphic to the monoid End(M,) of endomorphisms of M,. (c) Let N be the set of monogenous submagmas of M, distinct from M,. An element y of M, is called primitive if My is a maximal element of N. Show that XX, x(~(xx)) are primitive whereas x, (xx) (XX) are not. (d) Let P be the set of primitive elements of M,. Show that, ify, z E P, y # z, then M, n M, = ,@ (use Exercise 6). (e) Let Mo(P) be the free monoid constructed on P. If M, is given the monoid structure defined in (b), the injection P + M, can be extended to a homomorphismp: Mo(P) -+ M,. Show thatp is an isomorphism. (If t E M,, show by induction on 1 (z) that z E Im(p). On the other hand, if yl,. . ., y,, zl,. . ., z, E P are such that p(yl.. .y,) = p(zl.. .z,), then whence P(y,. . .yn) = p(z2. . .zm) and it follows that yl. . .y, = zl. . . z, arguing by induction on sup(n, m).) 8. Let X be a set; for every integer q > 0 let Mo(X), be the set of elements of Mo(X) of length q and let MO(~)(X) be the union of Mo(X),,, R E N. The injection Mo(X), -+ Mo(@(X) extends to a homomorphism Mo(Mo(X),) -+ Mo(@(X); Mu1 n Mq # a, whence ~1 = 21 and fy,(~(~z. . .~n)) =fy,(p(zZ. . .zm)); show that this homomorphism is bijective if q > 1. by {X, XY, yx} is not isomorphic to a free monoid. 9. Let X,Y E x with x # Y. Show that the submonoid of Mo(X) generated 10. Show that the group defined by the presentation (x, y; xy2 = y3x, yx2 = x3y) reduces to the identity element. (The first relation implies x2y8x-2 = y18 and x3y8x-3 = yZ7; use the second relation to deduce that y18 = yZ7, whence y9 = e; the fact that y2 is conjugate then implies y = e, whence x = e.) 161 EXERCISES presentation ((t,)l,I, (rj)JEJ) where 1 is a finite set (resp. where I and J arc finite sets). (a) Let G be a finitely presented group and x = (x,),€* a finite generating family of G. Show that there exists a finite family s = (s~)~~~ of relators ofthc family x such that (x, S) is a presentation of G. (Use the preceding exercise.: (b) Show that a finite group is finitely presented. (c) Give an example of a finitely presented group with a subgroup which i (d) If G is a finitely presented group and H a normal subgroup of G, shov (i) G/H is finitely presented. (ii) There exists a finite subset X of G such that H is the normal subgroup (iii) If (H,),,, is a right directed family of normal subgroup of G whose (Use (a) to prove that (i) implies (ii).) (e) If G is a finitely presented group, show that G/CT(G) is for all r 2 1. (f) Let (Ga),,A be a family of groups and let G = n G, be the product of the G,. Show that G is finitely generated (resp. finitely presented) if and only if each of the G, is finitely generated (resp. finitely presented) and G, = {e} for all but a finite number of a. fi 17. (a) Show that every subgroup of a finitely generated nilpotent group is finitely generated. (In the commutative case, argue by induction on the minimum number of generators of the group; then proceed by induction on the nilpotency class of the group.) Give an example of a finitely generated solvable group containing a sub. group which is not finitely generated. (b) Show that every finitely generated nilpotent group is finitely presented (Write the group in the form F(X)/R, with X finite, and choose r such tha R 3 Cr(F(X)) ; use (a) to show that R/C‘(F(X)) is finitely generated; observc that F(X)/Cr(F(X)) is finitely presented, cf. Exercise 16.) 18. Let S be a symmetric subset of a group G. Two elements g,, ga of G arc called S-neighbours if gZ1g1 belongs to S. A sequence (gl, . . ., g,) of element: of G is called an S-chain if g1 and g1 + are S-neighbours for 1 < i < n - 1 ; thc elements g, and g, are called respectively the beginning and the end of thc chain. (0) Let Y be a subset of G and R,{a, bJ the relation “there exists an S-chair in y beginning at a and ending at b”. Show that R, is an equivalence relatio! on y (cf. Set Themy, 11, tj 6, Exercise 10) ; the equivalence classes under thi relation are called the connected S-components of Y. Y is called S-connected if it ha at most one connected component. not finitely generated. the equivalence of the following properties : of G generated by X. union is equal to H, there exists i E I such that Hi = H. (Use tj 6, Exercise 15.) aEA 16: I ALGEBRAIC STRUCTURES 11. The group defined by the presentation (x, y; 2, y3, (xY)~) is isomorphic to 63. 7 12. The group defined by the presentation (x,y; x3, ya, (x~)~) is iso- morphic to 91,. (Show that the conjugates ofy form, with the identity element, a normal subgroup of order < 4 and index < 3.) 13. Let G be a group and X a subset of G disjoint from X-l. Let S = X u X-l. Show the equivalence of the two following properties: (i) X is a free family in G. (ii) No product s,. . . s,, with n 2 1, si E S and sisl+ , # e for 1 < i < n - 1, is equal to e. 14. A group G is calledfree if it possesses a basic family and hence is iso - morphic to a free group F(X) constructed on a set X. (a) Show that, if and (t;)j,J are two such families, then Card(1) = Card(J). (When Card(1) is infinite, show that Card(J) is also and that both are equal to Card(G). When Card(1) is an integer d, show that the number of subgroups of index 2 in G is 2d - 1.) The cardinal of a basic family of G is called the rank of G. (b) A free group is of rank 0 (resp. 1) if and only if it reduces to {e} (resp. is isomorphic to Z). (c) Let G be a free group of finite rank d and (xl,. . . , xd) a generating family of d elements of G. Show that this family is basic. (Use Exercise 34 and 5 5, Exercise 5.) (d) Show that a free group of rank 22 contains a free subgroup of given rank d for every cardinal d < x,. (Use Exercise 22.) 7 15. Let G be a group with presentation (t, r), where t = and r = (r,),,J; let F((Ti)i,I) denote the free group F(1) and T = (Ti). On the other hand, let x = be a generating family of G; write F(A) = F((Xa)aed, X = (X,) and R, the set of relators of x; it is a normal subgroup of F(A). For all i E I, let &(X) be an element of F(A) such that ti = +i(x) in G; for all a E A, let +,(T) be an element of F(1) such that x, = +,(t) in G. Show that R, is the normal subgroup of F(A) generated by the elements &-l+a(+i(X)lEI), aEA and r,(+i(X)i,I)jeJ. (If RL denotes the normal subgroup of F(A) generated by the elements in question, then RL c R,; on the other hand, show that the homomorphism F(1) -+ F(A) defined by the +i gives, when passing to the quotient, a homomorphism G -+ F(A)/RL the inverse of the canonical homomorphism F(A)/R: + F(A)/R, = G.) 16. A group is called finitely generated (resp. finitely presented) if it admits a 162 ALGEBRAIC STRUCTURES I EXERCISES (b) Show that G is S-connectcd if and only if S generates G. (cj Suppose that no element s E S satisfies the relation s2 = e; choose a subset X of S such that S is the disjoint union of X and X- l. Show the equiva- Iencz of the following conditions: _____. (i) The family X is free. (ii) There exists no S-chain (gl, . . . , g,) in G consisting of distinct elements n 2 3 in number such that g, and g, are S-neighbours. (Use Exercise 13.) T 19. Let X be a set, G = F(X) the free group constructed on X and s = xu x-1. (u) Ifg E G, every sequence (sl, . . . , s,) of elements of S such that g = sl. . . s, and spl+l # e for 1 < i < n is called a reduced decomposition of g. Show that every element of G admits one and only one reduced decomposition; the integer n is the length of g (cf. Exercise 26). (b) Let Y be a subset of G containing e. Show the equivalence of the follow - ing conditions: (b,) Y is S-connected (cf. Exercise 18). (b2) For all g E Y, if (s,, . . . , s,) is the reduced decomposition of g, then (c) Let Y, and Y, be two non-empty S-connected subsets of G such that s,...s,~Y for 1 < i < n. Y, n Y, = @. Show that there exists at most one ordered pair (Yl,Y,) EY1 x y2 such that y, and y, are S-neighbours. There exists one if and only if Y, U Y, is S-connected; Y, and Y, are then said to be neighbours. (d) Let Y,, . . . , Y, be a sequence of disjoint non-empty S-connected subscts of G. Suppose that Y, and Y,+l are neighbours for 1 < i < n. Show that, if n 2 3, Y, and Y, are not neighbours. Ti 20. We preserve the notation of the preceding exercise. Let H be a sub- group of G. - (u) Let R, be the set of S-connected subsets T of G such that hT n T = 0 if h E H, h # e. Show that RH is inductive for the relation of inclusion. I. (6) Let Y be a maximal element of R,. Show that G = hvH hY, in other words Y is a system of representatives of the right cosets modulo H. (If G = u hY is distinct from G, show that there exist x E G and y E Y which are S-neighbours and deduce that Y u {x} belongs to RH.) (c) Let S, be the set of elements h E H such that Y and hY are neighbours. It is a symmetric subset of H. Show that an element h E H belongs to SH if and only if h # e and there exist y, y’ E Y, s E S with ys = hy‘. Show that S, generates H. (If H’ is the subgroup of H generated by SH, prove that ,g, hY is a connected S-component of G.) Obtain this result by means of Exercise 14 of 3 4. (d) Let X, be a subset of S, such that S, is the disjoint union of X, and Xg1 (show that such a subset exists). Show that X, is a basic family for H and in particular that H is a free group (“Nielsen-Schreier Theorem”). (Apply the criterion of Exercise 18(c) to H to- gether with SH, noting that two elements h, h’ of H are S,-neighbours if and only if hY and h‘Y are neighbouring subsets of G. Use Exercise 19(d) to prove that S, satisfies condition (ii) in Exercise 18(c).) (e) Let d = (G:H) = Card(Y). Suppose d is finite. Show that: Card&) = Card(X) = Card(G) Card(X,) = 1 + d(Card(X) - 1) if Card(X) is infinite if Card(X) is finite. (Let T be a non-empty S-connected finite subset of G and T‘ (resp. T”) the set of ordered pairs of S-neighbouring elements whose first element (resp. both ofwhose elements) belongs (resp. belong) to T. Card(T’) = 2Card(X)Card(T) and it should be shown by induction on Card(T) that Card(T”) = 2Card(T) - 2. Apply the result to T = Y, noting that S, is equipotent to T’ - T”.) 21. Let H be a subgroup of a group G of finite index. Show that H is finitely presented if and only if G is. (It may be assumed that G is finitely generated, cf. $4, Exercise 14. Write G in the form G = F/R with F free and finitely generated; the inverse image F‘ of H in F is free and finitely generated, cf. Exercise 20, and H = F‘/R. If R is generated (as a normal subcrour, of F) bv I, -1 (T,),~~, show that it is generated (as-a normal’subgroup of F’) by the yr,y-l, 9 E Y, j E J, where Y denotes a representative system in F of the right cosets modulo F’.) 22. (a) Let (y,JnGz be a basic family of a group F. Let + be the automor- phism of F which maps y, toy, + , ; this automorphism defines as operation T of Z on F; let E = F x Z be the corresponding semi-direct product. Show that t!e elements x = (e, 1) and y = (yo, 0) form a basic family of E. (Let Fx,u be the free group constructed on {x, y} and letfbe the canonical homomorphism of F,, into E. Show that there exists a homomorphism g: E --f Fx, such that g((0, 1)) = x and g( (y,, 0)) = xnyx-” and f and g are inverses of one another.) (b) Deduce that the normal subgroup of Fxv generated by y is a free group With basic family (x~~x-~),,,. Obtain this result by applying Exercise 20 to the representative system consisting of the x“, n E Z. (c) Extend the above to free groups of arbitrary rank. 23. Let F be a free group with basic family (x, y), x # y. Show that the 164 165 I ALGEBRAIC STRUCTURES derived group of F has as basic family the family of commutators (XI, yj), i E Z, EXERCISES jEz, i 2 0,) # 0. (Use Exercise 20 or Exercise 32.) \- 24. Let G be a group, S = G - {e} and F = F(S). For all s E S, let x, denote the corresponding element of F. If s, t E S, let Y,,~ denote the element of F defined by : rs,t = x.ptx,;' Y,,~ = xpt if st # e in G if st = e in G. Let Q be the homomorphism of F onto G which maps x, to s and let R be the kernel of (b. Show that the family (r,J for s, 1 E S x S, is a basic family of the group R. (Apply Exercise 20 to the system of representatives of G in F consisting of e and the xr) 25. Let G be a group. Show the equivalence of the following properties: (a) G is a free group. (b) For every group H and extension E of G by H there exists a section (To prove that (b) implies (a), take E to be a free group and use Exercise 20.) 26. Let X be a set and g an element of the free group F(X). Let g = x!") be the canonical decomposition of g as a product of powers of elements of X (cf. no. 5, Proposition 7). The integer 1 (g) = 7 Irn(cc) I is called the length of g. (u) g is called cyclically reduced if either x, # x, or x1 = x, and rn(l)rn(n) > 0. Show that every conjugacy class of F(X) contains one and only one cyclically reduced element; it is the element of minimum length of the class in question. (b) An element x E F(X) is called primitive if there does not exist an integer n 2 2 and an element g E F(X) such that x = gn. Show that every element t # e of F(X) can be written uniquely in the form xn with n 2 1 and x primi - tive. (Reduce it to the case where z is cyclically reduced and observe that, if z = x", the element x is also cyclically reduced.) Show that the centralizer of z is the same as that of x; it is the cyclic subgroup generated by x. 27. Let G = G, be the sum of a family (G,),El of groups amalgamated by a common subgroup A. For all i E I, choose a subset Pt of G, satisfying G -+ E. n -~ condition (A) of no. 3. (a) Let x E G and let (a; i,, . . . , in; p,, . . . , p,) be a reduced decomposition of x; then - x = a n pa, with pa E P,,, - {eta}, ia # ia+,. a-1 Show that, if i, # in, the order of x is infinite. Show that, if i, = in, x is conju- gate to an element with a decomposition of length < n - 1. (6) Deduce that every element of G of finite order is conjugate to an element ofone of the Gi. In particular, if the GI have no element of finite order except e, the same is true of G. 28. We preserve the notation of the preceding exercise. (a) Let N be a subgroup of A. Suppose that, for all i E I, the group N is normal in GI. Show that N is normal in G = *A G, and that the canonical homomorphism of *A,N Gi/N into G/N is an isomorphism. (b) For all i E I, let Hi be a subgroup of G, and let B be a subgroup of A. Suppose that H, n A = B for all i. Show that the canonical homomorphism of*, H, into G is injective; its image is the subgroup of G generated by the T[ 29. Let G = G, *A G, be the sum of two groups G, and G, amalgamated (a) Show that G is finitely generated if G, and G, are finitely generated. (b) Suppose G, and G, are finitely presented. Show the equivalence of the (1) G is finitely presented. (2) A is finitely generated. (Show first that (1) implies (2). On the other hand, if A is not finitely generated, there exists a right directed family (Ai)ieI of subgroups of A with u A, = A and A, # A for all i. Let Hi (resp. H) be the kernel of the canonical homomorphism of GI * G, onto GI *A, G, (resp. onto G). Show that H is the union of the H, and that H, # H for all i; then apply Exercise 16(d).) (c) Deduce an example of a finitely generated group which is not finitely presented. (Take G, and G, to be free groups of rank 2 and A a free group of finite rank, cf. Exercise 22.) H,. by a subgroup A. two following properties : ,€I 30. Let A and B be two groups and G = A * B their free product. (a) Let Z be an element ofA\G/A distinct from A. Show that Z contains one and only one element z of the form b,a,b,a,. . . b, - lan - lb,, With n 2 1, a, E A - {e}, 6, E B - {e}. Show that ~ ... every element of Z can be mitten uniquely in the form aza' with a. a' E A. ,,. - (0) Let x E G - A. Letf, be the homomorphism of A * A into G whose restriction to the first (resp. second) factor of A * A is the identity (resp. the 0 I--+ xax-l). Show thatf, is injective. (Write x in the form ata' as above and reduce it thus to the case x = z; use the uniqueness of reduced decompositions in G.) In particular, A n'xh-1 = {e} and the normalizer of A in G is A. 166 167 I ALGEBRAIC STRUCTURES (c) Let T be an element of A\G/B. Show that ‘r contains one and only one element t of the form b,a,b,a,. . . bn-lan-l, with n 2 1, ai E A - {e}, b, E B - {e). Show that every element of T can be written uniquely in the form atb with a E A, b E B. (d) Let XEG and let h, be the unique endomorphism of G such that &(a) = a if a E A and hx(b) = xbx-l if b E B. Show that h, is injective. (Same method as in (b), using the decomposition x = atb of (c).) In particular, A n xBx-l = {e}. Give an example where h, is not surjective. 31. Let G be the free product of a family (GJiE1 of groups. Let S be the set of subgroups of G which are conjugate to one of the G, and 0 the union of the EXERCISES elements of S. (a) Let H, H‘ E S, with H # H’, and let x E H - {e}, x’ E H’ - {e}. Show that xx‘ E G - 0. (Reduce it to the case where H is one .. of the G,; write H‘ in the form yG,y- and proceed as in the preceding exercise.) fb) Let C be a subgroup of G contained in 0. Show that there exists H E S -_ sudh that C c H. (c) Let C be a subgroup of G all of whose elements are of finite order. Show that C is contained in 0 (cf. Exercise 27) ; deduce that C is contained in a conjugate of one of the G,. 1 32. Let A and B be two groups and G = A * B their free product. Let X be the set of commutators (a, b) with a EA - {e} and b EB - {e} and s = xvx-1. (a) Show that X n X- = @ and that, if x is an element of X, there exists a unique ordered pair a, b in (A - {e}) x (B - {e}) such that (a, b) = x. (b) Let n 2 1 and let sl,. . . , s, be a sequence of elements of S such that sis,+l # eforl < i < n;leta,EA -{e},b,EB -{e},c(n) = +I besuchthat S, = (a,, b,)€(,). Let g = sl.. .s,. Show, by induction on n, that the reduced decomposition of g is of length an + 3 and terminates either with anbn if c(n) = 1 or with bnan if c(n) = -1. (c) Let R be the kernel of the canonical projection A * B-tA x B corresponding (no. 3) to the canonical injections A -+ A x B and B -+ A x B. Show that R is a free group with basic family X. (Use (b) to prove that X is free: if R, is the subgroup of G generated by X, show that Rx is normal; deduce that it coincides with R.) 33. Let E = G *A G‘ be the sum of two groups amalgamated by a common subgroup A. Let P (resp. P’) be a subset of G (resp. G‘) satisfying condition (A) of no. 3. Let n be an integcr 2 1. Let L, denote the set of x E E whose reduced de- composition is of length 6271 - 1. Let M, denote the set of x E E of the form ap1P;p2p~. . .@,PA, with a E A, pi E P - {e}, pi E P’ - {e}; similarly, let M; denote the set of elements of the form ap;p1p;p2. . .&fin, where a, pi, pi satisfy the same conditions. (a) Show that L,, M, and MA are disjoint and that their union is the set of elements of E whose reduced decomposition is of length 6 2n. (6) Construct a bijection E of M, onto Mk such that E(UX) = a~(x) if a E A, x E M,. (c) Let X, = L, u M, and Xk = L, U Mk; show that G.X, = X, and G‘.x; = x;. (d) E extends to a bijection of X, onto Xk by setting E(X) = x if x E L,. Show that E can operate on X, so that g(x) = gx if g EG, x EX, and g’(X) = c-l(g’ E (x)) if g’ E G’, x E X,. Show that, if g E X,, then g(e) = g. (e) Let 7,: E + Qxn be the homomorphism defined by the action of E on X, described above and let R, be the kernel of T,. Show that R, n X, = {e}; deduce that the intersection of the R, reduces to e. (f) When G and G‘ are finite, show that E is residually finite (cf. 5 5, Exercise 5). (Note that the R, are then subgroups of finite index.) 7 34. (a) Let (H,) be a right directed family of normal subgroups of a group G. Suppose that (;I H, = {e}. Show that, for every group G’, the intersection of the kernels of the canonical homomorphisms G’ * G -+ G’ * (G/H,) is equal to {e}. (b) Let G be the free product of a family (G,) of groups. Show that, if all the G, are residually finite (cf. 5 5, Exercise 5), so is G. (Reduce it first to the case where the family is finite, then to the case where it has two elements; by (a) it may be assumed that the G, are finite, then apply the preceding exercise.) (c) Deduce that every free group is residually finite. 35. *Let G (resp. G’) be the additive group of fractions of the form a/b with a E Z, b E Z and b not divisible by 2 (resp. by 3). Let E = G *z G’ be the sum ofG and G’ amalgamated by their common subgroup Z. Show that G and G‘ are residually finite, but that E is not. (Prove that the only subgroup of E of finite index is E.)* 36. Let (Gl, . . ., G,) and (Al,. . ., A,-l) be two sequences of groups; for each i, let +,: A, --f G, and +,: A, -+ G,, be two injective homomorphisms; by these homomorphisms A, can be identified both with a subgroup of G, and with a subgroup of G1 + (a) Let HI = GI, H2 = HI *A1 G,, . . ., H, = H,-1 *An-l G,. The group % called the sum of the groups (G,) amalgamated by the subgroups (A,) ; it is denoted by G, *A1 G2 *A2 . . . *An-l G,. If T is an arbitrary group, de- fine a bijection between the homomorphisms of H, into T and the families 168 169 I ALGEBRAIC STRUCTURES (t,, . . ., tn) of homomorphisms ti: Gi 3 T such that ti 0 +, = ti+l 0 6, for 1~iGn-I. (b) Each G, is identified with the corresponding subgroup of H,. Show that G, Gi+l = A, and that G, n G,,, = A, n A,+, (the latter intersection being taken in G, + ,). 7 37. Let G be a group and S a finite symmetric subset of G generating G. Let 9: denote the set of finite subsets of G, ordered by inclusion; it is a directed set. (a) Let T E iX and let B, be the set of connected S -components of G - T (cf. Exercise 18); let B; = B, - (LY n B,). Show that B, is finite (note that, if T # .B and Z E B,, there exists t E T and z E Z which are S-neighbours). (b) Let T, T' E e with T c T'. If Z' E B,., show that there exists a unique element Z E B, such that Z' c Z; the mapping JTT,: BTt + BT such that fTT'(Z') = Z maps B;, onto BF. The inverse limit of the B,, with T running through e, is denoted by B. Show that B = lim t B; and that the image of B in B, is B;. The set B is called the set of ends of G; it is non-empty if and only if EXERCISES (a) If X E 3 and n E N, let d,,(X) denote the number of elements in G of the form xl. . .x, with xi E X. If X, Y E 5, show that there exists an integer a 2 1 such that d,,(X) < d,,(Y) for all n EN. Deduce that lim sup 10g(dn(x)) does not depend on the choice of X in 8; this n-m log(n) limit (finite or infinite) is denoted bv e(G1. G isinfinite. (c) Let T be a non-empty S -connected finite subset of G and let Z E B;. Show that there exist g E G such that gT n T = 0 and gT n Z # 0 and that then PT c Z. Show that there exists Z, E BT such that gZ, contains T and " all the Z' E B, such that Z' # Z (observe that T u zvz Z' is S-connected and does not meet gT) ; deduce that, if Z' E B, is different from Z1, then gZ' c Z. Show that, writing T' = T u gT, the inverse image of Z in consists of at least n - 1 elements, where n = Card(B,"). (d) Let S' be a symmetric finite subset of G generating G and B' the set of ends of G relative to S'. Define a bijection of B onto B' (reduce it to the case where S c S'). The cardinal of B is thus independent of the choice of S; it is called the number of ends of G. (e) Show, using (c), that the number of ends of G is equal to 0, 1, 2 or 2% (f) Show that the number of ends of Z is 2 and that that of Zn (n 2 2) is 1. 7 38. Let X be a finite set, F(X) the free group constructed on X and s = xux-1. (a) Let S, be the set of products sl. . .s,, s, E S, m < n. Show that every connected S-component (cf. Exercise 18) of F(X) - S, contains one and only oneelementoftheforms,...s,+lwiths,~Sands,si+l fefor 1 < i < n. (b) Deduce from (a) a bijection of the set B of ends of F(X) (cf. Exercise 37) onto the set of infinite sequences (s,, . . . , s,, . . .) of elements of S such that s,si + # e for all i. In particular, the number of ends of F(X) is equal to 2No if Card(X) 2 2. 39. *Let G be a finitely generated group and 8 the set of symmetric finite subsets of G containing e, which generate G. 170 (6) If X E 5 and G is infinite, show ;ha; d,(X) < d,+,(X) for all n. Deduce that e(G) is 2 1. (c) Let H be a finitely generated group. Show that e(H) < e(G) if H is isomorphic to a subgroup (resp. a quotient group) of G. If E is an extension of G by H, show that e(E) 2 e(H) + e(G), with equality when E = G x H. (d) Show that e(Zn) = n. (e) Let X E 5. Consider the following property: (i) There exists a real number a > 1 such that d,(X) 2 an for all sufficiently large n. Show that, if property (i) holds for oneelement of 3, it holds forevery element of &. G is then said to be ofexponential type; this implies e(G) = +LO. (f) Let GI and G, be two groups containing the same subgroup A and let H = G1 *A G, the corresponding amalgamated sum. Suppose G, and G, are finitely generated (hence also H is) and G, # A for i = 1,2. Show that H is of exponential type if at least one of the indices (G,:A), (G,:A) is 2 3. (g) Show that a free group of rank 22 is of exponential type., 40. *Let n be an integer 2 2 and T, the group of upper triangular matrices of order n over Z with all the diagonal terms equal to 1. Let e(T,) be the invariant of the group T,, defined in Exercise 39. Show that n 1. Determine all the ring structures on a set of n elements for 2 < n < 7 and also the ideals of these rings. 2. Let A be a ring. a (a) The mapping which associates with a E A the left homothety x ++ ax is an isomorphism of A onto the ring of endomorphisms of the additive group of A which commute with right homotheties. (6) Let M be the set of sequences (a, b, c, d) of four elements of A which may be represented in the form of a table or matrix (: t). With every element 171 I ALGEBRAIC STRUCTURES EXERCISES 7. For all n EN*, I“ b\ of M we associate the mapping (x, y) H (ax + by, cx + dy) of A x A \c dl into itself. Show that a bijection is thus obtained of M onto the ring B of endo- morphisms of the commutative group A x A which commute with the opera- tions (x, zi) - (xr, yr) for all r E A. Henceforth M is identified with the ring B. \ 1”, _- ~ Calculate the product of two matrices. (c) Show that the subset T of M consisting of the matrices of the form I. (’ ‘) is the subring of M which leaves stable the subgroup A x 0 of A x A. oc ,- I The mappings (: !) - a and ’) t+ c are homomorphisms of T into A; C \u ”/ \- -, let a and b be their kernels. Show that a + b = T, that ab = (0) and that ba = b n a is #(O) if A # (0) (cf. no. 9, Proposition 6). fd) If A = 2/22, T is not commutative, but the multiplicative group of \I invertible elements of T is commutative. 3. Let A be a ring and a E A. If there exists one and only one a’ E A such that aa’ = 1, a is invertible and a’ = a-l (show first that a is left cancellable, then consider the product aa’a) . 4. Let A be a pseudo -ring in which every additive subgroup of A is a left ideal of A. (a) If A is a ring, A is canonically isomorphic to Z/nZ for some suitable integer n. is isomorphic to a pseudo-subring of Z. (6) If A has no unit element and every non-zero element is cancellable, A 5. Let M be a monoid, Z(M) the free commutative group with basis M and u: M --f Z(’) the canonical mapping. (a) Show that there exists a unique multiplication on Z(M) such that Z(’’ is a ring and such that u is a monoid morphism of the monoid M into the multi- plicative monoid of the ring Z(M). (b) Let A be a ring and u: M --f A a monoid morphism of the monoid M in- to the multiplicative monoid of A. There exists a unique ring morphism f: Z(‘) -+ A such that fOU = v. 6. Let A be a commutative ring and M a submonoid of the additive group (a) If n! is not a divisor of zero in A, the product of a family of n elements of of A. Let n be an integer > 0 such that mn = 0 for m E M. M ’is zero. zero in A. (b) Show that the conclusion in (a) is not necessarily true if n! is a divisor of (use no. 2, Proposition 2). 8. In a ring, the right ideal generated by a left ideal is a two-sided ideal. 9. In a ring, the right annihilator of a right ideal is a two-sided ideal. 10. In a ring A, the two-sided ideal generated by the elements xy - yx, where x and y run through A, is the smallest of the two-sided ideals a such that A/a is commutative. fi 1 1. In a ring A, a two-sided ideal a is said to he irreducible if there exists no ordered pair of two-sided ideals b, c distinct from a and such that a = b n c. (a) Show that the intersection of all the irreducible two-sided ideals of A reduces to 0 (note that the set of two-sided ideals containing no element a # 0 is inductive and apply Zorn’s Lemma). (b) Deduce that every two-sided ideal a of A is the intersection of all the irreducible ideals which contain it. 12. Let A be a ring and (a,),,, a finite family of left ideals of A such that the additive group of A is the direct sum of the additive groups a,. If 1 = 2 e,(e, E a,), then e: = el, e,e, = 0 if i # j and a, = Ae, (write x = x . 1 for all x E A). Conversely, if (e,),,, is a finite family ofidempotents of A such that tiej = 0 for i # j and 1 = 2 e,, then A is the direct sum of the left ideals Ae,. 1EI For the ideals Ae, to be two-sided, 1 it is necessary and sufficient that the e, be in the centre of A. 7 13. Let A be a ring and e an idempotent of A. (a) Show that the additive group of A is the direct sum of the left ideal Ae and the left annihilator b of e (note that, for all x E A, x - xe E b). (b) Every right ideal b of A is the direct sum of b n a and 3 n 6. (c) If Ae = eA, a and b are two-sided ideals of A and define a direct de- a composition of A. 14. Let A be a commutative ring in which there is only a finite number n of ~Vkm of zero. Show that A has at most (n + 1)2 elements. (Let q,. . ., a, be the divisors of zero. Show that the annihilator & of a, has at most n + 1 dQmnts and that the quotient ring A/& has at most n + 1 elements.) 15- A ringoid is a set E with two laws of composition: (a) an associative 173 172 EXERCISES I ALGEBRAIC STRUCTURES multiplication xy; (6) a law written additively, not everywhere deJined (5 1, no. 1) and satisfying the following conditions : (1) it is commutative (in other words, if x + y is defined, so is y + x and x + y = y + x; x and y are then said to be addible) ; (2) ifx andy are addible, for x + y and z to be addible it suffices that x and y on the one hand, and y and z on the other, be addible; then x and y + z are addible and (x + y) + z = x + (y + z); (3) there exists an identity element 0; (4) ifx and z on the one hand, y and z on the other, are addible and (5) if x, y are addible, so are xz and yz (resp. zx and zy) and x + z = y + z, thenx = y; ’ (x + y)z = xz + yz (resp. z(x + y) = zx + zy) for all z E E. . Every ring is a ringoid. (a) Examine how the definitions and results of 5 8 and the above exercises extend to ringoids (a left ideal of a ringoid E is a subset a of E which is stable under additiin and such that E. a c a). (b) Let G be a group with operators and f and g two endomorphisms of G. For the mapping x -f(x)g(x) to be an endomorphism of G, it is necessary and sufficient that every element of the subgroup f (G) be permutable with every element of g(G) ; if this endomorphism is then denoted by f + g and fg is the composite endomorphism x -f( g(x)), show that the set E of endomorphisms of G with these laws of composition is a ringoid with unit element; for E to be a ring, it is necessary and sufficient that G be commutative. For an elementfE E to be addible to all the elements of E, it is necessary and sufficient thatf(G) be contained in the centre of G; the set N of these endomorphisms is a pseduo-ring called the kernel of the ringoid E. An endomorphismfof G is called normal if it is permutable with all the inner automorphisms of G; for every normal stable subgroup H of G,f(G) is then a normal stable subgroup of G. Show that the set D of normal endomorphisms of G forms a subringoid of E and that the kernel N is a two-sided ideal in D.7 16. Give examples of ideals a, b, c in the ring Z such that ab # a n b and (a + b)(a + c) # a + bc. 69 1. Which are the field structures amongst the ring structures determined in fj 8, Exercise 1 ? t See H. FITTING, Die Theorie der Automorphismenringe Abelscher Gruppen und ihre Analogen bei nicht kommutativen Gruppen, Math. Ann., 107 (1933), 514. 174 2. Afinite ring in which every non-zero element is cancellable is a field (cf. 3. Let G be a commutative group with operators which is simple (5 4, no. 4, 4. Let K be a field. Show that there exists a smallest subfield F of K and that F is canonically isomorphic either to the field Q or to the field Z/pZ for some prime number p. In the first case (resp. second case) K is said to be of charac - teristic 0 (resp. p) . 5. Let K be a commutative field of characteristic 22; let G be a subgroup ofthe additive group of K such that, if H denotes the set consisting of 0 and the inverses of the elements # 0 of G, H is also a subgroup of the additive group of K. Show that there exists an element a E K and a subfield K’ of K such that G = aK’ (establish first that, if x and y are elements of G such that y # 0, then xa/y E G; deduce that, if x, y, z are elements of G such that z # 0, then 5 2, Exercise 6). Definition 7). Show that the endomorphism ring A of G is a field. xy/z E GI. I 6. Let K be a commutative field of characteristic 22; let f be a mapping of K into K such thatf(x + y) =f(x) +f(y) for all x and y and for all x # 0 f(x)f(l/x) = 1. Show that f is an isomorphism of K onto a subfield of K (prove thatf(x2) = (f(x))”. 7. Let A be a commutative ring. (a) Every prime ideal is irreducible (5 8, Exercise 11). (b) The set of prime ideals is inductive for the relations c and 3. (c) Let a be an ideal of A distinct from A; let b be the set of x E A such that there exists an integer n > 0 with xn E a (n depends on x). Show that b is an ideal and is equal to the intersection of the prime ideals of A containing a. fi 8. A ring A is called a Boolean ring if each of its elements is idempotent (in other words, if x2 = x for all x E A). (a) In the set V(E) of subsets of a set E, show that a Boolean ring structure is defined by setting AB = A n B and A + B = (A n CB) u (B n CA). ms nng is isomorphic to the ring KE of mappings of E into the ring K = Z/(2) of integers modulo 2 (consider for each subset X c E its “characteristic func- (b) Every Boolean ring A is commutative and such that x + x = 0 for * E A (write x + x as an idempotent and then x + y as an idempotent). (c) Ifa Boolean ring A contains no divisor of 0, it is reduced to 0 or is iso - morphic to z/(2) (if x and y are any two elements of A, show that qb -!- Y) = 0). Deduce that in a Boolean ring every prime ideal is maximal. (d) In a Boolean ring A, every ideal a # A is the intersection of the prime containing a (apply Exercise 7(c)). Deduce that every irreducible ideal I iI 175 I I , tion” $X such that &(x) = 1 if x E X, +X(x) = 0 if x 4 X). I ALGEBRAIC STRUCTURES is maximal (in other words, that the notions of irreducible ideal, prime ideal and maximal ideal coincide in a Boolean ring). (8) Show that every Boolean ring is isomorphic to a subring of a product EXERCISES ringKE, where K = Z/(2) (use (d)). I f) If p, (1 < i < n) are n distinct maximal ideals in a Boolean ring A and ,./, ... a = n p,, show that the quotient ring A/a is isomorphic to the product ring K". Deduce that every finite Boolean ring is of the form K". (g) In a Boolean ring A, the relation xy = x is an order relation; if it is denoted by x ,< y, show that with this relation A is a distributive lattice (Set Theory, 111, 5 1, Exercise 16), has a least element ct and that, for every ordered pair (x, y) of elements such that x < y, there exists an element d (x, y) such that inf(x, d(x,y)) = m, sup(x, d(x, y)) = y. Conversely, if an ordered set A has these three properties, show that the laws of composition xy = inf(x,y) and x + y = d(sup(x, y), inf(x, y)) define a Boolean ring structure on A. 9. Let A be a ring such that x3 = x for all x E A. It is proposed to prove iSiQn that A is commutative. (a) Show that 6A = (0) and that 2A and 3A are two-sided ideals such that 2A + 3A = A and 2A n 3A = (0). Deduce that it can be assumed that either 2A = (0) or 3A = (0). (b) If 2A = {0}, calculate (1 + x)~, deduce that x2 = x for all x E A and \, conclude by means of Exercise 8. (c) If 3A = (0}, calculate (x + and (x - Y)~, show that xay + xyx + yx2 = 0 and left multiply by x; deduce that xy - yx = 0. (d) Let A be a ring such that 3A = (0) and x3 = x for all x E A; define a set I and an injective homomorphism of A into (Z/3Z)' (same method as in Exercise 8). 10. Let A be a commutative ring and U the intersection of its maximal ideals. (a) Show that the canonical homomorphism A* -+ (A/U)* is surjective and that its kernel is 1 + U. finite. Deduce that A is finite (apply Proposition 8 of 5 8, no. 10 to A/U). (b) Suppose that the set M of maximal ideals of A is finite and that A* is (c) Deduce that the set of prime numbers is infinite (note that Z* = (1, - 1)). (d) Suppose that A* is finite and that A/m is finite for every maximal ideal m. Can it be deduced that A is finite, that A is countable? 11. Let P be the set of prime numbers and A the product ring of the fields such ZlpZ, p E P. Let a be the subset of A consisting of the elements that a, # 0 only for a finite number of indices p. (a) a is an ideal of A. (b) Let B = A/a. For every integer n > 0 and every b # 0 in B, there exists one and only one element b' of B such that nb' = b (note that if p E P does 176 not divide n, multiplication by n in Z/pZ is bijective). Deduce that B contains a subfield isomorphic to Q. 12. Let A be a commutative ring and a, b E A. Show that the canonical image of ab in A/(a - 26) is an idempotent. Give an example where this idempotent is distinct from 0 and 1. 13. Determine the endomorphisms of the ring Z and the ring Z x Z. More generally, if I and J are finite sets, determine the homomorphisms of the ring Z1 into the ring ZJ. 14. Let A and B be two rings and f and p two homomorphisms of A into B. 1s the set of x E A such thatf(x) = g(x) a subring of A, an ideal of A? 7 15. Let A be a non-commutative ring with no divisors of 0. A is said to admit aJield glgt fractions if it is isomorphic to a subring B of a field K such that every element of K is of the form x-'y, where x E B, y E B. (a) Let A' be the set of elements #O in A. For A to admit a field of left fractions, it is necessary that the following condition be fulfilled: (G) for all x E A, x' E A', there exist u E A' and v E A such that ux = vx'. (6) Suppose conversely that condition (G) is fulfilled. Show that in the set A x A' the relation R between (x, x') and (y, y') which states "for every ordered pair (u, v) of elements #O such that ux' = vy', ux = y" is an equiva- lence relation. Let (x, x'), ( y, y') be two elements of A x A', 6 and q their respective classes (mod. R). For every ordered pair (u, u') E A x A' such that u'x = uy', show that the class (mod. R) of (uy, u'y') depends only on the classes 5 and q; if it is denoted by (q, a law of composition is defined on the set K = (A x A')/R. If K' is the set of elements of K distinct from the class 0 of elements (0, x') ofA x A', K', with the law induced by the above law, is a group. For every element x E A, the elements (x'x, x'), where x' runs through A', define an isomorphism of A (with multiplication alone) onto a subring of K. 1dentirt;ng A with its image under this isomorphism, the class (mod. R) of an ordered pair (x, x') E A x A' is identified with the element x'-'x. This being so, if ( = x'-lx is an element of K and 1 denotes the unit ele- ment of K, then E + 1 denotes the element x'-l(x + XI), which does not depend on the representation of c in the form X'-'X. We then write 6 + 0 = 6 and, for + 1). Show that the addition and multi- Plation thus defined on K determine on that set a field structure which the ring structure on A; in other words, condition (G) is mficient for A to admit a field of left fractions. # 0, E + q = 16- Let A be a ring in which every non-zero element is cancellable. For A to admit a field of left fractions (Exercise 15), it is necessary and sufficient that in A the intersection of two left ideals distinct from 0 never reduce to 0. 177 I ALGEBRAIC STRUCTURES 17. Let (KJIEI be a family of fields. For every subset J of I, let eJ denote the Kl whose i-th component is equal to 0 if element of the product K = EXERCISES (6) Show that the relation a # 0 implies a2 # 0 (note that if a2 = 0, it would follow that aAa = (0)). (c) Show that if ab = 0 and a # 0 (resp. b # 0) then b = 0 (resp. a =O). (Note that (ba)' = 0 and deduce that bAb = {O}.) 20. Let A be a non-zero pseudo-ring such that there exists a E A for which, for all b E A, one of the equations ax = 6, xu = b admits a solution x E A. Show that if A has no divisor of 0 other than 0, A admits a unit element. 21. Let A be a non-zero pseudo-ring in which, for all a # 0 and all b E A, one of the equations ax = b, xu = b admits a solution x E A. Show that A is a field (use Exercises 19 and 20). iEJandto1ifiEI-J. (a) Let x = (xl) be an element of K and J(x) the set of elements i E I such that x1 = 0. Show that there exists an invertible element u in K such that x = ueJ(,) = eJ(x)'. (6) Let a be a left (resp. right) ideal of K distinct from K. Let 5, be the set of subsets J of I such that e, E a. Show that 5, has the following properties: (b,) IfJE 5, and J' 5, J, then J' E 5,. (b2) IfJ1, J2 E $a, then J1 n JZ E 5,. (b3) 0 5,- Show that x E a is equivalent to J(x) E 5, (use (a)). Deduce that a is a two- sided ideal. (c) A set of subsets of I is called ajlter if it satisfies properties (b,), (b2), (b3) above (*cf. General Topology, I, fj 6*). Show that the mapping a H 5, is a strictly increasing bijection of the set of ideals of K distinct from K onto the set of filters of I. *Show that a is maximal if and only if 5, is an ultrafilter., (d) *Let 5 be a non-trivial ultrafilter on the set P of prime numbers, let Q be the corresponding ideal of the ring K = n PEP Z/pZ and let k = K/a. Show that k is a field of characteristic O., 7 18. (a) Let K be a field and a, b two non-permutable elements of K. Show that (1) a = (b - (a - I)-'b(a - l))(a-'ba - (U - l)-'b(a - l))-l (2) u = (1 - (a - l)-'b-'(a - l)b)(a-'b-'ab - (a - l)-'b-'(a - l)b)-'. (b) Let K be a non-commutative field and x an element not belonging to the centre of K. Show that K is generated by the set of conjugates axa-l of x. (Let K, be the subfield of K generated by the set of conjugates of x. Deduce from (a) that, if K, # K and a E K1 and b 4: K,, then necessarily ab = ba. Deduce a contradiction by considering in K1 two non-permutable elements a, a' and an element b $ K, and noting that ba $ Kl.) (c) Deduce from (b) that if Z is the centre of K and H is a subfield of K such that Z c H c K and aHa-l = H for all a # 0 in K, then H = Z or H = K (Cartan-Brauer-Hua Theorem). (d) Deduce from (a) that in a non-commutative field K, the set of commu - tators aba-lb-l of elements #O in K generates the field K. 19. Let A be a non-zero pseudo -ring such that, for all a # 0 in A and all b E A, the equation ax + ya = b has solutions consisting of ordered pairs (x, y) of elements of A; in other words, aA + Aa = A for a # 0 in A. (a) Show that A is the only non-zero two-sided ideal in A. 178 I § 10 1. Let X be the direct limit of a direct system (X&) of sets relative to a right directed set I. Let F, (resp. F) be the free group constructed on XI (resp. on X); eachJ.l extends to a homomorphism +ji:Fi + Fj. (a) Show that (F,, +jl) is a direct system of groups. (b) Let ci be the composite mapping X, + F, -+ lim Fi. Show that cj 0 +jl = E, ifj 3 i. Let E be the mapping of X into lim --f F, defined by the el. Show that E extends to an isomorphism of F onto lim --f F, (so that F(lim XI) can be identified with lim F(Xi)). --f 3 (c) State and prove --f analogous results for free magmas, free monoids, free commutative groups, free commutative monoids. a group with one element. 2. Show that a direct limit of simple groups is either a simple group or 1 79 HISTORICAL NOTE HISTORICAL NOTE (Numbers in brackets refer to the bibliography at the end of this Note.) There are few notions in Mathematics more primitive than that of a law of composition: it seems inseparable from the first rudiments of calculations on natural numbers and measurable quantities. The most ancient documents that have come down to us on the Mathematics of the Egyptians and Baby- lonians show that they already had a complete system of rules for calculating with natural numbers > 0, rational numbers > 0, lengths and areas; although the preserved texts deal only with problems in which the given quantities have explicit numerical values,? they leave us in no doubt concerning the generality they attributed to the rules employed and show a quite remarkable technical ability in the manipulation of equations of the first and second degree ([l], p. 179 et seq.). On the other hand there is not the slightest trace of a concern to justify the rules used nor even of precise definitions of the opera- tions which occur: the latter as the former arise in a purely empirical manner. Just such a concern, however, is already very clearly manifest in the works of the Greeks of the classical age; true, an axiomatic treatment is not found of the theory of natural numbers (such an axiomatization was only to appear at the end of the nineteenth century; see the Historical Note to Set Theory, IV) ; but in Euclid’s Elements there are numerous passages giving formal proofs of rules of calculation which are just as intuitively “obvious” as those of calcula- tion with integers (for example, the commutativity of the product of two rational numbers). The most remarkable proofs of this nature are those It must not be forgotten that Viete (16th century) was the first to use letters to denote all the elements (known and unknown) occurring in an algebraic problem. Until then the only problems solved in treatises on Algebra had numerical cO- efficients; when an author stated a general rule for treating analogous equations, he did so (as best he could) in ordinary language; in the absence of an explicit statement of this type, the way in which the calculations were carried out in the numerical cases dealt with rendered the possession of such a rule more or less credible. relating to the theory of magnitudes, the most original creation of Greek Mathe- matics (equivalent, as is known, to our theory of real numbers >O; see the Historical Note to General Topology, IV) ; here Euclid considers amongst other things the product of two ratios of magnitudes and shows that it is independent of the form of presentation of these ratios (the first example of a “quotient” of a law of composition by an equivalence relation in the sense of 3 1, no. 6) and that they commute ([2], Book V, Prop. 22-23).? It must not however be concealed that this progress towards rigour is ac- companied in Euclid by a stagnation and even in some ways by a retrogression as far as the technique of algebraic calculations is concerned. The overwhelm- ing preponderance of Geometry (in the light of which the theory of magnitudes was obviously conceived) paralyses any independent development of algebraic notation: the elements entering into the calculations must always be “repre - sented” geometrically ; moreover the two laws of composition which occur are not defined on the same set (addition of ratios is not defined in a general way and the product of two lengths is not a length but an area) ; the result is a lack of flexibility making it almost impracticable to manipulate algebraic relations of degree greater than the second. Only with the decline of classical Greek Mathematics does Diophantus return to the tradition of the “logisticians” or professional calculators, who had continued to apply such rules as they had inherited from the Egyptians and Babylonians : no longer encumbering himself with geometric representa - tions of the “numbers” he considers, he is naturally led to the development of rules of abstract algebraic calculation; for example, he gives rules which (in modern language) are equivalent to the formula xm + * = xmxn for small values (positive or negative) of m and n ([3], vol. I, pp. 8-13); a little later (pp. 12-13), the “rule of signs” is stated, the beginnings of calculating with nega- tive numbers: ; finally, Diophantus uses for the first time a letter to represent an unknown in an equation. In contrast, he seems little concerned to attach a general significance to the methods he applies for the solution of his prob- lems; as for the axiomatic conception of laws of composition as begun by Euclid, this appears foreign to Diophantus’ thought as to that of his immediate successors; it only reappears in Algebra at the beginning of the 19th century. There were first needed, during the intervening centuries, on the one hand the development of a system of algebraic notation adequate for the expression t Euclid gives at this point, it is true, no formal definition of the product of two ratios and the one which is found a little later on in the Elements (Book VI, Defini - tion 5) is considered to be interpolated: he has of course a perfectly clear con- ception of this operation and its properties. $ It seems that Diophantus was not acquainted with negative numbers; this rule can therefore only be interpreted as relating to the calculus of polynomials and &owing the “expansion” of such products as (a - b)(c - d). 180 181 HISTORICAL NOTE HISTORICAL NOTE of abstract laws and on the other a broadening of the notion of “number” sufficient to give it, through the observation of a wide enough variety of special cases, a general conception. For this purpose, the axiomatic theory of ratios of magnitudes created by the Greeks was insufficient, for it did no more than make precise the intuitive concept of a real number >O and the operations on these numbers, which were already known to the Babylonians in a more confused form; now “numbers” would be considered of which the Greeks had had no concept and which to begin with had no sensible “representation” : on the one hand, zero and negative numbers which appeared in the High Middle Ages in Hindu Mathematics; on the other, imaginary numbers, the creation of Italian algebraists of the 16th century. With the exception of zero, introduced first as a separating sign before being considered as a number (see the Historical Note to Set Theory, 111), the common character of these extensions is that (at least to begin with) they were purely “formal”. By this it must be understood that the new “numbers” appeared first of all as the result of operations applied to situations where, keeping to their strict definition, they had no meaning (for example, the difference a - b of two natural numbers when a < b): whence the names “false”, “fictive”, “absurd”, “impossible”, “imaginary”, etc., numbers attributed to them. For the Greeks of the Classical Period, enamoured above all of clear thought, such extensions were inconceivable ; they could only arise with calculators more disposed than were the Greeks to display a somewhat mystic faith in the power of their methods (“the generality of Analysis” as the 18th century would say) and to allow themselves to be carried along by the mechanics of their calcula- tions without investigating whether each step was well founded; a Confidence moreover usually justified a posteriori, by the exact results to which the exten - sion led, with these new mathematical entities, of rules of calculation uniquely valid, in all rigour, for the numbers which were already known. This explains how little by little these generalizations of the concept of number, which at first occurred only as intermediaries in a sequence of operations whose starting and finishing points were genuine “numbers”, came to be considered for their own sakes (independent of any application to concrete calculations) ; once this step had been taken, more or less tangible interpretations were sought, new objects which thus acquired the right to exist in Mathematics.? On this subject, the Hindus were already aware of the interpretation to be given to negative numbers in certain cases (a debt in a commercial problem, for example). In the following centuries, as the methods and results of Greek t This search was merely a transitory stage in the evolution of the concepts in question; from the middle of the 19th century there has been a return, this time fully deliberate, to a formal conception of the various extensions of the notion of number, a conception which has finally become integrated in the “formalistic” and axiomatic point of view which dominates the whole of modern mathematics. 182 and Hindu mathematics spread to the West (via the Arabs), the manipulation of these numbers became more familiar and they took on other “representa - tions” of a geometric or kinematic character. With the progressive improve- ment in algebraic notation, this was the only notable progress in Algebra during the close of the Middle Ages. At the beginning of the 16th century, Algebra took on a new impulse thanks to the discovery by mathematicians of the Italian school of the solution “by radicals” of equations of the 3rd and then of the 4th degree (of which we shall speak in more detail in the Historical Note to Algebra, V) ; it is at this point that, in spite of their aversion, they felt compelled to introduce into their calculations imaginary numbers ; moreover, little by little confidence is born in the calculus of these “impossible” numbers, as in that of negative numbers, even though no “representation” of them was conceived for more than two centuries. On the other hand, algebraic notation was decisively perfected by Viete and Descartes; from the latter onwards, algebraic notation is more or less that which we use today. From the middle of the 17th to the end of the 18th century, it seems that the vast horizons opened by the creation of Infinitesimal Calculus resulted in something of a neglect of Algebra in general and especially of mathematical reflection on laws of composition and on the nature of real and complex num - bers.? Thus the composition of forces and the composition of velocities, well known in Mechanics from the end of the 17th century, had no repercussions in Algebra, even though they already contained the germ of vector cal- culus. In fact it was necessary to await the movement of ideas which, about 1800, led to the geometric representation of complex numbers (see the His- torical Note to General Topology, VIII) to see addition of vectors used in Pure Mathematics.: It was round about the same time that for the first time in Algebra the notion of law of composition was extended in two different directions to elements which have only distant analogies with “numbers” (in the broadest sense of the word up till then). The first of these extensions is due to C. F. Gauss occasioned by his arithmetical researches on quadratic forms ax2 + bxy + cy2 with integer coefficients. Lagrange had defined on the set of forms with the - t The attempts of Leibniz should be excepted, on the one hand to give an algebraic form to the arguments of formal logic, on the other to found a “geometric calculus” operating directly on geometric objects without using coordinates ([4], VOl. v, p. 141). But these attempts remained as sketches and found no echo in his Contemporaries; they were only to be taken up again during the 19th century (see below). t This operation was moreover introduced without reference to Mechanics; the connection between the two theories was only explicitly recognized by the founders of Vector Calculus in the second quarter of the 19th century. 183 HISTORICAL NOTE Same discriminant an equivalence relation7 and had on the other hand proved an identity which provided this set with a commutative law of composition (not everywhere defined) ; starting from these results, Gauss showed that this law is compatible (in the sense of 3 1, no. 6) with a refined form of the above equivalence relation ([5], vol. I, p. 272): “This shows”, he then says, “what must be understood by the composite class of two or several classes”. He then proceeds to study the “quotient” law, establishes essentially that it is (in modern language) a commutative group law and does so with arguments whose gene- rality for the most part goes far beyond the special case considered by Gauss (for example, the argument by which he proves the uniqueness of the inverse element is identical with the one we have given for an arbitrary law of com- position in $2, no 3 (ibid., p. 273)). Nor does he stop there: returning a little later to this question, he recognizes the analogy between composition of classes and multiplication of integers modulo a prime number: (ibid., p. 371), but proves also that the group of classes of quadratic forms of giver1 discriminant is not always a cyclic group; the indications he gives on this subject show clearly that he had recognized, at any rate in this particular case, the general structure of finite Abelian groups, which we shall study in Chapter VII ([5], vol. I, p. 374 and vol. 11, p. 266). The other series of research of which we wish to speak also ended in the concept of a group and its definitive introduction into Mathematics: this is the “theory of substitutions”, the development of the ideas of Lagrange, Vandermonde and Gauss on the solution of algebraic equations. We do not give here the detailed history of this subject (see the Historical Note to Algebra, V); we must refer to the definition by Ruffini and then Cauchy ([GI, (2), vol. I, p. 64) of the “product” of two permutations of a finite set,§ HISTORICAL NOTE t Two forms are equivalent when one of them transforms into the other by a “change of variables” x‘ = ax + Py, y’ = yx + 6y, where a, p, y, 6 are integers such that a6 - py = 1. $ It is quite remarkable that Gauss writes composition of classes of quadratic forms additively, in spite of the analogy which he himself points out and in spite of the fact that Lagrange’s identity, which defines the composition of two forms, much more naturally suggests multiplicative notation (to which moreover all Gauss’s successors returned). This indifference in notational matters must further bear witness to the generality at which Gauss had certainly arrived in his concept of laws of composition. Moreover he did not restrict his attention to commutative laws, as is seen from a fragment not published in his lifetime but dating from the years 1819-1820, where he gives, more than twenty years before Hamilton, the formulae for multiplication of quaternions ([5], vol. VIII, p. 357). § The notion of composite function was of course known much earlier, at any rate for functions of real or complex variables; but the algebraic aspect of this law of composition and the relation with the product of two permutations only came to light in the works of Abel ([7], vol. I, p. 478) and Galois. 184 and the first notions concerning finite permutation groups : transitivity, primitivity, identity element, permutable elements, etc. But these first results remain, as a whole, quite superficial and Evariste Galois must be considered as the true initiator of the theory : having, in his memorable work [8], reduced the study of algebraic equations to that of permutation groups associated with them, he took the study of the latter considerably deeper, both with regard to the general properties of groups (Galois was the first to define the notion of normal subgroup and to recognize its importance) and with regard to the determination of groups with particular properties (where the results he ob- tained are still today considered to be amongst the most subtle in the theory). The first idea of the “linear representation of groups” also goes back to Gal- oist and this fact shows clearly that he possessed the notion of isomorphism of two group structures, independently of their “realizations”. However, if the ingenious methods of Gauss and Galois incontestably led them to a very broad conception of the notion of law of composition, they did not develop their ideas particularly on this point and their works had no immediate effect on the evolution of Abstract Algebra.: The clearest progress towards abstraction was made in a third direction: after reflecting on the nature of imaginary numbers (whose geometric representation had brought about, at the beginning of the 19th century, a considerable number of works), algebraists of the English school were the first, between 1830 and 1850, to isolate the abstract notion of law of composition and immediately they broadened the field of Algebra by applying this notion to a host of new mathe- matical entities: the algebra of Logic with Boole (see the Historical Note to Set The0 y, IV), vectors and quaternions with Hamilton, general hypercomplex systems, matrices and non-associative laws with Cayley ([lo], vol. I, pp. 127 and 301 and vol. 11, pp. 185 and 475). A parallel evolution was pursued inde - pendently on the Continent, notably in Vector Calculus (Mobius, Bellavitis), Linear Algebra and hypercomplex systems (Grassmann), of which we shall speak in more detail in the Historical Note to Algebra, 1II.s From all this ferment of original and fertile ideas which revitalized Algebra in the first half of the 19th century the latter emerged thoroughly renewed. Previously methods and results had gravitated round the central problem of the solution of algebraic equations (or Diophantine equations in the Theory t Here it is that Galois, by a daring extension of the “formalism” which had led to complex numbers, considers “imaginary roots” of a congruence modulo a prime number and thus discoversJiniteJields, which we shall study in Algebra, V. Moreover those of Galois remained unknown until 1846 and those of Gauss only exercised a direct influence on the Theory of Numbers. 8 The principal theories developed during this period are to be found in a re- markable contemporary exposition of H. Hankel [l 11, where the abstract notion of law of composition is conceived and presented with perfect clarity. 185 HISTORICAL NOTE of Numbers): “Algebra”, said Serret in the Introduction to his Course of Higher Algebra [12], “is, properly speaking, the Analysis of equations”. After 1850, if the treatises on Algebra still gave preeminence to the theory of equations for Some time to come, new research works were no longer dominated by the con- cern for immediate applications to the solution of numerical equations but were oriented more and more towards what today is considered to be the essential problem of Algebra, the study of algebraic structures for their own sake. These works fall quite neatly into three main currents which extend respec - tively the three movements of ideas which we have analysed above and pursue parallel courses without noticeably influencing one another until the last years of the 19th century.7 The first of these arose out of the work of the German school of the 19th century (Dirichlet, Kummer, Kronecker, Dedekind, Hilbert) in building up the theory of algebraic numbers, issuing out of the work of Gauss who was responsible for the first study of this type, that of the numbers a + bi (a and b rational). We shall not follow the evolution of this theory here (see the Historical Note to Commutatiue Algebra, VII) : we need only mention, for our purposes, the abstract algebraic notions which arose here. Right from the first successors of Gauss, the idea of ajeld (of algebraic numbers) is fundamental to all the works on this subject (as also to the works of Abel and Galois on algebraic equations) ; its field of application was enlarged when Dedekind and Weber [13] modelled the theory of algebraic functions of one variable on that of algebraic numbers. Dedekind [14] was also responsible for the notion of an ideal which provides a new example of a law of composition between sets of elements; he and Kronecker were responsible €or the more and more important role played by commutative groups and modules in the theory of algebraic fields; we shall return to this in the Historical Notes to Algebra, 111, V and VII. We shall also refer in the Historical Notes to Algebra, I11 and VIII to the development of Linear Algebra and hypercomplex systems which was pursued without introducing any new algebraic notion during the end of the 19th and the beginning of the 20th century, in England (Sylvester, W. Clifford) and America (B. and C. S. Pierce, Dickson, Wedderburn) following the path traced by Hamilton and Cayley, in Germany (Weierstrass, Dedekind, Fro- benius, Molien) and in France (Laguerre, E. Cartan) independently of the Anglo -Saxons and using quite different methods. HISTORICAL NOTE t We choose here to leave aside all consideration of the development, during this period, of algebraic geometry and the theory of invariants which is closely related to it; these two theories developed following their own methods oriented towards Analysis rather than Algebra and it is only recently that they have found their place in the vast edifice of Modern Algebra. 186 As far as group theory is concerned, this developed first of all mainly as the theory of finite permutation groups, following the publication of the works of Galois and their diffusion through the works of Serret [12] and above all the great ‘‘Trait6 des Substitutions” of C. Jordan [15]. The latter there de- veloped, greatly improving them, the works of his predecessors on the par- ticular properties of permutation groups (transitivity, primitivity, etc.), ob- taining results most of which have not been superseded since; he made an equally profound study of a very important particular class of groups, linear groups and their subgroups; moreover, he it was who introduced the funda - mental noticn of a homomorphism of one group into another and also (a little later) that of quotient group and who proved a part of the theorem known as the “Jordan -Holder Theorem”.f Finally Jordan was the first to study infinite groups [lS], which S. Lie on the one hand and F. Klein and H. Poin- car6 on the other were to develop considerably in two different directions several years later. In the meantime, what is essential in a group, that is its law of composition and not the nature of the objects which constitute the group, slowly come to be realized (see for example [lo], vol. 11, pp. 123 and 131 and [14], vol. 111, p. 439). However, even the research works on finite abstract groups had for a long time been conceived as the study of permutation groups and only around 1880 was the independent theory of finite groups beginning to develop con- sciously. We cannot pursue further the history of this theory which is only touched on very superficially in this Treatise; we shall just mention two of the tools which are still today among the most used in the study of finite groups and which both go back to the 19th century: the Sylow theorems: on p-groups which date from 1872 [17] and the theory of characters created in the last years of the century by Frobenius [19]. We refer the reader who wishes to go deeply into the theory of finite groups and the many difficult problems it rakes, to the monographs of Burnside 1201, Speiser [23], Zassenhais [24] and Gorenstein [27]. Around 1880 too the systematic study of group presentations was begun; previously only presentations of particular groups had been encountered, for example the alternating group 3, in a work of Hamilton [9 bis] or the monodromy groups of linear differential equations and Riemann surfaces (schwarz, Klein, Schlafli). W. Dyck was the first [I81 to define (without giving it a name) the free group generated by a finite number of generators, t Jordan had only established the invariance (up to order) of the orders of the quotient groups of a “Jordan -Holder series” for a finite group; it was 0. Holder who showed that the quotient groups themselves are (up to order) independent of the series considered. The existence of a subgroup of order p“ in a group whose order is divisible by 4“ 1s mentioned without proof in the papers of Galois [8]. 187 BIBLIOGRAPHY BIBLIOGRAPHY but he was less interested in it for its own sake than as a “universal” object allowing a precise definition of a group given “by generators and relations”. Developing an idea first set forth by Cayley and visibly influenced on the other hand by the works of Riemann’s school mentioned above, Dyck de- scribed an interpretation of a group of given presentation, where each generator is represented by a product of two inversions with respect to tangent or inter- secting circles (see also Burnside [20], Chapter XVIII). A little later, after the development of the theory of automorphic functions by PoincarC and his successors, then the introduction by PoincarC of the tools of Algebraic Topology, the first studies of the fundamental group would be on an equal footing with those of group presentations (Dehn, Tietze), the two theories lending one another mutual support. Moreover it was a topologist, J. Nielsen, who in 1924 introduced the term free group in the first deep study of its properties [21], almost immediately afterwards E. Artin (always with reference to questions in topology) introduced the notion of the free product of groups and 0. Schreier defined more generally the free product with amalgamated sub- groups [22] (cf. I, $7, no. 3, Exercise 20). Here again we cannot go into the history of the later developments in this direction, referring the reader to the work [26] for more details. There is no further room to speak of the extraordinary success, since the end of the 19th century, of the idea of group (and that of invariant which is intimately related with it) in Analysis, Geometry, Mechanics and Theoretical Physics. An analogous invasion by this notion and the algebraic notions related to it (groups with operators, rings, ideals, modules) into parts of Algebra which until then seemed quite unrelated is the distinguishing mark of the last period of evolution which we retrace here and which ended with the synthesis of the three branches which we have followed above. This unification is above all the work of the German school of the years 1900-1930: begun by Dede- kind and Hilbert in the last years of the 19th century, the work of axiomatiza- tion of Algebra was vigorously pursued by E. Steinitz, then, starting in 1920, under the impulse of E. Artin, E. Noether and the algebraists of their school (Hasse, Krull, 0. Schreier, van der Waerden). The treatise by van der Waerden [25], published in 1930, brought together these works for the first time in a single exposition, opening the way and serving as a guide to many research works in Algebra during more than twenty years. BIBLIOGRAPHY 1. 0. NEUGEBAUER , Vorlesungen iiber Geschichte der antiken Mathematik, Vol. I : 2. Euclidis Elementa, 5 vols., ed. J. L. Heiberg, Lipsiae (Teubner), 1883-88. 2 bis. T. L. HEATH , The thirteen books of Euclid’s Elements. . ., 3 vols., Cam- Vorgriechische Mathematik, Berlin (Springer), 1934. bridge, 1908. 188 3. Diophanti Alexandrini Opera Omnia. . ., 2 vols., ed. P. Tannery, Lipsiae 3 bis. Diophante d’Alexandrie, trad. P. Ver Eecke, Bruges (Desclte-de Brouwer), 4. G. W. LEIBNIZ , Mathematische Schriften, ed. C. I. Gerhardt, vol. V, Halle (Teubner), 1893-95. 1926. (Schmidt). 1858. 5. C.’F. GAUSS,’ Werke, vols. I (Gottingen, 1870), I1 (ibid., 1863) and VIII (ibid., 1900). 6. A. L. CAUCHY, Oeuvres complites (2), vol. I, Paris (Gauthier -Villars), 7. N. H. ABEL, Oeuvres, 2 vol., ed. Sylow and Lie, Christiania, 1881. 8. E. GALOIS , Ecrits et rnhnoires rnathe‘rnatigues, ed. R. Bourgne and J. Y. Azra, 9. W. R. HAMILTON , Lectures on Quaternions, Dublin, 1853. 9 bis. W. R. HAMILTON , Memorandum respecting a new system of roots of 10. A. CAYLEY , Collected rnathernaticalpapers, vols. I and 11, Cambridge (Univer - 11. H. HANKEL, Vorlesungen uber die complexen Zahlen und ihre Functionen, 1st 12. J. A. SERRET , Cours d’dlgibre suphieure, 3rd ed., Paris (Gauthier-Villars), 13. R. DEDEKIND and H. WEBER , Theorie der algebraischen Funktionen einer 1905. Paris (Gauthier-Villars), 1962. unity, Phil. Mag. (4), 12 (1856), p. 446. sity Press), 1889. part: Theorie der complexen Zahlensysteme, Leipzig (Voss), 1867. 1866. Veranderlichen, Crelle’s J., 92 (18821. DD. 181-290. -~ .. \ /’I& 14. R. DEDEKIND , Gesamrnelte mathematische Werke, 3 vols., Braunschweig (Vieweg.). 1932. 15. C. JORDA~,‘ Trait6 des substitutions et des kguations algbbrigues, Paris (Gauthier- Villars), 1870 (reprinted, Paris (A. Blanchard), 1957). 16. C. JORDAN, MCmoire sur les groups des mouvements, Ann. di Mat. (2), 2 (1868), pp: 167-215 and 322-345 (=Oeuvres, vol. IV, pp. 231-302, Paris (Gauthier-Villars), 1964). 17. L. SYLOW, ThCorkmes sur les groups de substitutions, Math. Ann., 5 (1872), pp. 584-594. 18. W. DYCK, Gruppentheoretische Studien, Math. Ann., 20 (1882), pp. 19. G. FROBENIUS , Uber Gruppencharaktere, Berliner Sitzungsber., 1896, pp. 985-1021 (= Gesamrnelte Abhandlungen, ed. J. P. Serre, Berlin- Heidelberg -New York (Springer), vol. I11 (1968), pp. 1-37). 20. w. BURNSIDE , Theory ofgroups ofjnite order, 2nd ed., Cambridge, 191 1. 21. J. NIELSEN , Die Isomorphismengruppen der freien Gruppen, Math. Ann., 22. 0. SCHREIER , Die Untergruppen der freien Gruppe, Abh. Harnb., 5 (1927), 1-44. 91 (1924), pp. 169-209. pp. 161-185. 189 BIBLIOGRAPHY 23. A. SPEISER, Theorie der Gruppen von endlicher Ordnung, 3rd ed., Berlin 24. H. ZASSENHAUS, Lehrbuch der Gruppentheorie, vol. I, Leipzig -Berlin (Teub- (Springer), 1937. ner), 1937. 2.5. B. L. VAN DER WAERDEN , Moderne Algebra, 2nd ed., vol. I, Berlin (Springer), CHAPTER I1 Linear Algebra -- - 1937; vol. I1 (ibid.), 1940. 96. W. MAGNUS, A. KARRASS and D. SOLITAR, Combinatorial group theory, --. . .. New York. (Interscience), 1966. 27. D. GORENSTEIN , Finite groups, New York (Harper and Row), 1968. This chapter is essentially devoted to the study of a particular type of com- mutative groups with operators (I, 5 4, no. 2) called modules. Certain properties stated in $5 1 and 2 for modules extend to all commutative groups with operators; these will be indicated as they occur. Moreover it will be seen in Chapter III,§ 2, no. 6 that the study of a commutative group with operators is always equivalent to that of a module suitably associated with the group with operators in question. 8 1. MODULES 1. MODULES; VECTOR SPACES; LINEAR COMBINATIONS DEFINITION 1. Given a ring A, a lgt module over A (or lgt A-module) is a set E with an algebraic structure dejined by giving: (1) a commutative group law on E (written additively in what follows) ; (2) a law of action (a, x) ++ a T x, whose domain of operators is the ring A and (MI) aT (x+y) = (aTx) + (aTy)forallaEA,XEE,yEE; (MII} (a + B) T x = (a T x) + (p T x) for all a E A, p E A, x E E; (MnI) a T (p T x) = (ap) T x for all a E A, p E A, x E E; (MIv) 1 T x = x for all x E E. which sathfis the following axiom : Axiom (MI) means that the external law of a left A -module E is distribu - tive with respect to addition on E; a module is thus a commutative group with operators. 190 If in Definition 1, axiom (MI,,) is replaced by (ML) a T (p T x) = (pa) T xfor all a E A, p E A, x E E, 191 I1 LINEAR ALGEBRA E with the algebraic structure thus defined is a right module ouer A or a right A-module. When speaking of A-modules (left or right), the elements of the ring A are often calIed scalars. Usually the external law of composition of a left A-module (resp. right A-module) is written multiplicatively, with the operator written on the left (resp. right) ; condition (MIII) is then written a(@) = (aP)x, condition (Mi,,) is written (xp)a = pa). If A0 denotes the opposite ring to A (I, 3 8, no. 3), every right module E over the ring A is a lejt module over the ring AO. It follows that an exposition can be given of the properties of modules whilst systematically confining atten- tion either to left modules or to right modules; in $9 1 and 2 we shall in general give this exposition for lgt modules and when we speak of a module (without specifying which type) we shall mean a left module whose external law will be written multiplicatively. When the ring A is commutative the notions of right module and left module with respect to A are identical. For all a E A, the mapping x +-+ ax of an A-module E into itself is called the homothety of ratio a of E (I, 3 4, no. 2); by (MI) a homothety is an endo- morphism of the commutative group structure (without operators) on E, but not in general of the module structure on E (I, $ 4, no. 2; cf. 11, 3 1, no. 2 and no. 13). Hence a0 = 0 and a( -x) = - (EX) ; if a is an invertible element of A, the homothety x F+ ax is an automorfihism of the commutative group structure (without operators) on E, for the relation y = ax implies by virtue of (MIv), x = .-'(ax) = a-ly. Similarly, by virtue of (MII), for all x E E, the mapping a H ax is a homo- morphism of the additive group A into the commutative group (without operators) E; hence Ox = 0 and (-a). = -(ax); moreover, by (MIv), for every integer n E Z, n.x = (n. 1)x. When the ring A consists only of the element 0, every A-module E con- sists only of the element 0, for then 1 = 0 in A, whence, for all x E E, Examples. (1) Let + be a homomorphism of a ring A into a ring B; the map- ping (a, x) ++ +(a). (resp. (a, x) ++ .+(a)) of A x B into B defines on B a left (resp. right) A-module structure. In particular if + is taken to be the identity mapping on A, a canonical left (resp. right) A-module structure is obtained on A; to avoid confusion, the set A with this structure is denoted by A, (resp. (2) On a commutative group G (written additively) the group with opera- tors structure defined by the external law (n, x) H n.x (I, § 3, no. 1) is a module structure over the ring Z of rational integers. (3) Let E be a commutative group written additively, d the endomorphism 7 - x = 1.x = 0.x = 0. Ad) * 192 3 1.1 MODULES ; VECTOR SPACES; LINEAR COMBINATIONS ring of E (I, 3 8, no. 3: recall that the product fg of two endomorphisms is by definition the composite endomorphism f 0 9). The external law (Jx) ++ f(x) between operators f €8 and elements XEE defines on E a canonical left &-module structure. Consider now a ring A and suppose there is given on E a left (resp. right) A-module structure; for all a E A, the homothety h,:x - ax (resp. x H xa) belongs to 8; the mapping +: a ++ h, is a homomorphism of the ring A (resp. the opposite ring A') into the ring d and by definition ax = (+(a))(x) (resp. Xa = (+(a))(x)). Conversely, giving a ring homomorphism +:A+& (resp. +:Ao -+ 8) defines a left (resp. right) A-module structure on E by the above formulae. In other words, being given a left (resp. right) A-module structure on an additive group E with additive law the given group law is equivalent to being given a ring homomorphism A -+ d (resp. Ao -+ 8). DEFINITION 2. A lefl (resp. right) vector space ouer ajeld K is a lejl (resp. right) K-module. The elements of a vector space are sometimes called vectors. Examples. (4) A field is both a left and a right vector space with respect to any of its subfields. *(5) The real number space of three dimensions R3 is a vector space with respect to the field of real numbers R, the product tx of a real number t and a point x with coordinates xl, x2, x3 being the point with coordinates txl, txZ, tx3. Similarly, the set of real-valued functions defined on an arbi- trary set F is a vector space with respect to R, the product tfof a real num- ber t and a function f being the real-valued function x H tf(x). * For two families (x,)~~~, (yJLEI of elements of an A-module E of finite sup- port (I, $2, no. I), the following equations hold: the equations are immediately reduced to the analogous equations for finite sum^ by considering a finite subset H of I containing the supports of (xJ and DEFINITION 3. An element x of an A-module E is said to be a linear combination with ~oefints in A of a family (a,), I of elements of E iJI there exists a Jamily ( A,) I of eht~ $A, $jnite support, such that x = 2 h,a,. (Yd. In general there are several distinct families (A,) satisfying this condition (cf. no. 11). 193 I1 LINEAR ALGEBRA Note that 0 is the only linear combination of the empty family of elements ofE (by the convention of I, 5 2, no. 1). 2. LINEAR MAPPINGS DEFINITION 4. Let E and F be two (left) modules with respect to the same ring A. A linear mapping (or A-linear mapping, or homomorphism, or A-homomorphism) of E into F is any mapping u : E -+ F such that: (3) (4G) satisfying (3) and (4D) u(x.h) = u(x).A for AEA,~EE. u(x + y) = ~(x) + u(y) for x E E, y E E; u(h.x) = A.u(x) for A E A, x E E. If E and F are two right A-modules, a linear mapping u : E -+ F is a mapping Remark. When E and F are two commutative groups considered as modules over the ring Z (no. I), every homomorphism u of the group E (without operators) into the group F (without operators) is also a linear mapping of E into F, since for n an integer > 0, the relation u(n . x) = n . u(x) follows from u(x + y) = u(x) + u(y) by induction on n and for n = -m < 0, u(n.x) = u(-(m.x)) = -u(m.x) = -(m.u(x)) = n.u(x). Examples. (1) Let E be an A-module and a an element of E; the mapping h H ha of the A-module A, into E is a linear mapping 8, such that O,( 1) = a. *(2) Let I be an open interval of the real line R, E the vector space of differentiable real -valued functions on I and F the vector space of all real- valued functions defined on I. The mapping XH x' which associates with every differentiable function x its derivative is a linear mapping from Note that a homothety x - MX on an A-module E is not necessarily a linear mapping: in other words, the relation a(h) = A(=) does not necessarily hold for all A E A and x E E. This relation is however true when a belongs to the centre of A; x E- ax is then called a central homothety (cf. no. 13). If u: E -+ F is a linear mapping, then, for every family (xJL I of elements of E and every family of elements of A such that the support of the family (h1xl), I is finite, E to Fa* 9 1.2 LINE.4R MAPPINGS PROPOSITION 2. Let E, F be two A-modules. (1) If u : E -+ F and u: F -+ E are two linear mappings such that u 0 u is the iden- tity mapping on E and u 0 u is the identity mapping on F, u is an isomorphism of E onto F and v is the inverse isomorphism. (2) Every bijective linear mapping u: E -+ F is an isomorphism of E onto F. These propositions follow immediately from Definition 4. Propositions 1 and 2 show that linear mappings can be taken as the morph- for the species of A-module structures (Set Theory, IV, 5 2, no. 1) ; we shall henceforth always assume that this choice of morphisms has been made. Given two left (resp. right) A-modules E and F, let Hom(E, F) or Hom,(E, F) denote the set of linear mappings of E into F. The set Hom(E, F) is a commutative ring, a subgroup of the product commu- tative group FE of all mappings of E into F (I, 5 4, no. 8) ; recall that for two elements u, v of FE and for all x E E, (u + v)(x) = U(X) + U(X), (-u)(x) = -Id(.) whence it follows immediately that, if u and v are linear, so are u + v and - u. If G is a third left (resp. right) A-module,f, fl, fz elements of Hom(E, F) and g, g,, g, elements of Hom(F, G), the following relations are immediately verified : (6) go(f1 +f2) =sof1 +goy2 (7) (Sl + sz) of = g1 of + s2 of (8) so(-f) = (-5)of = -(gof)* In particular, the law of composition (f, g) -fo g on Hom(E, E) defines with the above additive group structure a ring structure on Hom(E, E) whose unit element, denoted by 1, or Id,, is the identity mapping on E; the linear mappings of E into itself are also called endomorphisms of the A-module E and the ~ng Hom(E, E) is also denoted by End(E) or End,(E). The automorphisms of the A-module E are just the invertible elements of End(E) (Proposition 2) ; Cley form a multiplicative group, denoted by Aut(E) or GL(E) which is also dled the linear group relative to E. It follows from (6) and (7) that, for two A-modules E, F, Hom(E, F) has the canonical structure of a left module over the ring Hom(F, F) and of a right mMe over Hom(E, E). Let E, F, E', F' be four (left) A-modules, u : E' -+ E and v : F -+ F' A-linear mappings. If every element f E Hom(E, F) is associated with the element Ofo u E Hom(E', F'), a mapping Hom(E, F) + Hom(E', F') 195 (5) u( c LEI ALXL) = LEI c A,u(x1) as follows immediately from (3) and (4G) by induction on the cardinal of the support of the family (AlxL). PROPOSITION 1. Let E, F, G be three A-modules, u a linear mapping of E into F and v a linear mapping of F into G. Then the composite mapping v o u is linear. 194 I1 LINEAR ALGEBRA is defined, which is 2-linear and which is denoted by Hom(u, u) or Hom,(u, v). Ifu, ul, u2 belong to Hom(E', E) and u, ul, u2 to Hom(F, F'), then Hom(ul + u2, u) = Hom(ul, u) + Hom(u2, u) (9) Hom(u, u1 + u2) = Hom(u, vl) + Hom(u, v2) Let E", F" be two A-modules, u' : E" --f E' and v' : F' -+ F" linear mappings. Then (10) Hom(u o u', u' 0 u) = Hom(u', u') 0 Hom(u, u). If u is an isomorphism of E' onto E and u an isomorphism of F onto F', Hom(u, v) is an isomorphism of Hom(E, F) onto Hom(E', F') whose inverse isomorphism is Hom(u-l, v-') by (10). If h (resp. k) is an endomorphism of E (resp. F), Hom(h, lF) (resp. Hom(l,, k)) is just the homothety with ratio h (resp. k) for the right (resp. left) module structure on the ring End(E) (resp. End(F)) defined above. 3. SUBMODULES; QUOTIENT MODULES Let E be an A-module and M a subset of E; for the A-module structure on E to induce an A-module structure on M, it is necessary and sufficient that M be a stable subgroup of E (I, 3 4, no. 3), for if this is so, the structure of a group with operators induced on M obviously satisfies axioms (MII), (MIII) and (MIv) ; then M with this structure (or, by an abuse of language, the set M itself) is called a submodule of E; the canonical injection M --f E is a linear mapping. When E is a vector space, its submodules are called vector subspaces (or simply subspaces if no confusion can arise). Examples. (1) In any module E the set consisting of 0 is a submodule (the zero submodule, often denoted by 0, by an abuse of notation). (2) Let A be a ring. The submodules of A, (resp. Ad) are just the left ideals EXACT SEQUENCES 3 1.4 subgroup of E (I, 9 4, no. 4), that is a submodule of E. It is immediately verified that the structure of a group with operators on the quotient group E/M (I, 9 4, no. 4) is an A-module structure, under which the canonical mapping E --f E/M is linear; with this structure, E/M is called the quotient module of E by the submodule M. A quotient module of a vector space E is called a quotient vector space (or simply quotient space) of E. Example (6). Every left ideal a in a ring A defines a quotient module A,/a of the left A-module A,; by an abuse of notation, this quotient module is often denoted by A/a. Let E, F be two A-modules. It follows from the general properties of groups with operators (I, 9 4, no. 5) (or directly from the definitions) that if u: E --f F is a linear mapping, the image under u of any submodule of E is a submodule of F and the inverse image under u of any submodule of F is a submodule of ofE under u is a submodule of F (I, 9 4, no. 6, Proposition 7) ; by an abuse of language u(E) is called the image of.. The quotient module E/N is also called the coimage of u and the quotient module F/u(E) the cokernel ofu. In the canonical decomkosition of u (I, $4, no. 5) u:E E/N < u(E) F v is an iromorphism of the coimage of u onto the image of u (no. 2, Proposition 2). For u to be injective, it is necessary and sufficient that its kerneI be zero; for u to be surjective, it is necessary and sufficient that its cokernel be zero. The kernel, image, coimage and cokernel of u are denoted respectively by Ker u, Im u, Coim u, Coker u. Remark. Let M be a submodule of an A-module E and +:E -+ E/M the canonical homomorphism. For an A-linear mapping u: E --f F to be of the form v 0 +, where u is a linear mapping of E/M into F, it is necessary and sufficient that M = Ker(u) ; for, if this condition holds, the relation x - y E M implies ~(x) = u(y), hence u is compatible with this equivalence relation and clearly the mapping u: E/M + F derived from u by taking quotients is linear. E. In particular, the kernel N = -1 u (0) is a submodule of E and the image u(E) (11) \I (resp. right ideals) of ;he ring A. (3) Let E be an A-module, x an element of E and a a left ideal of A. The set of elements OIX, where dc runs through a, is a submodule of E, denoted by ax. (4) In a commutative group G considered as a Z-module (no. l), every sub- *(5) Let I be an open interval of the real line R; the set C of real-valued functions defined and continuous on I is a vector subspace of the vector space RI of all real -valued functions defined on I. Similarly, the set D of differentiable functions on I is a vector subspace of C.* 1, no. 6) with the module structure on E is of the form x - y E M, where M is a stable group of G is also a submodule. Let E be an A-module. Every equivalence relation compatible (I, 4. EXACT SEQUENCES DEFINITION 5. Let F, G, H be three A-modules; let f be a homomorphism of F kt0 G and g a homomorphism of G into H. The orderedpair (f, g) is called an exact squence if -1 g (0) =f(F) in 0th wora!s, ifthe kernel Ofg is equal to the image Of$ 196 197 I1 LINEAR ALGEBRA EXACT SEQUENCES 9 1.4 (e) Iff: E --z F is a honioniorphisni, the diagram f -1 0 f(0) A E - F 5 F/f(E) --to (where i is the canonical injection and p the canonical surjection) is an exact sequence. 2 of no. 2. Remarks. (2) To say that there is an exact sequence The proposition follows immediately from the definitions and Proposition f 0 ---f E --+ F & G - 0 means that f is injective, g surjective and that the canonical bijection associ - ated with g is an isomorphism of F/ f (E) onto G. It is also said that the triple (F, f, g) is an extension of the module G by the module E (I, 5 6, no. 7). (3) If there is an exact sequence with 4 terms f h EdF2G-H the cokernel off is F/ f (E) = FL1(0) and the kernel of h isg(F) ; the canonical bijection associated with g is therefore an isomorphism Coker f + Ker h. (4) Consider an ordered pair of A-module homomorphisms f (15) E-FLG. For diagram (1 5) to be an exact sequence, it is necessary and suficient that there exist two A-modules S, T and homomorphisms a:E -+ S, b:S -+ F, c:F -+ T, d :T --f G such that the three sequences The diagram f (12) FhG8’H is also called an exact sequence. homomorphisms : We consider similarly a diagram consisting of four A-modules and three f h E - F -% G -----f H. f (13) This diagram is called exact at F if the diagram E -+ F -% G is exact; it is called exact at G if F f G -+ H is exact. If diagram (13) is exact at F and at G, it is simply called exact, or an exact sequence. Exact sequences with an arbi- trary number of terms are defined similarly. h I Remark. (1) If the ordered pair (f, g) is an exact sequence, then g of = 0; but of course this property does not characterize exact sequences, for it only means that the image off is contained in the kernel of g. In the statements below, E, F, G denote A-modules, 0 the A-module reduced to its identity element; the arrows represent A-module homomorphisms. As there is only one homomorphism from the module 0 to a module E (resp. of E to 0), there is no point in giving these homomorphisms a name in the exact sequences where they appear. PROPOSITION 3. (a) For f 0hE-F to be an exact sequence, it is necessary and sufiient that f be injective. (b) For f E-F---tO to be an exact sequence, it is necessary and suficient that f be surjective. (c) For f O-E-F--+O to be an exact sequence, it is necessary and suficient that f be bijective (in other words (no. 2, Proposition 2) that f be an isomorphism of E onto F). (d) If F is a submodule of E and i: F -+ E is the canonical injection, p: E -+ E/F the canonical homomorphism, the diagram ELS-0 O-S-F~T--+O b O-T-G d areexactandf = boaandg = doc. -1 If (15) is an exact sequence, then take S = f (E) = g (0) and T = g(F), b and d being the canonical injections and a (resp. c) the homomorphism with the same graph as f (resp. g). Conversely, if S, T, a, b, c, d satisfy the above the exactness of (16) shows that f (E) = i’(0). -1 conditions, then f(E) = b(a(E)) = b(S) and-;(O) = -1-1 c(d(0)) = c(O), hence is an exact sequence. 198 The use of explicit letters to denote homomorphisms in an exact sequence is often dispensed with when it is not necessary for the arguments. 199 PRODUCTS OF MODULES 3 1.5 I1 LINEAR ALGEBRA Remark. (5) The definition of an exact sequence extends immediately to groups which are not necessarily commutative; in this case of course multi - plicative notation is used with 0 replaced by 1 in the formulae (if no con - fusion arises). Parts (a), (b), (c) of Proposition 3 are still valid and also (d) when F is a normal subgroup of E. Remark 2 and Proposition 3(e) hold on condition thatf(E) is a normal subgroup of F; Remarks 3 and 4 are valid without modification. 5. PRODUCTS OF MODULES Let (EJtEI be a family of modules over the same ring A. It is immediately verified that the product of the module structures on the E, (I, 3 4, no. 8) is an A-module structure on the product set E = n El. With this structure the set E is called the product module of the modules El; if x = (x,), y = (y,) are two elements of E, then (17) { A.x = (Ax,) for all A E A. mappings ; these mappings are obviously surjective. LEI x + y = (XI + YJ Formulae (17) express the fact that the projections pr,: E -+ E, are linear Recall that if the indexing set I is empty, the product set n El then consists of a single element; the product module structure on this set is then that under which this unique element is 0. LEI PROPOSITION 4. Let E = For every A-module F and every family of linear mappings fi: F -+ El there exists one and only one mapping f of F into E such that pr, 0 f =JJor all L E I and this mapping is linear. El be the product of a family of A-modules This follows directly from the definitions. Product of modules is “associative” : if ( Jh)hEL is a partition of I, the canoni- cal mapping is an isomorphism. PROPOSITION 5. (i) Let (El)LEI, (F,),,, be two families of A-modules with the same indexing set I; for every family of linear mappings fi: E, -+ F, (t E I), the mapping f: (x,) -+ (fr(xl)) of (ii) Let (Gl)lEI be a third family of A-modules with I as indexing set and, for all E, into F, (sometimes denoted by n 5) is linear. 200 t E I, let g, : F, + G, be a linear mapping; let g = g,. For each of the sequences E, 5 F, 2 G, to be exact, it is necessary and sujicient that the sequence be exact. Assertion (i) follows immediately from the definitions. On the other hand, to say that y = (y,) belongs to Ker(g) means that g,(yl) = 0 for all L E I and hence that yl E Ker(g,) for all t E I; similarly, to say that y belongs to Im( f) means that there exists x = (xl) E n E, such that y = f (x), which is equivalent to saying that y, = fi(x,) for all t E I or also that y, E Im(fi) for all L E I; whence (ii). COROLLARY . Under the conditions of Proposition 5, (i), Coim( f) + n Coim(f,), Coker( f) = n Coker(fi) LEI LEI 1 obtained by resjectively associating with the class of an element x = (x,) of n El, mod Ker( f), (resp. with the class of an element y = (y,) of F,, mod. Im( f)) th family of classes of the x, mod. Ker(fi) (resp. the family of classes of the y, mod. Im(f)). In particular, for f to be injective (resp. surjective, bijective, zero) it is necessary and m&nt that, for all t E 1,fi be injective (resp. surjective, bijective, zero). , If, for all c E I, we consider a submodule F, of El, the module n F, is a submodule of n El and by virtue of the Corollary to Proposition 5, LEI there is a canonical isomorphism LEI : (19) An important example of a product of modules is that where all the factor modules are equal to the same module F; their product F’ is then just the set Of mappings of I into F. The diagonal mapping F -+ F‘ mapping x E F to the constant function equal to x on I is linear. If (E,),,, is a family of A-modules I I 201 LINEAR ALGEBRA I1 and, for all L E I, f,:F -+ E, is a linear mapping, then the linear mapping x ++ (fi(x)) of F into EL is the composition of the mapping (xl) - (fr(x,)) of FI into EL and the diagonal mapping F -+ F'. 6. DIRECT SUM OF MODULES Let (EJlEI be a family of A-modules and F = EL their product. The set E of x E F such that pr, x = 0 except for afinite number of indices is obviously a submodule of F, called the external direct sum (or simply direct sum) of the family of modules (EL) and denoted by 9 EL (I, fj 4, no. 9). When I isjinite, then 2 EL = E,; if I = (p, q) (interval of Z), we also write BE, ,€I = Ep@Ep+l@ ... @E,. For all K E I, let j, be the mapping E, -+ F which associates with each X,EE, the element of F such that pr,(j,(x,)) = 0 for L # K and pr&,(x,)) = x,; it is immediate that jK is an injective linear mapping of E, into the direct sum E of the EL, which we shall call the canonical injection; the submodulej,(E,) of E, isomorphic to E,, is called the component submodule of E of index K. It is often identified with E, by means of j,. For all x E E = 3 EL, we have therefore (20) x = LEI C jL(prL x). PROPOSITION 6. Let (EJLEI be a family of A-modules, M an A-module and, for all t E I, let L:E, + M be a linear mapping. Then there exists one and only one linear mapping g: @ EL -+ M such that, for all t E I: &€I (21) g0.i =La By virtue of (20), if g exists, then necessarily, for all x E 3 E,, whence the uniqueness of g. Conversely, setting g(x) = c L(pr,(x)) for all x E @ EL, it is immediately verified that a linear mapping has been defined satisfying the conditions of the statement. ,€I DIRECT SUM OF MODULES 5 1.6 When no confusion can arise, we write g = c L (which is contrary to the conventions of I, 5 2, no. 1, when the family (A) is not of finite support). In particular, if J is any subset of I, the canonical injections j, for t EJ define a canonical linear mapping j,: @ EL -+ @ EL, which associates with each (xJLEI the element (x:),~~ such that x: = x, for t E J, x: = 0 for t J; this mapping is obviously injective. Moreover, if (JA)AEL is a partition of I, the mapping i: @ ( @ EL) -+ @ EL corresponding to the family (j,J by Proposi- tion 6 is an isomorphism called canonical ("associativity" of direct sums). COROLLARY 1. Let (EJLE1, (FA)AEL be two families ofA-modules. The mapping LEI IEJ LEI AEL LEJL LEI which associates with each g E HornA( @ EL, n FA) the family (pr, 0 g ojL), is a Z-module isomorphism (called canonical). LEI AEL This follows from Proposition 6 and no. 5, Proposition 4. COROLLARY 2. Let (EL),,, be a family of A-modules, F an A-module and, for each L E I, let A: EL -+ F be a linear mapping. For f = 2 fi to be an isomorphism of E = @ E, onto F, it is necessary and sujicient that there exist for each I E I a linear mapping g,: F --f EL with the following properties: LEI to1 (1) g, OL = l,, for all t E I. (3) For ally E F, the family (g,(y)) has Jinite support and (2) gt 0 f, = 0 for L # K. Note that if I is finite, the last condition may also be written as Obviously the conditions are necessary for they are satisfied by the defined and it is immediate that g is a linear mapping of F into E. For all YEF,f(g(y)) = ZA(,p,(y)) = y by hypothesis. On the other hand, for all & = pr, of. -1 Conversely if they hold, for all y E F, g(y) = j,(g,(y)) is XEEy gK(f ('1) = gK(TA(prL("))) = gK(fK(prK(X))) = PrK(X) by 202 203 I1 LINEAR ALGEBRA 0 1.7 COROLLARY 2. Let (EJIEI, (Fh)hsL, (Fl;)heL be four families of A-modules and, for each t E I (resp. each A E L), u,:EE -+ El (resp. vh:F, -+ FA) a linear mapping. Then the diagram INTERSECTION AND SUM OF SUBMODULES therefore g( f (x)) = Tj,(g,( f (x))) = c , j,(pr,(x)) = x, which proves the corollary. PROPOSITION 7. (i) Let (El), (F,), EI be two families of A-modules with the same indexing set I; for every family of linear mappingsf,: El + F, (t E I), the restriction to 63 E, of the linear mapping (x,) - (~(x,)) is a linear mapping f : 0, E, + 2 F, denoted by @J LEI or Of, (where f = f, @ f,+l @ . . . @ f, ;f I = (p, q) is an interval in Z). (ii) Let (G,),,, be a third family of A-modules with I as indexing set and, for all t E I, let g,:F, -+ G, be a linear mapping; we write g = 63 g,. For each of the se- quences El -+ F, % G, to be exact, it is necessary and suflcient that the sequence LEI ft BE, f_ OF, --% BG, be exact. El, the family (f,(x,)) has finite support, whence (i). On the other hand, to say that an element y = (y,) of @ F, belongs to Ker(g) means that y, E Ker(g,) for all E I (no. 5, Proposition 5); similarly, if y, E Im(f,) for all L E I, there exists for each L E I an x, E E, such that y, = fi (x,) and when y, = 0, it may be supposed that x, = 0; hence y E Im( f) and the converse is obvious. COROLLARY 1. Under the conditions $Proposition 7, (i), (25) Ker( f) = @ ,€I Ker(L), Obviously, for all (x,) E Im( f) = @ re1 Im(ft) and there are canonical isomorphisms Coim( f) + 9 Coim(f,), Coker( f) 3 @ 1EI Coker(f,) deJined as in no. 5, Corollary to Proposition 5. In particular, for f to be injective (resp. surjective, bijective, zero), it is necessary and suflcient that each of theA be injective (resp surjective, bijective, zero). If, for all t E I, we consider a submodule F, of El, the module 2 F, is a sub- module of % E, and, by virtue of Corollary 1 to Proposition 7, there is a canonical isomorphism 204 11 (where + and +‘ are the canonical isomorphism defilzed in Corollary 1 to Proposition 6) is commutative. The verification follows immediately from the definitions. When all the El are equal to the same A-module E, the direct sum @ El is also denoted by E“): its elements are the mappings of I into E with finite support. If, for all t,J is taken to be the identity mapping E -+ E, by Proposi - tion 6, a linear mapping E(I) -+ E is obtained, called codiagonal, which associ - ates with every family (xJlE1 of elements of E, of finite support, its sum 2 LEI LEI x,* Remark. Recall that the definition of direct sum extends immediately to a family (E,),,, of groups which are not necessarily commutative, multi - plicative notation of course replacing the additive notation; we then say “restsicted product” or “restricted sum” instead of “direct sum” (I, $ 4, no. 9). Note that E is a normal subgroup of the product F = n El and that each of the j,.(EK) is a normal subgroup of F; moreover, for two distinct indices A, p, every element of jA(Eh) is permutable with every element of j,,(E,,). Proposition 6 extends to the general case with the hypothesis that, for two distinct indices A, p, every element offh(Eh) is permutable in M with every element off,(E,,) (I, 4 4, no. 9, Proposition 12). The property of “associa - tivity” of the restricted sum follows immediately from this. Proposition 7 and its Corollaries 1 and 2 hold without modification. LEI 7. INTERSECTION AND SUM OF SUBMODULES For every family (M,),,, of submodules of an A-module E, the intersection n E, is a submodule of E. If, for each t E I, +, denotes the canonical homo- morphism E + E/M,, ,Q M, is the kernel of the homomorphism +:x ++ (+,(x)) Of E into ,€I (E/M,), in other words, there is an exact sequence LEI (27) o - n M, -+ E -L I3 (E/M,). LEI LEI 205 I1 LINEAR ALGEBRA The linear mapping + and the mapping LEI which is obtained by passing to the quotient, are called canonical. In particular: PROPOSITION 8. If a family (M,) , I of submodules of E has intersection reduced to 0 E is canonically isomorphic to a submodule ofn LEI (E/hf,). Given a subset X of an A-module E, the intersection F of the submodules of E containing X is called the submodule generated by X and X is called a generating set (or generating system) of F (I, 3 4, no. 3) ; for a family EI of elements of E, the submodule generated by the set of a, is called the sub- module generated by the family (a,). PROPOSITION 9. The submodule generated by a family (a,),,, of elements of an A- module E is the set of linear combinations of the family (aL). Every submodule of E which contains all the a, also contains the linear combinations of the a,. Conversely, formulae (1) and (2) of no. 1 prove that the set of linear combinations of the a, is a submodule of E which obviously contains all the a, and is therefore the smallest submodule containing them. COROLLARY 1. Let u:E + F be a linear mapping, S a subset of E and M the submodule ofE generated by S. Then u(M) is the submodule of F generated by u(S). In particular, the image under u of any finitely generated submodule of E is a finitely generated submodule of F. Remark. If U(X) = 0 for all x E S, then also u(x) = 0 for all x E M. We shall sometimes refer to this result as the “principle of extension of linear identities” or “principle of extension by linearity”. In particular, to verify that a linear mapping u:E 4 F is of the form u 0 +, where v:E/M -+ F is linear and +:E 3 E/M is the canonical projec - tion, it suffices to verify that u(S) = 0. An A -module is called finitely generated if it has aJinite generating set. COROLLARY 2. The submodule generated by the union of a family (M,)LEI of sub- modules of a module E is identical with the set of sums zI x,, where (x,) , € I runs through the set of families of elements of E offinite support such that x, E M, for all t E I. Clearly every linear combination of elements of ,VI M, is of the above form : the converse is obvious. 5 1.7 INTERSECTION AND SUM OF SUBMODULES The submodule of E generated by the union of a family (ML)LEI of sub- modules of E is called the sum of the family (M,) and is denoted by 2 M,. If for all t E I, h, is the canonical injection M, --f E and h: (x,) H 2 h,(x,) the linear mapping of @ M, into E corresponding to the family (h,) (no. 6, Proposition 6), 2 M, is the image of h; in other words, there is an exact sequcnce LEI LEI LEI COROLLARY 3. If(Mh)AEL is a right directed family of submodules of an A-module E, the sum zL MA is identical with the union u M,. AEL u MA c 2 MA is always true without hypothesis; on the other hand, for every finite subfamily (MA),,, of (MA)AEL,, there exists by hypothesis a p E L such that MA c M, for all A E J, hence c MA c M, and it thus fol- lows from Corollary 2 that 2 MA c AvL MA. h€L heL AcL AEL COROLLARY 4. Le! 0 -+ E + f F -% G + 0 be an exact sequence of A-modules, S a generating system of E, T a generating system of G. If T‘ is a subrct of F such that g(T’) = T, T’ v f (S) is a generating system of F. The submodule F’ of F generated by T’ uf(S) containsfiE) and, as %?(I;’) contains T, g(F’) = G; hence F’ = F. COROLLARY 5. In an exact sequence 0 + E + F -+ G + 0 of A-modules, if E and G arejnitely generated, so is F. PROPOSITION 10. Let M, N be two submodules of an A-module E. Tlien [here are two exact sequences ’ (29) o - M~N A M~N 2 M + N --+ o (30) 0 --+ E/(M n N) (E/M) @ (E/N) E/(M + N) --+ 0 where i:M -+ M + N, j:N -+ M + N are the canonical injections, the canonical surjections and where the homomorphisms u and u are dcjined ns follow : P:E/M -+ E/(M + N) and q:E/N --i E/(hf + N) 206 207 11 LINEAR ALGEBRA iff:MnN-+M-+M@N and g:MnN+N+M@N are the canonical injections, u = f + g, and ifr:E/(M n N) -+ E/M -+ (E/M) @ (E/N) and are the canonical mappings, u = r + S. We prove the exactness of (29) : obviously i - j is surjective and u is injec- tive. On the other hand, to say that (i -j)(x,y) = 0, where XEM and y EN, means that i(x) - j(y) = 0, hence i(x) = j(y) = z E M n N, whence by definition x = f (z), y = g(z), which proves that Ker(i -j) = Im u. We prove the exactness of (30) : clearly p - q is surjective. On the other hand, to say that v(t) = 0 for t E E/(M n N) means that r(t) = s(t) = 0, hence t is the class mod. (M n N) of an element z E E whose classes mod. M and mod. N are zero, which implies z E M n N and t = 0. Finally, to say that (p - q)(x, y) = 0, where x E E/M, y E E/N means that p(x) = q(y), or also that there exist two elements z', t" of E whose classes mod. M and mod. N are respectively x and y and which are such that z' - Z" E M + N. Hence there are t' E M, t" EN such that 2' - z" = t' - t", whence 2' - t' = 2" - t" = z. Let w be the class mod. (M n N) of z; r(w) is the class mod. M of z and hence also that of z', that is x; similarly s(w) = y, which completes the proof that Ker(p - q) = Imv. s:E/(M n N) -+ E/N --f (E/M) @ (E/N) DIRECT SUMS OF SUBMODULES 3 1.8 8. DIRECT SUMS OF SUBMODULES DEFINITION 6. An A-module E is said to be the direct sum of a family (MJle1 of submodules of E if the canonical mapping 9 M, -+ E (no. 6) is an isomorphism. It amounts to the same to say that every x E E can be written in a unique way in the form x = 2 x,, where x, E El for all t E I; the element x, thus corresponding to x is called the component of x in El; the mapping x H x, is linear. Remark. (1) Let (M,), (N,), EI be two families of submodules of a module E, with the same indexing set; suppose that E is both the direct sum of the family (M,) and of the family (N,) and that N, c M, for all LEI. Then N, = Ml for all L E I, as follows immediately from no. 6, Corollary 1 to Proposition 7 applied to the canonical injectionsf,: N, --f M,. PROPOSITION 11. Let (MJle1 be a family of submodules of an A-module E. The following properties are equivalent : LEI (a) The submodule 2 M, is the direct sum oftfie family LEI (b) The relation 2 x, = 0, where x, E M, for all t E I, implies x, = 0 for all (c) For all K E I, the intersection of M, and 2 M, reduces to 0. It is immediate that (a) and (b) are equivalent, since the relation 2 X, = 2 y, is equivalent to 2 (x, - yL) = 0. On the other hand, by virtue of Definition 6, (a) implies (c) by the uniqueness of the expression of an element of @ M, as a direct sum of elements x, E M,. Finally, the relation X, = 0, where x, E M, for all 1, can be written, for all K E I, x, = 2 ( -x,); condition (c) then implies x, = 0 for all K E I, hence (c) implies (b). DEFINITION 7. An endomorphism e of an A-module E is called a projector if e o e = e (in other words, if e is an idempotent in the ring End(E)). In End(E) a family (eJApL of projectors is called orthogonal zf e, 0 ep = 0 for A # p. PROPOSITION 12. Let E be an A-module. (i) If E is the direct sum of a family of submodules and, for all x E E, eh(x) is the component of x in MA, (eA) is an orthogonal family of projectors such that x = C eh(x)for all x E E. is an orthogonal family of projectors in End(E) such that x = 2 eA(x) for all x E E, E is the direct sum of the family of submodules Property (i) follows from the definitions and (ii) is a special case of no. 6, Corollary 2 to Proposition 6, applied to the canonical injections M, + E and the mappings e,: E --f M,. Note that when L is finite the condition x = 2 e,(x) for all x E E may L E I. l#K ,€I AsL (ii) Conversely, ;f hsL MA = e,(E). also be written in End(E). AeL (31) COROLLARY. For every projector e of E, E is the direct sum of the image M = e(E) andthekernelN=i1(0) ofe; forallx=x, +x,EE withx,EMandx,EN, XI = e(x) ; 1 - e is a projector of E of image N and kernel M. (1 - e), = 1 - 2e + e2 = 1 - e in End(E) and hence 1 - e is a pro- jector; as also e( 1 - e) = (1 - e)e = e - e2 = 0, E is the direct sum of the images M and N of e and 1 - e by Proposition 12. Finally, for all x E E, the relatior. x E M is equivalent to x = e(x); for x = e(x) implies by definition X E M and, conversely, if x = c(x') with x' E E, then "(x) = e2(x') = c(Y') = x; 209 208 I1 LINEAR ALGEBRA this shows therefore that M is the kernel of 1 - e and, exchanging the roles of e and 1 - e, it is similarly seen that N is the kernel of e. Remark. (2) Let E, F be two A-modules such that E is the direct sum of a jnite family (M,) , <1 of submodules and F the direct sum of a jnite family (N,)l<!qn of submodules. Then it is known (no. 6, Corollary 1 to Proposition 6) that Hom,(E, F) is canonically identified with the product HomA(Mt, N,) ; to be precise, to a family (u,,), where u,, E Hom,(M,, N,) there corresponds the linear mapping u: E --f F defined as follows. It suffices to define the restriction of u to each of the Mt and for each X~E Mi, 1. f Now let G be a third A-module, the direct sum of a jni&e family (Pk)lGkGp of submodules; lct u be a linear mapping of F into G and let (vk,) E n HomA(N,, Pk) be the family corresponding to it canonically. For all xt E Mi, j. k Thus it is seen that if we write the family (Wkt) corresponds canonically to the composile linear mapping w = u o u from E to G (cf. 9 10, no. 5). 9. SUPPLEMENTARY SUBMODULES DEFINITION 8. In an A-module E, two submodules M,, M, are said to be supple - mentary ifE is the direct sum of M, and M,. Proposition 11 of no. 8 shows that, for M, and M2 to be supplementary, it is necessary and sufficient that M1 + M, = E and M, n M, = (0) (cf. I, 9 4, no. 9, Proposition 15). PROPOSITION 13. Let M,, M, be two supplementary submodules in an A-module E. The restriction to M, of the canonical mapping E 3 E/M2 is an isomorphism of MI onto E/M,. This linear mapping is surjective since MI + M, = E and it is injective since its kernel is the intersection of M, and the kernel M, of E 3 E/M, and hence reduces to (0). COROLLARY . If M, and M; are two supplements of the same submodule M1 .f E, the set of ordered pairs (x, x') E M, x Ma such that x - x' E M, is the graph of an isomorphisln of M, onto M;. z 3 1.9 SUPPLEMENTARY SUBMODULES It is immediate that it is the graph of the composite isomorphism M, --t E/Ml + M;. DEFINITION 9. A submodule M of an A-module E is called a direct factor of E Ef it has a supplementary submodule in E. When this is so, E/M is isomorphic to a supplement of M (Proposition 13). A submodule does not necessarily admit a supplement (Exercise 11). When a submodule is a direct factor, it has in general several distinct supplements; these supplements are however canonically isomorphic to one another (Corollary to Proposition 13). PROPOSITION 14. For a submodule M of a module E to be a direct factor, it is neces- sary and sujicient that there exist a projector of E whose image is M or a projector of E whose kernel is M. This foIlows immediately from no. 8, Proposition 12 and Corollary. PROPOSITION 15. Given an exact sequence of A-modules O--+E--+F~G+O f (33) the following propositions are equivalent : (a) The submodule f (E) ofF is a direct factor. (b) There exists a linear retraction r:F + E associated with f (Set Theory, 11, (c) There exists a linear section s:G -+ F associated with g (Set Theory, 11, 3, When this is so, f + s: E @ G + F is an isomorphism. If there exists a projector e in End(F) such that e(F) = f(E), the homo- morphismfo e:F + E is a linear retraction associated with J; conversely, if there exists such a retraction r, it is immediate that f 0 r is a projector in F whose image is f(E), hence (a) and (b) are equivalent (Proposition 14). Iff@) admits a supplement E' in F andj:E' -+ F is the canonical injection, k? oj is an isomorphism of E' onto G and the inverse isomorphism, considered as a mapping of G into F, is a linear section associated with g. Conversely, if such a section s exists, s 0 g is a projector in F whose kernel isf(E), hence (a) and (c) are equivalent (Proposition 14). Moreover s is a bijection of G Onto s(G) and as s(G) is supplementary tof(E),f + s is an isomorphism. 5 3, no. 8, Definition 11). no. 8, Definition 11). -1 Note that being given r (resp. s) is equivalent to being given a supple - ment off@) in F, namely the kernel of r (resp. image of 5). When the exact sequence (33) satisfies the conditions of Proposition 15, it is said to split or (F,J g) is said to be a trivial extension of G by E (I, 3 6, no. 1). 210 21 1 I1 LINEAR ALGEBRA COROLLARY 1. Let u : E --f 1' be a linear mapping. For there to exist a linear mapping v:F + E such that u o v = 1, (the case where u is said to be right invertible and v is said to be the right inverse of u), it is necessary and su8cient that u be surjective and that its kernel be a direct factor in E. The submodule Im(v) of E is then a supple - ment of Ker(u). It is obviously necessary that u be surjective; as v is then a section associated with U, the conclusion follows from Proposition 15. COROLLARY 2. Let u: E -+ F be a linear mapping. For there to exist a linear mafiping v:F -+ E such that v 0 u = lE (the case where u is said to be left invertible and u is said to be the l~t inverse of u), it is necessary and sujicient that u be injective and that its image be a direct factor in F. The submodule Ker(v) ofF is then a supplement of Im(u). It is obviously necessary that u be injective; as v is then a retraction associ - ated with u, the conclusion also follows from Proposition 15. Remarks. (1) Let M, N be two supplementary submodules in an A-module E, p, q the projectors of E onto M and N respectively, corresponding to the decomposition of E as a direct sum of M and N. It is known (no. 6, Corollary 1 to Proposition 6) that, for every A-module F, the mapping (u, v) ++ u o p + v o q is an isomorphism of Horn,(M, F) 0 Hom,(N, F) onto Hom,(E, F). The image of Hom,(M, F) under this isomorphism is the set of linear mappings w: E 4 F such that w(x) = 0 for all x E N. (2) If M, N are two submodules of E such that M n N is a direct factor of M and N, then M n N is also a direct factor of M + N: if P (resp. Q) is a supplement of M n N in M (resp. N), M + N is the direct sum of M n N, P and Q, as is immediately verified. 10. MODULES OF FINITE LENGTH Recall (I, 3 4, no. 4, Definition 7) that an A-module M is called simple if it is not reduced to 0 and it contains no submodule distinct from M and (0). An A-module M is said to be offinite length if it has a Jordan-Holder series (MJostsn and the number n of quotients of this series (which does not depend on the Jordan -Holder series of M considered) is then called the length of M (I, 3 4, no. 7, Definition 11) ; we shall denote it by long(M) or long,(M). An A-module which is reduced to 0 is of length 0; if M is a non-zero A-module of finite length then long(M) > 0. PROPOSITION 16. Let M be an A-module and N a submodule of M; for M to be ofjnite length, it is necessary and su@cient that N and M/N be so, and then (34) long(N) + long(M/N) = long(M). The proof has been given in I, fj 4, no. 7, Proposition 10. 4 1.10 MODULES OF FINITE LENGTH COROLLARY 1. Let M be an A-module ofjnite length; for a submodule N of M to be equal to M, it is necessary and suficient that long(N) = long(M). COROLLARY 2. Let u:M -+ N be an A-module homomorphism. If M or N is of jnite length, so is Im(u). IfM is ofjinite length, so is Ker(u) and Iong(Im(u)) + long(Ker(u)) = Iong(M). (35) (36) IfN is ofjnite lmgth, so is Coker(u) and long(Im(u)) + long(Coker(u)) = long(N). COROLLARY 3. Let (M,)Obisn be a finite family of A-modules ofjnite length. rf there exists an exact sequence of linear mappings (37) 0 - Mo > Ml A M2 --+ -+ Mn.-l Iln-1 M, - 0 then 212 The corollary is obvious for n = 1 and is just Proposition 16 for n = 2; we argue by induction on n. If Mb-, = I~(U,,-~), then, by the induction hypothesis, c (- l)k long(M,) + (- 1)"-l long(MA-,) = 0. On the other hand, the exact sequence 0 -+ MA - -+ M, - -+ M, + 0 gives long(MA-,) + long(M,) = long(M,-,), whence relation (38). n-2 k=O COROLLARY 4. Let M and N be two submodules ofjinite length of an A-module E; M + N is offinite length and (39) long(M + N) + long(M n N) = long(M) + long(N). It suffices to apply Corollary 3 to the exact sequence (29) (no. 7). o+MnN+M@N+M+N+O uskg the fact that long(M 0 N) = long(M) + long(N) by (34). UQROLLARY 5. Let M be an A-module the sum of a family (N,) of submodules of bite length. Then M is ofjinite length and 213 LINEAR ALGEBRA FREE FAMILIES. BASES 3 1.11 j,(l) = e, so that e, = (8tt‘)t’ET) where is equal to 0 if t’ # t, to 1 if t’ = t (“Kronecker symbol”; (t, t‘) ++ at,, is just the characteristic function of the diagonal of T x T) ; every x = ( ti), E A:T’ may then be written uniquely: x = 2 Stet. The mapping 4 : t ++ e, of T into AiT’ is called canonical; it is injec- tive if A is non-zero. We shall see that the ordered pair (A:), 4) is a solution of a universal mapping problem (Set Theory, IV, 8 3, no. 1). PROPOSITION 17. For every A-module E and every mapping f:T --f E, there exists OM and only one A-linear mapping g:AiT’ + E such that f = g o +. The condition f = g 0 4 means that g(e,) = f (t) for all t E T, which is equivalent to g(Eet) = tf(t) for all 4 E A and all t E T and also means that g ~j, is the linear mapping 5 ++ S f (t) of A, into E for all t E T; the proposi - tion is therefore a special case of no. 6, Proposition 6. The linear mapping g is said to be determined by the family (f (t))tET of elements of E; by definition t€ T Moreover, for the two sides of (40) to be equal, it is necessary and su@cient that M It has been seen (no. 7, formula (28)) that there is a canonical surjective be the direct sum of the NL. linear mapping h: ON, -+ M; hence the corollary follows from (35). COROLLARY 6. Let M and N be two submodules Of an A-module E such that E/M and E/N are modules ofjnite length; then E/(M n N) is ofjnite length and (41) long(E/(M n N)) + long(E/(M + N)) = long(E/M) + long(E/N). It suffices to apply Corollary 3 to the exact sequence (30) 0 + E/(M n N) + (E/M) @ (E/N) + E/(M + N) + 0 long((E/M) @ (E/N)) = long(E/M) + long(E/N). COROLLARY 7. Let (M,) be ajnite family of submodules of an A-module E such that the E/Ml are modules Ofjnite length. Then E/(f) MI) is Ofjnite length and using the fact that (42) It has been seen (formula (27)) that there is a canonical injective linear mapping E/(Q Mi) + ? (E/Ml). Remark. With the exception of no. 7, Proposition 9, all the results of nos. 2 to 10 are valid for arbitrary commutative groups with operators, submodules (resp. quotient modules) being replaced in the statements by stable subgroups (resp. quotient groups by stable subgroups) ; we also make the convention of calling homomorphisms of groups with operators “linear mappings”. The corollaries to no. 7, Proposition 9 are also valid for commutative groups with operators: this is obvious for Corollaries 4 and 5, as also for Corollary 2, since ~(2 xL) = LEI 2 MX, for every operator a, and Corollary 3 follows from it. As for Corollary 1, it suffices to note that if N is a stable subgroup of F contain- ing u(S), u (N) is a stable subgroup of E containing S, hence L1(N) contains M and therefore u(M) c N. LEI -1 11. FREE FAMILIES. BASES Let A be a ring, T a set and consider the A-module A‘,T’. By definition, it is the external direct sum of a family (MJtoT of A-modules all equal to A, and for all t E T there is a canonical injection jt:A, -+ A(ST) (no. 6). We write (43) The kernel R of g is the set of (t,) E A?’ such that 3 tt f (t) = 0; it is sometimes said that the module R is the module of linear relations between the elements ofthe family (f (t))teT. The exact sequence 0 -+ R - A:T’ E (4‘4) is said to be determined by the family (f(t))teT. COROLLARY 1. Let T, T’ be two sets, g:T --f T’ a mapping. Then there exists one and only one A-linear mapping f: A(T’ -+ A(T’) which renders commutative the diagram T -% T‘ where (p and (p‘ are the canonical mafipings. 0’ It suffices to apply Proposition 17 to the composite mapping T 5 T’ -+ A‘”’. COROLLARY 2. For a family (at)tpT of elements of an A-module E to be a generating ?istern of E, it is necessary and suficient that the linear mapping AkT’ --t E determined bY thG family be surjective. This is just another way of expressing Proposition 9 of no. 7. 214 215 I1 LIKEAlZ ALGEBRA DEFINITION 10. A family (aJtET of elements of an A-module E is called a free family (resp. a basis of E) if the linear mapping ALT' + E determined by this family is injective (resp. bijective). A module is called free $it has a basis. In particular, a commutative group G is called free if G (written additively) is afree Z-module (CL I, 8 7, no. 7). Definition 10, together with Corollary 2 to Proposition 17, show that a basis of an A-module E is a free generating family of E. Every free family of elements of E is thus a basis of the submodule it generates. By definition, the A-module A:) is free and the family (et)tET is a basis (called canonical) of this A -module. When A # {0}, T is often identified with the set of e, by the canonical bijection t - e,; this amounts to writing tET 2 6,. t nstead oft& <,at for the elements of A:'. When this convention is adopted, the elements of A:) are called formal linear combinations (with coefficients in A) of the elements of T. Definition 10 and Proposition 17 give immediately the following result: COROLLARY 3. Let E be a free A-module, (aJiET a basis of E, F an A-module and (bJteT a family of elements of F. There exists one and only one linear mapping f: E -+ F such that (45) For f to be injective (resp. surjective), it is necessary and sujicient that (b,) be a free family in F (resp. a generating system of F). When a family (a,) teT is not free, it is called related. Definition 10 may also be expressed as follows: to say that the family is free means that the relation 2 A a - 0 (where the family (A,) is of finite support) implies A, = 0 for all t E T; to say that (aJtET is a basis of E means that every x E E can be f (a,) = 6, for all t E T. tBT - - FREE FAMILIES. BASES 3 1.11 2 written in one and only one way in the form x = 2 Eta,; for all t E T, 5, is then called the component (or coordinate) ofx of index t with respect to the basis (a,) ; the mapping x --f tt from E to A, is linear. Suppose A # (0); then, in an A-module E, two elements of a free family (a,)tET whose indices are distinct are themselves distinct: for if a,. = at" for t' # t", then 2 A , a - - 0 with At. = 1, A,,, = - 1 and A, = 0 for the elements of T distinct from t' and t". A subset S of E will be called a free subset (resp. a basis of E) if the family defined by the identity mapping of S onto itself is free (resp. a basis of E) ; every family defined by a bijective mapping of an indexing set onto S is then free (resp. a basis). The elements of a free subset of E are also called linearly independent. tET 2 If a subset of E is not free, it is called related or a related system and its elements are said to be linearly dependent. Every subset of a free subset is free; in particular, the empty subset is fi-ee and is a basis of the submodule {0} of E. PROPOSITION 18. For a family (at)tGT of a module E to be jree, it is necessary and suficient that everyjnite subfamily of (at)tET be free. This follows immediately from the definition. Proposition 18 shows that the set of free subsets of E, ordered by in- clusion, is inductive (Set Theory, 111, § 2, no. 4); as it is non-empty (since 0 belongs to it), it has a maximal element (aJIEI by Zorn's Lemma. It follows (if A # {0}) that for all x E E there exist an element p # 0 of A and a family (El) of element A such that px = Elat (cf. 5 7, no. 1). PROPOSITION 19. Let E be an A-module, the direct sum of a family (M,),,, of submodules. g, for each A E L, S, is a free subset (resp. generating set, basis) of MA, then S = AvL S, is afree subset (resp. generating set, basis) of E. The proposition follows from the definitions and the relation ASS' = @ A? h€L (associativity of direct sums, cf. no. 6). Remark. (1) By Definition 10, if A # {0} and (al)lEI is a free family, no element a, can be equal to a linear combination of the a, of index L # x. But conversely, a family (al) satisfying this condition is not necessarily a free family. For example, let A be an integral domain and a, b two distinct non-zero elements; in A, considered as an A-module, a and b form a related system, since (- b)a + ab = 0. But in general there does not exist an element x E A such that a = xb of b = xu (cf. however 9 7, no. 1, Remark). An element x of a module E is called free if {x} is a free subset, that is if the relation ax = 0 implies u = 0. Every element of a free subset is free and in particular 0 cannot belong to any free subset when A # {O}. Remarks. (2) A free module can have elements #O which are not free: for example, the A-module A, is free but the right divisors of zero in A are not free elements of A,. (3) In the additive group Z/(n) (n an integer 22) considered as a Z- module, no element is free and afurtiuri Z/(n) is not a free module. (4) It can happen that every element # 0 of an A-module is free without the module being free. For example, the field Q of rational numbers is a 2-module with this property, for two elements #O of Q always form a 217 216 I1 LINEAR ALGEBRA related system and a basis of Q could therefore only contain a single element a; but the elements of Q are not all of the form na with n E Z (cf. VII, 9 3). PRoPoSITloN 20. Every A -module E is isomorphic to a quotient module of a free A-module. If T is a generating set of E, there exists a surjective linear mapping A:T) --f E (Corollary 2 to Proposition 17), and if R is the kernel of this mapping, E is isomorphic to AiT’/R. In particular we may take T = E; then there is a surjective linear map- ping AkE) -+ E, called canonical. In particular, to say that an A-module E isjnitely generated (no. 7) means that it is isomorphic to a quotient of a free A -module with ajnite basis or also that there exists an exact sequence of the form A: -+ E ---f 0 (n an integer > 0). Note that if A # {0} every basis of ajnitely generoted free module E is neces- sarilyjnite, for if S is a finite generating system and B a basis of E, each ele- ment of S is a linear combination of a finite number of elements of B and if B‘ is the set of all the elements of B which figure thus in the expression for the elements of S, B is finite and every x E E is a linear combination of elements of B, hence B’ = B. PROPOSITION 2 1. Every exact sequence of A-modules f O--+G~E-+F---tO in which F is a free A-module, splits (no. 9). To be precise, if (b,JhsL is a basis ofF and, for each h E L, a,, is an element of E such that f (a,,) = b,,, the family (ah)hs is free and generates a suptlementay submodule to g(G). There exists one and only one linear mapping h:F -+ E such that h(bJ = ah for all h E L (Corollary 3 to Proposition 17). As h is a linear section associated withf, the proposition follows from I, 5 4, no. 9, Proposition 15. Remark. (5) Let (at)l4tt,, be a basis of an A-module E and let (bt)l<t<n be a family of elements of E given by the relations (46) 6, = AHal + + biai (1 < i < n) where A,, is invertible in A; then (b,)14iG,, is a basis of E. It suffices to argue by induction on n, the proposition being obvious for n = 1. If E‘ is the submodule of E generated by the family (u,)~,,,~-~, it follows from the induction hypothesis that (b,) , , <, - is a basis of E’; on the other hand, it follows from (46) that if pb,, E E’ with p E A, then also p.h,,,,a,, E E’, ANNIHILATORS. FAITHFUL MODULES. MONOGENOUS MODULES 9 I. 12 whence p = 0 since A,, is invertible. The family (b,),,,,,, is thus free and, as a,, = -An-,,%lnal - . . . - A~~An-l,nan-l + h;:b,, it is seen that (b,),,,,,, is a generating system of E, which completes the proof. This result is easily generalized to a family (aJreI whose indexing set I is well ordered. 12. ANNIHILATORS. FAITHFUL MODULES. MONOGENOUS MODULES DEFINITION 11. The annihilator of a subset S of an A-module E is the set of elements u E A such that M = 0 for all x E S. The annihilator of S is usually denoted by Ann(S) ; for a subset S consisting of a single element x, we write Ann(x) instead of Ann({x}) and call Ann(x) the annihilator of x. The relation ax = 0 may also be expressed by saying that x is annihiiated by a. It is immediate that the annihilator of an arbitrary subset S of E is a lgt ideal of A; for it to be equal to A, it is necessary and sufficient (by virtue of (MIv)) that S = (0). If two subsets S, T of E are such that S c T, the anni- hilator of T is contained in the annihilator of S. If (S,),,, is an arbitrary family of subsets of E, the annihilator of the union S, is the intersection of the annihilators of the S,. In particular, the annihilator of a subset S of E is the intersection of the annihilators of the elements of S. To say that an element of E is free is equivalent. to saying that its annihilator is {O}. For all x E E and all 0: E A, the annihilator of M is the set of p E A such that Pa E Ann(x). for all x E M, then also a@) = 0 for all x E M and all P E A, hence ap be- longs to the annihilator of M for all p E A. In particular, the annihilator of E is a two-sided ideal of A. For all u E A, let ha be the homothety x ++ M; it is known that the mapping a ++ ha of A into the endomorphism ring 6‘ = Hom,(E, E) of the commu - tative group (without operators) E, is a ring homomorphism ($2, no. 5). The inverse image of 0 under this homomorphism is the annihilator a of E; the image of A under the homomorphism tc H ha is therefore isomorphic to the quotient ring A/a. The module E is called faithful if its annihilator a reduces to 0. I ~ The annihilator of a submodule M of E is a two-sided ideal of A; for, if M = 0 I Let E be any A-module, a a two-sided ideal of A contained in Ann(E) and let ti be an element of the quotient ring A/a; for all x E E, the element cuc is the same for all the a E A belonging to the class a mod. a; if this ele- ment is denoted by ax, it is immediately seen that the mapping 219 218 I1 LINEAR ALGEBRA (a, x) H bx defines (with addition on E) an (A/a)-module structure on E. When a = Ann(E), the (A/a)-module E thus defined isfaithful; we shall say that it is the faithful module associated with the A-module E. Observe that every submodule of an A-module E is also a submodule of the associ - ated faithful module and conversely. DEFINITION 12. A module is called monogenous ;fit is generated by a single element. Proposition 9 of no. 7 shows that, if E is a monogenous A-module and a is an element generating E, E is identical with the set A.a of ta, where t runs through A. Examples. (1) Every monogenous group, being commutative (I, 4 4, no. 10, Proposition 18), is a monogenous Z-module. (2) If A is a commutative ring, the monogenous submodules of the A-module A are just the principal ideals (I, 0 8, no. 6) of the ring A. (3) Every simple A-module E is monogenous, since the submodule of E generated by an element # 0 of E is necessarily equal to E. PROPOSITION 22. Let A be a ring. Every quotient module of A, is monogenous. Con- versely, let E be a monogenous A-module, c a generator of E and a its annihilator; the linear mapping H tc defines, when passing to the quotient, an isomorphism of A,la onto E. As A, is itself monogenous, being generated by 1, the first assertion follows from no. 7, Corollary 1 to Proposition 9. The second is obvious, since 5 H tc is by hypothesis surjective and has kernel a. Note that, if A is not commutative, the annihilators of two distinct gener- ators c, c' of a monogenous A-module E are in general distinct and are also distinct from the annihilator of the module E. On the other hand, if A is commutative, the annihilator of a generator c of E is contained in the annihilator of every element of E and hence is the annihilator of the whole of E. COROLLARY . Every submodule of a monogenous A-module E is isomorphic to a quo- tient module b/a where a and b are two left-ideals of A such that a c 6. Every quotient module of a monogenous A-module is monogenous. The second assertion is immediate and the first follows from Proposition 22 and I, $ 4, no. 6, Theorem 4. Note on the other hand that a submodule of a monogenous module is not necessarily monogenous. For example, if A is a Commutative ring in which there exist non-principal ideals (VII, $ 1, no. l), these ideals are non-monogenous submodules of the monogenous A-module A. It follows from the definitions that a submodule of an A-module E gener- ated by a family (0,) of elements of E is the sum of the monogenous submodules 220 CHANGE OF RING OF SCALARS $ 1.13 Aa, of E; for (a,) to be a basis of E, it is necessary and sufficient that each of the a, be a free element of E and that the sum of the Aa, be direct. PROPOSITION 23. Let E be an A-module, the direct sum of an infinite family (MJlE1 of non-zero submodules. For every generating system S of E, Card(S) For all x E S, let C, be the finite set of indices t E I such that the component ofx in M, is #O and let C = *Vs C,. Every x E S belongs by definition to the submodule of E the direct sum of the M, for t E C and the hypothesis that S generates E implies therefore that C = I; as I is by hypothesis infinite, so is S (Set Theory, 111, $ 5, no. 1, Corollary 1 to Proposition 1); therefore Card(1) = Card(C) 6 Card(S) (Set Theory, 111, $6, no. 3, Corollary 3 to Theorem 2). COROLLARY 1. Under the hypotheses of Proposition 23, suppose that each M, is monogenous and that E is the direct sum of a second family (NA)heL ofnon -zero mono - genous submodules. Then Card (L) = Card (I). If b, is a generator of N,, the set of bh is a generating system of E, hence Card(L) 2 Card(1). In particular L is infinite and, exchanging the roles of (M,) and (NA), similarly Card(1) 2 Card(L), whence the corollary. COROLLARY 2. If a module E admits an infinite basis B, every generating system of E has cardinal 13. CHANGE OF RING OF SCALARS Let A, B be two rings and p a homomorphism of the ring B into the ring A. For every A-module E, the external law (p, x) - p (p)x defines (with addition) a B-module structure said to be associated with p and the A-module structure on E; this B-module is denoted by p,(E) or EL,] (and even simply E if no confusion can arise). In particular, if B is a subring of A and p : B -+ A is the canonical injection, E,,, is called the B-module obtained by restricting the ring of scalars A to B; by an abuse of language, this expression is also used when the homomorphism p is arbitram. Card(1). Card(B) and every basis of E is equipotent to B. If F is a submodile of the A-moduli E, p,(F) is a submodule of p*(E) and P*(E/F) is equal to p*(E)/p*(F). Let E, F be two A -modules; every A-linear mapping u: E + F is also a B-linear mapping EL,] -+ FIB] denoted by p*(u); in other words, there is a canonical injection of Z-modules (47) F, -+ HomB(ELBl, FfBl)- This mapping is not necessarily bijective; in other words a B-linear mapping EL,, -+ FIB, is not necessarily A -linear. For example, a sub-B- module of EL,, is not necessarily a sub -A-module of E: if A is a field and B 22 1 LINEAR ALGEBRA XI a subfield of A, the vector subspace B, of the vector B-space (As)LBl is not a vector sub-A-space if B # A. It is immediate that, for every family (EJLEI of A-modules, the B-module 3 1.13 CHANGE OF RING OF SCALARS p*(g EL) (resp. P@ EL)) is equal to rI LEI p*(EL) (resp- 0, P~E,)). Every generating system of p*(E) is a generating system of E but the con- verse is not necessarily true. PROPOSITION 24. Let A, B be two rings and p:B -+ A a ring homomorphism. (i) If p is surjective, the canonical mapping (47) is bijective. For every A-module E, every sub-B-module of p*(E) is a sub-A-module of E; every generating system of E is a generating system of p*(E). (ii) If p is injective, every free family in the A-module E is a free family in the B-module p*(E). The proposition follows immediately from the definitions. Note that even if p is injective, a free family in p,(E) is not necessarily frce in E. *For example, 1 and 42 do not form a free system in R considered as a vector R-space, although they form a free system in R considered as a vector Q-space (cf. Remark l).* PROPOSITION 25. Let A, B be two rings, p:B +A a ring homomorphism and E an A-module. Let (ah)AEL be a generating system (resp. free family of elements, basis) of A considered as a 14t B-module. L.et (a,JUe be a generating system (resp. free family of elements, basis) of the A-module E. Then (ahaU) is a generating system (resp. (when p is injective) free family of elements, basis) ofthe B-module p*(E). v, If x = 2 n where yU E A and (aA) is a generating system of A, we may write yw = 2 p(Phu)ah, with PhU E B, for all p E M, whence x = 2 p(PhU)ahav. ?.EL A, U On the other hand, if (ak) and (a,) are free families, a relation UE M YLI w' , ... . .. 1 p(P,,)a,a, = 0, with PAW E B, may be written VEM 2 (2 hEL p(PhU)ah)aU = 0; h. u - it thus implies A, p. if p is injective. COROLLARY . If A is a jinitely generated Ldt B-module and E a jinitely generated left A-module, p * (E) is a jinitely generated le$t A-module. Let C be a third ring, p' : C -+ B a ring homomorphism and p" = p o p' the composite homomorphism. It follows immediately from the definitions that p(PhU)ah = 0 for all p E M and therefore Phn = 0 for all &(E) = p;(p*(E)) for every A -module E. In particular, if p is an isomorphism of B onto A, then E = p;(p*(E)), where p' denotes the inverse isomorphism of p. Remarks. (1) Let K be a field and A a subring of K with the following pro - perty: for every finite family ( clSn of elements of K, there exists a y E A which is non-zero and such that yEl E A for 1 < i < n (a hypothesis which is always satisfied when A is commutative and K is the field of fractions of A). Let E be a vector space over K and EIA1 the A-module obtained by restricting the ring of scalars to A. Then, if a family (xh) is free in E,,] it is also.free in E. Attention may be confined to the case where L = (1, n) ; if there were a rela- tion 2 tixt = 0 with ti E K, the ti not all zero, it would follow that for all PEA, 5 (PF,Jxl = 0. By hypothesis we can suppose P # 0 in A such that Psi = at belongs to A for all i; but the relation aixf = 0 is contrary to the hypothesis, the ai being not all zero. (2) If the ring homomorphism p:B + A is surjective and b is its kernel (so that A is canonically identified with B/b), then, for every A -module E, b is contained in the annihilator of p,(E) and E is the A-module derived from p*(E) by the process defined in no. 12. Let A, B be two rings and p : B -+ A a homomorphism. Let E be an A-module and F a B-module; a B-linear mapping u:F -+ p,(E) (also called a B-linear mapping of F into E if no confusion arises) is also called a semi-linear mapping (relative to p) of the B-module F into the A-module E; it is also said that the ordered pair (u, p) is a dimorphism of F into E; this therefore means that, for x E F, y E F and P E B, i=l i=l The set Hom,(F, p*(E)) of B-linear mappings of F into E is also written as Hom,(F, E) if no confusion can arise. When p is an isomorphism of B onto A, the relation u(Px) = p(P)u(x) for all p E B may also be written as u( p'(a)x) = ax for all a E A, where p' denotes the inverse isomorphism of p; to say that u is semi -linear for p is then equivalent to saying that u is an A-linear mapping of &(F) into E. Example. It has been seen (no. 1) that a honiothety h,: x H ax on an A-module E is not necessarily a linear mapping. But if a is invertible, ha is a semi-linear mapping (which iy moreover bijective) relative to the inner automorphism 5- ata-l ofA, for a(h) = (aka-')(ax). 222 223 n LINEAR ALGEBRA Let C be a third ring, p‘: C + B a homomorphism and G a C-module. If v:G -+ F is a semi-linear mapping relative to p’, the composition w = u 0 u is a semi-linear mapping of G into E relative to the homomorphism p” = p o p’. If p is an isomorphism and u : F -+ E is a bijective semi-linear mapping relative to p, the inverse mapping u’ : E -+ F is a semi-linear mapping relative to the inverse isomorphism p’ : A -+ B of p. It is thus seen that, for the species of structure defined by giving on an ordered pair (A, E) of sets a ring structure on A and a left A-module structure on E, the dimorphism (u, +) can be taken as morphism (Set Theory, IV, 5 2, no. 1) ; we shall always assume in what follows that this choice of morphisms has been made. Remark (3). Let A,, A, be two rings, A = A, x A, their product and let el = (I, 0), e, = (0, 1) in A, so that A, and A, are canonically identified with the two-sided ideals Ae, and Ae, of A. For every A-module E, e,E and e,E are sub-A -modules El, E, of E, annihilated respectively by e, and el, so that, canonically identifying AIAe, with A, and AIAe, with A,, El (resp. E2) is given an A,-module (resp. A,-module) structure. Moreover, E is the direct sum of El and E,, for every x E E can be written as x = e,x + e,x and the relation e,x = e,y implies e,x = e:x = e,e,y = 0. Conversely, for every ordered pair consisting of an A,-module F, and an A,-module F,, let El be the A-module (PI) * (Fl), E, the A-module (p,) * (F,), p1 and p2 being the projections of A onto A, and A, respectively; then in the A-module E = E, @ E,, El = e,E, E, = e,E. The study of A-modules is thus reduced to that of A,-modules and that of A,-modules. In particular, every submodule M of E is of the form M, @ M,, where M, = e,M and M, = e,M. 14. MULTIMODULES Let A, B be two rings and consider on a set E two left module structures with the same additive law and whose ring of operators are respectively A and B; let d be the endomorphism ring of the additive group E and for all a E A (resp. p E B) let h, (resp. hb) denote the element x H ax (resp. x H (3.) of 8. Clearly the three following properties are equivalent: (a) ha o hb = $0 ha for all a and (3; (b) the image of A under the homomorphism a H h, is con- tained in Hom,(E, E) ; (c) the image of B under the homomorphism p ++ hb is contained in Hom,(E, E). When the A-module (resp. B-module) structure in question is a right module structure, the ring A (resp. B) must be replaced in (b) (resp. (c)) by Ao (resp. BO). The above properties can be expressed by saying that the two (left or right) module structures defined on E are compatible. DEFINITION 13. Let (AJhEL, (B,JWEM be two families of rings; an ((A,), (B,))- multimodule (or multimodule over the families of rings (A,)hEL, (B,JVEM) is a set E with, for each h E L, a left A,-module structure and, for each p E M, a right B,- module structure, all these module structures being compatible with one another. 5 1.14 When the family (B,) (resp. (A,)) is empty, E is called a le$t (resp. right) multimodule. When Card(L) + Card(M) = 2, we say “birnodule” instead of “multimodule”; it is then often convenient to consider (as can always be done by replacing a ring of operators by its opposite, cf. no. 1) a bimodule as having a l$t module structure with respect to a ring A and a right module structure with respect to a ring B, the permutability of the laws then being expressed by the relation MULTIMODULES a(xp) = (ax)p It is then also said that E is an (A, B)-bimodule. Two multimodule structures on a set E are said to be compatible if all the module structures on E which define one or the other of these multimodule structures are compatible with one another. for x E E, a E A, p E B. (49) Examples. (1) On a ring A the module structures of A, and A, are com- patible and A can therefore be considered canonically as an (A, A)-bimo - dule. (2) A left A-module E has a canonical left module structure over the ring End,(E) and the A-module and End,(E)-module structures on E are compatible. Clearly when E is a multimodule over two families (B,),€M of rings, E is also a multimodule over any two subfamilies (Ah)hEL, (B,JSEM,, where the A,-module and B,-module structures for A E L’ and p E M’ are those initially given. Since multimodules are particular examples of commutative groups with operators, the results of nos. 2 to 10 (cf. no. 10, Remark) can be applied to them; in particular, if E, F are two ((A,), (B,))-multimodules, a homomm- @him u: E -+ F is a mapping which is an A,-homomorphism for all h E L and a B,-homomorphism for all p E M. The stable subgroups of an ((A,), (B,))- multimodule are ((A,), (B,))-multimodules (called submultimodules), as also are the quotients by such subgroups (called quotient multimodules) ; similarly for products and direct sums. Let E be an ((A,), (B,))-multimodule and for each h E L (resp. p E M) let -+ A, (resp. 9,: BL -+ B,) be a ring homomorphism; clearly the 4-module structures associated with the +, and the A,-module structures given on E and the BL-module structures associated with the +,, and the %-module structures given on E are compatible with one another and hence define on E an ((A;), (B;))-multimodule structure, said to be associated with the g;ven ((A,), (B,))-multimodule structure and the +, and 9,. If E, Fare two ((A,), (B,))-multimodules, the additive group ofhomomorph- kms of E into F is denoted by Hom(Ah),(B,)(E, F) (or simply Hom(E, F)). Formulae (6) to (8) of no. 2 are obviously valid for ((Ah), (B,))-multimodule homomorphisms and, in particular, Hom(E, E) = End(E) has a ring structure; 225 224 LINEAR ALGEBRA XI moreover Hom(E, F) has a canonical ldt End(F)-module structure and right End(E)-module structure, these two structures being compatible; in other words, Hom(E, F) has a canonical (End(F), End(E))-bimodule structure, Suppose now that E has a multimodule structure whose rings of left (resp. right) operators are on the one hand the A, for A E L (resp. the B, for IJ. E M) and on the other the rings of another family (resp. (Bh,),,,Ew). Suppose similarly that F has a multimodule structure whose rings of left (resp. right) operators are on the one hand the A, for AEL (resp. the B, for p E M) and on the other the rings of another family (Aie)hwELM (resp. (B;v),,,EM,,); to abbreviate we shall say that E is an ((Ah), (A;,); (B,,), (BL))- multimodule and F an ((Ah), (AX-) ; (B,,), (B’;“))-rnultimodule. Consider E and F as ((Ah), (B,))-multimoduIes, thus restricting the operators to the subfamilies (Ah) and (B,,). By what was said at the beginnings of this no., the multimodule structures given on E and F define canonically ring homomorphisms A;, -+ (B,) (E), B? + Endw,), (B,) (E) 7 + End(,,). (B,,)(F), B’;” -+ (B,,,(F) ; moreover, two elements of Endo,,, (B,,,(E) (resp. End(,,,, (B,,)(F)) respective images of elements of two distinct rings among the A;, or the BL? (resp. the Ail or the BF) are permutable; it follows that the above homomorphisms define on Homo,,, (B,)(E, F) a multimodule structure whose rings of left operators are the A’;. (A” E L”) and the BL, (p’ E M’) and whose rings of right operators are the A;. (A‘ E L’) and the BE” (p” E M”). If now E’ is an ((Ah), (A;,); (B,,), (B;,))-multimodule and F’ an ((Ah), (AX-) ; (B,,), (B’;.))-multimodule, Horn(,,), (B,)(E’, F’) is an ((A:*), (BL,) : (A;,), (BE.))-multimodule; if u : E’ + E, v: F -+ F’ are multimodule homomorphisms, Hom(u, u) :Hom(AA).(B,)(E, F) + Hom(A,).(B,)(E’, F’) is defined as in no. 2 and is a multimodule homomorphism. Remarks. (1) Let F be an A-module and C the centre of the ring A; as central homotheties commute with all homotheties, F has a bimodule structure whose rings of left operators are A and C. If E is another A-module, Hom,(E, F) has therefore a canonical C-module structure (where, forfE Hom,(E, F) and y E C, y f is the homomorphism x H yf(x)) ; if E’, F’ are two A-modules, u: E’ -+ E, u:F -+ F’ two A-homomorphisms, the mapping Hom(u, v) is C-linear. (2) Let E be a left A-module; as A has a canonical (A, A)-bimodule struc - ture, so has the direct sum A(T) for any indexing set T; by the above, PROPERTIES OF Hom,(E, F) RELATIVE TO EXACT SEQUENCES 5 2.1 Hom,(Ay’, E) has a canonical ldt A-module structure arising from the right A-module structure on A:*’: forfE HomA(ALT’, E) and a E A, ajis the linear mapping x i-> f (xa). Corollary 2 to Proposition 17 of no. 1 1 defines a canonical mappingj,, from the product module ET to Hom,(AiT’, E), the image under jE, being the linear mapping f: AiT) --f E such that f(~) = xt for all t E T (where (PJ is the canonical basis of A$T)) ; it is known (loc. cit.) thatj,, is bijective and it follows from the definition given above of the A-module structure on HomA(AjT), E) thatjE, is A-linear. Finally, if u : E += F is an A-module homomorphism, the diagram of a family (xJto ET 5 E) , I J. FT + Hom,(Ay’, F) IF. T is commutative. Note that when T consists of a single element, jn:E --* Horn,(&, E) is just the mapping x ++ Ox defined in no. 2, Exurnfile 1. $2. MODULES OF LINEAR MAPPINGS. DUALITY 1. PROPERTIES OF Hom,(E, F) RELATIVE TO EXACT SEQUENCES THEOREM 1. Let A be a ring, E‘, E, E” three A -modules and u : E’ --f E, u : E + E” two homomorphisms. For the sequence E’ E E” -0 (1) (2) to be exact, it is necessary and sujicient that, for every A-module F, the sequence 0 --+ Hom(E”,F) Hom(E,F) Hom(E’,F) (where ii = Hom(u, lF), ij = Hom(v, lF)) be exact. Suppose that sequence (1) is exact. Ifw E Hom(E”, F) and U(w) = w 0 u = 0, then w = 0 since v is surjective. Sequence (2) is therefore exact at Hom(E”, F). we show that it is exact at Hom(E, F). ii 0 6 = Hom(u 0 u, lF) (3 1, no. 2, formula (10)) and u 0 u = 0 since sequence (1) is exact at E. Therefore 60 t7 = 0, that is Im(ij) c Ker(ii). On the other hand, if w ~Ker(i), then WoU = 0 and hence Ker(w) 3 Im(u). But as sequence (1) is exact at E, Im(U) = Ker(u) and hence Ker(w) 2 Ker(v); as u is surjective, it follows from 9 1, no. 3, Remark that there exists a w’ E Hom(E”, F) such that 226 227 I1 LINEAR ALGEBRA = W’ v = V(w’). Therefore Ker(ii) c Im(V), which completes the proof I that sequence (2) is exact. Conversely, suppose that (2) is exact for every A-module F. As ii 6 = Hom(v 0 u, lF) = 0, w 0 v 0 u = 0 for every homomorphism w: E” --f F. Taking F = E” and w = lEw, it is seen first that v o u = 0 and hence up’) c Ker(v). Now take F = Coker(u) and let +:E -+ F = E/u(E’) be the canonical mapping. Then U(+) = + 0 u = 0 by definition and hence there exists a + E Hom(E”, F) such that + = V(+) = + 0 V; this obviously implies u(E‘) = Ker(+) 3 Ker(v), which proves that sequence (1) is exact at E. Finally, let 0 be the canonical homomorphism of E” onto F = E”/v(E) ; then 6tO) = 0 o v = 0, hence 0 = 0; therefore, F = (0) and v is surjective. Sequence (i) ’is therefore exact at E”. COROLLARY . For an A-linear mappirlg u:E + F to be surjective (resp. bijective, resp. zero), it is necessary and su8cient that, for every A-module G, the mapping Hom(u, lG) :Hom(F, G) --f Hom(E, G) be injective (resp. bijective, resp. zero). It suffices to apply Theorem 1 to the case where E” = {0} (resp. E’ = (0) resp. E“ = E and v = lE). Note that starting from an exact sequence o -+ E! -4 E -A E” -+ o the corresponding sequence I 0 + Hom(E”, F) A Hom(E, F) d Hom(E’, F) -+ 0 is not necessarily exact, in other words, the homomorphism ii is not necessarily surjective. If E’ is identified with a submodule of E, this means that a linear mapping of E’ into F cannot always be extended to a linear mapping of E into F (Exercises 11 and 12). However: PROPOSITION 1. If the exact sequence of linear mappings (3) splits (in other words, if u(E’) is a direct factor of E) the sequence (4) 0 -+ Hom(E”, F) A Hom(E, F) A Hom(E’, F) -3 0 is exact and splits. Conversely, ;f, for every A-module F, sequence (4) is exact, sequence (3) splits. If the exact sequence (3) splits, there exists a linear retraction u’:E -+ E’ associated with u (fj 1, no. 9, Proposition 15) ; if 0 --3 E’ A E ”, E” --+ 0 - ii‘ = Hom(u‘, lF) :Hom(E’, F) --f Hom(E, F), PROPERTIES OF Hom,(E, F) RELATIVE TO EXACT SEQUENCES 4 2.1 the fact that u‘ o u is the identity implies that U 0 ii‘ is the identity (5 1, no. 2, formula (10)) and hence the first assertion follows from 5 1, no. 9, Proposition 15. Conversely, suppose sequence (4) is exact for F = E’. Then there exists an element f E Hom(E, E’) such thatfo u = lE,, and the conclusion follows from 9 1, no. 9, Proposition 15. Note that the first assertion of Proposition 1 can also be considered as a special case of 5 1, no. 6, Corollary 1 to Proposition 6, canonically identifying Hom(E’, F) @ Hom(E”, F) with Hom(E’ @ E”, F) by means of the Z-linear mapping Hom(p’, lF) + Hom(p”, lF), where b‘: E @ E” -+ E’ and p“:E‘ @ E” ---f E“ are the canonical projections. THEOREM 2. Let A be a ring, F’, F, F” three A-modules and u: F‘ --f F, v: F + F” two homomorphism. For the sequence 0 ---f F’ F L+ F” (5) (6) to be exact, it is necessary and sufficient that, for every A-module E, the sequence - 0 ---+ HomjE,F’) A Hom(E,F) & - Hom(E,F”) (where ii = Hom(lE, u), V = Hom(lE, v)) be exact. Suppose that the sequence (5) is exact. Note first that COG = Hom(l,,vou) = 0 (11, fj 1, no. 2, formula (10)) since v 0 u = 0. The image of Hom(E, F’) under Cis therefore contained in the kernel N of V; let f be the homomorphism of the Z-module Hom(E, F’) into N whose graph is equal to that of ii; it is neces- sary to prove that f is bijective and hence to define a mapping g:N -+ Hom(E, F’) such that f 0 g and g 0 fare the identity mappings. For this, let w be an element of N, that is a linear mapping w : E --f F such that VOW = 0. The latter relation is equivalent to w(E) c Ker(v) = u(F’) by hypothesis, hence, since u is injective, there exists one and only one linear mapping w’: E -+ F’ such that w = u 0 w‘ and we take g(w) = w’; it is immedi- ately verified that g satisfies the desired conditions. conversely, suppose that sequence (6) is exact for every A-module E. Hom(l~, v 0 U) = 6 0 ii = 0, then v 0 u 0 w = 0 for every homomorphism w:E +- F‘. Taking E = F’ and w = lF, it is seen first that v 0 u = 0 and hence u(F’) = Ker(v). Now we take E = Ker(v) and let +:E+F be the canonical injection. Then V(+) = v o + = 0 by definition and hence there exists $E Horn@, F’) such that 4 = U(+) = u 0 4, which obviously implies Ker(v) C u(F‘) and completes the proof of the exactness of (5) at F. Finally, if 8 is the identity mapping of Keru, then ii(0) = 0, hence 0 = 0 and Ker = {O}, which proves the exactness of (5) at F’. 228 229 I1 LINEAR ALGEBRA Remark. (1) Theorem 2 allows us, for every submodule F' of F, to identify Hom(E, F') with a sub-Z-module of Hom(E, F). When this identification is made, then, for every family (M,) of submodules of F Hom(E, M,) = I? Hom(E, M,) for if u E Hom(E, F) belongs to each of the Hom(E, M,), then, for all x E E, U(X) E M, for all A and hence u maps E into MA. COROLLARY . For an A-linear mapping u : E + F to be injective, it is necessary and SUB- cient that, for every A-module G, the mapping Hom( 1 G, u) : Hom(G, E) -* Hom(G, F) be injective. It suffices to apply Theorem 2 to the case where F' = (0). Starting from an exact sequence U D 0 -+ F' --+ F -----f F" --+ 0 the corresponding sequence 0 ---f Hom(E, F') > Hom(E, F) 5 Hom(E, F") --+ 0 is not necessarily exact, in other words V is not necessarily surjective. If F' is identified with a submodule of F and F" with the quotient module F/F', this means that a linear mapping of E into F" is not necessarily of the form u o w, where LU is a linear mapping of E into F. However: Hom(l,, u) :Hom(P, E) -+ Hom(P, E") 4 2.2 PROJECTIVE MODULES Remark (2). The results of this no. are valid without modification for all commutative groups with operators. PROPOSITION 2. If the exact sequence (7) 0 F' A F A F" --+ 0 splits (in other words, if u(F') is a direct factor of F), the sequence (8) 0 --+ Hom(E, F') 5 Hom(E, F) 6_ Hom(E, F") -+ 0 is exact and splits. Conversely, if sequence (8) is exact for every A-module E, the exact sequence (7) splits. The first assertion follows from the fact that Hom(E, F') @ Hom(E, F") is canonically identified with Hom(E, F' @ F") by means of the Z-linear mapping Hom( 1 E, j') + Hom( 1 E, j"), j' : F' + F' @ F" and j" : F" + F' @ F" being the canonical injections (3 1, no. 6, Corollary 1 to Proposition 6). Con- versely, if sequence (8) is exact for E = F", there is an element g E Hom(F", F) such that v o g = l,.. and the conclusion follows from fj 1, no. 9, Proposition 2. PROJECTIVE MODULES DEFINITION 1. An A-module P is called projective ;f, for every exact sequence F' + F -+ F" of A-linear mapping, the sequence Horn (P, F') -+ Horn (P, F) -+ Horn (P, F") is exact. PROPOSITION 3. For an A-module P, the direct sum ofa.family of'submodules (M,), to be projective, it is necessary and sujicient that each of the M, be projective. For every A -module homomorphism u : E + F, Hom( l,, u) :Hom(P, E) + Hom(P, F) is identified with conclusion thus follows from Definition 1 and 4 1, no. 5, Proposition 5 (ii). COROLLARY . Every free A-module is projective. Horn( lML, u) (4 1, no. 6, Corollary 1 to Proposition 6) ; the 230 I1 LINEAR ALGEBRA homomorphism v:E + P and the homomorphism 1,: P -> P; the existence ofa homomorphism g:P 4 E such that 1, = v o g implies that the sequence O--+E’---+E~P-+O splits (3 1, no. 9, Proposition 15). As for every A-module M there exist a free A-module L and an exact sequence 0 -+ R --+ L --+ M --f 0 (9 1, no. 1 1, Propo- sition 20), clearly (d) implies (e). Finally (e) implies (a) by virtue of Proposition 3 and its Corollary. COROLLARY 1. For an A-module to be projective andjinitely generated, it is necessary and sufient that it be a direct factor of a free A-module with ajinite basis. The condition is obviously sufficient; conversely, a finitely generated pro - jective module E is isomorphic to a quotient of a free module F with a finite basis (3 1, no. 11) and E is isomorphic to a direct factor of F by virtue of Pro- position 4 (d). COROLLARY 2. Let C be a commutative ring and E, F two jinitely generated projec- tive C -modules; then Hom,(E, F) is ajnitely generated projective C-module. It may be assumed that there are two finitely generated free C -modules M, N such that M = E @ E’, N = F @ F‘ ; it follows from $ 1, no. 6, Corollary 1 to Proposition 6 that Hom,(M, N) is finitely generated and free and on the other hand that Hom,(M, N) is isomorphic to Hom,(E, F) @ Homc(E’, F) @ Hom,(E, F‘) @ Hom(E’, F’), whence the corollary. 3. LINEAR FORMS; DUAL OF A MODULE Let E be a le$t A-module. As A is an (A, A)-bimodule, Hom,(E, A,) has a canonical right A-module structure (5 1, no. 14). DEFINITION 2. For every right A -module E, the right A-module Hom,(E, A,) is called the dual module ofE (or simply the dualt of E) and its elements are called the linear forms on E. If E is a right A-module, the set Hom,(E, Ad) with its canonical ldt A- module structure is likewise called the dual of E and its elements are called LINEAR FORMS; DUAL OF A MODULE $2.3 In this chapter, E* will be used to denote the dual of a (left or right) A- module E. Example. *On the vector space (with respect to the field R) of continuous real-valued functions on an interval (a, 6) of R, the mapping XH J: x(t)dt is a linear form., Let E be a left A-module and E* its dual; for every ordered pair of elements x E E, x* E E*, the element x*(x) of A is denoted by (x, x*). Then the rela- tions (9) SO) (x + Y,X*) = (x,x*) + (Y,X*> (x, x* + y*) = (x, x*> + (x, y*> (11) (ax,x*) = a(x,x*) (12) (x,x*cc) = (x,x*)a hold for x, y in E, x*, y* in E* and a E A. The mapping (x, x*) H (x, x*) of E x E* into A is called the canonical bilinear form on E x E* (the notion of bilinear form will be defined generally in IX, 3 1). Every linear form x* on E can be considered as the partial mapping x ++ (x, x*) corresponding to the canonical bilinear form. When E is a right A-module, the value x*(x) of a linear form x* E E* at an element x E E is denoted by (x*, x) and the formulae corresponding to (1 1) and (12) are written as <x*,xa) = <x*,x)a <ax*, x) = a<x*, x). When A is commutative, either notation is permissible. PROPOSITION 5. For every ring A, the mapping which associates with every 5 E A the linear form q H 14 on A, is an isomorphism of& onto the dual of&. It is the particular case of the canonical isomorphism E + HomA(A,, E) of§ 1, no. 14, Remark 2, corresponding to E = A,; the commutativity of dia- gram (50) of 3 l, no. 14, shows that we have here an isomorphism of right A- modules. If Ad is identified with the dual of A, by means of the isomorphism of Proposition 5, the canonical bilinear form on A, x A, is then expressed by (13) <E, E,*) = EF;* for 5, t* in A. form on Ad x A, being expressed by Ssarly, the dual of A, is canonically identified with A,, the canonical (14) (E*, t) = S*E for 5, E,* in A. 233 LINEAR ALGEBRA I1 TRANSPOSE OF A LINEAR MAPPING 6 2.5 4. ORTHOGONALITY DEFINITION 3. Let E be an A-module and E" its dual; an elenzclit x E E and an elment x* E E* are called orthogonal if (x, x*) = 0. A subset M of E and a subset M' of E* are called orthogonal sets if, for all x E M, x* E M', x and x* are orthogonal. In particular, x* E E* (resp. x E E) is called orthogonal to M (resp. M') if it is orthogonal to every element of M (resp. M'). If x* and y* are orthogonal to M, so is x* + y* and x*a for all GC E A by virtue of (10) and (1 2) (no. 3), which justifies the following definition: DEFINITION 4. Given a subset M of E (resp. a subset M' ofE*), the set of.* E E* (resp. the set of x E E) which are orthogonal to M (resp. M') is called the submodule totally orthogonal to M (resp. M') (or simply the submodule orthogonal to M (resp. M') if no confusion can arise). By definition of a linear form, the submodule of E* orthogonal to E is PROPOSITION 6. Let M, N be two subsets of E such that M c N; if M' and N' are the submodules of E* orthogonal to M and N respectively, then N' c M'. PROPOSITION 7. Let (M,) be a family of subsets of E; the submodule orthogonal to the union of the M, is the intersection of the submodules MI which are respectively orthogonal to the M,; this submodule is also the submodule orthogonal to the submodule of E generated by the union of the M,. reduced to 0; the submodule of E* orthogonal to (0) is identical with E*. These results are immediate consequences of the definitions. There is an analogous proposition (which we shall leave to the reader to If M is a submodule of E, M' the submodule of E* orthogonal to M and M" the submodule of E orthogonal to M', then M c M" but it may be that M # M' (Exercise 9). Note however that if M" is the orthogonal of M" in E*, then M" = M'; for M' c M" and on the other hand the rela- tion M c M' implies M" c M'. state) for submodules of E orthogonal to subsets of E*. 5. TRANSPOSE OF A LINEAR MAPPING Let E, F be two left A-modules; for every linear mapping u:E + F, the map- ping Hom(u, lA,) is a linear mapping of the right A-module F* into the right A-module E* (5 1, no. 2), called the trampose of u. DEFINITION 5. For every linear mapping u of an A-module E into an A-module F, the linear mapping y* H y* o u of the dual F* of F into the dual E* of E is called the trampose of u and is denoted by 'u. In other words: The transpose t~ is therefore defined by the relation (u(x),y*) = (x, %(y*)) for all x E E and ally* E F*. (15) Definition 5 applies without alteration to right A-modules and is then equivalent to the relation (y*, u(x)) = (%(y*), x) for all x E E and ally* E F*. Formulae (9) and (10) of 0 1, no. 2 here give t(U1 + u2) = 541 + tu2 "v 0 u) = tU 0 LU (16) (17) for two elements ul, u2 of Horn,& F) and for u E Horn,& F) and u E Hom,(F, G), G being a third A-module; finally, clearly '1, = l,.. (18) Remark. From (1 7) and (18) it follows that if u is left (resp. right) invertible, t~ is right (resp. left) invertible. PROPOSITION 8. Let u:E --f F be an A -linear mapping, M a submodule of E and M' the orthogonal of M in E*; the orthogonal of u(M) in F* is the inverse image tu- (M') . This follows immediately from (15). COROLLARY . The orthogonal ofthe image u(E) in F* is the kernel f~-l(0) of%. The orthogonal of E in E* is 0. If u:E --f F is an isomorphism, h: F* -+ E* is an isomorphism and if u:F + E is the inverse isomorphism of u, tv is the inverse isomorphism of t~ (formulae (1 7) and (18)). DEFINITION 6. Given an isomorphism u of an A-module E onto an A-module F, the trrnfiose of th inverse isomorphism of u (equal to the inverse isomorphism of the transpose of u) is called the contragredient isomorphism of u and denoted by 6. (19) The isomorphism u' is thus characterized by the relation If U:F -+ G is an isomorphism, the contragredient isomorphism of v 0 u (u(x), u'(y*)) = (x, x*) for x E E, x* E E*. isv'ou'. In particular, the mapping u H u' is an isomorphism of the linear group GL(E) Onto a subgroup of the linear group GL(E*). 235 234 I1 LINEAR ALGEBRA DUAL OF A QUOTIENT MODULE. DUAL OF A DIRECT SUM. DUAL BASES 3 2.6 This is a particular case of 3 1, no. 6, Corollary 1 to Proposition 6, applied to the case where If, by means of the canonical injectionsj,, the EL are identified with sub- modules of their direct sum E and if, by means of the product mapping x* ++ (yr(x*)), E* is identified with n E:, it can then be said that n Er is the dual of @ E,, the canonical bilinear form being given by F, = A,. LEI LEI LEI Let Q: A -+ B be an isomorphism of a ring A onto a ring B, E a left A-module, F a left B -module and u:E --f F a semi-linear mapping (3 1, no. 13) relative to Q. Let 0-l be the inverse isomorphism of 0; for all y* E F*, the mapping x H (~(x), y*)"-' of E into A is a linear form; if it is also denoted by 'u(y*), a mapping 'U : F* -+ E* is defined which is also called the transpose of the semi- linear mapping u; it is thus characterized by the identity (20) (u(.),y*> = (x, 'U(Y*))" for x E E, y* E F*. It is immediately verified that 'U is a semi-linear mapping relative to 0-l. If v denotes the mapping u considered as an A-linear mapping of E into a,(F) (3 1, no. 13), we may write u = $ 0 v, where + is the identity mapping o,(F) --f F, considered as a semi-linear mapping relative to Q. It is immediate that 'U = tv 0 '+ and ('+, 0-l) is a di-isomorphism of F* onto (a,(F))* relative to the isomorphism 0-l; this relation allows us immedi- ately to extend the properties of transposes of linear mappings to transposes of semi-linear mappings. 6. DUAL OF A QUOTIENT MODULE. DUAL OF A DIRECT SUM. DUAL BASES We apply Theorem 1 of no. 1 to the case where F = A,: PROPOSITION 9. Let E', E, E" be A-modules and (21) E' E E" --f 0 an exact sequence of linear mappings. Then the sequence of transpose mappings 0 --3 E"* -% E* -% E'* is exact. COROLLARY . Let M be a submodule of an A-module E and +:E -+ E/M the canon - ical homomorphism. Then '+ is an isomorphism of the dual of E/M onto the submodule M' ofE* orthogonal to M. Ifj: M -+ E is the canonical injection, the kernel of 5 is by definition the orthogonal of M in E*. Moreover, in the notation of the corollary, an injective homomorphism E*/M -+ M* is obtained from 5 when passing to the quotient. PROPOSITION 10. Let (EJLE1 be a family of A-modules and for all t E I let j,: EL -+ E = @ ,€I EL be the canonical injection. Then theproduct mapping x* H ("j,(x*)) is an isomorphism of the dual E* of E onto the product 4 E:. 236 COROLLARY . Let M, N be two supplementary submodules in an A -module E and p: E -+ M, q : E -+ N the corresponding projectors; then 9 + 'q : M* 0 N* -+ E* is an isomorphism and 9 (resp. '4) is an isomorphism of M* (resp. N*) onto the submodule of E* orthogonal to N (resp. M). Moreover, if i: M --f E and j: N -+ E are the canonical injections, 9 0 'i and 'q 0 7 are the projectors E* -+ 'p(M*), E* -+ 'q(N*) corresponding to the decomposition of E* as the direct sum of V(M*) and 'q (N* ) . poi = I,, qoj = I,, poj = qoi = 0, iop +jog = l,, whence, by transposition (no. 5, formulae (16), (17) and (18)), 'i 0 = I,., Yo tq = IN., ti o 9 = ti o tq = 0. 9 o 'i + 'q 0 7 = 1,. and the proposition follows from 4 1, no. 6, Corollary 2 to Proposition 6. Under the hypotheses of the Corollary, M* (resp. N*) is often identified with the orthogonal V(M*) (resp. 'q(N*)) of N (resp. M) in E*, thus identi- fying every linear form u on M (resp. N) with the linear form on E extending u and which is zero on N (resp. M). When an A-module E admits a basis (eJtET, it has been seen that giving this basis defines canonically an isomorphism u :A:) -+ E. By virtue of Proposition 10 and no. 3, Proposition 5, the dual of A',T' is canonically identified with the product A:; consider the contragredient isomorphism u': A;f --f E*. If, for all t E T,f, is the element of A;f all of whose projections are zero with the exception ofthat ofindex t, which is equal to 1, and if we write e; = a(&), the elements e; ofE* are, by (19) and (22), characterized by the relations (23) or t' # t 1 for t' = t. <et, e?) = (" It amounts to the same to say that, for all x = 2 Etet E E, e,*(x) = Et; et* is called the coordinate form of index t on E. It tET follows from (23) that (e?) is afiee system in E*. In particular, ifT isjnite, the ef form a basis of E*, theft then forming the canonical basis of A:. Hence : 237 I1 LINEAR ALGEBRA PROPOSITION 11. The dual of a free module with a basis Of n elements is a free module with a basis of n elements. Note that the dual of a free module with an infinite basis is not neces - sarily a free module (VII, $ 3, Exercise 10). DEFINITION 7. IfE is a free module with aJinite basis (e,), the basis (e:) Ofthe dual E* ofE dejned by relations (23) i.i called the dual basis of (eJ. Relations (23) can also be written in the form (24) (et, e:> = stt, where 8,,. is the Kronecker symbol on T x T. Note that if T is finite and (ef) is the dual basis of (et), then, for (x, x*) = t ttt:. tsT The dual basis of a finite basis of a right A-module is of course defined similarly. COROLLARY . The dual of a finitely generated projective module is a finitely generated projective module. A finitely generated projective left A -module can be identified with a direct factor M of a free A -module A: with a finite basis (no. 2, Corollary 1 to Pro- position 4). Then (Proposition 11 and Corollary to Proposition 10) M* is iso - morphic to a direct factor of A$, whence the corollary. PROPOSITION 12. Let E be an A-module and (a,)t6T a generating system of E. The followine conditions are equivalent : .I (a) is a projective A-module. (b) There exists a family (a:)t ST .f linear forms on E such that, for all x E E, thefamily ((x, a:))t ET hasjnite support and x = tET C (x, a:>at. There exists a surjective homomorphism u:L -+ E, where L = AP), such that if (et)tET is the canonical basis ofL then u(eJ = a, (5 1, no. 11, Proposition 17) ; for E to be projective, it is necessary and sufficient that there exist a linear mapping u: E -+ L such that u o'v = 1, (no. 2, Proposition 4 and 1, no. 9, Proposition 15). If such a mapping exists and we write %(e:) = a:, the11 (x, af> = (x, %(e:)) = (v(x), P:), hence the family ((x, a:)) has finite sup - BIDUAL § 2.7 port and x = u( ((x), &et) = 2 (x, at*)at for all x E E. Conversely, if condition (b) of the statement is fulfilled, the sum C (x, a:)e, is defined for all x E E and x --f 2 (x, @)e, is a linear mapping v: E --f L such that u o v = 1,. tET tET 7. BIDUAL Let E be a left A-module. The dual E** of the dual E* of E is called the bidual of E; it is also a 1eJt A-module (no. 3). For all x E E, it follows from no. 3, formulae (10) and (12), that the mapping x* - (x, x*) is a linear form on the right A-module E*, in other words an element of the bidual E**, which we shall denote by 2; moreover, it follows immediately from (9) and (1 1) (no. 3) that the mapping cE: x H i of E into E** is limar; this mapping will be called canonical; in general, it is netither injective nor surjective, even when E is finitely generated (cf. Exercise 9(e) and 3 7, no. 5, Theorem 6). An A -module E is called rejexive if the canonical homomorphism c, :E --f E** is bijective. Let F be a second left A-module; for every linear mapping u: E -+ F, the mapping '(%) :E** -+ F**, which will also be denoted by %, is linear and the diagram . (27) E** + F** is Commutative, as follows immediately from the definitions and formula (15) giving the transpose of a linear mapping. PROPOSITION 13. If E is a free module (resp. a free module with a Jinite basis), the canonical mapping c,: E -+ E** is injective (resp. bijective). Let (eJtET be a basis of E and let (ef) be the family of corresponding co- ordinate forms; by definition, if x E E is such that x" = 0, then (x, e:) = 0 for all t E T, in other words all the coordinates of x are zero, hence x = 0. Suppose hrther that T is finite; since (%, e:) = st,,, (Z,) is the dual basis of (e:) in E** and, as cE transforms a basis of E into a basis of E**, C, is bijective (3 1, no. 11, Corollary 3 to Proposition 17). We have moreover proved : ll" COROLLARY 1. Let E be afree A-module with afinite basis; for every basis (e,) .f E, (Cdet)) is the dual basis of the basis (ef) of E* dual to (e,). In this case it is said that (e,) and (e:) are two dual bases ofone another. 238 I 239 I1 LINEAR ALGEBRA COROLLARY 2. If E is a free A-module with a jinite basis, every jinite bask of E* is the dual basis of a basis of E. It suffices to consider in E** the dual basis of the given basis and canonically to identify E with E**. COROLLARY 3. Let E, F be two A-modules each with afinite basis, E (resp. F) being canonically ident$ed with its bidual E** (resp. F**). Then, for eve? linear mapping u:E -+ F, tt~ = U. This follows immediately from the commutativity of diagram (27). COROLLARY 4. IfP is a projective module (resp. ajnitely generatedprojective module) the canonical mapping c,:P --f P** is injective (resp. bijective). We shall use the following lemma: Lemma 1. Let M, N be two supplementary submodules in an A-module E and i: M --f E, j : N -+ E the canonical injections. Then the diagram CM 0 CN M@N - M**@N** 11, + 11, I E - E** CE I (28) i+j is commutative. By definition, for x E M, y E N, z* E E*, <cE(i(x) +j(y)), z*> = (i(x) +j(y), 2*> = <w, z*> + (JY), z*> = (x, “;(z*)> 3- (Y, “jz*)> = (cM(x), ti(z*)> + (‘N(Y), ”jz*)> = <tti(CM(x)) + ‘?“j(N(Y)), 2*>* This being so, if E is a free module (resp. a free module with a finite basis), cE is injective (resp. bijective) ; on the other hand, it follows from no. 6, Pro- position 10, that tti @ ty is bijective; the commutativity of diagram (28) then implies that cM @ cN is injective (resp. bijective) and so therefore are cM and CN (9 1, no. 6, Corollary 1 to Proposition 7), whence the corollary, taking account of no. 2, Proposition 4. LINEAR EQUATIONS 5 2.8 8. LINEAR EQUATIONS Let E, F be two A-modules. Every equation of the form ~(x) =yo, where u: E 4 F is a given linear mapping, yo a given element of F and the unknown x is subjected to the condition that it take its values in E, is called a linear equation; yo is called the right hand side of the equation; if yo = 0, the equation is called homogeneous linear. Every elemefit xo E E such that u(xo) = yo is called a solution of the linear equation u(x) = yo.? It is often said, loosely speaking, that a problem is linear if it is equivalent to determining the solutions of a linear equation. Given a linear equation u(x) =yo, the equation u(x) = 0 is called the homogeneous linear equation associated with u(x) = yo. PROPOSITION 14. rfxo is a solution Ofthe linear equation ~(x) = yo, the set ofsolu- tiom of this equation is equal to the set of elements xo + Z, where z runs through the set ofsolutions of the associated homogeneous equation U(X) = 0. The relation u(x) = yo may be written as ~(x) = u(xo), which is equivalent to u(x - xo) = 0. In other words, if the equation U(X) = yo has at least one solution xo, the set of its solutions is the set xo + il(0), obtained by translation from the kernel u (0) of u. Observe that u (0), being a submodule, is never empty, since it con- tains 0 (called the zero solution, or trivial solution, of the homogeneous equation By virtue of Proposition 14, for a linear equation u(x) = yo to have exactly one solution, it is necessary and sufficient that it have at least one solution and that u (0) = (0} (in other words, that the associated homogeneous equation have no non-zero solution, or also that u be injective); in this case, for all 9 E F, the equation u(x) = y has at most one solution. PROPOSITION 15. Let u be a linear mapping of a module E into a module F. If the equation ~(x) = yo has at least one solution, yo is orthogonal to the kernel oftu. TO say that ~(x) =yo admits a solution means that yo EU(E) and the Proposition follows from no. 5, Corollary to Proposition 8. -1 -1 u(x) = 0). -1 f’ This is in fact an abuse of language; from the logical point of view, we are not here defining the word “solution”, but simply the sentence “x,, is a solution of the equation u(x) = yo” as equivalent to the relation “xO E E and u(xo) = yo”. Observe that in a mathematical theory F where the relation “A is a ring, E and F are A- modules, u is a homomorphism of E into F, yo an element of F” is a theorem, every T of F such that the relation “T E E and u( T) = yo” is true in F is a solution of the equation u(x) = yo in the sense of Set Theory, I, 9 5, no. 2; this justifies the above abuse of language. 24Q I 24 1 I1 LINEAR ALGEBRA Observe that the necessary criterion for the existence of a solution of ~(x) = yo, given by Proposition 15, is sufficient when A is ajeld (3 7, no. 6, Proposition 12), but not in general (Exercise 10). Remarks. (1) Let E be an A-module, (FL),EI a family of A-modules and for all , E I let u, : E -+ F, be a linear mapping. Every system of linear equations (29) U'M = Y, (, E 1) where they, E F, are given, is equivalent to a single linear equation u(x) = y, where u is the mapping x H (u,(x)) of E into F = n LEI F, and y = (y,). The system (29) is called homogeneous if yL = 0 for all , E I. (2) Suppose that E admits a basis (ah)heL; ifwe set u(ah) = 6, for all A E L, to say that x Ehah satisfies the equation U(X) = yo is equivalent to say- ing that the family (of finite support) of elements of A satisfies the relation (30) hE~ CEb h h - - YO' Conversely, looking for families (Eh)hEL of elements of A of finite support satisfying (30), is equivalent to solving the linear equation u(x) = yo, where u is the unique linear mapping of E into F such that u(a,) = bh for all h E L (Q 1, no. 11, Corollary 3 to Proposition 17). (3) A linear equation u(x) = yo is called scalar when F = A, and therefore u is a linear form on E and yo a scalar. If E admits a basis (ah)hEL, it follows from Remark (2) that such an equation may also be written as (31) c LEh'h = YO E A where the family of scalars (ah) is arbitrary and where it is understood that the family (Eh) must have finite support. In general, by the solution (in A) of a system of scalar linear equations (32) hzLEhah6 = .I)& (L I) where ah, E A and 3' E A, is understood a family (Eh)hEL of elements of A of Jinite support and satisfying (32) ; the ah, are called the coeficients of the system of equations and the .I)~ the right hand sides. The solution of such a system is equivalent to that of the equation ~(x) = y, where y = (.I),) and u:A(?) ---f A: is the linear mapping (EX) (,zLchmh$* (4) A linear system (32) is also called a system of left scalar linear equations when it is necessary to avoid confusion. A system of equations (33) Xy L'Ar :A = ?L (I I) 242 9 3.1 is likewise called a system of right scalar linear equations; such a system can immediately be transformed into a system (32) by considering the Eh, qL and ah, as belonging to the opposite ring Ao to A. TENSOR PRODUCT OF TWO MODULES $3. TENSOR PRODUCTS 1. TENSOR PRODUCT OF TWO MODULES Let G,, G, be two Z-modules; a mapping u of the set G = G, x G, into a Z-module is called biudditive (or Z-bilineai) if u(x,, x,) is "additive with respect to x, and with respect to x,"; to be precise, this means that, for x,, y, in G17 x2, y2 in (32, 4x1 + Y,, x2) = u(x1, x2) + U(Y1, x2) 4x1, x2 + Y2) = 4x1, x2) + 4Xl,Y,). Note that this implies in particular that u(0, x,) = u(x,, 0) = 0 for all Let A be a ring, E a right A-module and F a lgt A-module. We are going to consider the universal mapping problem (Set Theory, IV, 9 3, no. 1) where C is the species of Z-module structure (the morphisms then being Z-linear mappings, in other words, additive group homomorphisms) and the a-mappings are the mappings f of E x F into a Z-module G which are Z-bilinear and further satisfy, for all x E E, y E F and h E A x1 E G,, x2 E G,. We show that this problem admits a solution. For this we consider the Z-module C = Z(E F, of formal linear combinations of the elements of E x F with coefficients in Z (3 1, no. 1 l), a basis ofwhich can be considered to consist of the ordered pairs (x, y), where x E E and y E F. Let D be the sub-Z-module of c gemrated by the elements of one of the following types : where x, xl, x, are in E, y, y,, y, are in F and h is in A. DERNITION 1. The tensor product of the right A-module E and the lgt A-module F, dcnotedby E (8 F or E F (or simply E @ F if no confusion is to be feared) is the quotient A Z-module C/D (the quotient of the Z-module C of formal linear 243 LINEAR ALGEBRA I1 TENSOR PRODUCT OF TWO LINEAR MAPPINGS 4 3.2 combinations of elements of E x F with coefficients in Z, by the submodule D generated by the elements of one of the types (2)). For x E E and y E F, the element of E @A F which is the canonical image of the element (x, y) 0 f C = Z‘E F, is denoted by x @ y and called the tensor product of x and y. The mapping (x, y) -X @ y of E x F into E @A F is called canonical, It is a Z-bilinear mapping which satisfies conditions (1). We show that the tensor product E @A F and the above canonical mapping form a solution of the universal mapping problem posed earlier. To be pre- cise : PROPOSITION 1. (a) Let g be a Z-linear mapping of E F into a Z-module G. The mapping (x, y) H~(x, y) = g(x @ y) of E x F into G is Z-bilinear and satisJies conditions (1). (b) Conversely, let f be a Z-bilinear mapping of E x F into a Z-module G satis- fying conditions (1). Then there exists one and only one Z-linear mapping g of E @A F into G such that f (x, y) = g(x @ y) for x E E, y E F. If (b denotes the canonical mapping of E x F into E @A F, then f = g o +; whence (a). To show (b), we note that, in the notation of Definition 1, f extends to a Z-linear mapping f of C into G (5 1, no. 11, Proposition 17). By virtue of relations (l),Jis zero for all the elements of C of one of the types (2) and hence on D. There therefore exists a Z-linear mapping g of C/U = E @A F into G such thatf = g 0 +, where + : C -+ C/D is the canonical homomorphism (5 1, no. 8, Remark). The uniqueness of g is immediate since E @A F is generated, as a Z-module, by the elements of the form x @ y. Proposition 1 defines a canonical isomorphism of the Z-module of Z-bilinear mappings f of E x F into G, satisfying conditions (l), onto the Z-module @A F, G). When A = Z, conditions (1) are automatically satisfied for every Z- bilinear mapping f and the submodule D of C is already generated by the elements of the first two types in (2). If now we return to the general case and E‘ and F’ denote the under- lying Z-modules of E and F respectively, the above remark and Definition 1 show immediately that the Z-module E F can be canonically identi - fied with the quotient of the Z-module E’ BZ F’ by the sub -Z-module generated by the elements of the form (xh) @ y - x @ (hy), where x runs through E, y runs through F and A runs through A. COROLLARY 1. Let H be a Z-module and h: E x F -+ H a Z-bilinear mapping satisfiing conditions (1) and such that H is generated by h(E x F). Suppose that for every Z-module G and every Z-bilinear mapping f of E x F into G satkfiing (1) 7 there exists a Z-linear mapping g:H -+ G such that f = g 0 h. Then, if + denotes the canonical mapping of E x F into E @A F, there exists one and only one isomorph - ism 0 of E @A F onto H such that h = 0 o +. The hypothesis that h(E x F) generates H implies the uniqueness of g; the corollary is then just the general uniqueness property of a solution of a universal mapping problem (Set Theory, IV, 3 3, no. 1). COROLLARY 2. Let Eo (resp. Fa) denote the module E (resp. F) considered as a lgt (resp. right) module over the opposite ring A’; then there exists one and only one Z- module isomorphism o:E @A F + Fa @AQ Eo such that o(~ @ y) = y @ x for x E E and y E F (‘‘commutativity’’ of tensor products). By definition of the Ao-module structures on Eo and Fa, the mapping (x, y) -y @ x of E x F into Fa BAo Eo is Z-bilinear and satisfies condi- tions (l), whence the existence and uniqueness of the Z-linear mapping o. Similarly a Z-linear mapping T:FO BAO EO + E @A F is defined such that ~(y @ x) = x @ y and clearly o and T are inverse isomorphisms. Remark. The tensor product of non -zero modules may be zero: for example, taking the two Z -modules E = 2/22 and F = 2/32, 2x = 0 and 3y = 0 for all x E E and y E F; therefore, in E BZ F, x @ y = 3(x B y) - 2(x B y) = x 63 (3y) - (2x1 8 y = 0 for all x and y (cf. no. 6, Corollary 4 to Proposition 6). 2. TENSOR PRODUCT OF TWO LINEAR MAPPINGS Let A be a ring, E, E’ two right A-modules, F, F’ two left A-modules and u:E --f E’ and v:F + F’ two A-linear mappings. It is easily verified that the mapping b,Y) 4%) @ V(Y) of E x F into E’ @A F’ is Z-bilinear and satisfies conditions (1) of no. 1. By Proposition 1 of no. 1 there thus exists one and only one Z-linear mapping W: E @A F -+ E’ @A F’ such that (3) w(x 63Y) = u(x) 63 V(Y) for x E E, y E F. This mapping is denoted by u @ v (when no confusion can arise) and is called the tensor product of the linear mappings u and v. It follows immediately from (3) that (u, v) H u @ v is a Z-bilinear mapping called canonical HomA(E, E’) x HomA(F, F’) -+ Hom,(E @A F, E’ @A F’). 244 245 I1 LINEAR ALGEBRA There corresponds to it by Proposition 1 of no. 1 a Z-linear mapping called canonical (4) HornA(% E’) @Z HomA(F, F’) -+ Hom,(E @A F, E’ @A F’) which associates with every element u @ v ofthe tensorproduct the linear mapping @ v:E @A F -+ E’ @A F’. Note that the canonical mapping (4) is not necessarily injective nor surjective. The notation u @ v can therefore lead to con- fusion and it will be necessary for the context to indicate whether it denotes an element of the tensor product or a linear mapping. Further, let E” be a right A-module, F” a left A-module and u‘: E’ --t E”, v’:F’ 4 F” A-linear mappings; it follows from (3) that (5) (U’otl)@(v’ov) = (U‘@U’)O(U@U). OPERATORS ON A TENSOR PRODUCT; TENSOR PRODUCTS AS MULTIMODULES $3.4 v: F’ -+ F relative to the homomorphism p: B -> A; u (resp. u) can be considered as a B-linear mapping E’ --t p*(E) (resp. F’ --f p*(k)), whence a Z-linear mappingw:E’ BBF’-+ p,(E) gB p,(F) such that w(x’@y’) = u(x’) @u(y’) for x’ E E’, 9’ E F’; composing the canonical mapping (6) with this mapping, a Z-linear mapping w‘:E‘ BB F’ -+ E @A F is hence obtained such that w’(x‘ @ y‘) = u(x‘) @ u(y’) for x’ E E’, y‘ E F’; this is the mapping which will normally be denoted by u @ v if no confusion can arise. Clearly (u, v) ++ u @ v is a Z-bilinear mapping HomB(E’, p*(E)) x HomB(F’, p,(F)) -+ Hom,(E’ BB F’, E @A F). Moreover, if C is a third ring, CT: C -+ B a homomorphism, E” a right C-module, F” a left C-module, u‘ : E” -+ E’ and v’ : F” + F’ semi-linear mappings rela- tive to 0, then (u 0 u’) @ (v 0 v’) = (u @ v) 0 (u‘ @ v’). 4. OPERATORS ON A TENSOR PRODUCT; TENSOR PRODUCTS AS MULTI- MODULES 3. CHANGE OF RING PROPOSITION 2. Let A, B be two rings, p :B 4 A a ring homomorphism and E (resp. F) a right (resp. left) A-module. Then there exists one and only one Z-linear mapping such that, for all x E E and y E F, the image under $ of the element x @ y of p*(E) BB p,(F) is the element x @ y of E @A F; this Z-linear mapping is surjec - tive. We consider the mapping (x,y) ++x @ y of p,(E) x p*(F) into E BAF; it is Z-bilinear and, for all p EB, by definition (xp(p)) @ y = x @ (p(p)y), hence conditions (1) of no. 1 hold, whence the existence and uniqueness of 4 (no. 1, Proposition 1). The latter assertion follows from the fact that the ele- ments x @ y generate the Z-module E @A F. COROLLARY . Let 3 be a two-sided ideal of A such that 3 is contained in the anni- hilator of E and in the annihilator of F, so that E (resp. F) has a canonical lgt (resp. right) (A/3)-module structure ( 3 1, no. 12). Then the canonical homomorphism (6) +:E @AF-tE @ApF (6) $:p*(E) @B p*(F) @A The mapping (6) is called canonical. corresponding to the canonical homomorphism p :A -+ A18 is the identity. For all GL E A/8, all x E E and all y E F, xGc = XQ (resp. Zy = ay) for all Q such that p(a) = a. If C = Z(ExF), the submodule of C generated by the elements (XQ, y) - (x, ~y) is then equal to the submodule generated by the elements (xE, y) - (x, Zy). With the hypotheses and notation of Proposition 2, let E‘ be a right B- module, F’ a left B-module and consider two semi-linear mappings u : E’ 3 E, With the hypotheses and notation of no. 1, for every endomorphism u (resp. U) of the A-module E (resp. F), u @ 1, (resp. 1, @ V) is an endomorphism of the Z-module E @A F; it follows immediately from (5) (no. 2) that the map - ping u &+ u @ 1, (resp. v ++ 1, @ u) is a ring homomorphism EndA(E) -+ End,(E @A F) (resp. EndA(F) -+ End,(E @A F)) ; moreover, (u @ IF) (lE @ v, = @ v, (u @ ‘F) = @’ (7) and therefore (4 1, no. 14) E @A F has a canonical left bimodule structure with respect to the rings EndA(E) and EndA(F). This being so, suppose given on E a ((B;); A, ((2;))-multimodule structure and on F a (A, (Ba); (Ci))-multimodule structure (3 1, no. 14); it amounts to the same to say that ring homomorphisms BI -+ EndA(E), clo -+ EndA@) are given with pairwise permutable images, and ring homomorphisms Bg -+ End,(F), (2fl.O -+ EndA(F) with pairwise permutable images. If these homomorphisms are composed respectively with the canonical homomorph - isms EndA(E) -+ End,(E @A F) and EndA(F) --+ End,(E @A F) defined above, it is seen (taking account of (7)) that ring homomorphisms B; -+ End,(E @A F), B(: -+ End,(E @A F), Cio -+ End,(E @A F) Cio -+ End,(E @A F) are defined with pairwise permutable images; in other words, there has been defined on E @A F a ((B:), (B;); (Ci), (Ci))-muZtimoduZe structure; it is this multhodule which is also called the tensor product (relative to A) of the 247 246 I1 LINEAR ALGEBRA (pi); A, (C;))-multimodule E and the (A, (Bi); ((2;))-multimodule F. This multi - module is the solution of a universal mapping problem analogous to that considered in no. 1 ; to be precise: PROPOSITION 3. Let G be a ((Bi), (B;); (Ci), (C;))-multimodule. (a) Let g be a linear mapping of the multimodule E @A F into G. The mapping f: (x, y) H g(x @ y) of E x F into G is Z-bilinear and satisj5es relations (1) of no. 1 and the conditions f (PIX, Y) = P;f (x, Y), f (-4 Y) = f (x, Y)v; f (x, Yv:) = f(X, !I).: { f(X, Plld = pllf(x, Y) , (8) for all x E E, y E F, pi E BI, v; E C;, p; E B;, v: E C:, i, j, h, k arbitrary. (b) Conversely, let f be a Z-bilinear mapping of E x F into G satisfying conditions (1) (no. 1) and (8). Then there exists one and only one linear mapping g of the multi- module E @A F into the multimodule G such that f (x, y) = g(x @ y) for x E E, y E F. Assertion (a) follows immediately from the definition of the multimodule structure on E @A F, since for example (x @ y)v; = (xv;) @ y. To prove (b), we note first that Proposition 1 of no. 1 gives the existence and uniqueness of a Z-linear mapping g such that g(x @ y) = f (x, y) for x E E, y E F; all that is needed is to verify that g is linear for the multimodule structures. As the elements x @y generate the Z-module E @A F, it suffices to verify the relations g(p'(x @ y)) = &g(x @ y) and their analogues; but this follows immediately from the formula g(x @ y) = f (x, y) and relations (8). SCHOLIUM . An element of E @A F may in general be written in several ways in the form 2 (xi @ yi), where xi E E and yi E F; but to define a linear mapping g of the multimodule E BAF into a multimodule G, there is no need to verify that, if c 1 (xi @ yi) = 4 (xi @ y;), then 7 g(xi @ yi) = 7 g(x; @ y;) ; it suffices to be given g(x @ y) for xeE and y EF and to verify that (x, y) I-+ g(x @J y) is Z-bilinear and satisfies conditions (1) (no. 1) and (8). Let E' be a ((Bi), A, ((2;))-multimodule, F' an (A, (Bi); ((2:))-multimodule and u: E -+ E', v: F --f F' linear mappings of multimodules; it follows immedi - ately from the definitions (no. 2) that u @ u is a linear mapping of the multimodule E @A F into the multimodule E' @A F'. With E always denoting a right A-module, let A, denote the ring A con- sidered as an (A, A)-bimodule (§ 1, no. 14, Example 1); by the above, the tensor product E @A (sAd) has a canonical right A -module structure such that (x @ h)p = x @ (hp) for x E E, h E A, p E A. The mapping (x, h) H xh of E x (a,) into E is Z-bilinear and satisfies conditions (1) (no. 1) and (8) TENSOR PRODUCT OF TWO MODULES OVER A COMMUTATIVE RING $ 3.5 248 (where, in the latter, the B:, Cl and B{ are absent and the family (Ci) reduces to A) ; hence (Proposition 3), there exists an A-linear mapping g (called canoni - cal) of E @A (sAd) into E such that g(x @ A) = xh for x E E, A E A. PROPOSITION 4. If E is a right A-module, the mapping h:x ++ x @ 1 of E into E @A (ad) is a right A -module isomorphism, whose inverse isomorphism g is such that g(x @ h) = xh for x E E, h E A. If g is the canonical mapping, g 0 h is the identity mapping 1, and h 0 g coincides with the identity mapping of E @A (&) onto itself for elements of the form x @ y, which generate the latter Z-module; hence the conclusion. We shall normally write E @A A instead of E @A ($Ad) and shall often identify E @A A with E by means of the above canonical isomorphisms. Observe that, if E also has a (left or right) B-module structure which is com- patible with its right A-module structure, g and h are also isomorphisms for the B-module structures on E and E @A A (and hence multimodule isomorphisms). Now let F be a left A-module; (&) @A F (also denoted by A @A F) then has a canonical left A-module structure and as in Proposition 4 a canonical isomorphism is defined of A BAF onto F mapping A @ x to Ax, and its inverse isomorphism is x - 1 @ x. In particular there exists a canonical isomorphism of the (A, A)-bimodule (A,) (Jd) onto ,Ad which maps A @ p to hp. 5. TENSOR PRODUCT OF TWO MODULES OVER A COMMUTATIVE RING Let C be a commutative ring; for every C-module E, the module structure on E is compatible with itself(§ 1, no. 14). If E and F are two C-modules, the considera - tions of no. 4 then allow us to define two C-module structures on the tensor product E F, respectively such that y(x @y) = (yx) @y and such that y(x my) = x @J (yy); but as, by Definition 1 of no. 1, in this case (yx) @ y = x @ (yy), these two structures are the same. Henceforth when we speak of E F as a C-module, we shall mean with the structure thus defined, unless otherwise stated. The canonical isomorphism. o:F &E+E BcF (no. 1, Corollary 2 to Proposition 1) is then a C-module isomorphism. It follows from this definition that, if (ah)hEL (resp. (bu)uEM) is a generating system of the C-module E (resp. F), (ah @ 6,) is a generating system of the C-module E BC F; in particular, if E and F are jnitely generated C-modules, so is E BC F. For every C -module G, the Z-bilinear mappings f of E x F into G for which (9) f(yx, = f(X, YY) = Yf (x, Y) for x E E, y E F, y E c 249 I1 LINEAR ALGEBRA are then called C-bilinear and form a C-module denoted by -Y2(E, F; G); Proposition 3 (no. 4) defines a canonical C-module isomorphism (cf. 4 1, no. 14, Remark 1). 5 3.6 defined in 4 1, no. 14, Remark I), whence we also derive a C-linear mapping, called canonical (13) HornA(& E') @c HOmA(F, F') -+ Homc(E @A F, E' @A F'). two vector K-spaces, the canonical mapping PROPERTIES OF E @A F RELATIVE TO EXACT SEQUENCES (3) Let A be an integral domain and K its field of fractions. If E and F are (ECAI) @A (FIA1) --f @K (no. 3 and fj 1, no. 13) is bijective. It suffices (no. 4) to prove that iff is an A- bilinear mapping of E x F into a vector K-space G, f is also K-bilinear. Now, for all a # 0 in A. whence since G is a vector K-space. af (a-%y) =f(x,y) = af(x, .-'!I) f(a-'x,y) =f(x, .-'!I) = a-'f(.,y) 6. PROPERTIES OF E @,, F RELATIVE TO EXACT SEQUENCES PROPOSITION 5. Let E, E', E" be right A-modules, F a lt$t A-module and E'& E u\ E" 0 an exact sequence of linear mappings. Writing U = u @ l,, U = v @ l,, the sequence (14) (15) E' @AF A E @AF E" @AF - 0 of Z-homomorphisms is exact. By virtue of no. 2, formula (5), fio u = (uou) @ 1, = 0; the image H = n(E' @ F) is contained in the kernel L = Ker(5); by passing to the quotient, we therefore derive from 5 a Z-linear mapping f of the cokernel M = (E @ F)/H of ii into E" @ F; it must be proved that f is bijective and it will hence suffice to define a Z-linear mapping g:E" @ F -+ M such that g 0 f and f 0 g are the identity mappings. Let x" E E", y E F; by hypothesis there exists x E E such that u(x) = x". We show that, if xl, x2 are two elements of E such that u(xl) = u(x2) = X" and +: E @ F -+ M is the canonical mapping, then +(xl @ y) = +(x2 @ y). It suffices to prove that if v(x) = 0 then +(x @ y) = 0, which follows from the fact that x = u(x') with x' E E', whence x @J y = u(x') @ y = U(x' @ y) E H. If (x", y) is mapped to the unique value of +(x @ y) for all x E E such that o(x> = x", a mapping is defined of E" x F into M; this mapping is Z-bilinear ad satisfies conditions (1) (no. 1)) since ~(xh) = x"h and (xh) @ y = x 8 (~y) x E E; hence there is a Z-linear mapping g of E" @ F into M such that &X" @3 !I) = +(x @ y) for y E F, x E E and X" = ~(x). This definition further Proves that fog coincides with the identity mapping for the elements of 5?,(E, F; G) -+ Homc(E gC F, G). Let E', F' be two C-modules and u: E -+ E', v: F --f F' two C-linear map- pings; then (no. 4) u @ v is a C-linear mapping of E & F into E' & F'. Further, it is immediate that (u, v) H u @ v is a C-bilinear mapping of Homc(E, E') x Homc(F, F') into Homc(E BC F, E' BC F'); hence there canonically corresponds to it a C-linear mapping, called canonical : (1 1) Hom,(E, E') gC Hom,(F, F') -+ Homc(E gC F, E' gC F') which associates with every element u @ v of the tensor product Homc(E, E') BC Hom,(F, F') the linear mapping u @ v. Note that the canonical mapping (1 1) is not necessarily injective nor surjective (4 4, Exercise 2). Remarks. (1) Let A, B be two commutative rings, p : B -+ A a ring homomorph- ism and E and F two A-modules; then the canonical mapping (6) of no. 3 is a B-linear mapping (12) p*(E) @ p*(F) --f p*(E @A F)* (2) What has been said in this no. can be generalized to the following case: let E be a right A-module, F a left A-module, C a commutative ring and p: C -+ A a homomorphism of C into A such that p(C) is contained in the centre of A (cf. 111, 3 1, no. 3). We may then consider the C-modules p*(E) and p*(F) and the hypothesis on p implies that these C-module structures are compatible respectively with the A-module structures on E and F (4 1, no. 14). The tensor product E @A F is thus (by virtue of no. 4) given two C-module structures such that y(x @ y) = (xp(y)) @ y and Y(X@Y) = x @ (P(Y)Y) respectively for y E C, x E E, y E F and Definition 1 (no. 1) shows also that these two structures are identical. If E' (resp. F') is a right (resp. left) A-module and u: E -+ E', v: F -+ F' are two A-linear mappings, then u @ v:E @A F -+ E' @A F' is C -linear for the C-module structures just defined; the mapping (u, u) H u @ v: HornA@, E') x HomA(F, F') --f Homc(E @A F, E' @A F') is C-bilinear (for the C-module structures on HomA(E, E') and HomA(F, F') 250 25 1 3 3.6 Now u @ t = (u @ IF“) o (1, @ t) (no. 2, formula (5)) and u €3 t is therefore surjective, being the composition of two surjective homomorphisms by virtue of Proposition 5 and its Corollary. On the other hand, for z E E @ F to be in the kernel of u @ t, it is necessary and sufficient that (1, @ t) (z) belong to the kernel of u @ lF,,, that is, by virtue of (15) to the image of u @ lFn: E’ @ F” -+ E @ F”. PROPERTIES OF E @A F RELATIVE TO EXACT SEQUENCES But as the homomorphism t: F -+ F” is surjective, so is l,, @ t: E’ @ F-+ E’ @F” by the Corollary to Proposition 5, hence the condition on z reduces to the existence of an a E E’ @ F such that (lE @ t)(z) = (’ @ t)(a)* Let b = z - (u @ lF)(a); then (1, @ t)(b) = 0, and by virtue of (17), b belongs to the image of 1, @ s, which proves the proposition. In other words: COROLLARY 1. Let E’ be a submodule of a right A -module E, F’ a submodule of a leJt A-module F and Im(E‘ @A F) and Im(E @A F‘) the sub-Z-modules ofE @A F, the respective images of the canonical mappings E‘ @A F -+ E F, E @AF’-+E @AF. Then there is a canonical Z-module isomorphism (20) such that& t E E/E’, q E F, x( E @ q) is the class of all elements x @ y E E @A F such that x E 5 and y E q. Note that when E is a ((B;) ; A, (C;))-multimodule, F a (A, (Bjl) ; ((2;))- multimodule and E‘ and F’ submultimodules of E and F respectively, the iso- morphism (20) is an isomorphism for the ((Bi), (Bjl); (C;), (Ci))-multi- module structures of the two sides (no. 3). COROLLARY 2. Let a be a right ideal of A, F a lejl A-module and aF the sub-Z- module of F generated by the elements of the form h, where A E a and x E F. Then thme iS a canonical Z-module isomorphism (21) X: (A/a) @A F + F/aF such that, for all X E A/a and all x E F, n(X @ x) is the class mod. aF of Ax, where A E x. In particular, for A = Z, it is seen that for every integer n and every Z- module F, (Z/nZ) @z F is canonically identified with the quotient Z-module F/nF. x :(E/E’) @A (F/F’) -+ (E @A F)/(Im(E’ @A F) + Im(E @A F’)) I1 LINEAR ALGEBRA E” @ F of the form X“ @ y and hence f o g is the identity mapping of E” @ F; on the other hand, for x E E and y E F, f (+(x @ y)) = ~(x) @ y by definition, hence g( f (+(x By))) = +(x @y) and, as the elements of the form +(x my) generate M, g 0 f is the identity mapping of M. COROLLARY . Let F, F’, F“ be lgt A-modules, E a right A -module and F’ F f, F” + 0 (16) an exact sequence of linear mappings. Writing S = 1, @ s, i = 1, @ t, the sequence of Z-homomorphisms i (17) E @AF’ --& E @AF E @AF” d 0 is exact. When E (resp. F) is considered as a left (resp. right) Ao-module, F @A~ E is identified with E @A F and there are analogous identifications for F‘ BAo E and F” BA0 E (no. 1, Corollary 2 to Proposition 1); the corollary then follows immediately from Proposition 5. Remark. Note that in general, if E’ is a submodule of a right A-module E and j: E’ --f E the canonical injection, the canonical mapping j @ 1,: E’ @ F -+ E @ F is not necessarily injective. In other words, for an exact sequence (18) o E’ -L E D_ EM -+ o (19) it cannot in general be concluded that the sequence o--+E*@F-A E~F -2 E”@F-+ o is exact. Take for example A = Z, E = Z, E‘ = 22, F = 2/22. As E‘ is isomorphic to E, E‘ €3 F is isomorphic to E €3 F which is itself isomorphic to F (no. 4, Proposition 4). But for all x’ = 2x E E’ (where ICE E) and all y E F, j(x‘) @ y = (2x) @ y = x @ (2y) = 0, since 2y = 0, and the canonical image of E’ @ F in E @ F reduces to 0. In other words, care must be taken to distinguish, for a submodule E‘ of E and an element x E E, between the element x @ y “calculated in E’ @ F” and the element x @ y “calculated in E @ F” (in other words, the element j(x) €3 y). We shall study later, under the name of Jut modules, the modules F such that the sequence (19) is exact for every exact sequence (18) (Corn- mutative Algebra, I, § 2). PROPOSITION 6. Given two exact sequences (14) and (16), the homomorphism u @ t:E @A F + E” @A F” is surjective and its kernel is equal to Im(u @ IF) + Im(1, @ $1 252 253 I1 LINEAR ALGEBRA COROLLARY 3. Let A be a commutative ring, a an ideal of A and E and F two A-modules such that a is contained in the annihilator of F. Then the (A/a)-modules E @A F and (E/aE) @A,o F are canonically isomorphic. F and E @A F are annihilated by a and hence have canonical (A/a)-modulc structures (9 1, no. 12) and ifwe write E’ = ctE, then Im(E’ @A E’) = 0; then there is a canonical isomorphism (20) of E @A F onto (E/aE) @A F and the latter is itself identical with (E/aE) @A,a I; (no. 3, Corollary to Proposition 2). COROLLARY 4. Let a, b be two ideals in a commutative ring C; the C-module (C/a) @c (C/b) is then canonically isomorphic to C(a + 6). 7. TENSOR PRODUCTS OF PRODUCTS AND DIRECT SUMS Let (EJhoL be a family of right A-modules, (F,,),,, a familyof left A-modules and consider the product modules C = ,EL n E,, D = ,EM n F,. The mapping ((xh), (y,)) ++ (x, @ y,) of C x D into the product Z-module 3 3.7 TENSOR PRODUCTS OF PRODUCTS AND DIRECT SUMS moreover, as the families (x,) and (y,) have finite support, so does (x, @ y,) and hence finally g is a canonical homomorphism is Z-bilinear and obviously satisfies conditions (1) (no. 1). Thus there exists (no. 1, Proposition 1) a Z-linear mapping, called canonical such thatf((x,) @ (Y,)) = (xh @Y,). When C = RL, D = SM, R (resp. S) being a right (resp. left) A-module, the canonical mapping (22) associates with every tensor product u @ v, where u is a mapping of L into R and u a mapping of M into S, the mapping (A, p)i+u(h) @ u(p) of L x M into R gAS; even in this case the canonical mapping (22) is in general neither injective nor surjective (Exercise 3; cf. Corollary 3 to Proposition 7). P I When the E, are ((BE); A, ((2;))-multimodules and the F, (A, (BL); ((2:))- multimodules, the homomorphism (22) is also a homomorphism for the ((B:), (€3;) ; (Ci), (C:))-multimodule structures of the two sides. E, (resp. ,FM F,) of C (resp. D) ; the canonical injections E -+ C, F -+ D define canonically a Z-linear mapping E @J~F--+ C @AD which, composed with the mapping (22), gives a Z- linear mapping g of E @A F into IT A, Ir (EX @A F,) such that Consider now the submodule E = d(.^) 63 (Yu)) = (X^ @!I,); 254 which is a multimodule homomorphism under the same conditions as (22). PROPOSITION 7. The canonical maffing (23) is bijective. To prove this it suffices to define a Z-linear mapping h of the direct sum G = (E, @A F,) into E @A F such that g 0 h and h 0 g are the identity mappings. But, to define a Z-linear mapping of G into E @A F, it suffices (3 1, no. 6, Proposition 6) to define a Z-linear mapping for every ordered pair (A, p), and we take h,, = i, @ j,, where i,: E, + E and j, : F, -+ F are the canonical injections. Then clearly h 0 g coincides with the identity mapping for the elements of the form (7 x,) @ (T y,) which generate the Z-module E @A F; similarly g 0 h coincides with the identity mapping for the elements of the form 2 (x, @ y,) which generate the Z- @ (A. W)E L x M hAu:E,k @AFv-tE @AF module G, since for each ordered pair (A, h, II p) the products xA @ !/, (xA EA, Y, Fu) generate the Z-module E, @A F,. Whence the proposition. Let u,, : E, + EL, vu : F, + F; be A-homomorphisms; clearly the diagram @A J. (p F;) - @ (Ei @AF;) 4 h. u is commutative. CGROLLARY 1. If the lefit A-module F admits a basis (b,),EM, every element of E @A F can be written uniqdely in the form 2 (x, @ b,), where x, E E and the family (x,) hasjnite support. The Z-module E @A F is isomorphic to E(‘) considered as a Z-module. The basis (6,) defines an isomorphism ofF onto @ Ab,, whence there is an isomorphism E @A F -+ O (E @A Ab,) by virtue of Proposition 7; as 255 ,EM ,EM I1 LINEAR ALGEBRA t ++ @, is an isomorphism of A, onto Ab,, x H x €3 b, is an isomorphism of E onto E €3, (Ab,) by virtue of Proposition 4 of no. 4, whence the corollary. If E is a ((B;); A, ((2;))-multimodule, the canonical isomorphism E @, F --f E(’) is a ((B;) ; ((2;))-multimodule isomorphism. In particular, if also E admits a basis (aJhsL, every z E E @A F may be written in one and only one way in the form h. 2 U (ah?,,) €3 b,, where the Eh, belong to A (and form a family of finite support); the mapping is an isomorphism of E €3, F onto A‘L M, for the Z-module structures (and even the module structures over the centre of A). More par - ticularly : COROLLARY 2. IfE and F are two free modules over a commutative ring C and (ah) (resp. (b,)) is a basis of the C-module E (resp. F), then (a, €3 b,) is a basis of the C-module E gC F. By an abuse of language, the basis (a, @ b,) is sometimes called the tensor product of the bases (ah) and (b,). Remark (1). Let E be a free right A-module, F a free left A-module, (ah)heL a basis of E and (b,),EM a basis of F. Every element z E E €3, F may be written uniquely as a, @ y,, where y, E F, and also uniquely as 2 x, @ b,, where x, E E. If we write y, = 2 qh,bU, x, = 7 a,?,,, where the &,, and q,, belong to A, then ch, = q,, for ill (A, p), for ++ (thy)(L, ,) 3 3.7 TENSOR PRODUCTS OF PRODUCTS AND DIRECT SUMS F = A,; but then the two sides are canonically identified with n E, (no. 4, comes the identity. COROLLARY 4. Let A be a ring with no divisor of zero, E a free right A -module and F a free lejl A-module. Then the relation x €3 y = 0 in E @, F implies x = 0 Let (ah) be a basis of E, (6,) a basis of F and let x = 7 ahEh, y = 7 q,b,; then x €3 y = 2 ((ahthq,) @ b, and the relation x €3 y = 0 implies t,q, = 0 for every ordered A, w pairs of indices (A, p) (Corollary 1). Hence, if x # 0, that is Eh # 0 for at least one A, it follows that q, = 0 for all p, whence y = 0. COROLLARY 5. Let E be a right A-module, F a left A-module, M a submodule of E and N a submodule of F. If M is a direct factor of E and N a direct factor of F, the canonical homomorphism M @A N -+ E @, F is injective and the image of M @, N under this homomorphism is a direct factor of the Z-module E @A F. Proposition 4) and after these identifications the canonical mapping ,EL (22) be- or y = 0. This follows immediately from Proposition 7. Note that if E is a ((Bi); A, ((2;))-multimodule and F an (A, (B;I); (C;S))- multimodule and M and N direct factors in these multimodules, M @ N is a direct factor of the ((Bi), (BA); (C;), (C;S))-multimodule E @ F. COROLLARY 6. Let P be a projective l@ A-module and E, F two right A -modules. For every injective homomorphism u : E -+ F, the homomorphism u@ 1,:E @,P+F @*P is injective. Proposition 4) and u @ 1, is identified (Proposition 7) with it therefore suffices to prove the corollary when P is free (4 1, no. 6, Corollary 1 to Proposition 7). The same argument reduces the problem to the case where P = A,, which follows immediately from no. 4, Proposition 4. There exists a left A -module Q such that L = P @ Q is free (4 2, no. 2, (u @ 1P) 0 (u @ la) ; COROLLARY 3. Let jinitely generated free) lt$t A-module. Then the canonical maFping (22) be a family of right A -modules and F a free (resp. is injective (resp. bijective). If (b,) is a basis of F, every element of (a Ex) @, F can be written uniquely as z = 2 ((xp)) €3 b,) (Corollary 1) ; to say that its canonical image is zero means that, for all A E L, c (xp) €3 b,) = 0, hence xy) = 0 for all A E L and all p (Corollary 1) and therefore t = 0. Showing that the canonical mapping is bijective when F admits a finite basis is immediately reduced, by virtue of Proposition 7, to the case where COROLLARY 7. Let C be a commutative ring. IfE and F are two projective C-modules, E I&. F is a projective C-module. This follows immediately from Corollary 5 and the fact that the tensor product of two free C-modules is a free C -module (Corollary 2). Remark 2. Under the hypotheses of Proposition 7, let EL be a submodule of Eh, FL asubmodule of F, and let E’ = @ EL, F’ = @ FI. Let Im(E’ @A F’) 257 ,EM h€L 256 LINEAR ALGEBRA I1 4 3.9 TENSOR PRODUCT OF FAMILlES OF MULTIMODUL,ES (resp. Im(EI, BAF;)) denote the image of E’ BAF’ (resp. EI, BAF;) in E @A F (resp. EA @A F,) under the canonical mapping; then the isomorph - ism (23) identifies the sub-Z-modules Im(E‘ @A F’) and this follows immediately from the commutativity of the diagram where the vertical arrows are the canonical isomorphisms. 8. ASSOCIATIVITY OF THE TENSOR PRODUCT PROPOSITION 8. Let A, B be two rings, E a right A-module, F an (A, B)-bimodule and G a lejit B-module. Then E @A F is a right B-module, F BB G a lt$t A-module and there exists one and only one Z-linear mapping +:(E @AF) @BG+E @A(F @BG) such that +((x @I y) @ t) = x @ (y @ z) for XEE, ~EF, ZEG; moreover this Z-linear mapping is bijective (“associativity” of the tensor product). F and left A-module structure on F BB G have been defined in no. 4. The uniqueness of + is obvious since the elements (x @ y) €3 z generate the Z-module (E @A F) BB G. To show the existence of +, we note that, for all z E G, h,: y H y @ z is an A-linear mapping of the left A-module F into the left A-module F BB G. We write g, = 1 @ h,, which is therefore a Z-linear mapping of E @A F into E @A (F BB G) and con- sider the mapping (t, z) ++ gZ(t) from (E @A F x G into E @A (F & G); as h,,, = h, + h2, for z E G, 2’ E G, it is immediate that the above mapping is Z-bilinear. Further, we show that for all p E B, g,,(t) = gZ(tp) ; it is obviously sufficient to do this for t = x @ y where x E E and y E F; now and The right B-module structure on E gv& @Y) = x @ (Y 63 pz) &((X 63 YIP) = €3 YP) = x 0 (YP 63 2). Proposition 1 (no. 1) then proves the existence of a Z-linear mapping +:(E @AF) @BG+E 63A(F @BG) such that +(t @ z) = g,(t), hence +((x @ y) €3 z) = x 63 (y 63 2). Simi- larly a Z-linear mapping $:E @A (F @B G) -% (E @A F) @B G is defined such that +(x @ (y €3 2)) = (x @ y) @ z and clearly + 0 + and +o II, are the identity mappings of (E QAF) @BG and E @A (F BBG) respectively, since they reduce to the identity mappings on generating sys- tems of these Z-modules. It is immediate that, if E is a ((C;); A, (D;))-multimodule, F an (A, ((2:); B, (D;))-multimodule and G a (B, (Cy); (D:))-multimodule, the canonical isomorphism defined in Proposition 8 is a ((Ci), (C:), ((2:) ; (Di), (Di), (D;))-multimodule isomorphism. In particular, if C is a commutative ring and E, F, G three C-modules, there is a canonical C-module isomorphism (E @cF) @cG+E @c (F @‘cG). We shall see below that, under certain conditions, the definition of tensor product can be generalized to a family of multimodules, which will in par- ticular give us under the hypotheses of Proposition 8 a Z-module E F gB G, which is canonically isomorphic to each of the Z-modules (E @A F) gB G and E @A (F gB G) and with which the latter will be identified. 9. TENSOR PRODUCT OF FAMILIES OF MULTIMODULES Let (GA),eL be a family of Z-modules; a mapping u of the set G = n G, into a Z-module is said to be multiadditive (or Z-multilinear) if (x,) ++ u( AQL (xA)) is additive with respect to each of the variables x,; to be precise, this means that, for all p E L and every element (aA) E n GA, canonically identifying G with ,#I4 Gw x &u (24) u(x, + Y,, (ah)) = u(xfi, + u(Yu, (‘A)) for x~, YIA in Gu* This implies in particular that u((xA)) = 0 if one of the xA is zero. We consider also the universal mapping problem where I; is the species of Z-module structure and the a-mappings are the multiadditive mappings of G into a Z-module. A solution is still obtained by considering the Z-module C = Z‘G) of formal linear combinations of elements of G with coefficients in Z and the sub-Z-module D of C generated by the elements of the form (x, + Y,, (zA)*+~ - (xw (ZA)A+,) - (Y,, (zA)x+J where PEL, x,EG,, ~,EG, and the tA~GA (A # p) are arbitrary. The quotient Z-module C/D is called the tensor product (over Z) of the family (GA)AEL of Z-modules and is denoted by A(L G,; for every element (xJA~L of G which is an element of the canonical basis of C, @ x, denotes the canonical image ?.EL P 259 LINEAR ALGEBRA I1 TENSOR PRODUCT OF FAMILIES OF MULTIMODULES a-mappings the Z-multilinear mappings u of G into a Z-module which fur - ther satisfy the conditions 5 3.9 of his element in C/D. It follows from the above definitions that the mapping 4: (x,) w ,FL x, of G into ,% G, is Z-multilinear and that, for every Z-multi- linear mapping f of G into a Z-module H, there exists one and only one Z-linear mapping g : hEL @ G, -+ H such that f = g o 4; the ordered pair ( AFL Gh, 4) thus resolves the universal mapping problem in question. Let (Gi),EL be another family of Z-modules and, for all h E L, let v,: G, --f GL be a Z-linear mapping (in other words a homomorphism of commutative groups). Then the mapping of G into AqL GE, is Z-multilinear and hence defines canonically a Z-linear map - ping of A% G, into @ GE, denoted by ,’ u, and such that hsL In particular, we consider, for some p. E L, an endomorphism 8 of G,; we denote by 6 the endomorphism of ,FL G, equal to ,FL u, where v, = 0 and u, = l,, for A # p. Then we suppose given a set Q, a mapping c:w - (p(w), .(a)) of R into L x L and, for all w E Q, an endomorphism pw of Gp(w) and an endomorphism qO of Gut,,; there correspond to them two endomorphisms jO and go of P = ,% G,. Let R be the sub-Z-module of P generated by the union of the images Df the endomorphisms fiw - fw when w runs through Q. The quotient Z-module P/R is called the tensor product of the family (G,JAEL relative to c, p, q and is denoted by @ G,; composing the canonical homomorphism P -+ P/R with the mapping +: G -+ ,’ GA defined above, we obtain a Z- multilinear mapping +(c, p, q): G -+ (C. @ P. q) G, and write +(c, p, q)( (x,)) = ,cqq, x, or simply @ x,. The ordered pair consisting of (cFq, x, and +(c, p, 4) resolves the following universal mapping problem: let j,,, (resp. qw) denote the mapping (C. P, 4) (C) (‘P(O), (‘A) A # P(W)) (PO(’P(0)) 9 (‘h)h+ P(W)) (resp. (XU(OD (x,) h # U(WJ (4O(XLT(O,) 9 (4 h # a(oJ 1 of G into itself. Then Z is taken to be the species of Z-module structure and the (26) u OFw = u 0 for all o E Q. The proof is obvious from the above definitions. This construction recovers in particular that of E @A F described in no. 1: in this case we take L = {I, 2}, GI = E, G2 = F, R = A; further, for all o E A, we must have p(o) = 1, U(W) = 2, is the endomorphism x H xw of the Z-module E and qw the endomorphism z/ +> oy of the Z- module F. Let (GL),eL be a second family of Z-modules; keeping the mapping c the same, suppose given for all w E Q, an endomorphism pk of G;,,, and an endomorphism qk of G&w). For all h E L, then let u,:G, + GL be a Z-linear mapping such that, for all w E a, (27) ‘D(W) ‘PW pk ‘P(W) and ’U(W) qW = qk uO(0) (in other words, for all A E L, u, is a morphism for the laws of action on G, (resp. G,) defined on the pc and q,, (resp. pi and 44) with and q such that p([) = h and o(q) = A). Then the mapping of G into @ GL is 2-multilinear and satisfies conditions (26) and hence defines a Z-linear mapping of @ 8, GL, which we shall denote simply by 9 v, if no confusion can arise. We shall now give an “associativity” property for the general tensor products thus defined. Let (Li)lG,.,, be ajnite partition of L; for every index i, let Q, denote the subset of Q consisting of the elements such that p(w) EL, and .(a) EL,; clearly the Q, are pairwise disjoint; we set R’ = Q - (u a{). For each index i, c(*) will denote the mapping w - ( p(w), o(w)) of Q, into L, x L,; for a E Q,, we write@‘:’ and 9:) instead ofpw and qw. Then for each i there is a “partial” tensor product F - @ G,. - (cw,p~~~,,$o) (C. P’. ll’) G, into (C.P 94) (C. P. a) We shall further make the following “permutability” hypothesis : (p) If w E R’, PO (resp. go) permutes with each of the endomorphisms pc and qn of GP(0) (resp. G,(,)) such that 5 $ Q‘, q $ Q’ and p(w) = p(5) = o(q) (resp. =(a) = P(E) = a(?)). 260 261 LINEAR ALGEBRA I1 5 3.9 TENSOR PRODUCT OF FAMILIES OF MULTIMODULES For each w E Q', let i be the index such that p(w) E L,; then consider the family (u,),~~, where up(w) = Po and u, = IGA for A # p(w) ; hypothesis (P) implies that the family (u,) satisfies conditions (27) (where p' and p must be replaced by p('), q' and q by qCi), w by an element 4 running through R,) ; thus an endomorphism,@ u, = ro is derived of the Z-module F,. Similarly, an (C\", endomorphism s, is defined of the Z-module F, starting with q,, j being the index such that G(W) E L,; finally let d(w) = (i,j). Then we can define the tensor product @ F, and the corresponding canonical mapping (d, r. s) n On the other hand, for each i, the canonical mapping +, = +(c(o, p, (I) . n G, -+ F,; using the associativity of the product of sets, a Z-multilinear mapping Jc = +(d,,, s) o (+,) of G into (dqs, Fi is derived. We show that the ordered pair ( @ F,, Jc) is a solution of the same universal problem as ( (C. @ P. 4) G,, +(C,B,9)), whence will follow the existence of a unique Z-module isomorphism )* hfL1 (d. r, s) such that Jc = 0 o +(c,p,q) (Set Theory, IV, 3 3, no. 1). By induction on n, the proof is reduced to the case n = 2; for simplicity we write F, @ F, and y1 @ y, instead of (dFs) Fi and (dqs)yl. Consider the mapping from G to F1 @ F2 (d) (d) (d) It is obviously Z-multilinear; we show that it satisfies conditions (26) for all w E Q. This is obvious if w E SZ, or w E Q,; otherwise, supposing, in order to fix the ideas, that p(w) E L, and ~(w) E L,, the values of h 0 jjo and h 0 4- at (x,) are respectively which are also equal by definition of F1 @ F,. This being so, let u be a Z-multilinear mapping of G into a Z-module H, satisfying conditions (26) ; we shall define a Z-linear mapping v: F, @ (d) F, -+ H (d) such that u = u 0 ,'r and that will prove our assertion (repeating the argument of no. 1, Corollary 1 to Proposition 1). For all z, = (xJAEL2 we consider the "partial " linear mapping of II G, into H (28) u(., -4 :(xdhsL1 ++ ~((XA)~EL~, 22) = ~((xk)hEL). Clearly it is Z-multilinear and satisfies conditions (26) for w E R,; by defini - tion there thus exists a Z-linear mapping y1 - zq(yl, 2,) of F, into H such that AELl (29) W1( ($$'A, .2) = U((XA)h~L1, .2) We next consider the mapping "2:(xA)AsL2 - W1(,, (xA)heL2) of hg2 G, into Hom,(F,, H) ; it is obviously Z-multilinear and satisfies conditions (26) for w E R,, by virtue of the hypothesis on u and relations (28) and (29) and taking account of the fact that the elements of the form @ xh generate the Z-module F,. Hence there is a Z-linear mapping w,:F, -+ Hom,(F,, H) such that ~z( @ ~k) = u~((~JAEL~) (CW or also (30) (%( (9) 'A))( (9) 'A) = u((xA)kEL). We now consider, for y1 E F,, y, E F,, the element of H (31) W(Y1,Yz) = (WZ(YZ))(Yl). (32) 4ro(Y1),!/2) = W(Y1, &/2)). Clearly w is a Z-bilinear mapping of F, x F, into H. We show further that, for all w E Q', (assuming, to fix the ideas, that p(w) E L, and .(a) E L,) It suffices to verify this relation when y, (resp. yz) is of the form @ x, (resp. (8 x,), since these elements generate the Z-module F, (resp. F,). But definition, To( (8 %A) = ((8) x;, where xL(,) = po(xo(,,) and x; = x, for (C(1)) (CW) (C(1)) 262 263 I1 LINEAR ALGEBRA 3 3.9 TENSOR PRODUCT OF FAMILIES OF MULTIMODULES When we speak of the tensor product (34) as a C-module, we shall always mean with this structure, unless otherwise mentioned, and the tensor product (34) is also denoted by 8 E, if no confusion can arise. For every C-module G, the Z-multilinear mappings of n E, into G which, for every index i, 16iGn n satisfy the relation I=, f(x1, . . . > xi - 1, YX,, xi t 1, . . * , xn) = yf(x1, . . . , ~n) (35) for y E C and (x,) E El are then called C-multilinear and form a C-module denoted by 9,(E1, . . . , En; G) ; the universal property of the tensor product (34) then allows us to define a canonical C-module isomorphism (36) =%(El,. . .> En; G) + HO~C(E, @C E2 @ . . CZJC En, G) I which associates with every C-multilinear mapping f the C-linear mapping g such that I ~ f(~1,. * .J xn) = g(x, @ x2 @ . . . @ Xn). A C-multilinear mapping of El x . . . x En into C is also called an n-linear Let (Fl)lGis, be another family of C-modules; for every system of n C- linear mappings u,: E, -+ F,, u1 @ u2 @ . . . @ u, (also denoted by (8 ul) form. i I is a C-linear mapping of ldisn El BC E2 @ . + + BC En into F, BC F, @ . . RC F,. 1 # p(w) in L,; similarly so( (8 xh) = (C(2)) (8 XI, where = qw(&w,) and xz = xh for A # G(W) in L,; using (30) and (31), relation (32) then follows from (26). Hence there exists a Z-linear mapping u of F, @ F, into H such that u(y, g)y2) = w(y,,y2) and it then follows from (30) and (31) that oh = U. The most important special case of the general tensor product defined above is the following: we start with a family of rings and a family (E,) , ,,, where El is a riglit A,-module, En is a left A, - module and for 2 < i < n - 1, E, is an A,)-bimodule. Then the above defini - tion is applied as follows: L is the set (1, n), Gi = E,, SZ is the set the sum of the A, (1 < i < n - 1). For w €A, (1 < i < n - l), take p(w) = i, G(W) = i + 1, pw is the endomorphism x H xw of the Z-module E, and qo the endomorphism y H wy of the Z-module Ei + , ; the corresponding tensor product is denoted by (CW (d) (33) El @A1 E2 @AZ E, €3 *. . @A,-z En-, En (a notation where the A, may occasionally be suppressed) and the elements 8 xi of this tensor product, for a family (x,) such that x, E E, for 1 < i < n, may be written x, @ x, @ . . . @ x, if no confusion can arise; an analogous notation is used for a Z-linear mapping @ uI. Hypothesis (P) holds for every partition of (1, n), since the E, are bimodules for 2 < i 6 n - 1. When n = 3, we have thus defined the Z-module E @A F @= G alluded to in no. 8, and re- covered Proposition 8 (no. 8). When each of the E, is a multimodule (with, for 2 < i < n - 1, A, - , one of the rings operating on the left and A, one of the rings operating on the right and analogous conditions for i = 1 and i = n), as in no. 4, a multimodule structure is defined on El BA1 E2 @ . . . @An-l En with respect to all the rings except the A, which operate on the E, (1 < i < n). a family of C-modules. By giving El and En two C-module structures identical with the given struc - ture and E, for 2 < i < n - 1 three C-module structures identical with the given structure, we define on the tensor product n C-modules structures which are compatible with one another and whichare in fact identical, since for all y E C and (xi) E a E,, by definition (C.P.P) (C) In particular, let C be a commutative ring, (E,) , , (34) El @c E2 €3c E, @ . * @c En-, €3~ En n (yx,) @ x2 €3 * * . @ xn =x,@(yx,)@ *.-@xn= =x,@x2€3**. @(yx,). Further, (ul,. . ., u,) H u, @ u, @ . . @ u, is a C-multilinear mapping of Homc(E,, F,) into I Homc(El @c E €3 . * * @c En, F, €3~ F2 @ * * @c Fn). Hence there corresponds canonically to the latter mapping a C-linear mapping called canonical (37) I Homc(E,, FJ @c Homc(E,, F2) €3 . . . @c Homc(En, FJ -+ Homc(El @c E, @ . * . @c En, F, @c F2 @ . * * @c Fn) The general associativity property seen earlier can be specialized here as E,, €3 . . . @o EL,, where (i,, . . ., ir) is 265 I generalizing that defined in no. 5 for n = 2. k, Fk be the tensor product Ei, I follows. Given a partition (Jk)lakhm of the interval (1, n) of N, let, for each 1 264 LINEAR ALGEBRA I1 the strictly increasing sequence of elements of Jk. Then there is a canonical C-module isomorphism (called “associativity isomorphism”) Fl@cF2 @ ... @cFrn+E1@cE2@ ... @cEn which, in the above notation, maps the tensor product y1 By2 @ . . . Bym, where yk = xi1 @ xi2 63 *. . @ xt,, to the tensor product x1 @ x2 @ . . . @ x, (where xi E E, for all i). we obtain a canonical (“commutativity”) isomorphism In particular, ifsc is apermutation of (1, n), writing Jk = {~(k)) for 1 < k < n, E,(i) @c En(,) @ . . . @C Ex(n) + El @C E2 8 * . . @C En which maps xnc1) @ x,(~) 63 . . . @ xn(,,) to x1 @ x2 @ . . . @ x,. We shall often identify the various tensor products which correspond to one another under these canonical isomorphisms. For 1 < i < n, suppose that E, admits a basis (b!:)kieLi; by induction on n, it follows from no. 7, Corollary 2 to Proposition 7 that the family (by: @ b‘,”,‘ @ . .. @ b:;), where (Al,. . ., A,) runs through 1Bt4n n L,, is a basis of @ Et, sometimes called the tensor product of the bases (b(hll)) in ques- l<l<n fj 4.1 THE ISOMORPHISMS tion. Remarks. (1) The above remarks concerning the case of modules over a commutative ring generalize as in no. 5, Remark 2 when there is a tensor product El BA1 E, @ . . . @An-l En where the rings Ai are not necessarily commutative and where, for each i, there is a homomorphism pi: C + A of the same commutative ring C such that: (1) p,(C) is contained in the centre of Ai; (2) for 2 < i < n - 1, the C-module structures on E, obtained using the homomorphisms and p, coincide. Then a C-module structure is obtained on El BA1 E, @ . . @An-l E, and canonical mappings analogous to (13) (no. 5), which will be left to the reader to describe. (2) Let A, B be two rings, E a right A-module, E’ a left A-module, F a right B-module and F‘ a left B-module. The Z-bilinear mappings of (E @A E‘) x (F @, F’) into a Z-module G are then in one-to-one correspondence with the Z-multilinear mappings f of E x E‘ x F x F’ into G satisfying the conditions xA, x‘, y, y’) = f(X, h‘, Y, y’) f (x, x’, yp, Y’) = f(X, x’, Y> w’) (38) for A E A, p E B, x E E, x’ E E’, y E F, y‘ E F’. The general constructions given in this no. reduce the proof of this to defining a canonical Z-module isomorph- ism between (E BAE’) & (F C3,F’) and E @AE’ &F @BF’, which follows from the associativity property of tensor products of the form (33). 3 4. RELATIONS BETWEEN TENSOR PRODUCTS AND HOMOMORPHISM MODULES 1. THE ISOMORPHISMS Hom,(E F, G) + HomA(F, Hom,(E, G)) AND Horn& @A F, G) + HomA(E, Hom,(F, G)) Let E be a right A-module, F a left A-module, G a Z-module and H the Z-module of mappings f : E x F --f G which are Z-bilinear and satisfy f (xh, y) = f (x, Ay) for x E E, y E F, A E A. (1) It has been seen (5 3, no. 1, Proposition 1) that there exists a canonical Z- module homomorphism H -+ Homz(E @A F, G). (2) On the other hand, a left A-module structure has been defined on Homz(E, G) and a right A-module structure on Hom,(F, G) (5 3, no. 3); we may therefore consider the Z-modules HomA(E, Hom,(F, G)) and Hom,(F, Hom,(E, G)). A mapping f of E x F into G is canonically identi- fied with a mapping of E into the set GF of mappings of F into G (Set Theory, 11, 3 5, no. 2) ; by expressing the fact that the latter mapping belongs to HomA(E, Homz(F, G)), we obtain precisely the fact thatfis biadditive and conditions (1) ; whence there is a canonical isomorphism H --f Hom,(E, Hom,(F, G)) (3) (4) and similarly there is defined a canonical isomorphism H + HomA(F, Hom,(E, G)). Suppose now that E and G also have left (resp. right) B-module structures and that the A-module and B-module structures on E are compatible. Then E @A F has canonically a left (resp. right) B-module structure (5 3, no. 4) and on the other hand Hom,(E, G) has canonically a left A-module structure (5 1, no. 14). We may therefore consider the Z-modules Hom,(E @A F, G) and Hom,(F, Hom,(E, G)), which are submodules of Hom,(E BA F, G) and HomA(F, Homz(E, G)) respectively (3 2, no. 1, Theorem 2). We ex- amine under what condition a mapping f E H has as image under the iso- morphisms (2) and (4) an element of Horn,(E @A F, G) and an element of Hom,(F, G)) respectively; in each of the two cases we find the Same condition f (Px, y) = Pf (x, y) (resp. f (4, Y) = f (x, y) B) for x E E, y E F, p E B. 266 267 LINEAR ALGEBRA I1 Similarly, suppose that F and G are left (resp. right) C-modules and that the A-module and C-module structures on F are compatible. Then, for a mapping f EH to have as image under (2) or (3) an element of Homc(E @A F, G) or HornA@, Hom,(F, C)) respectively, it is necessary and sufficient that it satisfy the same condition f (x, YY) = Yf(X,Y) (resp-f(x, YY) = f(X, Y)Y) for x E E, y E F, y E C. We have therefore established the following result (in the notation intro - duced above) : PROPOSITION 1. (a) Let E be a (B, A)-bimodule, F a Zdt A-module and G a lt$t B-module. For every mapping g E HomB(E @A F, G), let g' be the mapping of F into Hom,(E, G) deJined by (g'(y))(x) = g(x C3 y) for x E E, y E F. The mapping -. g H g' is an isomorphism (5) p :Hom,(E gA F, G) -+ HomA(F, HornB@, G)). (b) Let E be a right A -module, I: an (A, C)-bimodule and G a right C -module. For every mapping h E Hom,(E @A F, G), let h' be the mapping of E into Homc(F, G) d&ed by (h'(x))(y) = h(x @ y) for x E E, y E F. The mapping h H h' is an isomorphism (6) y:Hom,(E @A F, G) --f HomA(E, Hom,(F, G)). ring A; then for every I?-module G, the three I'-modules Homr(E @A F, G), Homr(F, G)), Ho~A(F, Homr(E, G)) are canonically isomorphic to the I?-module of I?-bilinear mappings of E x F into G which satisfy (1). More particularly: COROLLARY . If C is a commutative ring and E, F, G three C-modules, then the C-modules In particular B and C may be taken to be a subring I? of the centre of the Hornc@ gC F, G), Horn#, Homc(E, G)), Horn@, HomdF, G)), %(E, F; G) are canonically isomorphic. 2. THE HOMOMORPHISM E* @A F -+ HomA(E, F) Let A, B be two rings, E a left A-module, F a left B-module and G an (A, B)- bimodule. The Z-module HomA(E, G) has canonically a right B-module struc- 268 I I THE HOMOMORPHISM E* F ---f Hom,(E, F) 9 4.2 ture (0 1, no. 14) such that (up)(.) = u(x)p for p E B, u E HomA(E, G), x E E. On the other hand, G BB F has canonically a Zgt A-module structure (9 3, no. 4). We shall define a canonical 2-homomorphism v:HomA(E, G) @B F + HOmA(E, G @B F). ( 7) To this end, we consider, for all y E F and all u E HomA(E, G), the mapping ~'(u, y) :x H ~(x) @ y of E onto G @B F. It is immediately verified that v'(u, y) is A-linear and that v' is a Z-bilinear mapping of HornA@, G) x F into HomA(E, G gB F) ; moreover, for all p E B, v'(up, y) and v'(u, py) are equal, for (u(x)P) @ y = u(x) @ (py). We conclude (9 3, no. 1, Proposition 1) the existence of the desired homomorphism v such that v(u @ y) is the A-linear mapping x ++ u(x) @ y. It is immediately verified that, if E is an (A, ((2:); (D;))-multimodule, F a (B, (C;) ; (D[))-multimodule and G an (A, (Cy) ; B, (DG))-multimodule, the mapping (7) is a ((Di), (C;), (Cy); (Ci), (D[), (Dz))-multimodule homo - morphism. PROPOSITION 2. (i) When F is a projective (resp. JiniteZy generated projective) B-module, the canonical homomorphism (7) is injective (resp. bijective). (ii) When E is a jinitely generated projective A-module, the canonical homomorphism (7) is bijective. (i) Fixing E and G, for every left B-module F, we write T(F) = HOmA(E, G) @B F, T'(F) = HornA@, G @B F); for every left B-module homomorphism u:F -+ F', we write T(u) = 1 @ u (1 here denoting the identity mapping of Horn,(E, G)), T'(u) = Hom(l,, 1, @ u); on the other hand we write vF instead of v. Then we have the following lemmas: &ma 1. For every homomorphism u : F -+ F', the diagram T(F) & T'(F) (8) IT'(u1 T(F') --+ T'(F') VF' is commutative. The verification is immediate. 269 I1 LINEAR ALGEBRA Lemma 2. Let M, N be two supplementary submodules in F and i:M -+ F, j:N --f F the canonical injections. The diagram T(M) @ T(N) - T'(M) @ T'(N) T(i) +T(j) VM@VN T'(i) +T'(j) I 'JF T'P) I (9) T(F) is commutative and the vertical arrows are bijective. The commutativity follows from Lemma 1, the other assertions from 3 1, no. 6, Corollary 2 to Proposition 6 and 3 3, no. 7, Proposition 7. Lemma 3. Under the hypotheses of Lemma 2, for vF to be injective (resp. surjective), it is necessary and sujicient that vM and vN be so. This follows from Lemma 2 and 3 1, no. 6, Corollary 1 to Proposition 6. Then Lemma 3, together with 3 2, no. 2, Proposition 4, shows that it suffices to consider the case where F is a free module. But, if (b,) is a basis of F, every element of Hom,(E, G) gB F may then be written uniquely as 2 u, @ b,, where u, E Hom,(E, G) (3 3, no. 7, Corollary 1 to Proposition 7) ; the image of this element under v is the A-linear mapping x ++ 2 uv(x) @ b,; it cannot be zero for all x E E unless u,(x) = 0 for all x E E and all p, which is equivalent to saying that u, = 0 for all p.; hence v is injective. When also F admits aJnite bark, Lemma 3 shows (by induction on the number of elements in the basis of F) that to prove that v is surjective, it suffices to do so when F = B,; but in this case the two sides of (7) are canonically identified with Hom,(E, G) (3 3, no. 4, Proposition 4) and v becomes the identity. (ii) To show the proposition when E is projective and finitely generated, this time we fix F and G and write, for every left A-module E, T(E) = Hom,(E, G) @* F, T(v) = Hom(v, la) @ I,, T'(E) = Hom,(E, G BB F) T'(v) = Hom(v, 1, @ 1,) ; and, for every left A -module homomorphism v: E -+ E', on the other hand, we write vE instead of v. Then we have the two lemmas: Lemma 4. For every homomorphism v : E + E', the diagram T(E') 5 T'(E') 3 4.2 THE HOMOMORPHISM E* 8, F -+ Hom,(E, F) Lemma 5. Let M and N be two supplementary submodules in E and p: E + M, q : E -+ N the canonical projections. The diagram is commutative. 270 T(M) @ T(N) - 'JM @ 'JN T'(M) 0 T'(N) I I is Commutative and the vertical arrows are bijective. They are proved as Lemmas 1 and 2, taking account of 5 1, no. 6, Corollary 2 to Proposition 6, fj 2, no. 1, Proposition 1 and 3 3, no. 7, Proposition 7. The remainder of the proof then proceeds as in (i) and is reduced to the case where E = A,; the two sides of (7) are then canonically identified with G @B F and v becomes the identity. In particular take B = A and G the (A, A)-bimodule ,Ad (5 3, no. 4), so that the right A-module Hom,(E, ,Ad) is just the dual E* of E and (,Ad) 8, F is canonically identified with F (5 3, no. 4, Proposition 4). Homomorphism (7) then becomes a canonical Z-homomorphism 0:E* @A F + Horn,@, F) (11) and 0(x* @ y) is the linear mapping of E into F x CJ (x, x*)y. Remark (1). The characterization of projective A-modules given in 5 2, no. 6, Proposition 12, can also be expressed as follows: for a left A-module E to be projective, it is necessary and sufficient that the canonical homomorphism 0,:E* @, E + Hom,(E, E) = End,(E) be such that 1, belongs to the image of 0,. COROLLARY . (i) When F is a projective (resp. Jnitely generated projective) module, the canonical homomorphism (1 I) is injective (resp. bijective). (ii) When E is a Jinitely generated projective module, the canonical homomorphism (1 1) is bijective. Even when E and F are both finitely generated, 0 is not necessarily sur- jective, as is shown by the example A = 2, E = F = 2/22; the right hand side of (11) is non-zero but E* = 0. On the other hand, examples can be given where E is free, but (11) is neither injective nor surjective (Exercise 3(b)). t 271 11 LINEAR ALGEBRA s 4.3 3. TRACE OF AN ENDOMORPHISM Let C be a commutative ring and E a C-module. The mapping (x*, x) H (x, x*) of E* x E into C is then C-bilinear, since, for all y E C, (yx, x*) = y(x, x*) and (x, x*y) = (x, x*)y; we derive a canonical C-lineay mapping T:E* @,E-+C such that T(X* @ xi = (x, x*) (5 3, no. 5). Suppose now also that E is ajnitely generated projective C-module; the canonical isomorphism (1 1) of no. 2 is then a C-module isomorphism and we can therefore define by transporting the strut- ture a canonical linear form Tr = 7 0 BE1 on the C-module End,(E). For all u E End,(E) the scalar Tr(u) is called the trace of the endomorphism u; every u E End,(E) can be written (in general in an infinity of ways) in the form x H 2 (x, x:)y, where xi* E E* and y, E Ei by virtue of no. 2, Corollary to Tr(u) = 2 (yi, x:) (cf. § 10, no. 11). TRACE OF AN ENDOMORPHISM (16) Proposition f 2; then (17) L By definition, Tr(u + v) = Tr(u) + Tr(v) (18) When E admits a finite basis (eJ, the inverse isomorphism 8-1 of 8 can be found explicitly as follows. Let(e:) be the dual basis of (e,) (9 2, no. 6); for all u E Hom(E, F) and all x = 7 tie, with tt E A, and therefore u = 2 t B(eF @ u(ei)), in other words In particular, if further F = E, it is seen that the image under 0,’ of the identity mapping lE is the element 2 e: @ e,, which is therefore independent of the basis (e,) considered in E. Note on the other hand that when E is a finitely generated projective module the ring structure on End,(E) can be transported by 0,’ to E* @, E; it is immediately verified that, for x, y in E, x*, y* in E*, in the ring End,(E), (13) oE(x* @ x, eE(y* @y) = eE((y*(y, .*>) @ .). Remark (2). Let E be a right A-module; replacing E by E* in (1 1) , we obtain a canonical Z-homomorphism (14) E** @, F --f Hom,(E*, F). On the other hand, there is a canonical A-homomorphism cE: E --f E**, whence there is a Z-homomorphism cE @ 1,:E @, F --f E** @, F; com- posing the latter with the homomorphism (14), we hence obtain a canonical Z-homomorphism (15) 8’: E 8, F -+ Horn,@*, F) such that O’(x @ y) is the linear mapping x* H (x, x*)y. If E and F are projective modules, the mapping (15) is injective. For cE is then injective (4 2, no. 7, Corollary 4 to Proposition 13) and as F is projective, the Z-homomorphism cE @ 1,: E @A F + E** F is also injective (4 3, no. 7, Corollary 6 to Proposition 7) ; finally, it has been seen (Proposition 2) that the homomorphism (14) is injective, whence the conclusion. If E is projective and finitely generated, the mapping (15) is bijective for the two mappings of which it is composed are then bijective (9 2, no. 7, Corollary 4 to I Proposition 13 and Proposition 2 above). ‘I Tr(yu) = yTr(u) for u, v in End,(E) and y E C. Moreover: PROPOSITION 3. Let C be a commutative ring, E, F two finitely generated projective C-modules and u : E -+ F and v: F + E two linear mappings; then Tr(v 0 u) = Tr(u 0 u). (20) The two mappings (u, v) H Tr(u 0 u), (u, v) H Tr(u 0 u) of Hom,(E, F) x Hom,(F, E) into C are C-bilinear; it therefore suffices to verify (20) when u is of the form *I+ (x, a*)b and v of the form y H (y, b*)a, with a E E, a* E E*, b E F, b* E F*. But then v 0 u is the mapping x H (x, a*)(b, b*)a and u 0 v the map- Ping y - (y, b*)(a, a*)b. Formula (1 7) shows that the values of the two sides of (20) are equal to (a, a*)<b, b*). COROLLARY. If ul, . . . , up are endomorphisms of E, then O 4-1) Tr(ul o u2 o . . o up) = Tr(u, o u$+~ o . . . o up o u1 o . . . .h 1 < i < p (“invariance of the trace under cyclic permutation”). 272 I 273 4.4 THE HOMOMORPHISM Hom,(E,, F,) BC Hom(E,, F,) are then both canonically identified with Hom(E,, F,) and, after these identi - fications, it is verified that A becomes the identity. We now suppose that F, is projective and finitely generated; the argument of no. 2 (depending this time on Lemmas 4 and 5) reduces the proof for El any finitely generated projective module to the case where El = C, that is to the first case dealt with. For the ordered pair (El, E,), the procedure is similar, this time applying Lemmas 4 and 5 twice; we leave the details to the reader. I1 LINEAR ALGEBRA It suffices to apply (20) to the product (u, 0 u, 0 *. . 0 Ui-,) 0 (Ui 0 ui+, 0 f 0 up). Note that on the other hand it is not necessarily true that Tr(u o u o w) = Tr(u o w o u) for three endomorphisms u, u, w of E. 4. THE HOMOMORPHISM Homc(El, F,) BC Homc(Ez, F,) -+ Let C be a commutative ring and El, E,, F,, F, four C-modules; in 9 3, no. 5, formula (1 3) we defined a canonical C -module homomorphism (21) h:Hom(E,, F,) €3 Hom(E,, F,) --f Hom(E, €3 E,, F1 €3 F,). PROPOSITION 4. When one of the ordered pairs (El, E,), (El, F,), (E,, F,) con- sists of finitely generated projective C-modules, the canonical homomorphism (2 1) is bijective. It is obviously sufficient to perform the proof for the ordered pairs (El, F,) We consider first the case of the ordered pair (El, F,) ; we fix E,, F,, F, and write for every C-module T(E) = Hom(E, F,) BC Hom(E,, F,) and T’(E) = Hom(E @ E,, F, €3 F,) and, for every C-homomorphism u: E --f E’, + Homc(El @C Ez, FI @C Fz) . and (El, E,). h) are valid and are every C-module F, T(v) = Hom(v, IF,) ‘8 lHom(Ez.Fz) T’(v) = Hom(v €3 l~,, ~F~~FJ- and Then Lemmas 4 and 5 (no. 2) (where v is replaced by proved by completely analogous methods. T(F) = Hom(C, F) BC Hom(E,, F,) and T’(F) = Hom(C @ E,, F €3 F,) and, for every C-homomorphism u : F -+ F’, T(‘) = Hom(lC, @ lHom(Ez,Fz) and T’(u) = Hom(1, €3 I,,, u @ 1F.J. This time it is immediately verified that Lemmas 1 and 2 (no. 2) (where h always We next fix E, and F, and this time write, for replaces v) are valid. This being so, we show the proposition first when El = C and F1 is projec - tive and finitely generated. The argument of no. 2 (which rests on Lemmas 1 and 2), together with the above remarks, reduces this to proving the proposition when also F, = C; then Hom(E,, F,), El @ E, and F1 @ F, are identified with C, E, and F2 respectively (5 3, no. 4, Proposition 4) ; the two sides of (21) Note that when El = C“’, EZ = CCJ’ are free (finitely gcnerated or not), then Hom(E,, F,) = F:, Horn(&, Fz = Fi and C3 Ez, Fi C3 Fz) = (Fi C3 F2)IXJ to within canonical isomorphisms and (2 1) is then identical with a special case of the canonical homomorphism (22) of 4 3, no. 7. When E, = C, the canonical homomorphism (21) gives, after identifying Hom(E,, F,) with F, and El @ E, with El, a canonical homomorphism Hom(E, F) €3 G + Hom(E, F €3 G) (22) for for C-modules E, F, G C. which is just the homomorphism (7) of no. 2 Note that when F = C the canonical homomorphism (22) again gives (1 1) (no. 2) for the case of a commutative ring. Suppose now that F, = F, = C; as F, @ F, is identified with C, there is this time a canonical homomorphism p:E* @ F* -+ (E €3 F)* for two C-modules E, F; for x* E E*, y* E F*, the image of x* @ y* under the canonical homomorphism (23) is the linear form u on E @ F such that (23) u(x @ Y) = (x, x*>(Y,Y*>. (24) Moreover, if El, E,, F,, F, are four C-modules, f :El + E,, g:F, -+ F, two linear mappings, it follows immediately from (24) that the diagram (25) i *I 8 fg is commutative. 274 2 75 _______________j I1 LINEAR ALGEBRA COROLLARY 1. If one of the modules E, F is projective and finitely generated, the canonical homomorphism (23) is bijective. COROLLARY 2. Let El, E, be two finitely generated projective C-modules, u1 an endomorphism of El and u2 an endomorphism of E,; then (26) Tr(ul @ uz) = Tr(ul)Tr(uz). x1 H (xl, xT) yl. and u, of the form x, H (xz, xg) yz; then the image of x1 under u1 @I uz is by definition By linearity, it suffices to consider the case where u1 is of the form x, (x1, xT>(xz, x:>(Yl @ Yz) = (x1@ xz, xT 63 X3(Yl@ Y2) xT @ xg being canonically identified under p with an element of (El @ E2)*. As (yl @ y,, xT @ x:) = (yl, xT)(yz, x:), formula (26) follows in this case from (1 7). Remark. If E, F, G are any three C-modules, it is immediately verified that the diagram w@ 1 E*@F*@G* --+ (E@F)*@G* (27) is commutative, by virtue of formula (24). We note also that, without any hypothesis on the C-modules E, F, there are canonical isomorphisms (E @ F)* 3 Hom(E, F*) (E @ F)* -+ Hom(F, E*) which are just the isomorphism (6) and (5) of no. 1 for G = C, A=B=C. Thus a canonical one -to-one correspondence has been defined between the bilinear forms on E x F, the homomorphisms of E into F* and the homomorph - isms of F into E*: if u (resp. v) is a homomorphism of E into F* (resp. of F into E*), the corresponding bilinear form is given by 276 4 5.1 EXTENSION OF THE RING OF SCALARS OF A MODULE $5. EXTENSION OF THE RING OF SCALARS 1. EXTENSION OF THE RING OF SCALARS OF A MODULE Let A, €3 be two rings and p:A -+ B a ring homomorphism; we consider the right A -module p*(B,) defined by this homomorphism (9 1, no. 13); this A-module also has a left B-module structure, namely that of B, and, as b‘(bp(a)) = (b’b)p(a) for a E A, 6, b‘ in B, these two module structures on B are compatible (4 1, no. 14). This allows us, for every left A-module E, to define a left B-module structure on the tensor product p,(B,) @A E such that p’(p 8 x) = (@’@) @ x for @, p‘ in B and x E E (4 3, no. 3). This left B-module is said to be derived from E by extending the ring of scalars to B by means of p and it is denoted by p*(E) or E(B) if no confusion arises. PROPOSITION 1. For every lejl A-module E, the mapping +:x ++ 1 @ x of E into the A-module p,(p*(E)) is A-linear and the set +(E) generates the B-module p*(E). Further, for every lejt B-module F and every A-linear mapping f of E into the A- module p*(F), there exists one and only one B-linear mappingf of p*(E) into F such thatf( 1 @ x) = f(x) for all x E E. B can be considered as a (B, A)-bimodule by means of p; then there is a canonical Z-module isomorphism Horn,@ @A E, F) -+ Hom,(E, Hom,(B,, F)) as has been seen in 5 4, no. 1, Proposition 1. But the left A-module Hom,(B,, F) is canonically identified with p*(F) : for, by definition (4 1, no. 14), there corre- sponds to an element ~EF the homomorphism B(y): B,+F such that (O(y))(l) = y; for all A EA, there thus corresponds to p(h)y E F the homo - morphism p ++ pp(A)y of B, into F, which is just AB(y) for the left A-module structure on HornB& F) (5 1, no. 14). Using this identification, we obtain therefore a canonical Z-module isomrophism, the inverse of (1) (1) s:Hom~(E, P*(F)) -+ HOm,(P*(E), F) and it follows immediately from the definitions that if 8(f) =J then f(1 @ x) =f(x) for all x E E. In particular, the mapping +E:~ H 1 @I x is just (2) (3) +E = 8-1(1p*CE)). Proposition 1 is therefore proved. The mapping +E : E -+ p* (p* (E)) is called canonical. 277 I1 LINEAR ALGEBRA Remarks. (1) Proposition 1 shows that the ordered pair consisting of E(B) and is a solution of the universal mapping problem (Set Theory, IV, Q 3, no. l), where C is the species of left B -module structures (the morphisms being B-linear mappings) and the &-mappings are the A-linear mappings 3 5.1 C-module C @, B is canonically identified with the C-module C, under the isomorphism which maps y @ @ to yo@) (5 3, no. 4, Proposition 4) and this is0- morphism is also an isomorphism for the right A-module structure on c 8, B defined by p and the right A-module structure on C defined by 0 o p. Thus a canonical isomorphism EXTENSION OF THE RING OF SCALARS OF A MODULE (c @BB) @AE-tC @AE is obtained and, composing it with the isomorphism @B (B @AE) --f (c @BB) @A defined earlier, the desired canonical isomorphism is obtained. If +, 9’ and +” denote the canonical mappings E-t p* (E), p* (E -+) G* (p* (E)) and E -+ (0 0 p)*(E), +’ 0 + is identified with 4’’ under the canonical iso- morphism of Proposition 2. PROPOSITION 3. Let A, B be two commutative rings, p :A -+ B a ring homomorphism and E, E’ two A-modules. There exists one and only one B-homomorphism E(B) @B -+ (E @A ”)(B) (5) mapping (1 @ x) @ (1 @ x‘) to 1 @ (x @ x’) for x E E, x’ E E’, and this homo- morphism is bijective. (B @A E’) and is identified with (E @A B) @, (B @A E’) since A and B are commutative; the latter product is identified successively with E @A (B B) @A E’, E @A (B @A E‘), E @A (E’ @A B) and finally (E @A E’) BAB, using the associativity of the tensor product (4 3, no. 8, Proposition 8), Proposition 4,s 3, no. 4, and the commutativity of A and B. The desired isomorphism is the com- position of the successive canonical isomorphisms. Clearly if S is a generating system of E, the image of S under the canonical mapping E --f E(,) is a generating system of Eo,; in particular, if E is a finitely generated A-module, E(,) is a finitely generated B-module. PROPOSITION 4. Let E be an A-module admitting a basis (aA)AcL; ;f 4: x H 1 @ x is the canonical mapping of E into p*(E), then (+(aA))AEL is a basis of p*(E). If p is injective, so is +. The first assertion follows immediately from 4 3, no. 7, Corollary 1 to Pro - Position 7. Also, for every family (E,),,, of elements of A of finite support, (b(zL !$a,) = 2 p(E,)+(a,) and the relation +( 2 &,a,) = 0 is therefore COROLLARY. For every projective A-module E, the B -module p*(E) is projective. If further p is injective, the canonical mapping OfE into p*(E) is injective. The left hand side of (5) may be written (B @A E) equivalent to p(Sh) ,EL = 0 for all A E L, whence the second AsL assertion. 2 79 of Ernto a B-mod&. ~ (2) If E is an (A, (Ci); (D;))-multimodule and F a (B, (CL); (DI))- multimodule, then the isomorphism (2) is linear with respect to the ((Di, ((2:); (Ci), (D;))-multimodule structures of the two sides (Q I, no. 14 and Q 3, no. 4). (3) Let E be a left A -module, a a two-sided ideal of A and p:A+ A/a the canonical homomorphism. In the notation of Q 3, no. 6, Corollary 2 to Proposition 6, the A-module E/aE is annihilated by a and therefore has canonically a left (A/a)-module structure (3 1, no. 12) ; it is immediate that the canonical mapping x: p*(E) -+ E/aE defined in Q 3, no. 6, Corollary 2 to Proposition 6 is an isomorphism for the (A/a)-module structures. COROLLARY . Let E, E’ be two l$t A-modules; for every A-linear mapping u : E --f E‘, u = 1, @ u is the unique B-linear mapping which renders commutative the diagram 6E - E(B) where +E and +E, are the canonical mappings. It suffices to apply Proposition 1 to the A-homomorphism +E, 0 u: E --f E&). The mapping v defined in the above corollary is denoted by p* (u) or q,). If E” is a third left A-module and v:E‘ --f E” an A-linear mapping, it is I immediate that (’ u)(B) = u(B) u(B). Extending the ring of operators of a module is a transitive operation; to be precise : PROPOSITION 2. Let p : A -+ B, 0: B -+ C be ring homomorphisms. For every l& A-module E, there exists one and only one C-homomorphism (4) C*(P*(E)) -+ (0 O P)*(E) mapping 1 @ (1 @ x) to 1 @ x for all x E E and this homomorphism is bij,,tive. The underlying Z-modules of o* ( p* (E)) .and (0 0 p) * (E) are respectively C @B (B @A E) E. There exists a canonical Z-isomorphism C @, (B @A E) 3 (C @, B) @A E (4 3, no. 8, Proposition 8), which is also a C-isomorphism for the left C-module structures on both sides. Moreover, the 278 and C ) I1 LINEAR ALGEBRA By hypothesis there exists a free A-module M containing E and in which E admits a supplement F. It follows immediately from 9 3, no. 7, Proposi - tion 7 that Mo, is identified with the direct sum of Eo, and F,, and if + and ,+ are the canonical mappings E -+ EcB) and F -+ F(B), the canonical mapping M -+ MCB) is just x + y H +(x) + $(y). The corollary follows immediately from Proposition 4 applied to the A-module M. When E is a right A-module, we write similarly p*(E) = E @A p,(B,), B being considered this time as an (A,B)-bimodule and the right B-module structure on p*(E) being such that (2 @ p)p' = x @ (pp') for p E B' p' E B and x E E. We leave to the reader the task of stating for right modules the results corresponding to those of this no. and the following. Remark (4). Consider the Zeft A-module p*(B,) defined by p and for every left A -module E, consider the Z-module (6) i;(E) = HO~A(P*PA, El. As p,(B,) has a right B-module structure, a 14t B-module structure is derived on g(E) (9 1, no. 14) such that, if u E $(E) and b'E B, b'u is the homo - morphism bH u(bb') of p*(B,) into E. We further define an A-finear mapping, called canonical associating with every homomorphism u E $(E) the element u(1) in E. As B can be considered as an (A, B)-bimodule by means of p, for every left B-module F, there is a canonical Z -module isomorphism (5 1, no. 1, Proposition 1). As the left A-module p*(B,) gB F is canonically identified with p,(F) by virtue of 9 3, no. 4, Proposition 4, we obtain a canonical Z -module isomorphism, the inverse of the above (7) ?:P*($(W) -E HomA(p*(Bs) @B F, HomA(P*(B8), (8) g(E)) - HomA(p*(F), which associates with every B -linear mapping g of F into p(E) the com- posite mapping 71 o g, considered as an A-linear mapping of p*(F) into E. In particular, under the hypotheses of Proposition 2, if F is replaced by o*(C,), we obtain a canonical C-isomorphism (9) WE)) - (. O P)"(E). 2. RELATIONS BETWEEN RESTRICTION AND EXTENSION OF THE RING OF Let p :A -+ B be a ring homomorphism. For every left A-module E, a canonical A-linear mapping SCALARS (10) +E:E -+ p*(FJ*(E)) was defined in no. 1 such that +E(x) = 1 @ x. We consider now a left B- RELATIONS BETWEEN RESTRICTION AND EXTENSION OF RING OF SCALARS 9 5.2 module F and apply Proposition 1 (no. 1) to the A homomorphism lP+(F): p* (F) -+ p*(F) : we obtain a B-linear mapping (11) +F: P*(P*(F)) -+ F equal to 8(lPacFJ and such therefore that, for all y€F and all p EB, $F(P = py. PROPOSITION 5. Let E be a 1eJt A-module avd F a lej5t B-module; the comfosite mappings (12) P*(E) P*(P*(P*(E))) -JIO'o- P*W are respectively equal to the identity mappings of p* (E) and p* (F), We give the proof, for example, for (1 2) ; for all x E E, the mapping p* (c$~) maps 1 @ x to the element 1 @ (1 @ X) and the mapping +P*(E) maps 1 @ (1 @ x) to the element 1 @ X; the conclusion follows from the fact that the elements of the form 1 @ x generate the B-module p*(E); the proof is even simpler for (1 3). COROLLARY . The mappings p* (+E) and +P*(F, are injective and respectively identqy p*(E) with a direct factor of p*(p,(p*(E))) and p,(F) with a direct factor of P*(P*(P*(F)))* This is a consequence of Proposition 5 and 3 1, no. 9, Corollary 2 to Proposition 15. PROPOSITION 6. Let E be a ldt A-module and F a right B-module. There exists one and only one Z-homomorphism (14) p*(F) @A @B p*(E) mapping y @ x to y 8 (1 @ x) for all x E E and all y E F and this homomorphism is bijective. By definition the right hand side of (14) is F gB (B @A E), where B is considered as a (B, A)-bimodule, and there is a canonical Z-isomorphism (F QB B) @A E -+ F BB (B @A E) defined in 3 3, no. 8, Proposition 8; on the other hand, the canonical isomorphism F -+ F BB B of fj 3, no. 4, Proposition 4 is an isomorphism for the right A-module structures on the two sides, defined by p. Whence the desired isomorphism. When A and B are commutative, the isomorphism (14) is an A-module isomorph- ;sm p*(F) @A --f p*(F @B p*(E)). 281 280 I1 LINEAR ALGEBRA 3. EXTENSION OF THE RING OF OPERATORS OF A HOMOMORPHISM Let A be a commutative ring, B a ring, p :A + B a ring homomorphism and E, F two A-modules; as B is an (A, A)-bimodule (by means of p) and F can be considered as an (A, A)-bimodule, there are on the Z-module B gA F two A-module structures, under which respectively a(b @ y) = (p(a)b) @ y and a(b @ y) = b @ (ay) for a E A, b EB, y E F. We shall denote the two A-modules thus defined by G' and G"; G' is moreover just the A-module This being so, in the definition of the canonical homomorphism of 3 4, no. 2, formula (7), we replace B by A, the B-module F by the ring B considered as an A-module by means of p and G by F considered as an (A, A)-bimodule; as A is commutative, we may write the canonical Z-homomorphism obtained as MODULE P* (P* (F) ) * 3 5.4 DUAL OF A MODULE OBTAINED BY EXTENSION OF SCALARS (i) If B is a projective (resp. jinitely generated projective) A-module, the homomorph - (ii) If E is a jinitely generated projective A-module, the homomorphism (1 7) is As (16) is bijective, the proposition follows from 3 4, no. 2. Proposition 2, ism (1 7) is injective (resp. bijective). bijective. applied to the canonical homomorphism (1 5). 4. DUAL OF A MODULE OBTAINED BY EXTENSION OF SCALARS Let A, B be two rings, p: A -+ B a ring homomorphism, E a left A -module and E* its dual. We shall define a canonical B-linear mapping uE: (E*)(B) --f (E(B))*. (19) The left hand side of (19) may be written as HomA(E, A) @A p*(B,), where, in Hom,(E, A), A is considered as an (A, A)-bimodule. Then there is a canonical Z-homomorphism (3 4, no. 2, formula (7)) V:HomA(E, A) @A p*(Bs) --f A @A p*(Bs)) = P*(Bs)) with the identification given by the canonical isomorphism of 3 3, no. 4, Proposition 4. On the other hand, the right hand side of (19) may be written as HomB(p,(B,) @A E, B,); as B is a (B, A)-bimodule, there is a canonical Z-isomorphism (4 4, no. 1, Proposition 1) P:HomB(p*(B,) @A E, Bs) --f HomB(BS, Bs)) and HornB@,, B,) is canonically identified, as an A-module, with p,(B,) (see the proof of no. 1, Proposition 1). Taking account of these identifi - cations, we obtain the homomorphism uE; it is easily verified that this homo - morphism is characterized by the equation (20) (E @ x, uE(x* @ 3)) = EP((~, x*))?, for x E E, x* E E*, E,, q in B, which shows immediately that uE is B-linear. More - over, for every A -linear mapping u: E -+ F the diagram B @A HomA(E, F) + HornA@ G"). On the other hand (no. 1, formula (2)), there is a canonical Z-isowlorphism (16) Horn,(& G') = HO~A(E, P*(P*(F))) +H~~B(P*(E), P*(F)). Suppose now that p(A) is contained in the centre of B, in which case p is also called a central homomorphism * (or p is said to define an A-algebra struc - ture on B, cf. 111, 5 1, no. 3)*. Then the A-module structures of G' and G" are identical and composing the homomorphisms (1 6) and (1 5) we thus obtain a canonical Z -homomorphism (17) a:B @A HomA(E, F) -+ HO~BRB), F(B)) which is characterized by the fact that, for all u E HomA(E, F) and all b E B (18) a(b @ U) = rb @ u, where rb denotes right multiplication by b in B. Moreover, the hypothesis that w is a central homomorphism implies that (bb')p(a) = bp(a))b' for b, b' in B and a E A; in other words the right B-module structure of B, is compatible with its A-module structure; it thus defines on B @, HomA(E, F) a right B-module structure (3 3, no. 4) and also on F(B) = B @A F, and finally, as the left and right 13-module structures on F(B) are compatible, a right B-module structure is also obtained on HOmg(E(B), F(B)) (3 1, no. 14). Then it is immediately verified that (17) is a right B-module homomorphism for these structures. PROPOSITION 7. Let A be a commutative ring, 13 a ring, p:A +- B a central homo- morphism and E, F two A-modules. PROPOSITION 8. If one of the A-modules E, p* (B,) is projective andfinitely generated, the homomorphism uE is bijective. This follows from the above and 5 4, no. 2, Proposition 2. 282 283 I1 LINEAR ALGEBRA Suppose in particular that E is a jnitely generated free A-module and let <,<,, be a basis of E and (e?) the dual basis; then the canonical isomorph - ism (19) maps the basis (e: @ 1) of (E*)(B) to the dual basis of the basis (1 €9 8,) of E,,,. 5. A CRITERION FOR FINITENESS PROPOSITION 9. Let B be a ring, A a subring of B and P a projective ldt A-module. Then, if P,, is a jnitely generated B-module, P is itself a Jinitely generated A-module. We know ($2, no. 6, Proposition 12) that there exists a family (ah)hsL of elements of P and a family (a:)heL of elements of the dual P* such that, for all x E P, the family ((x, a:)) is of finite support and x = c (x, a:)ah. Since P(B) is finitely generated, there exists a finite family ( y,), I of elements of P such that Po, is generated by the elements 1 @ y,. For each index i, the family ((y,, a:)) has finite support. Hence there exists a finite subset H of L such that ( y,, a,*) = 0 for i E I and A $ H. Since (1 c3 yt, Id, 8 a,*> = (Yl, a,*>, it follows that Id, @I a: = 0 for A $ H. Hence, for all x E P, (x, a:) = (I @ X, 1, @ a:) = 0 for A # H. This shows that the A-module P is generated by the ah such that AEH. INVERSE LIMITS OF MODULES Q 6.1 shall say that (E,, fa,) is an inverse system of l$t A,-modules and that the A- module E is its inverse limit. Let (Eh,fdo) be another inverse system of left & -modules and, for all a, let u,: E& 4 E, be an A,-linear mapping, these mappings forming an inverse system; then u = lirn u, is an A-linear mapping of lim E& into lim E,. t t c Moreover: PROPOSITION 1. Let (E,, fa,), (EL,f&,), (El, flD) be three inverse systems of A,- modules and (u,), (va) two inverse systems ofA,-linear mappings such that the sequences US 0 - Eh E, --f E: are exact for all a. Then, writing u = lirn u,, v = lim u,, the sequence t- c o + lim~; ft, lim~, --L lim~: t. t t is exact. As ul'(0) = (0) for all a, it follows from Set Theory, 111, 5 7, no. 2, Proposi - tion 2 that u-l(O) = {0}, hence u is injective; further, the u,(EL) form an inverse system of subsets of the E, and thus u(1im E&) = limu,(Ek). As u,(E@ = ul'(0) by hypothesis, v-l(O) = lirn u,(E&) = u(1im E&) (Set Theory, 111, 3 7, no. 2, Proposition 2), which completes the proof. t t c t $6. INVERSE AND DIRECT LIMITS OF MODULES Throughout this paragraph, I will denote a non-empty preordered set and a 6 p the preordet relation on I. Unless otherwise mentioned, the inverse and direct systems have indexing set I. 1. INVERSE LIMITS OF MODULES Let (A,, +,,) be an inverse system of rings (I, 5 10, no. l), (E,,f,,) an inverse system of commutative groups (written additively) (I, 3 10, no. 1) and suppose that each E, has a ldt Admodule structure; moreover suppose that for a < p (fa,, +,,J is a dimorphism of E, into E, (Q 1, no. 13), in other words that (1) fa,(ht.x,) = +a,(ht.)fat.(Xo) , for xp E ED, h, E A,; then it follows from I, 3 10, no. 2 that E = lirn E, has a Zejit module structure over A = lim A,. For all a E I, letf: E --+ E,, +,:A + A, be the canonical mappings; then (fa, +,) is a dimorphism of E into E,. We t t 284 2 Remarks. (1) Proposition 1 and its proof are valid for arbitrary groups, except for change of notation. (2) Note that if there are exact sequences 0 + Ek % E, E: -+ 0 it does not necessarily follow that the sequence U o - lim E& --+ lim E, -L lim E: --+ o is exact; in other words, the inverse limit of an inverse system of surjective linear mappings is not necessarily surjective (cf. Exercise 1). t t t Suppose now that the A, are equal to the same ring A and the br*, to 1,; then for every inverse system (Ea,fao) of A-modules, E = lirn E, is an A- mapping such that (u,) is an inverse system of mappings; then u = Emu, module. Let F be an A-module and, for all a, let u,: F -+ E, c be an A-linear is an A-linear mapping of F into lirn E,. Conversely, for every A -linear mapping t D:F+limE,, the family of u, t =fa o u is an inverse system of A-linear + 285 LINEAR ALGEBRA I1 3 6.2 let u,: Ek --f E, be an A,-linear mapping, these mappings forming a direct system; then u = lim u, is an A-linear mapping of lirn Ek into lim E,. PROPOSITION 3. Let (E,, fBa), (EL, f;,), (E:, f;,) be three direct systems of A,- modules and (u,), (v,) two direct systems of A,-linear mappings such that the sequences EL --!% E, EE DIRECT LIMITS OF MODULES Moreover: + 3 4 are exact for all c(. Then, writing u = lirn u,, v = lirn v,, the sequence 4 ----f lim E; A lim E, --L lim E: + -+ 3 is exact. u(lim E;) = lim u,(EL) and v-I(O) = lim vi '(0) (Set Theory, 111, 5 7, no. 6, Loosely speaking, Proposition 3 can also be expressed by saying that passing to the direct limit preserves exactness. PROPOSITION 4. Let (E,, fBa) be a direct system of A,-modules, E = lirn E, its rf, for all a E I, S, is a generating system of E,, then S = ,VI f,(S,) is a generating system of E. Every x E E is of the formfa(xa) for some a E 1 and some x, E E, and by hypothesis x = 7 ADf'y:', where h:) E A, and y&" E S,; writing A(!) = +,(A:)), y"' =f,(y!'), we obtain x = 2 h"h~(~). PROPOSITION 5. With the hypotheses and notation of Proposition 4, suppose that for all a E 1, E, is the direct sum of a family (M,")aEL of submodules (the indexing set L being independent of a) and that f,,(M;) c M: for a < p and for all h E L. Then E is the direct sum of the submodules Mh = lim M,h (A E L). Corollary --+ to Proposition 4 7). --+ direct limit and +,:A, -+A and fa:E, --f E the canonical mappings for + all a E I. 2 It follows from Proposition 4 that E is the sum of the MA. Let (yh)heL be a family such that yh E MA for all A E L and whose support is finite and suppose that yh = 0. By virtue of Set Theory, 111, 5 7, no. 5, Lemma 1, there exist an a E I and a family (x,"),,, of finite support consisting of elements of E, such that x," E M," and yh = fa(x,") for all h E L. The relation fa( c x,") = 0 implies the existence of a 2 a such that f,,(ZLx,") = 0 (Set Theory, 111, 287 AeL mappings such that v = lim v,. We note on the other hand that for cc < mapping Hom(lF,fap) = fal,:HomA(F, ED) + HomA(F, Ea) is a Z-module homomorphism such that (Hom,(F, Ea),faP) is an inverse system of Z-modules; as faD(vD) = fap 0 up, the above remarks can therefore be expressed as follows : PRO~OSITION 2. For every inverse system (E,, fap) of A-modules and every A-module F, the canonical mapping u I-+ (fa 0 u) is a Z-module isomorphism (2) lF:HomA(F, lim c- E,) -+ lirn f- Hom,(F, E,). COROLLARY . For every A-module homomorphism v:F --f F', the the t g, = Hom(v, lEU) : Hom(F', E,) + Hom(F, E,) form an inverse system Z-linear mappings and the diagram IF' Hom(F', lirn E,) --+ lirn +-- Hom(F', E,) c lim i, i- IF f- (3) Hom(u, 1~) Hom(F, lim E,) --+ lirn Hom(F, E,) +-- is commutative. For all u E Hom(F', lim E,), IF(u o v) = (fa 0 u 0 v) by definition and the f- commutativity of diagram (3) follows immediately from the definitions. 2. DIRECT LIMITS OF MODULES Henceforth I is assumed to be right directed. Let (A,, +oa) be a direct system of rings (I, 5 10, no. 3), (E,, foa) a direct system of commutative groups (written additively) (I, 3 10, no. 3) and sup- pose that each E, has a lej5t A,-module structure; further, suppose that, for cc < @, ( f,,, is a dimorphism of E, into E, (3 1, no. 13), in other words that (4) foa(haxa) = +,,(ha) fDa(Xa) for x, E E,, h, E A,; then E = lim E, has a le$t module structure over A = lirn 4. A, (I, 3 10, no. 4). For all a E I, let fa:E, + E, +,:A, -+ A be the canonical mappings; then (fa, 4,) is a dimorphism of E, into E. We shall say that (E,, fpa) is a direct system of lej5t A,-modules and that the A-module E is its direct limit. Let (EL, f;,) be another direct system of left & -modules and, for all a, + 286 LINEAR ALGEBRA I1 9 7, no. 5, Lemma l), which may be written as zL xt = 0, where =foa(x~) E Mk by hypothesis; hence xt = 0 for all A E L and therefore y~ =~,(xt) = 0 for all A E L, which proves that the sum of the MA is direct. COROLLARY. Let (Pa) be a direct system Of subsets of E, and let P = 3 lim Pa. u, for all E I, pa is afree subset (resp. basis) Of E,, then P is a free subset (resp. basis) ofE. The second assertion follows immediately from the first and Proposition 4. It is therefore sufficient to prove that if the Pa are free every subset {Y({))~~~~~ consisting of distinct elements of P, is free. There exists an a E I and elements x:) 8 Pa such that y“) = fa(xc)) for 1 < i < n (Set Theory, 111, 3 7, no. 5, Lemma 1) ; if c A(f~y“) = 0, it may be assumed that A“) = +,(A:)) for 1 6 i 6 n and hence fa(? A:’x:)) = 0; this implies A:)x:’ = 0 for some p a, where A:’ = +oa(A:)), xr = foa(x2)) and the x:) belong to P, and are distinct since y(f) =fo(xg)); then A:) = 0 for 1 < i < n, whence f A‘{’ = +@(AX)) = 0 for 1 < i < n. Suppose now that all the rings A, are equal to the same ring A and the (boa to 1,; then, for every direct system (E,, fDa) of A-modules, E = 3 lirn E, is an A-module. Let F be an A-module and for all a let u,: E, -+ F be an A-linear mapping such that (u,) is a direct system of mappings; then u = lirn u, is an A-linear mapping of E into F. Conversely, for every A-linear mapping v:lim E, --f F, the family of u, = v 0 fa is a direct system of A-linear mappings such that v = limv,. On the other hand we note that for a < p the mapping + 3 3 Hom(foa, 1,) =fa~:~~~A(Ep, F) --f HomA(E,, F) is a Z-module homomorphism such that (HomA(E,, F),fao) is an inverse system of Z-modules; asfa,(vo) = vo 0 fDa, the above remarks can be expressed as follows : PROPOSITION 6. For every direct system (E,, f,,) of A-modules and every A-module F, the canonical mapping u H (u 0 fa) is a Z-module isomorphism dF:HomA(lim E,, F) -+ lim HornA@,, F). COROLLARY 1. For eve y A-module homomorphism v : F + F’, the (5) ---f t 5, = Hom(lE,, v):Hom(E,, F) +Hom(E,, F’) TENSOR PRODUCT OF DIRECT LIMITS $6.3 form an inverse system of 2-linear mappings and the diagram Hom(1im E,, F) --+ dF lirn Hom(E,, F) Horn( IE, U) dF# t (6) Hom(1im -1 E,, F’) --+ lim Hom(E,, F’) --+ is commutative. For all u E Hom(1im E,, F), dF,(v 0 u) = (u 0 u 0 fa) by definition and the commutativity of diagram --f (E:, yo,) is an inverse sgstcm Ofright A-modules and lirn E,* is canonically isomorphic 3 (6) then follows immediately from the definitions. COROLLARY 2. If (E,, fo,) is a direct system of 1eJt A-modules and E = lim E,, to E*. t Remark. Let E be an A-module and (M,),,, an increasing family of sub- modules of E such that E is the union of the Ma; ifj,,: hf, -+ M, (for C( < p) andj,: Ma ---f E are the canonical injections, it is immediate that j = limj, is an isomorphism of lirn Ma onto E (Set Theory, 111, 3 7, no. 6, Remark + 1). In particular, every A-module + is the direct limit of the right directed family of its finitely generated submodules. 3. TENSOR PRODUCT OF DIRECT LIMITS Let (A,, poa) be a direct system of rings and (Ea,fDa) (resp. (F,,g,,)) be a direct system of right (resp. left) A,-modules. For a < p, there is a Z-module homomorphism fDa @ @A, Fa -+ (EO)[A.l @A, (FO)[A,l and on the other hand there is a canonical Z-module homomorphism @A, (Ft3)[Am1 --f EO @AD FO ’ corresponding to the ring homomorphism pp4 (5 3, no. 3, Proposition 2); whence by composition we obtain a Z-module homomorphism hoa:Ea @A% Fa + E, @Aq F, which maps the tensor product x, @ y, to f,,(x,) @ gD,(y,). Clearly Pa @A, Fa7 hBa) is a direct system of Z-modules. Let A = lirn A,, E = lim E,, F = lirn Fa and 289 3 4 ----f 288 LINEAR ALGEBRA I1 let p,:A, 3 A,fa: E, + E, g,: Fa --f F be the canonical mappings. As above, a Z-linear mapping x,:E, BAa Fa -+ E @A F is defined, which maps the tensor product x, @ y, to fa(xa) @ g,(y,), and it is immediate that these mappings form a direct system. Thus we obtain a Z-linear mapping x = lim x,:lim (E, BAa Fa) + E @A F. (7) --+--+ PROPOSITION 7. The Z-linear mapping (7) is bijective. We set P = lim(E, gAa Fa) and, for all a E I, let h,:E, mAa Fa -+ P be --f the canonical mapping. On the other hand, for all a E I, let ta:Ea x Fa-tEa@AaFa be the canonical Z-bilinear mapping; for a < p, tP(fpa(xa), gDa(Ya)) =fpa(xa) @ g,a(Ya) = hBa(ta(xa,Ya)) and hence (ta) is a direct system of mappings. Canonically identifying lim(E, x Fa) with E x F (Set Theory, 111, 8 7, no. 7, Proposition lo), we derive a mapping t = lirn t,: E x F + P such that --f --+ t(fa(xa), ga(ya)) = ha(ta(xa,Ya)) = ha(xa @Ya)* Taking account of Set Theory, 111, 3 7, no. 5, Lemma 1, it is immediately seen that t is Z-bilinear; moreover, for x E E, y E F, A E A, there exists a E I such that x =f,(x,), y = g,(y,), A = p,(A,) with A, E A,, x, E E,, y, E Fa (Set Theory, 111, 3 3, no. 7, Lemma 1) ; whence t(xky) = ha((xaAa) @ Ya) = ha(xa @ (hay,)) = t(x, AY). Hence there exists one and only one Z-linear mapping d: E @A F + P such that x'(x @ y) = t(x, y) (3 3, no. 1, Proposition 1). Moreover, by defini - tion x'(x(fia(xa @ ya))) = x'(fa(xa) 8 gab,)) = ha(xa @Ya> x(x'(L(xa) @ ga(Ya))) = x(ha(xa 8 ya)) =fa(xa> @ ga(Ya) and as the elements of the form fa(xa) @ ga(ya) (resp. ha(xa @ y,)) generate the Z-module E @A F (resp. P), x' o x and x o x' are the identity mappings. Loosely speaking, Proposition 7 may be expressed by saying that tensor products commute with direct limits and usually the two sides of (7) are identified by means of the isomorphism x. COROLLARY 1. Let (EL, f;,) (resp. (FL, g&)) be another direct system of right (resp. left) A,-modules; for all a E I, let u,: E, -+ EL (resp. v,:F, -+ FA) be an 290 4 6.3 TENSOR PRODUCT OF DIRECT LIMITS A,-linear mapping such that (u,) (resp. (v,)) is a direct system. Then (u, @ u,) is a direct system of Z-linear mappings and the diagram lim(E, gAa Fa) - (lim E,) @A (lim Fa) 1yu, c3 u.) lim(E: gAa F:) ---+ (lim EL) @A (lim F:) (lim u.) --f @ (lim u.) (8) - 1 is commutative. --+ The verification is immediate. Let (A:, p;,) be another direct system of rings and suppose that each E, is an (A:, A,)-bimodule, the fBa being (A:,A,)-linear for a 6 p. Then if we write A' = limA&, the isomorphism (7) is linear with respect to the left A'-module structures on the two sides by virtue of Corollary 1. This can be immediately generalized to arbitrary multimodules. In particular, if the A, are commutative, A = lirn A, is commutative and ---z isomorphism (7) is an A-module isomorphism. --f COROLLARY 2. Let (E,, foa) be a direct system of right A,-modules and let EL = E, gAa A be the A-module obtained by extending the ring of scalars to A = lirn A, by means of the canonical homomorphism pa:A, +A. Then isomorphic to lim E,. (EL, fo, + @ lA) is a direct system of right A -modules, whose direct limit is canonically 3 It suffices to apply Proposition 7 with Fa the ring A considered as an (A,, A)-bimodule by means of pa. COROLLARY 3. Let A be a ring, (E,, fo,) a direct system of right A -modules and F a left A-module. Then the Z-modules lirn (E, @A F) and (lim E,) @A F are canonic- ally isomorphic. --f --+ It suffices to take A, = A and Fa = F for all a E I in Proposition 7. In particular, if p:A -+ B is a ring homomorphism, lirn p*(E,) and COROLLARY 4. Let M be a right A -module, N a lgt A-module, (x,),~~~,, a family of elements of M, ( y,) ,, a family of elements of N, such that 2 (x, @ y,) = 0 in N) confuining the xi (resp. the y,) and such that 7 (x, 8 y,) = 0 in MI @A N,. M (resp. N) is canonically identified with the direct limit of the right direc - ted family of its finitely generated submodules containing the x, (resp. the yl) and it suffices to apply Set Theory, 111, § 7, no. 5, Lemma 1. + * lim E,) are canonically isomorphic. (-+ M BA N. Then there exists ajnitely generated submodule M, (resp. 1 N,) of M (resp. 29 1 I1 LINEAR ALGEBRA 9 7.2 Remark. For a family (at)lsI of elements of a vector space E over a field K to be free, it is necessary and sufficient that, for all K E I, aK belong to no subspace of E generated by the a, of index c # K. We know that this con - dition is necessary in any module (§ 1, no. 11, Remark 1). It is sufficient by virtue of Lemma 1, as is seen immediately arguing by reductio ad absurdurn and considering a minimal related subfamily of (a‘). DIMENSION OF VECTOR SPACES $7. VECTOR SPACES 1. BASES OF A VECTOR SPACE THEOREM 1. Every vector space over ajield K is a free K-module. It must be proved that every vector space admits a basis; this will follow from the following more precise theorem : THEOR~M 2. Given a generating system S of a vector space E over ajield K and a free subset L of E contained in S, there exists a basis B of E such that L c B c S. Theorem 1 follows from this statement by taking L = 0. To prove Theorem 2, we note that the set 2 of free subsets of E contained in S, ordered by inclusion, is an inductive set (Set Theory, 111, 3 2, no. 4), by virtue of 5 1, no. 11; so is the set 91 of free subsets containing L and con- tained in S. By Zorn’s Lemma, 2ll admits a maximal element B and it suffices to prove that the vector subspace of E generated by B is equal to E. This follows immediately from the definition of B and the following lemma: Lemma 1. Let be a free family of elements of E; zf b E E does not belong to the subspace F generated by (a,), the subset ofE consisting ofthe a, and b is free. Suppose that there were a relation pb + c h,a, = 0 with p E K and )\, E K for all c E I, the family (A,) having finite support; if p # 0, it would follow that b = -c (p-lh,)a, and hence b E F contrary to the hypothesis; hence p = 0 and the relation becomes 2 h,a, = 0, which implies A, = 0 for all I E I by hypothesis; whence the lemma. COROLLARY . For a subset B of a vector space E, the following properties are equiva - lent: (a) B is a basis ofE. (b) B is a maximal free subset of E. (c) B is a minimal generating system of E. This follows immediately from Theorem 2. Example. Given a ring A and a subjeld K of A, A is a (right or left) vector space over K and therefore admits a basis; in particular, every extension jeld of a field K has a basis as a left (resp. right) vector space over K. *Thus the field R of real numbers admits an (infinite) basis as a vector space over the field Q of rational numbers; such a basis of R is called a Hamel busis.* 2. DIMENSION OF VECTOR SPACES THEOREM 3. Two bases of the same vector space E ouer ajield K are equipotent. We note first that if E admits an inznite basis B, it follows from 5 1, no. 12, Corollary 2 to Proposition 23 that every other basis of E is equipotent to B. We may therefore confine our attention to the case where E has a finite basis of n elements. We note that every monogenous vector space over K, not reduced to 0, is a simple K-module (I, 5 4, no. 4, Definition 7), for it is gener - ated by each of its elements #O, by virtue of the relation pa = (ph)(A-la) for p E K, A E K and A # 0. Hence if (a,),,,,, is a basis of E, then E = @ Ka, up to isomorphism and the subspaces E, = @ Ka, for 0 6 k 6 n form a Jordan -Holder series of E, E,/E,-, being isomorphic to Ka,. Theorem 3 then follows in this case from the Jordan -Holder Theorem (I, 5 4, no. 7, Theorem n 1=1 X 1=1 6) * A proof can be given independent of the Jordan -Holder Theorem, by showing by induction on n that, if E admits a basis of n elements, every other basis B’ has at most n elements. The proposition is obvious for n = 0. If n 2 1, B‘ is non-empty; then let a E B’. By Theorem 2 (no. 1) there exists a subset C of B such that {a} U C is a basis of E and a 6 C, since {a} U B is obviously a generating system of E. As B is a basis of E, C = B is impossible (no. 1, Corollary to Theorem 2) and hence C has at most n - 1 elements. Let V be the subspace generated by C and V‘ the subspace generated by B’ - {a}; V and V‘ are both supplementary to the subspace Ka of E and hence are isomorphic (9 1, no. 10, Proposition 13). As V admits a basis with at most n - 1 elements, B - {a} has at most n - 1 elements by the induc - tion hypothesis and hence B’ has at most n elements. DEFINITION 1. The dimension of a vector space E ouer a jield K, denoted by dim, E or [E:K] (or simply dim E) is the cardinal ofany ofthe bases of E. rfM is a subset of E, the rank of M (ouer K), denoted by rg M or rg, M, is the dimension of the vector subspace of E generated by M. TO say that E is finite-dimensional is equivalent to saying that E is a K- module ofjinite length and dim, E = long, E. 293 292 I1 LJNEAR ALGEBRA COROLLARY. For every subset M of E, the rank of M is at most equal to dim E. If v is the vector subspace of E generated by M, M contains a basis B' ofV (no. 1, Theorem 2) and as B' is a free subset of E, it is contained in a basis B of E (no. 1, Theorem 2); then Card(B) < Card(B), whence the corollary. Theorems 2 and 3 immediately imply the following proposition : PROPOSITION 1. (i) For a lefi vector space over K to be of Jinite dimension n, it is necessary and suficient that it be isomorphic to KT. (i;) For two vector spaces KT and K: to be isomorphic (m and n integers >O), it is necessary and su&cient that m = n. (iii) In a vector space E offinite dimension n, every generating system has at least n elements; a generating system of E with n elements is a basis of E. (iv) In a vector space E oj'jnite dimension n, every free subset has at most n elements; a free subset with n elements is a basis of E. PROPOSITION 2. Let (EL)LEI be a family of vector spaces over K. Then dim,(@ LEI EL) = LEI 2 dim, EL. If the EL are canonically identified with subspaces of E = 2 EL and B, is a basis of EL (L E I), then B = cI B, is a basis of E (3 1, no. 11, Proposition 19) ; whence relation (1) since the B, are pairwise disjoint. Remark. (1) Examples can be given of modules admitting two finite bases not having the same number of elements (3 1, Exercise 16(c)). How - ever : PROPOSITION 3. Let A be a ring such that there exists a homomorphism p of A into afield D; then for every free A-module E, any two bases ofE are equipotent. Consider the vector space p*(E) = D @A E over D obtained by extending the ring of scalars to D (4 5, no. 1) and let +:x H 1 @ x be the canonical mapping of E into p*(E); if (ah) is a basis of E, (+(ah)) is a basis of p*(E) (4 5, no. 1, Proposition 4) ; the proposition then follows from Theorem 3. COROLLARY . If A is a commutative ring #O and E a free A-module, any two bases of E are equipotent. There exists in A at least one maximal ideal m (I, 5 8, no. 6, Theorem 1) and, as A/m is a field, the conditions of Proposition 3 are fulfilled. Remarks. (2) When a free A-module E is such that any two bases of E are equipotent, the cardinal of an arbitrary basis of E over A is also called the dimension or rank of E and denoted by dimA E or dim E. DIMENSION AND CODIMENSION OF A SUBSPACE OF A VECTOR SPACE 5 7.3 (3) Let A be a ring such that any two bases of a free A-module are equi- potent and let K be a subfield of A, so that A can be considered as a l$t vector space over K by restricting the scalars. Every free A-module E can simi- larly be considered as a left vector space over K and it then follows from 4 1, no. 13, Proposition 25 that dim, E = dim, E . dim, A,. (2) (4) In Chapter VIII we shall see examples of rings satisfying the con- clusion of Proposition 3 but not the hypothesis. 3. DIMENSION AND CODIMENSION OF A SUBSPACE OF A VECTOR SPACE PROPOSITION 4. Every subspace F of a vector space E is a direct factor of E and dimF + dim(E/F) = dimE. As the quotient vector space E/F is a free module, we know (3 1, no. 11, Pro- position 21) that F is a direct factor of E; relation (3) is then a special case of formula (1) (no. 2). COROLLARY 1. If E, F, G are vector spaces over a Jield K, every exact sequence of linear mappings 0 -+ E -+ F -+ G --f 0 splits. (3) This is another way of expressing Proposition 4 (3 1, no. 9). COROLLARY 2. Let (Ei)oGiGn be a finite family of vector spaces over aJield K. If there exists an exact sequence of linear mappings ufl-1 U" (4) 0 --j Eo uo El u1 E2 - ... ---+ En-l En -> O the relation 2 dim E2,ct1 = 2 dim E,, Zkhn (5) 2k+lGn holds, or, ;fall the spaces arejnite dimemional, Let I, = Im uk = Keru,., for 0 < k < n - 1; Ik+1 is therefore iso- morphic to E,,,/I,, hence (formula (3)) dim 1, t- dim Ikfl = dim Ek+l for 0 < k < n - 2 and moreover dim I, = dim Eo and In-l = En, hence dim I, - = dim En. Replacing dim E, by its expression as a function of the n-1 k=O dim I, in the two sides of (5), we obtain on each side 2 dim I,, whence the corollary. 294 295 I1 LINEAR ALGEBRA COROLLARY 3. If M and N are two subspaces of a vector space E, then (7) dim(M + N) + dim(M n N) = dim M + dim N. It suffices to apply Corollary 2 to the exact sequence O+MnN-+M@N+M+N+O ($ 1, no. 8, Proposition 10) taking account of the fact that dim(M @ N) = dim M + dim N (no. 2, Proposition 2). COROLLARY 4. For every subspace F of a vector space E, dim F < dim E; if E is finite dimensional, the relation dim F = dim E is equivalent to F = E. The first assertion is obvious from (3) ; further, if dim E is finite, the rela- tion dimF = dimE implies dim(E/F) = 0 by (3) and a vector space of dimension 0 reduces to 0. COROLLARY 5. If a vector space E is the sum of a family (F,) of vector subspaces, then (8) dim E < 2 dim F,. If further dim E isjnite, the two sides of (8) are equal if and only if E is the direct sum of the family (F,). The inequality (8) follows from (3) and the fact that E is isomorphic to a quotient of @ F, (5 1, no. 7, formula (28)). The second assertion is a particular case of 5 1, no. 10, Corollary 5 to Proposition 16, for the equality of the two sides of (8) implies that dim F, = 0 except for a finite number of indices. DEFINITION 2. Given a vector space E, the codimension (with respect to E) of a sub- space F of E, denoted by codim, F, or simply codim F, is the dimension of E/F (equal to that of any supplement of F in E) . Relation (3) may then be written (9) dim F + codim F = dim E. PROPOSITION 5. Let F, F‘ be two subspaces of a vector space E, such that F c F’. Then codim,F’ < codim,F < dim E. If codim, F is Jinite, the relation codim, F‘ = codim, F implies F = F’. The inequality codim,F < dimE is obvious from (9) and if dimE is 296 DIMENSION AND CODIMENSION OF A SUBSPACE OF A VECTOR SPACE $ 7.3 finite the relation codim, F = dim E implies dim F = 0 and hence F = (0). The remainder of the proposition follows from this, for codim, F’ = codim,,,(F’/F), since E/F’ is canonically isomorphic to (E/F)/(F’/F) (I, 3 4, no. 7, Theorem 4). PROPOSITION 6. If M and N are two subspaces of a vector space E, then (10) codim(M + N) + codim(M n N) = codim M + codim N. It suffices to apply Corollary 2 of Proposition 4 to the exact sequence 0 + E/(M n N) --f (E/M) 0 (E/N) + E/(M + N) .+ 0 (4 1, no. 7, Proposition 10) and use no. 2, Proposition 2. Note that if E is finite dirncnsional, (10) is a conscqucnce of (7) and (9). PROPOSITION 7. If (F,) is a finite family of subspaces of a vecfor space E, then codim(q F,) < codim F,. E/F, (5 1, no. 7, formula (27)). Vector subspaces of dimension 1 (resp. dimension 2) of a vector space E are often called lines passing through 0 (rcsp. planes passing through 0) (or simply lines (resp. planes) if no confusion arises (cf. $9, no. 3)), by analogy with the language of Classical Geometry; a subspace of E is called a hyperplane passing through 0 (or simply a hyperplane) if it is of codimension 1. Hyperplanes can also be defined as the maximal elements of the set 6 of vector subspaces of E distinct from E, ordered by inclusion. There is a one-to-one correspond- ence between the subspaces of E containing a subspace H and the subspaces of E/H (I, 5 4, no. 7, Theorem 4) ; if E is of dimension 2 1, 8 is non-empty and to say that H is maximal in 6 means that E/H contains no subspace distinct from (0) and E/H, which implies that E/H is generated by any of its elements # 0, in other words it is of dimension 1. 1, the hyperplanes are the sub- spaces of dimension n - 1 by formula (3). PROPOSITION 8. In a vector space E over a jield K, every vector subspace F is the intersection of the hyperplanes which contain it. It suffices to show that for all x $ F there exists a hyperplane H containing F and not containing x. By hypothesis F n Kx = (0) and hence the sum M OfF and Kx is direct. Let N be supplementary to M in E; E is then the direct Sum ofH = F + N and Kx and H is therefore a hyperplane with the desired Property. If F = F,, E/F is isomorphic to a subspace of the direct sum of the In a vector space of finite dimension n 297 I1 LINEAR ALGEBRA Remark. Most of the properties proved in this no. for subspaces of a vector space do not hold for submodules of a free module whose dimension (8 2, no. 7, Remark 2) is defined. *For example, an ideal of a commutative ring does not necessarily admit a basis, for there are integral domains A in which cer- tain ideals are not principal (VII, 5 1, no. 1) and any two elements of such a ring are linearly dependent (9 1, no. 11, Remark l),* A submodule of a free A-module E may be free, distinct from E and have the same dimension as E, as is shown by principal ideals in an integral domain A; the same example proves moreover that a free submodule of a free A-module does not neces - sarily admit a supplement. 4 7.5 DUAL OF A VECTOR SPACE If E is an infinite -dimensional vector space, there are injective (resp. surjective) endomorphisms of E which are not bijective (Exercise 9). Let K, K' be two fields, o:K + K' an isomorphism of K onto K', E a vector K-space, E' a vector K'-space and u : K -+ K' a semi-linear mapping relative to c (5 1, no. 13); the dimension of the subspace u(E) of E' is also called the rank of u. It is also the rank of u considered as a linear mapping of E into o* (E'), for every basis of u(E) is also a basis of o,(u(E)). 5. DUAL OF A VECTOR SPACE I I * THEOREM 4. The dimension of the dual E* of a vector space E is at least equal to the dimemion of E. For E* to befinite -dimensional, it is necessary and suj'icient that I E be so, and then dim E* = dim E. If K is the field of scalars of E, E is isomorphic to a space KY) and therefore E* is isomorphic to K', (5 2, no. 6, Proposition 10). As KS' is a subspace of Ki, dim E = Card(1) < dim E* (no. 3, Corollary 4 to Proposition 4); further, if I is finite, K: = K:) (cf. Exercise 3(d)). COROLLARY . For a vector space E, the relations E = (0) and E* = (0) are equiva - lent. THEOREM 5. Given two exact sequences of vector spaces (over the samejeld K) and linear mappings 0 + E'+ E --+ E" -+ 0 0 + F'+ F + F" --f 0 I I I I l and two vector spaces G, H over K, the carrespoilding sequences I 0 -+ Hom(E", G) -+ Hom(E, G) --+ Hom(E', G) + 0 I I 0 -+ Hom(H, F') + Hom(H, F) -+ Hom(H, F") -+ 0 are exact and split. (no. 3, Proposition 4) and from 5 2, no. 1, Propositions 1 and 2. COROLLARY. For every exact sequence This follows from the fact that every vector subspace is a direct factor o -+ E' E I EM -.+ 0 0 ---+ E"" --+ tu E" --f tU I of vector spaces over the same field K and linear mappings the sequence E'* --+ 0 is exact and splits. 4. RANK OF A LINEAR MAPPING DEFINITION 3. Let E, F be two vector spaces over afield K. For every linear mapping u of E into F, the dimension of the subspace u(E) of F is called the rank of u and denoted by rg(u)* If N = Ker(u), E/N is isomorphic to u(E), whence the relation (11) rg(u) = codim,(Ker(u)) and therefore (12) Moreover, by formula (3) (13) rg(u) + dim(Coker(u)) = dimF. PROPOSITION 9. Let E, F be two vector spaces over a jield K and u: E + F a linear mapping. rg(u) + dim(Ker(u) = dimE. (i) rg(u) < inf(dim E, dim F). (ii) Suppose that E is jinite-dimensional; in order that rg(u) = dim E, it is mcessary and sujicient that u be injective. (iii) Suppose that F is finite-dimensional; in order that rg(u) = dim F, it is necessary and suj'icient that u be surjective. This follows immediately from relations (1 2) and (1 3). COROLLARY . Let E be a vector space offinite dimension n and u an endomorphism of E. The following properties are equivalent: (a) u is bijective; (b) u is injective; (c) u is surjective; (d) u is right invertible; (e) u is lejit invertible; (f) u is of rank n. 298 299 I1 LINEAR ALGEBRA It follows in particular that for every vector subspace M of E the canonical homomorphism E*/M‘ --f M*, where M’ is the subspace of E* orthogonal to M (3 2, no. 4), is bijective. THEOREM 6. For every vector space E over a jeld K, the canonical mapping cE:E -+ E** (3 2, no. 7) is injective; for it to be bijective, it is necessary and sujicient that E be finite-dimensional. The first assertion and the fact that if E is finite-dimensional c, is bijective are special cases of 9 2, no. 7, Proposition 14. Suppose that E is infinite-dimen - sional, so that we may assume that E = K!L), where L is an infinite set and therefore E* = Kk. Let (eh)hsL be the canonical basis of E and let (e:)hsL be the corresponding family of coordinate forms in E* (9 2, no. 6) ; the vector sub- space of E* generated by the e: is just the direct sum F’ = K(dL) and the hypo - thesis that L is infinite implies that F‘ # E*. Then there exists a hyperplane H’ of E* containing F’ (no. 3, Proposition 8) and, as E*/H’ is non-zero, its dual is also non-zero (Corollary to Theorem 4), which is identified with the orthogonal H” of H’ in E** (3 2, no. 6, Corollary to Proposition 9). But H” n cE(E) is contained in the image under c, of the orthogonal of F’ in E, which is by de - finition 0; hence cE(E) = E** is impossible. E will usually be identiJied with the subspace of E** the image of cE. Let E, F be two vector spaces over a field K and u: E -+ F a linear mapping. (1) Of the dual ofIm(u) = u(E) onto Im(%) = %(F*). (2) Of the dual OfKer(u) = ul(0) onto Coker(%) = E*/tu(F*). (3) Ofthe dual ofCoker(u) = F/u(E) onto Ker(%) = %-l(O). We write I = Im(u), N = Ker(u), C = Coker(u) ; from the exact sequences We shall define canonical isomorphism: P j (14) 0 ---+ N --+ E -+ I 0, 0 ---z I -+ F--+ C 0 we derive, by transposition (Corollary to Theorem 5), the exact sequences ‘P 0 --+ I* --+ E* --+ N* -+ 0, 0 --+ C* ---f F* - I* ---f 0. Moreover, as u = j o p, t~ = 9 0 y; the exact sequences (15) thus define canonical isomorphisms of C* onto Ker(%), of I* onto Im(%) and of N* onto Coker(%), since 9 is injective and surjective. To be precise, let y E Im(u), t E Ker(u), t E Coker(u), y’ c Im@), 2’ E Coker(h), t‘ E Ker(%); when y’, z’, t‘ are canonically identified with linear forms on Im(u), Ker(u) and Coker (u) respectively, then (16) ‘j (15) (y, y’) = (x, y’) for all x E E such that U(X) = y; 3 7.5 DUAL OF A VECTOR SPACE (17) (18) (2, 2’) = (2, x*) (t, t’) = (s, t’) for all x* E E* whose class mod. %(F) is equal to 2’; for all s E F whose class mod. u(E) is equal to t. In particular we derive from these results: PROPOSITION 10. Let E, F be two vector spaces over the samejeld K and u:E + F a linear mapping. (i) For u to be injective (resp. Jurjective), it is necessary and suficient that t~ be surjective (resp. injective). (ii) rg(u) < rg(tu) and rg(u) = rgfu) ;frg(u) isjnite. The second assertion follows from the above and Theorem 4. THEOREM 7. Let E be a vector space over ajcld K, F a subspace of E and F’ the orthogonal of F in E * . (i) dim F’ 2 codim, F; for dim F’ to bejnite, it is necessary and suficient that codim, F bejnite, and then dim F’ = codim, F. (ii) The orthogonal of F‘ in E is equal to F. (iii) Everyjnite -dimensional subspace G‘ of E* is the orthogonal of some subspace of E, necessarily equal to the orthogonal of G‘ in E and ofJinite codimension. (i) We know that F’ is isomorphic to the dual (E/F)* (9 2, no. 6, Corollary to Proposition 9) and hence the assertion follows from Theorem 4, since dim(E/F) = codim, F by definition. (ii) Let F, be the orthogonal of F’ in E; clearly I; c F, and the ortho - gonal F; of F, is equal to F’ (9 2, no. 4) ; the canonical linear mapping (E/F1)* -+ (E/F)*, the transpose of E/F -+ E/F, is therefore bijective (3 2, no. 6, Corollary to Proposition 10); it then follows from Proposition 10 that the canonical mapping E/F -+ E/F, is bijective, which implies F, = F. (iii) Let G’ be a subspace of E* of finite dimension p and let F be its ortho- gonal in E; then codim, F < dim G‘. For, if is a basis of G‘, F is the kernel of the linear mapping x ++ ((x, a:)) from E to Kf whose rank is at mostp (no. 4, Proposition 9), whence the conclusion (no. 4). Then let F’ be the orthogonal of F in E*; it follows from (i) that dim F‘ < dim G’; but on the other hand obviously G’ c F’, whence F’ = G’ (3 2, no. 3, Corollary 4 to Pro- position 4). Remark. An inznite-dimensional subspace G’ of E* is not necessarily the orthogonal of a subspace of E, in other words, if F is the orthogonal of G‘ in E, the orthogonal F’ of F in E* may be distinct from G’ (Exercise 20(b)).t 2 t By giving E and E* suitable topologies and only considering in E and E* subspaces closed with respect to these topologies, it is possible to re-establish a Perfect symmetry between the properties of E and E* when E is infinite -dimen - sionaI (cf. Topological vector spaces, 11, $ 6). 300 I 30 1 I1 LINEAR ALGEBRA COROLLARY 1. Let (X:)~ be the subspace of E consisting of the x such that be a jnite sequence of linear forms on E and let F (x, x:) = O for 1 < i < p. Then codim, F is equal to the rank ofthe set ofthe xi* and every linear form on E which is zero on F is a linear combination of the x:. Then codim, F < p and in order that codim, F = p, it is necessary and suflcient that the xi* be linearly indepen - dent. The set G’ of linear combinations of the x: is a subspace of E* and F is the orthogonal of G‘ in E, hence codim, F = dim G‘ by Theorem 7; further dim G’ < p and the relation dim G’ = p means that (x:) is a free system (no. 2, Proposition 1) ; hence the corollary. COROLLARY 2. (i) Let (x:)lGtsp be a jnite sequence of linear form on E. For ($) to be a free system, it is necessary and suficient that there exist a sequence (xi) I, of elements of E such that (xi, x:) = &, (Kronecker index). (ii) Let (x~)~~~~~ be ajnite sequence of elements of E. For (xi) to be a free system, it is necessary and suficient that there exist a sequence (x:) QiQp of linear forms on E such that (xi, xT) = aij. Clearly (ii) follows from (i) by considering E as identified with a subspace of E** by means of c, (Theorem 6). Let G’ be the subspace of E* generated by the x: and F its orthogonal in E; E/F and G’ can each be canonically identified with the dual of the other; if the family (x:) is free, there is in E/F a basis (ii) the dual of (xf) and every representative system (xi) of the classes .ti has the required properties. Conversely, the existence of the system (xi) such that (xi, xT) = Bij implies that for all i the subspace of E* orthogonal to K.x, contains the XT of indexj # i but does not contain x:, hence the sys- tem (x:)lciQp is free. COROLLARY 3. Let S be a set and V a vector subspace of the right vector K-space K: of mappings of S into K. In order that dim V 2 p (where p is an integer), it is necessary and suficient that there existp elements si ofS andp elementsf, ofV (1 < i < p) such thatf,(sj) = aij. The space KZ is canonically identified with the dual of E = K&S’ and f(s) = (e,,f) for s E S and f EK:, (eJsEs being the canonical basis of E. Corollary 2 then shows that the condition is sufficient. Conversely, suppose that dimV 2 p, so that there exists a subspace G‘ of V of dimension p; let F be the orthogonal of G‘ in E, so that dim(E/F) = p. It follows from no. 1, Theorem 2 that there existp elements si E S such that the e,, (1 < i < p) form a basis of a supplement F of E (applying Theorem 2 (no. 1) to a free subset con - sisting of a basis of E and the generating system the union of this free subset and 5 7.5 DUAL OF A VECTOR SPACE the canonical basis of E) ; we then take thef, to be the elements of a basis of G‘ dual to the basis of E/F consisting of the classes of the e,, mod. F. COROLLARY 4. Let E be a vector space and M, N two subspaces of E ofjnite co- dimension; if M’, N’ are the orthogonals of M and N in E*, the orthogonal of M n N in E* is M’ + N’. As M (resp. N) is the orthogonal of M‘ (resp. N’) in E (Theorem 7), M n N is the orthogonal of M’ + N‘ in E and hence M’ + N’ is the orthogonal of M n N in E* (Theorem 7 (iii)). COROLLARY 5. Let E be a vector space ofjnite dimension n. For every subspace F of E of dimension p, the orthogonal F‘ of F in E* is of dimension n - p. For every subspace G’ of E* of dimension q, the orthogonal G ofG’ in E is of dimension n - q and G’ is the orthogonal of G in E*. Theorem 7 gives another characterization of hyperplanes in E: PROPOSITION 11. For every hyperplane H in a vector space E, there exists a linear form x: on E such that H = x,* - (0). Given such a form xz, for a linear form x* on E to sutisjy H = x*-’(O), it is necessary and suficient that x* = x;a, where a is a scalar #O. Conversely, for every linear form x* # 0 on E, the subspace x*-l(O) is a hyperplane of E. This statement merely expresses Theorem 7 for subspaces of E of codimen- sion 1 and subspaces of E* of dimension 1. If H is a hyperplane and a linear form such that H = x*-l(O), the relation (x, x,*> = 0 which characterizes the elements x E H, is called an equation ofH. More generally, if (x:) is a family of linear forms on E and F denotes the vector subspace the intersection of the hyperplanes x?-l(O), the relation “for all 1, (x, x:) = 0” characterizes the elements of F; the relations (x, XI”) = O for all c form a system of equations of the subspace F. Theorem 7 (ii) expresses the fact that every vector subspace ofE can be defined by a system ofequations. Theorem 7 (i) and (ii) ?roves moreover that a subspace F ofjnite codimen - sion p can be defined by a system ofp equations , (19) (x,x:) = 0, 1 < i <p, where the forms x: are linearly independent. Conversely, Corollary 1 to Theorem 7 shows that a subspace F defined by a system ofp equations (19) is of co- dimension <p and that it is of codimension p if and only if the x? are linearly ~ I 303 302 LINEAR ALGEBRA I1 LINEAR EQUATIONS IN VECTOR SPACES 9 7.6 to have at least one solution, it is necessary that, for every family (p,) of scalars ofjnite support such that 7 xTpt = 0, 7 qtpt = 0. If the rank ofthe system (21) isjnitc, this GO ndition is also suficient. The condition is obviously necessary. It says that, if F' is the subspace of E* generated by the family (x;"), there exists a linear mappingf:F'-+ K, such that f (x:) = qr for all t E I. If F' is of finite dimension r, F' is the ortho - gonal of a subspace F of E of codimension r (no. 5, Theorem 7) and F' is identi - fied with the dual orE/F (4 2, no. 6, Corollary to Proposition 9) ; f is therefore an element of the bidual (E/F)**. As E/F is finite -dimcnsional there exists one and only one element y E E/F such that f (x*) = (y, x*) for all x* E F' (no. 5, Theorem 6). The solutions of (21) are then the x E E whose canonical image in E/F is y. independent; it amounts to the same to say that F cannot be defined by a system consisting of at most p - 1 of the equations (19). 6. LINEAR EQUATIONS IN VECTOR SPACES PROPOSITION 12. Let E, F be two vector spaces over ajeld K and u: E -+ F a linear mapping. For the linear equation (20) 4.) =Yo to have at least one solution x E E, it is necessary and suficient that yo be orthogonal to the kernel of the transpose mapping 'u. The orthogonal ofu(E) in F* is 'u-l(O) (9 2, no. 5, Corollary to Proposition 8) and the orthogonal of tu-l(0) in F is therefore u(E) (no. 5, Theorem 7 (ii)). We shall obtain a more convenient criterion for systems of scalar linear equa- tions (21) (x, = ?r (1 E 1) where the unknown x takes its values in a vector space E over a field K, the x: are linear forms on E and the right hand sides qt elements of K. If we consider a basis (ah)hoL of E, the system (21) is equivalent to the sys- tem of equations with x = 1 Shah, the solutions of (22) being of necessity families (Eh) of elements of K ofjnite support. DEFINITION 4. The dimension of the subspace of E* geizerated by the family (x:) is called the rank of the system (2 1). PROPOSITION 13. For the system (21) to be of finite rank r, it is necessary and su@- cient that the linear mapping u:x H ((x, x;")) ofE into K: be ofrank r. If F' is the subspace of E* generated by the $, the kernel of u is the ortho - gonal F of F' in E ; if F' is of dimension r, F is of codimension r and conversely (no. 5, Theorem 7) and rg(u) = codim, F (no. 4, formula (1 1)). THEOREM 8. Let &EL (21) <x, $7 = qr (L E 1) be a system of scalar linear equations on a vector space E over aJield K. For this system 304 Remark. When the rank of the system (21) is inznite, the condition of Theorem 8 is no longer suficient. For example, suppose that the x: are the coordinate forms on the space E = K1', I being infinite (4 2, no. 6); as the x: are linearly independent, the condition of Theorem 8 holds for every family (qL) but the system (21) only has solutions if the family (qt) has finite support. A system (21) is always of finite rank if it has only afinite number of equations and its rank is then at most equal to the number of equations (no. 2, Proposition 1). Similarly, if E is offinite dimension n (which for a system (22) corresponds to the case where there is only ajnite number n of unknowns), its dual E4' is of dimen - sion nand hence the rank ofsystem (21) is at most equal to n (no. 3, Corollary 4 to Proposition 4). From this we deduce: COROLLARY 1. A system of scalar linear equations in a vector space, consisting of a Jinite number of equations whose le$t hand sides are linearly indcpendcnt forms, always admits solutions. COROLLARY 2. For a homogeneous linear system (22) of equations in n unknowns with coej'icients in a Jield K to admit non -trivial solutions consisting of elements of K, it is necessary and suficient that its rank be < n. This will always be the case if the number of equations is finite and < n. COROLLARY 3. For a linear system (22) with coejicients and right hand sides in a field K, consisting of n equations in n unknowns, to have one and only one solution con- sisting of elements of K, it is necessary and su@cient that the associated homogeneous system have no non-trivial solution (or, what amounts to the same, that the left hand sides of this system be linearly independent forms). 305 I1 LINEAR ALGEBRA 7. TENSOR PRODUCTS OF VECTOR SPACES The results of $3 3, 4 and 5 relating to free or projective modules apply in particular to vector spaces and give the following properties : PROPOSITION 14. Given an exact sequence (23) 0 4 E'+ E -+ E" -+ 0 of right vector spaces over afield K and linear mappings and a lejit vector space F over K, the corresponding sequence of Z-linear mappings O-tE' @,F+E @,F+E" BKF-+O is exact and splits. As the sequence (23) splits, this is a particular case of 5 3, no. 7, Corollary 5 to Proposition 7 and fj 3, no. 6, Proposition 5. Because of Proposition 14, when E' is a vector subspace of E, j:E' + E the canonical injection, E' BKF is usually identiJied with a sub-Z-module of E mK F by means of the injectionj @ 1,. With this convention: COROLLARY . Let K be ajeld, E a right vector space over K, F a lejit vector space over K, (Ma)aeA a family of vector subspaces of E and (Np)BEB a family of vector subspaces of F. Then It is obviously sufficient to prove the particular case (25) (a9A Ma) @K F = a$A (Ma @K F)* Clearly the left hand side of (25) is contained in the right hand side. To of F. Every element of E BK F prove the converse, we consider a basis ( f,) , no. 7, Corollary 1 to Proposition 7); if E' is a vector subspace of E, the relation 2 xh @ fh E E' gK F is equivalent, by Proposition 14, to xh E E' for all A E L. To say that zL xh @ fh belongs to each of the Ma BK F thus means that for all A E L and all tc E A, xh E Ma, that is x, E afA Ma for all A E L, which proves that the right hand side of (25) is contained in the left hand side. PROPOSITION 15. If (Eh),,L is a family of right vector spaces over a field K and (FJUEM a family oflejit vector spaces over K, the canonical mapping can then be expressed uniquely in the form c xh @ fh, where xh E E (3 3, heL (5 3, no. 7, formula (22)) is injective. 306 3 7.7 TENSOR PRODUCTS OF VECTOR SPACES We write F = n F,; the mapping (26) is the composition of the canonical mappings UEM and as F and the E, are vector spaces over K, this reduces to 3 3, no. 7, Corol - lary 3 to Proposition 7. When the conditions of Proposition 15 are fulfilled, the tensor product (a Eh) (,E FU) is often identified with its canonical image in (Eh @K 'u). h. U With this convention : COROLLARY . Let F be a lgt vector space over K; for every set X, the left vector space KZ @K F is ident$ed with the subspace of the space FX of all mappings of X into F, consisting of the mappings u such that u(X) is offinite rank in F. If (fh) is a basis of F, the element 2 vh @ fh of K: BK F is identified under (26) with the mapping x H v,(x) f,. As uh = 0 except for the indices h belonging to a finite subset H of L, the image of X under the above mapping is contained in the finite-dimensional subspace of F generated by the fh of indices h E H. Conversely, let u:X + F be a mapping such that u(X) is con- tained in a finite-dimensional subspace G of F and let (bJIGtGn be a basis of G. For all x E X, we may write ~(x) = 2 vi(x)bi, where the vt(x) are well u is then identified with the element AsL ' determined elements of K; thus n mappings i=l vi : X + K are defined and clearly vi @ b,. t=1 Similarly, for a right vector space E over K and a set Y, E BK K: is identi - fied with a subspace of the space EY, consisting of the mappings v:Y -+ E such that v(Y) is of finite rank. More particularly, for every field K, K: mK KZ is identified with a subspace of the space KXxY of mappings of X x Y into K (K being considered as a (K, K)-bimodule); an element 2 ui @ vi, where ui is a mapping of X into K and vi a mapping of Y into K, is identi - fied with the mapping (x, y) H ut(x)vi(y) of X x Y into K. 307 I1 LINEAR ALGEBRA PROPOSITION 16. (i) Let K, L be twojelds, E a lejl vector space over K, F a left vector space over L and G a (K, L)-binzodule. Then the canonical Z-homomorphism (27) v: Hom,(E, G) F --f Horn,@, G BL F) (6 4, no. 2, formula (7)) is injective; it is bijective when one of the vector spaces E, 9 7.8 RANK OF AN ELEMENT OF A TENSOR PRODUCT (ah) a basis of E and (b,) a basis of F; then (ah €9 6,) is a basis of the vector K-space E 8, F (11, 3 3, no. 7, Corollary 2 to Proposition 2) and therefore dim,(E 8, F) = dim, E .dim, F. (30) (2) Let K be a commutative field, El, E,, F,, F, four vector spaces over K and u:E, + F,, v: E, -+ F, two linear mappings; then rg(u €9 v) = rg(.) .rg(v). (31) It is immediate that (u €9 v)(E, €9 E,) is the canonical image of u(E,) €9 v(E,) in F, €9 F, and hence (Proposition 14) is isomorphic to u(E,) @ v(E,); the conclusion then follows from (30). (3) Under the same hypotheses as in Remark 1, dim,(Hom, (E, F)) 3 dim, E . dim, F. If E is isomorphic to K(I), Hom(E, F) is isomorphic to (Hom(K, F))' (4 1, no. 6, Corollary 1 to Proposition 6) and hence to F' (4 1, no. 14); as F(I) is a subspace of F' and dirn(F(')) = Card(1) .dim F = dim E. dim F (no. 2, Propo- sition 2), the inequality (32) follows from no. 3, Corollary 4 to Proposition 4. The same argument shows that the two sides of (32) are equal when dim E isfinite (cf. 9 10, nos. 3 and 4). 8. RANK OF AN ELEMENT OF A TENSOR PRODUCT Let E be a right vector space and F a left vector space over the same field K; to every element u E E @, F there corresponds canonically under (29) a homomorphism u1 E Hom,(E*, F) ; if u = c xi €9 y, with xi E E, yi E F, the element u, is the linear mapping (32) \Y F is jnite-dimensional. (ii) Let El, E,, F,, F, be four vector spaces over a commutative jeld K; then -. - \I the canonical K-homomorphism (28) A:Hom(E,, F,) €9 Hom(E,, F,) +Hom(E, €9 E,, F1 C3 F,) (4 4, no. 4, formula (21)) is injective; it is bijective ;f one of the ordered pairs (El, E,), (El, F,), (E,, F,) consists ofjnite-dimemionnl spaces. Assertion (i) is a particular case of 3 4, no. 2, Proposition 2. Similarly the second assertion of (ii) is a particular case of§ 4, no. 4, Proposition 4. Finally, to see that the homomorphism (28) is always injective, observe that Hom(E,, F,) is a vector subspace of FH' (i = 1,2) and that Horn(& €9 E,, Fl @ F,) is canonically identified with a vector subspace of the space (F, @ FZ)EIXEa (11, 9 3, no. 1, Proposition 1); when these identifications are made and also the left hand side of (28) is identified with a subspace of FY1 @ FF (Proposi- tion 14), the canonical mapping (28) becomes the restriction to this subspace of the canonical mapping (26) and it has been seen (Proposition 15) that the latter is injective. COROLLARY 1. Let E and F be two vector spaces over afield K; the canonical mapping E* @I, F -+ Hom,(E, F) (4 4, no. 2, formula (1 1)) is injective; it is bijective when E or F is finite-dimen- sional. This is a special case of Proposition 16 (i) . COROLLARY 2. Let E be a right vector space and F a le$t vector space over the same jield K; the canonical mapping (29) E @, F -+ Hom,(E*, F) (3 4, no. 2, formula (15)) is injective; it is bijective when E is finite-dimensional. This is a special case of 3 4, no. 2, Remark 2. Remarks. (1) Let K be a commutative field, E, F two vector spaces over K, (33) On the other hand, E €9, F is canonically identified with F @,o E, where E is considered as a left vector space and F as a right vector space over the opposite field KO ; thus there corresponds canonically to u a homomorphism 6 Hom,(F*, E) given by (34) u1 (resp. u2), considered as a mapping of E* into F** (resp. of F* into E**) is just the transpose of u2 (resp. u,). The ranks of u, and u2 are thus equal to the mmejinite number r, the common dimension of the subspaces ul(E*) of F and u,(F*) of E, each of which is canonically isomorphic to the dual of the other (no. 5) ; we shall say that r (denoted rg(u)) is the rank of the element u of E I& F and that u,(E*) and u,(F*) are the subspaces (ofF and E respectively) associated with u. 308 309 4 7.9 EXTENSION OF SCALARS FOR A VECTOR SPACE LINEAR ALGEBRA I1 PROPOSITION 17. Let u be an element of E @K F and M c E and N c F its associated subspaces. For every expression u = i=l 5 xi @ yi of u, where xi E E and yi E F for 1 < i < s, the subspace M (resp. N) is contained in the subspace of E (resp. F) generated by the xi (resp. the yi). Moreover, the following properties are equivalent : (a) The integer s is equal to the rank of u. (b) The family (xi) <i is a basis of M. (c) The family ( yi) &, is a basis of N. (d) The families (xt)lGi<s and (yi)lgits are bothfree. By (33) (resp. (34)) each element of N = ul(E*) (resp. M = u,(F*)) is a linear combination of the y, (resp. the xi) ; whence the first assertion. If s = r, the subspace generated by the xi (resp. yi) with dimension <dim M (resp. <dim N) and containing M (resp. N) is identical with it and hence (a) implies (b) and (c) and a fortiori (d). Conversely, each of conditions (11) and (c) implies (a) by definition of rg(u). Finally if (d) holds, there exists a family (xf)lGrcs ofelements ofE* such that (xi, x;) = aij (no. 5, Corollary 1 to Theorem 7) and hence it follows from (33) that (yt) is a basis ofN, which com - pletes the proof. COROLLARY 1. The rank ofu is the smallest integer s such that there exists an expres - xi €3 yt, where xi E E and y, E Ffor 1 < i < s. i=l This follows immediately from Proposition 17 and no. 2, Proposition 1. COROLLARY 2. Let K be a commutativefield, E, F two vector spaces over K and L a commutative extension Jield of K. Let u be an elemcnt of E QK F, M and N its associated subspaces and u' the canonical image of u in (E @K F),,, (canonically identified with E,,, €3, F(,), cf. 3 5, no. 1, Proposition 3); then rg(u') = rg(u) and the subspaces associated with u' are canonically identijed with M(L) and No). If u = 2 xi @ yi, where the families (xi) and (yi) are free, then i=l PROPOSITION 18. Let p be a homomorphism of a field K into a ring A. For every exact sequence of vector K-spaces and K-linear mappings E' E -&+ E" the sequence E&) E(A) -% EYA) is exact. This is a special case of no. 7, Proposition 14, taking account of 3 1, no. 4, Remark 4. 1 COROLLARY. For every K-linear mapping f: E' -+ E, Im(JA,) = (Im( f))(A), Ker(AA,) = (Ker( f))(A), Coker(JA,) = (Coker( f)),A), to within canonical isomorphisms. PROPOSITION 19. Let p be an injective homomorphism of a jield K into a ring A. For every le$t vector space E over K, the cunonical mapping 4 : E -+ p* (E) = A gK E is injective. Moreover, for every vector subspace E' of E, p*(E') = A BK E' is canonically identijed with a direct factor sub-A-module of A BK E and, with this identification, (A @K E') n +(El = +(E'). (35) The first assertion is a special case of 5 5, no. 1, Proposition 4; the second is a special case of no. 7, Proposition 14; finally, to show (35), it suffices to take in A (considered as a right K -module) a basis (aJhp , such that ah,, = 1 for some in- dex A0 (no. 1, Theorem 2) ; the elements of A gK E can be written uniquely as E', it is necessary and sufficient that E E' for all A. On the other hand, the elements of NE) are those for which xh = o for A z ho; for an element C a, €3 xh to belong #no E E', whence the conclusion. ah €3 xh with xh E E and, for such an element to belong to A to (A BK E') n $(E), it is necessary and sufficient that xh h = 0 for A # A. and 11 LINEAR ALGEBRA PROPOSITION 20. Let p be a homomorphism of a field K into a ring A. For every ldt vector space E over K, the canonical right A -module homomorphism u: (E*) (A) (E(A)) * (5 5, no. 4) is injective; it is bijective when E isjnite-dimensional. The second assertion follows from 5 5, no. 4, Proposition 8. To prove the first, we note that every element of (E*)(A, may be written as 2 f x: @ ai, where ai E A and (xi*) siG,, is a free family in E*; there corresponds to it in (E(A,)* the linear form y* such that y*(l @ x) = 7 p((x, x:))ai for all x E E. But, there exists in E a family (xi)lQi4n such that (xi, x:) = ai, (no. 5, Corollary 2 to Theorem 7), whence y*(l @ xi) = ai; the relation y* = 0 therefore implies ai = 0 for all i, which proves our assertion. PROPOSITION 21. Let R be ajeld and L an extensionjeld of K. (i) For every vector space E over K, dimL(Eo,) = dim, E. (ii) For every K-linear mapping u:E -+ F, where E and I: are vector spaces over If (eJLEI is a basis of E over K, (1 @ eJrEI is a basis of Eo, over L (4 5, no. 1, Proposition 4), whence the first assertion; the second follows from the first and the fact that u(,(Eo)) is canonically identified with (u(E))(~) by the Corollary to Proposition 18. PROPOSITION 22. Let K be a commutative ficld, p:K --f A an injective central homomorphism and E, F two vector spaces over K. Then the canonical homomorphism (36) w:A @K Hom(E, F) 4 HomA(E(A), F(AJ (5 5, no. 3, formula (17)) is injective; it is bijective ifA or E is afinite-dimensional vector space over K. K, rgb(L)) = rg(u). This is a particular case of 3 5, no. 3, Proposition 7. 10. MODULES OVER INTEGRAL DOMAINS PROPOSITION 23. In a module E over an integral domain A, the set T of nongree elements is a submodule of E. If x and y are not free, there exist two non-zero elements a, p in A such that OLX: = 0 and By = 0. Then ap # 0 since A is an integral domain and ag(b + py) = 0 for all A and p in A since A is commutative, hence Ax + Ay is not free. Remark. Let E be a module over any commutative ring A. If x is a non-free element of E, every element of the submodule Ax is non-free. On the 3 7.10 other hand, if A contains divisors of 0, the sum of two non -free elements of E may be free; for example, in Z/GZ considered as a module over itself, 3 and 4 are not free, but 3 + 4 = 1 is free. MODULES OVER INTEGRAL DOMAINS Proposition 23 leads to the following definition : DEFINITION 5. In a module E over an integral domain A, the torsion submodule of E is the submodule of E consisting of the non-jiree elements (also called the torsion elements of E) . When E is equal to its torsion submodule (that is when every element of E is annihilated by an element #O of A) E is called a torsion module. When the torsion submodule of E is reduced to 0 (that is every non-zero element ofE is free) E is called (by an abuse of language) a torsion-jiree module. Every submodule of a free A-module (and in particular every projective A-module) is torsion-free. The Z-module Q is torsion-free. PROPOSITION 24. Let A be an integral domain. For every A-module E, let T(E) denote the torsion submodule of E. Let f: E -+ E’ be an A-linear mapping, E and E‘ being A-modules. 6) f (V)) = T(E’). (ii) Iff is injective, f (T(E)) = T(E’) n f (E). (iii) Iff is surjective and Ker( f) c T(E), then f (T(E)) = T(E’). Assertions (i) and (ii) are obvious. On the other hand, iff is surjective and x’ E T(E’), then x’ = f (x), where x E E, and by hypothesis there exists a # 0 in A such that f (ax) = ax’ = 0; whence ax E Ker( f) and by the hypothesis there exists p # 0 in A such that p(ax) = 0; as Pa # 0, x ET(E). COROLLARY 1. For every A-module E, E/T(E) is torsionyree. If f:E-+E‘ is an A-linear mapping, let fT denote the mapping T(E) -+T(E’) with the same graph as the restriction off to T(E). With this notation: COROLLARY 2. For every exact sequence of A-modules and A-linear mappings the sequence f 0 + E’ --+ E 4 E” is exact. 0 ---+ T(E’) fT_ T(E) % T(E”) Ker(g,) = Ker(g) n T(E) = f (E’) n T(E) =f(T(E‘)) = Im(fT). PROPOSITION 25. Let A be an integral domain and (E,) a family of A-modules; then 312 T(? El) = @ T(E,). 313 LINEAR ALGEBRA I1 $ 7.10 M E 2ll such that 1 E M and 1 @ x = 0 in the tensorproduct M @A E (Set Theory, 111, $ 7, no. 5, Lemma 1). It may further be supposed (replacing if need be M by a monogenous subnioduIe M' 3 M of K) that M = A. y-l, where y E A and y # 0. Now the mapping E, H yE, is an isomorphism of M onto the A- module A; on the other hand, the canonical isomorphism A @* E + E ($3, no. 4, Proposition 4) maps 4 @ x to the element cx of E; thus there exists an komorphism M @A E -+ E which maps the tensor product E, @ x to the element (yE,)x of E. The hypothesis 1 @ x = 0 in M @A E thus implies yx = 0. Remark. Let a-l+(x), P-'+(y) be two elements of E(K), with aEA, p EA, XEE, ~EE, ap # 0. In order that a-'+(x) = p-'+(y), it is necessary and sufficient that Px - ay be a torsion element of E, for this relation is equivalent to P+(x) = a+(y), which may also be written +(px - ay) = 0. COROLLARY 1. If E is a torsion-free A-module, the canonical mapping + : E -+ E,,, is injective. MODULES OVER INTEGRAL DOMAINS Recall (3 5, no. 1) that for every A-linear mapping of E into a vector space F over K, there exists one and only one K-linear mappingf: Eo, -+ F such that f = fo +; we shall say thatfis associated withf. COROLLARY 2. Let f be an A-linear mapping of E into a vector space F over K; ifKer( f) c T(E), the K-linear mappingfassociated with f is injective. We write an element of Ker(f) in the form A-l+(x), where A E A, A # 0, x E E; the relationY(h-l+(x)) = 0 is equivalent to A-tf(+(x)) = 0 in F and hence tof(x) =f(+(x)) = 0. By hypothesis, this implies x E T(E) and hence +(x) = 0, which proves the corollary. COROLLARY 3. Let E be an A-module and g an A-linear mapping of E into a vector space F over K such that g(E) generates F and Ker(g) c T(E). Then the K-linear mapping 2 associated with g is an isomorphism of Eo, onto F. 2 is injective by Corollary 2 and the hypothesis that g(E) generates F implies that .j is surjective. For every A-module E, the vector space E(K) is said to be associated with E. For every subset s of E the rank of S over K (or by an abuse of language, the rank of S) is the rank of the canonical image +(S) of S in E,K), in other words (no. 2, Definition 1) the dimension over K of the vector subspace of E,K) generated by 4 (S) . When E is a torsion-fee A-module, it is usually identified with its canonical image +(E) in E(K). With this convention, every generating system of E con- tains a basis of Eo, (no. 1, Theorem 2). In particular: COROLLARY 4. Every finitely generated A-module is offinite rank. 315 Let (x,) be an element of @ E, such that x, E T(E,) for all c; then each ofthe X, is annihilated by an element a, # 0 of A and it may be assumed that a, = 1 when x, = 0; as the family (x,) has finite support, the element a = n at of A is defined and #O; it obviously annihilates @ x, and hence @ T(E,) c T(? EL) ; the converse is immediate. L If E and F are two A-modules, clearly T(E @A F) contains the canonical images of T(E) BAF and E gAT(F); but examples can be given of torsion -free A-modules E, F such that T(E Note that an inznite product of torsion modules is not necessarily a tor- sion module; for example, in the Z-module n=l n (Z/p"Z) (p an integer > l), the element all of whose coordinates are 1 is free. F) # 0 (Exercise 31). m PROPOSITION 26. Let A be an integral domain, K its Jield of fractions, E an A- module and Eo, = K @* E the vector space over K obtained by extending the ring of operators; let 4 denote the canonical A-linear mapping x n 1 @ x of E into E,,,. (i) Every element ofEo, is of the form A-l+(x) for h E A, A # 0 and x E E. (ii) The kernel of+ is the torsion submodule T(E) of E. n (i) Every element of EcK, is of the form z = t=1 2 &+(xi) with ti E K and x, E E; for all i, there exists ai E A such that at # 0 and a1c1 E A; if a = n t=1 at, then 0: # 0 and at, = p, E A for all i, whence, in E(K), n since + is A-linear. (ii) If 3~ # 0 is not free in E, there exists a # 0 in A such that ax = 0, whence a+(x) = +(ax) = 0 in E(K), which implies +(x) = 0. Conversely, sup- pose that, for some x E E, 1 @ x = 0 in E(K); we show that x is a torsion element of E. We consider the set 92 of monogenous sub-A-modules of K; this is a right directed set under the relation of inclusion, for any two elements a, (3 of K can be written as a = C-lE,, p = C-lq, where 4, q, Z: belong to A and C # 0, hence A.a c A.<-l and A.P c A.C-l. Moreover K is the union of the modules M E 92 and can therefore be considered as the direct limit of the direct system de- fined by the modules M E 92 and the canonical injections (3 6, no. 2, Remark). Hence also, to within a canonical isomorphism, ECK) = lim --+ (M @A E) (3 6, no. 3, Proposition 7) and the relation 1 @ x = 0 in E,K) implies that there exists an 314 11 LINEAR ALGEBRA DEFINITION OF K'-STRUCTURES 5 8.1 Note that the converse of this corollary is not necessarily true; for example Q is a Z-module of rank 1 but is not finitely generated over Z. Recall (3 5, no. 1) that for every linear mapping f :E + E' (where E and E' are A-modules) ,AK) denotes the K-linear mapping 1 PROPOSITION 27. For every exact sequence @ f : E(K) + E&). f E' -+ E --% E" of A-linear mappings, the corresponding scquence of K-linear mappings &K) E{K) 3 E(K) -----f E&) is exact. Suppose that g(K)(A-l @ x) = 0, with A E A, A # 0, x E E; this is equiva- lent to h-l @ g(x) = 0 in E:K) and hence also to 1 @g(x) = A(A-1 @g(x)) = 0; by Proposition 26, there exists a # 0 in A such that ag(x) = 0 in E", or also g(w) = 0. By hypothesis, there is therefore an x' E E' such that CIX = f (x') and therefore A-l @ x =AK)(x-lA-l @ x'), which proves the proposition. COROLLARY 1. If E' is a submodule of E, E{K) is canonically identified with a vector subspace OfE(K) and (E/E')(K) with E(K)/E&). It suffices to apply Proposition 27 to the exact sequence 0 -+ E' -+ E -+ E/E' -+ 0. COROLLARY 2. For every A-linear mapping f :E --f F, Ker(f,,)) = (Ker(f))(K), Im(fo,) = (Im(f))(K,, Coker (AK)) = (Coker( f))(K) to within canonical iso- morphisms. In particular, for AK) to be injective (resp. surjective, resp. zero), it is necessary and suficient that Ker( f) c T(E) (resp. that Coker( f) be a torsion module, resp. that Im( f) c T(F)). This follows from Corollary 1 and Proposition 26. COROLLARY 3. Let E be an A-module and (x,JhfL a family of elements of E. For (x,,) to be a free family, it is necessary and sujicient that in the vector K-space E(K) the family (1 @ xA) be free. The family (xJ defines an A-linear mappingf:A(L) -+ E such that f (eh) = xh for all A E L ((eh) being the canonical basis of A(L)) and to say that (xh) is free means thatfis injective. It suffices to apply Corollary 2 tof, observing that A(L) is torsion -free (Proposition 25). $8. RESTRICTION OF THE FIELD OF SCALARS IN VECTOR SPACES Throughout this paragraph, K denotes a jeld and K' a subjeld of K. On a set V, a fight (resp. left) vector space structure over K defines, by restriction ofscalars, a right (resp. left) vector space structure over K'. 1. DEFINITION OF K'-STRUCTURES PROPOSITION 1. Let V be a right vector space over K and V' a subset of V which is a vector subspace over K'. The following conditions are equivalent: (a) The K-linear mapping A of V:K) = V' BK> K into V such that h(x' @ E,) = x'E, for all x' E V', E, E K, is bijective. (b) Every K'-linear mapping f' of V' into a vector K-space W can be extended uniquely to a K-linear mapping f of V into W. (c) Every basis of V' over K' is a basis of V over K. (d) There exists a basis of V' over K' which is also a basis of V over K. (e) The vector K-space V is generated by V' and every subset of V' free over K' is free over K. We know (5 5, no. 1) that V{K) has a right vector K-space structure under which (x' @ S)q = x' @ (E,?) (S, r, in K, x' E V') and that for every K'-linear mapping f' of V' into a vector K -space W, there exists one and only one K -linear mapping $' of V[K) into W such that f'(x' @ 1) = f '(x') for X' E V'. Ifj is the canonical injection of V' into V, A is just the corresponding K-linear mapping j. If 1 is bijective, then for every K'-linear mapping f ':V' -+ W,f' 0 1-l is the unique K-linear mapping of V into W extending f'; in other words, (a) implies (b). Conversely, if (b) holds, there exists in particular a K-linear mapping p of V into V(K) such that p(x') = X' 8 1 for all x' EV'; it is immediate that p o A = on the other hand, A(p(x')) = x' for all x' E V' and, as by hypothesis j :V' --t V can be extended uniquely to an endomorphism of V, of necessity A o p = l,, which completes the proof that (a) and (b) are equivalent. For every basis B' of V' over K', the set B of elements u' @ 1 of V&), where 0' runs through B', is a basis of V(K) over K (5 5, no. 1, Proposition 4) and = €3'. For A to be Sijective, it is necessary that the image under A of every basis of V&) be a basis of V over K and it is sufficient that it be so for a single basis of V&) (5 1, no. 1 1, Corollary 2 to Proposition 17). This proves the equivalence of (a), (c) and (d). As every subset of V' which is free over K' is contained in a basis of V' Over K' (8 7, no. 1, Theorem 2), (c) implies (e). Finally, suppose that (e) holds; if B' is a basis of V' over K', it is a free subset. of V over R; on the other hand B' 317 316 LINEAR ALGEBRA I1 5 8.3 The assertion concerning sums is obvious. On the other hand, if (W[)LEI is K) RATIONALITY FOR A LINEAR MAPPING a family of vector sub-K'-spaces of V', then (tc Wl) @= K = tG (W, (5 7, no. 7, Corollary to Proposition 14), which proves the corollary. which are rational over K'. A basis B of V over K is said to be rational over K' if it consists of vectors generates V' over K' and hence generates V over K by hypothesis; therefore B' is a basis of V over K, which proves that (e) implies (c). DEFINITION 1. Let V be a right vector space over a field K and K' a subfield of K. Every vector sub-K'-space V' of V satisfying the equivalent conditions of Proposition 1 is called a K'-structure on V. Example. Let B be a basis of V over K. For every subfield K' of K, the vector sub-#'-space of V generated by B admits B as a basis over K' and hence is a K'-structure on V. *For example, if K is commutative and V is taken to be the polynomial K-algebra KIXl, . . . , X,], then, for every subfield K' of K, K'[X,, . . . , X,] is a K'-structure on V., 2. RATIONALITY FOR A SUBSPACE DEFINITION 2. Let V be a right vector space over K, with a K'-structure V'. A vector ofV is said to be rational over K' ;fit belongs to V'. A vector sub-K-space W of V is said to be rational over K' if it is generated (over K) by vectors rational over K'. Let (v;)~ EI be a basis of V' over K', which is therefore also a basis of V over K (no. 1, Proposition 1). For a vector x = 2 v:Et of V to be rational over K', it is necessary and sufficient that 5, E K' for all t E I. If W is a vector sub-K-space of V which is rational over K', it follows from Definition 2 that W' = W n V' is a vector sub-K'-space of W, which generates W over K; on the other hand every subset of W' which is free over K' is also free over K since it is contained in V' (no. 1, Proposition 1). It follows (no. 1, Proposition 1) that W' is a K-structure on W said to be induced by the K'-struc- ture V' on V. For every vector sub-K'-space W' of V', we shall denote by W'.K the vector sub-K-space of V consisting of the linear combinations of elements of W' with coefficients in K. PROPOSITION 2. Let V be a right vector space over K and V' a K'-structure on V. The mapping W' H W'.K is a bijection of the set of vector sub-K-spaces of V' onto the set of vector sub-K-spaces of V which are rational over K and the inverse bijection is W H W n V'. Clearly the bijection A-l:V --f V' BK, K, the inverse of the bijection A defined in no. 1, Proposition 1, maps every vector sub-K'-space W' of V' onto its image under the canonical injection x' H x' @ 1 and W' . K onto W' gK, K; the assertions of Proposition 2 are thus consequences of Definition 2 and 3 7, no. 9, Proposition 19. COROLLARY 1. Every sum and every intersection of uector sub-K-spaces of V which are rational over K' is a rational subspace over K'. 318 COROLLARY 2. Every basis of V over K which is rational over K' is a basis of V' over K'. If W' is the vector sub-K'-space of V' generated by B, then W' . K = V = V' . K, whence V' = W' by virtue of Proposition 2. 3. RATIONALITY FOR A LINEAR MAPPING DEFINITION 3. Let V,, V, be two right vector spaces over K with K'-structures V;, V; respectively. A K-linear mapping f :V, -+ V, is said to be rational over K' ij If V, is a third right vector space over K, with a K'-structure V; and a K-linear mapping g:V, --f V3 is rational over K', clearly g 0 f :V, -+ V3 is rational over K'. PROPOSITION 3. Let V1, V, be two right vector spaces over K and V;, V; K'- structures on V,, V, respectively. V, (resp. V,) is canonically identijed with Vi BK, K (resp. V; gK, K) (no. 1, Proposition 1). (i) The mapping f' H f' @ 1, =hfc, is a bijection of HOmK,(V;, v;) onto the set of K-linear mappings of Vl into V, which are rational over K'; the inverse bijec - tion associates with every K-linear mapping f :V, -+ V, rational over K' the K'- linear mapping f':V; -+ VL with the same graph as the restriction off to V;. f(Vl) c v;. (ii) For every K-linear mapping f :V, --f V, rational over K', (i) Clearly with the above identifications, iff ':V; -+ Vh is a K'-linear mapping,xk, = f' 8 1, is rational over K' and f is the mapping with the same graph as the restriction of j& to V;. Conversely, iff :V, -+ V, is a K-linear mapping rational over K' andf':V; -+ V; has the same graph as the restriction off to V;, f andhfc, coincide on V;, which is a generating system of V, over K, hence f =At,. K and, as f'(V;) c Vh, f (Vi) = f '(Vi) = f(V,) n VL (9 7, no. 9, Proposition 19) ; the (ii) If f = f' 8 l,, then f (V,) = f (V; gK, K) = f'(V;) fonnulaf -1 (V;) = V; + Ker(f) follows immediately. 319 LINEAR ALGEBRA I1 8 8.5 K‘-structures on V and K,. By virtue of no. 3, Proposition 3, the set R’ of these linear forms is the image of the dual V’* of V’ under the composite mapping APPLICATION TO LINEAR SYSTEMS 6 v’* --f K @K, v’* v* (1) where +(x’*) = 1 @I x’* and u(< @ x’*) is the linear form y* on V such that y* (x’) = c(x’*, x’) for all x’ E V’ ($ 5, no. 4). We know that this mapping is in- jective ($ 7, no. 9, Propositions 20 and 19) and clearly R‘ is a left vector sub-K’ - space of V*; moreover every subset of R free over K‘ is free over K. But in general R‘ does not necessarily generate V* over K and does not therefore define a K’-structure on V* (Exercise 2). However, if V is ofjnite dimension n over K, v* is of dimension n over K‘ and R’ then defines canonically a K’-structure on V* . PROPOSITION 5. Let V be a right vector space over K, V’ a K‘-structure on V and W a vector sub-K-space of V. For W to be rational over K‘, it is necessary and sugicient that there exist a set H c V* of rational linear forms over K’ such that W is the orthogonal of H in V ($ 2, no. 4). Let H be a subset of V* whose elements are rational linear forms over K’. For all x* E H, the kernel of x* is a vector sub-K-space of V, rational over K‘ (no. 3, Corollary 2 to Proposition 3) ; the intersection of these kernels is therefore also a vector sub-K -space of V, rational over K’ (no. 2, Corollary 1 to Proposition 2). Conversely, let W be a vector sub-K-space of V rational over K’, so that W is identified with W‘ @IK, K, where W‘ = W n V‘ (no. 2, Proposition 2). For a linear form x’* E V’* to be zero on w’, it is necessary and sufficient that the linear form x* E V* which corresponds to it under (1) be zero on W, for by no. 3, Corollary I to Proposition 3, Ker(x*) = (Ker(x’*)) @IK, K and Let H’ be the orthogonal of W‘ in V‘*; we know (9 7, no. 5, Theorem 7) that W’ is the orthogonal of H’ in V’; if H is the image of H’ in V* under the mapping (l), it follows from the above that W is the orthogonal of H in V, faking account of no. 7, Corollary to Proposition 14. 5. APPLICATION TO LINEAR SYSTEMS Ker(x*’) = (Ker(x*)) n V‘. PRoPo~oN 6. (i) Given 2 system of homogeneous linear equations COROLLARY 1. In the notation ofl‘roposition 3, Im(f) = (Im(f’))(K), Cok4f) = (Coker(f’)),K,. Ker(f) = (Ker(f’))(K), In particular, for f to be injective (resp. surjective, zero), it is necessary and su3cient This is a particular case of $ 7, no. 9, Corollary to Proposition 18. that f ’ be so. Iff is bijective, its inverse mapping is rational ouer K’. COROLLARY 2. Let f :V1 --f V, be a K-linear mapping rational over K’. For every vector sub-K-space W1 of V, (resp. W, of V,) rational ouer K’, f (W,) (resp. ?‘(w,)) is a vector sub-K-space ofV, (resp. V,) rational over K‘. In the notation of Proposition 3, for every vector sub-K‘-space W; of v;, xK,(w; @= K) = f ‘(W;) gK, K; whence the assertion relating to W, (no. 2, Proposition 2). On the other hand, let Wk be a vector sub-K’-space of V& and let g‘ be the canonical K‘-linear mapping V;-+V;/W;; then f’(W&) = Ker(g‘ of ’) ; hence, by Corollary 1, -1 xi)(wh @K, K) =f’(w;) @K* K, whence the assertion relating to W,. Let V,, V, be two right vector K-spaces with K’-structures V;, Vh respec - tively. It is immediate that V; x V& is a K-structure on V, x V,, called the product of the K’-structures V; and V;. PROPOSITION 4. For a K-linear mapping f :V, + Vz to be rational ouer K‘, it is necessary and suj%ient that its graph I’ be rational over K’for the product K’-structure onV, x Vz. Let g be the mapping x, H (xl, f (xl)) from V, to V1 x Vz; this is a K- linear mapping such that I’ = g(V,) ; iff is rational over K’, so is I’ by virtue of Corollary 2 to Proposition 3. Conversely, suppose that I’ is rational over K’ and let it have the K-structure induced by that on V, x Vz; it follows immediately from the definitions that the restrictions pl, pz to J? of the pro- jections pr,, pr2, are K-linear mappings, rational over K’ of I’ into V1 and V, respectively. As p, is bijective, its inverse mapping q1 is rational over K‘ (Corollary 1 to Proposition 3) and hence so is f = pz 0 ql. 4. RATIONAL LINEAR FORMS Let V be a right vector space over K with a K’-structure V’. As K& is a K’- structure on the right vector K-space Kd, we may define linear form x* E V*, rational ouer K’, as the linear mappings of V into Kd, rational over K’ for the LEI CCL E-0 - EM) whose coefiients uIll belong to K‘, every solution (EL) ofthis system consisting ofelements of K a linear combination with coqqicients in K of solutions (5:) of (2) consisting ofehents 0fK‘. 320 32 1 I1 LINEAR ALGEBRA Ss 8.7 CRITERIA FOR RATIONALITY and sufficient that the Eb belong to L (no. 2) and hence the smallest field L with this property is the field geiierated by K', and the Eb for b E B. We next prove (ii). Let B,, B, be bases of V,, V, respectively, which are rational over K' and write, for all b, E B,, f (b,) = 2 baabzbl (*the family (abzbl) is just the matrix off with respect to the bases B, and B,; cf. 9 10, no. 4,). As B, (resp. B,) is a basis of (V,), (resp. (VJL) over L for every field such that K' c L c K, for f to be rational over L, it is necessary and sufficient that the abzbl belong to L; the smallest field with this property is therefore the field generated by K' and the abzbl for b, E B,, 6, E B,. Finally, to establish assertion (i) for a subspace W of V, we shall first prove the following lemma : Lemma 1. Let V be a right vector K-space with a K'-structure V' and W a vector sub-K-space of V. There exist two vector sub-K-spaces W,, W, of V, rational over K', such that V is the direct sum of W1 and W, and, ;fV is ident$ed with W, x W,, W is the graph of a K-linear mapping g of W, into W,. Let B be a basis of V rational over K'. Applying Theorem 2 of 4 7, no. 1, to a basis of W over K, considered as a free subset of V, and the generating system the union of this free subset and B, it is seen that there exists a subset C of B such that V is the direct sum of W and the subspace W, of V generated by C. Also let W, be the subspace of V generated by B - C. As B c V', clearly W, and Wz are rational over K'. Moreover, for all x E W,, there exists one and only one vector g(x) of W, such that x + g(x) E W, since V is the direct sum of W and W,; then W is the graph of g and g is K-linear since W is a vector sub-K-space of V. Having proved this lemma, it is known that W is rational over a subfield L of K containing K' if and only if g is rational over L (no. 3, Proposition 4). The smallest field K'(g) such that g is rational over K(g) is therefore also the smallest field over which W is rational. bz E B2 (ii) Given a system of linear equations (3) -=.- LEI 2 aIIlEl = B, (PEW whose coeficients aUL and right hand sides p, belong to K', ;f there exists a solution to the system consisting of elements of K, there also exists a solution consisting of elements of K'. (i) For every set S, let the right vector K-space K(dS) be given the K'- structure K;'". Let f be the K-linear mapping of KZ) into KLM' mapping every vector (QLeI to the vector (C,),EM defined by C, = LEI 2 aPlEl for all PEM. Clearly f is rational over K'; its kernel V, which is the set of solu- tions in K of the system (2), is a subspace of Kz) rational over K' (no. 3, Corol - lary 2 to Proposition 3) and hence generated by the solutions of (2) in K'. (ii) We consider K as a left vector K'-space; there exists a K'-linear pro- jector p of K onto its vector subspace K: ($7, no. 3, Proposition 4); if (EL) is a solution of (3) in K, then LEI 2 a,,P(E1) = p( 2 a,,S,) = P(p,) = p,, which proves that @(El)) is a solution of (3) in K'. A ring K is called (left) faithfullyjat over a subring K if Proposition 6 holds for K and K'; we shall study this notion in detail later (Commutatiue Algebra, I, 5 3). 6. SMALLEST FIELD OF RATIONALITY Let V be a right vector K -space with a K'-structure V'. For every field L such that K' c L c K, we write V, = V'.L; clearly every basis of V' over K' is a basis of V over K and a basis of V, over L. Hence V, is an L- structure on V and V' a K'-structure on V,. PROPOSITION 7. (i) Let V be a right vector K-space with a K'-structure V'. For every vector x E V (resp. every vector sub-K-space W of V), the set of subjields L of containinp K' and such that x (resp. W) is rational over L has a least element K'(x) .-, (resp. K(w)). (ii\ Let V,. V, be two ripht vector K-sbaces with K-structures V;, VL respec- \I I, 1 tively. For every K-linear ma$ing f of V, >nto V,, the set of subfelds L fl K con- taining K' and such that f is rational over L has a least element K'( f ). We first prove assertion (i) for a vector x E V. Let B be a basis of V rational over K'; B is a basis of V' over K' and a basis of V, over L for every field L such that K' c L c K; for x = 2 bEb to be rational over L, it is necessary beB 7. CRITERIA FOR RATIONALITY For every subfield L of K, let End,(K) denote the endomorphism ring of K considered as a lgt vector space over L; if L contains K', End,(K) is a subring af End,.(K). For every subset dl of End,.(K), there exists a largest subfield L of K containing K' and such that dl is contained in End,(K), namely the set of E E K such that +(Eq) = E. +(q) for all q E K and all 4 E JZ (it is im- mediately verified that this set is a ring and, on the other hand, replacing q by 5-l-q in the preceding relation, we obtain +(E-lq) = E-l.+(q) when 4 # 0). We shall call this field the centralizer ofd in K and denote it by x(A). Now let V be a right vector K -space with a K'-structure V'. For all +EEnd,.(K), there exists one and only one endomorphism +v of the Z- 322 323 I1 LINEAR ALGEBRA module V such that +v(x’.t) = x’.+(t) for ~’EV‘ and ~EK: for, in no. 1, a Z-isomorphism A of V’ BK, K onto V was defined which maps x’ @ E to x’. 6 and +v is necessarily equal to ?, o (lv, @ +) o A-l. THEOREM 1. Let A be a subset of End,.(K) and L = x(A) the subjield of K the centralizer of A. (i) Let V be a right vector K-space with a K’-structure. For a vector x E V to be rational over L, it is necessary and suficient that bv(x.q) = x. +(q) for all + E A and all -q E K. For a vector sub-K-space W of V to be rational over L, it is necessary and suficient that +v(W) c W for all + E A. (ii) Let V,, V2 be two right vector K-spaces each with a K’-structure. For a K- linear mapping f of V1 into V2 to be rational over L, it is necessary and suficient that We first prove assertion (i) for x. Let B be a basis of V rational over K’ f (+vl(xl)) = +v,(f (x1)) for all x1 E v1 and all + E 4. and write x = 1 b. tb; then, for + E A’ and q E K, bsB +V(‘.?) - x.+(?) = 2B b.(+(cb?) - tb.+(?)) and therefore, the relations “for all + EA and all q E K, bV(x.q) = x.+(q)” “for all + EA, all b E B and all q E K, +(cbq) = cb.+(q)” are equivalent. The second of these relations means that for all b E B, tb E ~(d), which proves the first assertion of (i). We next prove (ii) . For f to be rational over L, it is necessary and sufficient that, for all xi E V, rational over K’, f (xi) is a vector in V2 rational over L; this will imply that f (x,) is rational over L for every vector x1 of V, rational over L, such a vector being a linear combination with coefficients in L of vectors rational over K’. The above condition is equivalent, by the first part of the argument, to the relation and (4) (5) f(+v,(xi.q)) = +v,(fW.r)) for+Edand?EK* f (4 .+(?) = +v,(f (4) .?) for 4) E At and y1 E K which may also be written As every element of V1 is a linear combination with coefficients in K of elements of V, rational over K’, condition (5) is equivalent to f (+vl(xl)) = +v,(f (x1)) for all x, E V, and all 4) E A. DEFINITION OF AFFINE SPACES 9 9.1 Finally, to prove the second assertion in (i), we use no. 6, Lemma 1 : W is the graph of a K-linear mapping g: W, --t W, and W is rational over if and only if the mapping g is rational over L (no. 3, Proposition 4). By (ii), for g to be rational over L, it is necessary and sufficient that g(+w,(x,)) = +wz(g(x,)) for all x1 E W1 and all + E A; as +v = bWl x +wz, the above condition means that the graph W of g is stable under +v for all + E A. $9. AFFINE SPACES AND PROJECTIVE SPACES 1. DEFINITION OF AFFINE SPACES DEFINITION 1. Given a left (resp. right) vector space T over ajeld K, an afine space attached to T is any homogeneous space E of the additive group T (I, 4 5, no. 5) such that 0 is the only operator in T leaving invariant all the elements of E (that is, T oper- ates faithfully and transitively on E). Under these conditions, T is called the transla - tion space of E and its elements are called the translations of E (or free vectors of E) . In what follows we shall confine our attention to the case where T is a left vector space over K. The dimension (over K) of the translation vector space T of an affine space E is called the dimension of E (over K) and is de- noted by dim E or dim, E. An affine space of dimension one (resp. two) is called an afine line (resp. an afine plane). The elements of an afine space are also called points. Under the conditions of Definition 1, for t E T and a E E we shall denote by t + a or a + t the transform of the point a under t. Then the relations (1) s+(t+a)=(s+t)+a, 0 +a=a hold for s E T, t E T, a E E. The mapping x H x + t is a bijection of E onto itself, which we identify with t. Definition 1 moreover implies that, for all a E E, the mapping t H t + a is a bijection of T onto E. In other words, given two points a, b of E, there exists one and only one translation t such that b t + a; we shall denote this translation by b - a; then the formulae (2) a-a==, a-b=-(b-a), b=(b-a)+a (6-b) + (b-a) =c-a hold for aEE, b EE, GEE. If four points a, 6, a’, b‘ of E are such that b - a = b’ - a‘, the formula b‘ = (6‘ - b) + (b - a) + a = (b’ - a’) + (a’ - a) + a ad the commutativity of addition in T show that b’ - b = a’ - a. 324 325 I1 LINEAR ALGEBRA Given a point a E E, the mapping x H x - a is a bijection of E onto T; when E is identified with T under this mapping, we say that E is considered as the vector space obtained by taking a as origin in E. Conversely, every vector space T has canonically the structure of an affine space attached to T, namely the homogeneous space structure corresponding to the subgroup (0) of T (I, 3 5, no. 6). Remark. The definitions of this no. and some of the results which follow extend immediately to the case where, instead of a vector space T, we consider an arbitrary commutative group with operators T. 3 9.3 AFFINE LINEAR VARIETIES language) the baycentre of the points at (1 < i < m) (for m = 2, we say “midpoint” instead of “barycentre”) ; it is characterized by the relation 2. BARYCENTRIC CALCULUS PROPOSITION 1. Let (x~)~~~ be a family of points in an afine space E and a family of elements of K ofJinite support such that 2 A, = 1 (resp. zI 1, = 0) Ifa is any point of E, the point x E E defined by x - a = LEI C h,(x, - a) (resp. thefree vector 2I A,(x, - a)) is independent .f the point considrred. If a’ is another point of E, then If 2 A, = 1, then hl(xl - a) = (x - a) + (a - a’) = x - a’; if C hl = 0, then 2 h,(x - a‘) = C Al(x - a) ; whence the proposition. Under the conditions of Proposition 1, the point x defined by x - a = 1EI C hl(x, - a) (resp. the free vector 2 Al(x, - a)) is denoted by 2 h,xl. When 2 A, = 1, the point x = given the masses A,. Thus in particular the notation b - a introduced in no. 1 is recovered. Alxl is called the barycentre of the points X, Given m points al, . . ., a, of E, whose number m is not a multiple of the characteristic of K (V, 4 l), the point g = 2 - a, is called (by an abuse of t=l m m1 3. AFFINE LINEAR VARIETIES DEFINITION 2. Given an a8ne space E, a subset V of E is called an afine linear variety (or simply a linear variety or an afine subset of E) ;f, for every family (xJLGI of points of V and every family of elements of K of Jinite support such that 2 A, = 1, the barycentre 2 A,x, belongs to V. re1 LEI It amounts to the same to say that the condition of Definition 2 holds for The empty set is a linear variety; every intersection of hear varieties is a Let V be a non-empty subset of E and a a point of V; the relation every jnite family of points of V. linear variety. 1=1 means that x is a barycentre 2 A1xl + (1 - 5 A)a of the family consisting of the xi and a. Therefore: PROPOSITION 2. For a non-empty subset V of an afine space E to be a linear variety, it is necessary and suficient that V be a vector subspace for the vector space structure on E obtained by taking a point ofV as origin. In particular, the non-empty affine linear varieties of a vector space T (considered as an affine space) are just the translates of the vector subspaces of T; the vector subspaces of T are therefore the linear varieties containing 0. Let V be a non-empty linear variety of the afline space E; the set of free vectors x - y, where x and y run through V, is a vector subspace D of the translation space T of E called the direction of V: for, if a E V, then t=1 1=1 x-y=(x-a)-(y-a) ad our assertion follows from Proposition 2. It is immediate that D operates faithfully and transitively on V, which therefore has canonically the structure of an a&ne space attached to D. By the dimension of the affine variety V, we mean the dimension of V with this affine space structure, that is the dimension of the vector space D. The linear varieties of dimension 0 are the points of E; those of dimension 1 (resp. 2) are called lines (resp. planes) of E. 327 326 , I1 LINEAR ALGEBRA Every vector #O belonging to the direction of a line is called a direction vector of this line; its components with respect to a basis of T form what is called a system of direction parameters of the line in question. The codimension of a linear variety V in E is the codimension of its direction D in T; a linear variety of codimension 1 in E is called an (affine) hyperplane of E. Two linear varieties with the same direction are called parallel; it amounts to the same to say that one is derived from the other by translation. If V is a linear variety in T (considered as an affine space), its direction is the linear variety parallel to V and containing 0. PROPOSITION 3. Given a family (aJrEI of points of an afine space E, the set V of barycentres zI A,a, ((A,) ofjnite support, zI A, = 1) is a linear variety of E. If the family (a,) is empty, then V = because of the condition A, = 1. It may therefore be assumed that the family (a,) is non-empty and in this case the proposition is obvious, taking one of the a, as origin in E. The variety V is obviously the smallest linear variety containing the a,; it is said to be generated by the family (a,) and this family is called a generating system of V. In the notation of Proposition 3, assuming the family (a,) is non-empty, for the expression for every point x E V in the form x = ?,,a, to be unique, it is necessary and sufficient that, denoting an arbitrary index of I by K, the family of vectors a, - aK, where t runs through the set of indices #K, be free in T. Then the family of points of E is said to be affinely free (or that its elements form an afinely free system, or are afinely independent) and that A, is the barycentric coordinate of x of index L with respect to the affinely free family A family (aJrEI of points of E which is not affinely free is said to be affinely related. PROPOSITION 4. For a non-empty family (aJLEI of points in an afine space E to be afinely related, it is necessary and sufficient that there exist a family (A,) , I of elements not all zero in K, ofjnite support, such that 2 A, = 0 and LEI 1 h,a, = 0. Given an index K E I, to say that the family of vectors (a, - aK), where L runs through the set of indices #K, is related in T, means that there exists a family of scalars (A,)~#~ not all zero such that l#K 2 &(a, - a,) = 0, which may also be written zI i,a, = 0, with A, = - (a,) * A,, in other words A, = 0. AFFINE LINEAR MAPPINGS fj 9.4 i PROPOSITION 5. For a non-empty family of points OJ' an afine space E to be ajinely free, it is necessary and sufficient that, for every index K E I, aK do not belong to the linear variety generated by the a, of index #K. I The proposition is obvious if I has only a single element. Otherwise, taking ~ 1 1, Remark. as origin in E one of the a, of index #K, the proposition follows from fj 7, no. 4. AFFINE LINEAR MAPPINGS DEFINITION 3. Given two affine spaces E, E' attached to two vector spaces T, T' over the samejeld K, a mapping u of E into E' is called an affine linear mapping (or an ajine mapping) ;f, for every family (xJIEI of points of E and every family (h,),,Isuch that 2 A, = 1, LEI u( c A,x,) = c A,u(x,). LEI LEI I PROPOS~TION 6. Let u be an afine mapping of E into E'. There exists one and only one linear mapping u ofT into T' such that u(x + t) = u(x) + v(t) for all x E E, t E T. Let a be any point of E. The mapping I t - u(a + t) - u(a) is a linear mapping u, of T into T', for we may write a + At = A(a + t) + (1 - A)a a+s+t=(a+s) + (a+t) - a and it follows from (3) that v,(At) = hva(t) and u,(s + t) = ua(s) + v4(t). Moreover, if b is another point of E, then u, = v,; for the relation (a+t) -a+ b = b +t implies 1 u(a + t) - .(a) + u(b) = u(b + t) that is u(a + t) - u(a) = u(b + t) - u(b). Whence the existence of v; the uniqueness is immediate. ZJ is called the linear mapping of T into T' associated with u. Conversely, for every linear mapping u of T into T' and every ordered pair of points a E E, a' E E', it is immediately verified that x H a' + u(x - a) 328 329 I1 LINEAR ALGEBRA SS 9.5 DEFINITION OF PROJECTIVE SPACES exists an affine mapping u of E into E' such that u(V) = V': taking as origins in E and E' points of V and V' respectively, then taking in E (resp. E') a basis whose first m vectors form a basis of V (resp. V'), the proposition follows immediately from 3 1, no. 11, Corollary 3 to Proposition 17. As the field K has canonically a left vector space structure (of dimension 1) Over K, it can be considered as an affine space of dimension 1. An affine mapping of an affine space D (over K) into the affine space K is also called an ajine linear function (or an a&e function). If a point a is taken as origin in E, every affine function on E can then be written uniquely as x H cc + u(x), where cc E K and v is a linear form on the vector space E thus obtained; the afine functions on E therefore form a right vector space over K of dimension 1 + dim E. If u is a non-constant affine function on E and A E K, the set of x€E satisfying the equation U(X) = A is a hyperplane; conversely, for every hyperplane H in E, there exists an affine function uo on E such that H = "'(0) and every affine function u such that H = u (0) is of the form uop, where ~EK (3 7, no. 5, Proposition 11). If u is an affine function on E, the hyperplanes with equations U(X) = cc and u(x) = p are parallel. 5. DEFINITION OF PROJECTIVE SPACES DEFINITI ON 4. Given a 1eJt (resp. right) vector space V over ajeld K, the lgt (resp. right) projective space derived from V, denoted by P(V), is the quotient of the comple - ment V - (0) of(0) in V by the equivalence relation A(V) "there exists A # 0 in K such that y = hx (resp. y = xh) between x and y in V - (0). When V = K:", we also write P,(K) instead of P(K:+') and A,,(K) instead of A(V). Definition 4 can also be expressed by saying that P(V) is the set of lines (passing through 0) in V with the origin removed; P(V) is therefore canonic - ally identified with the set of lines (passing through 0) in V. The elements of a projective space are called the points of that space. When V is of dimension n, the integer n - 1 is called the dimension of the Projective space P(V) if n is finite, and the cardinal n otherwise; this cardinal is denoted by dim, P(V) or dim P(V). Thus a projective space of dimension - 1 is empty and a projective space of dimension 0 is a single point. A pro- jective space of dimension 1 (resp. 2) is called a projective line (resp. projective Plane). 6. HOMOG ENEOUS COORDINATES -1 1 1 I 1 Henceforth we shall only consider left projective spaces. 1 Let v be a vector space of finite dimension n + 1 over K, P(V) the projective of dimension n derived from V and (e,) <, ,, a basis of V. Let x denote the canonical mapping of V - (0) onto the quotient set P(V). For every 33 1 I I is an afKne mapping of E into E' whose associated linear mapping is u. To say that u is an affine mapping of E into E' therefore also means that, if an arbitrary point a in E and the point u(a) in E' are taken as origins, u is a linear mapping for the two vector spaces thus obtained. Let E" be a third affine space, T" its translation space, u' an affine mapping of E' into E" and v' the linear mapping of T' into T" associated with u'. Clearly u' 0 u is an affine mapping of E into E"; moreover, for a E E and tET, u'(u(a + t)) = u'(u(a) + v(t)) = u'(u(a)) + v'(v(t)) and hence u' 0 u is the linear mapping of T into T" associated with u' o U. For an affine mapping u to be bijective, it is necessary and sufficient that the associated linear mapping u be so, and u-l is then an affine mapping whose associated linear mapping is u-,. In particular, the affine bijections of E onto itself form a group G, called the ajine group of E. The mapping which associates with u E G the linear mapping u associated with u is, by the above, a homomorphism of G onto the linear group GL(T). If u is a translation, v is the identity and conversely. Hence, the kernel of the above homomorphism is the translation group T of E which is therefore a normal subgroup of G. If u E G, the automorphism t H utu-I of T (where t is identified with the translation x H x + t) is the linear mapping v associated with u. For x E E and t E T, by definition x + utu-1 = u(u-'(x) + t) = u(u-'(x)) + v(t) = x + v(t) and hence utu-l = v(t). Let a E E and G, be the subgroup of G consisting of the u E G such that .(a) = a. If E is identified with T by taking a as origin, G, is identified with GL(T). Every u E G can be expressed uniquely in the form u = t,u, (resp. in the form u = u,t,), where u,, u2 are in G, and t,, t, in T: for, writing t, = .(a) - a, u-lt, E G,, whence the existence of u1 and t,; the existence of u2 and t, is obtained analogously. The uniqueness follows from the fact that G, n T reduces to the identity element of G. Moreover whence u2 = u,, t, = u;'t,u,. Finally, the linear mappings associated with u and u, are the same and hence, if as above G, is identified with GL(T), u, is the linear mapping from T to itself associated with u. It is thus seen that G is the semi-direct product of G, by T (I, 3 6, no. 1). Let E, E' be two affine spaces over K. The direct (resp. inverse) image of a linear variety of E (resp. E') under an affine mapping u of E into E' is a linear variety of E' (resp. E) ; the rank of u is by definition the dimension of u(E) ; it is equal to the rank of the linear mapping associated with u. If V, v' are linear varieties of the same finite dimension m in E, E' respectively, there t,u, = u,(u;lt,u,) 330 LINEAR ALGEBRA I1 4 9.8 PROJECTIVE COMPLETION OF AN AFFINE SPACE point x = 5 tiet of V - {0), (to, El, . . . , 5,) is called a system of homogeneous coordinates of the point X(X) with respect to the basis (ei) of V. Every system (&) of n + 1 elements not all zero of K is therefore a system of homogeneous coordinates of a point of P(V) with respect to (e,); for two such systems (Et), (&) to be systems of homogeneous coordinates of the same point of p(v) with respect to the same basis (eJ, it is necessary and sufficient that there exists an element A # 0 of K such that 5; = Ati for 0 < i < n. This definition is immediately generalized to the case where V is infinite- dimensional. Given another basis (it) of V such that et = j=O 2 ai,ij (0 < i < n) and a system (&) of homogeneous Coordinates of X(X) with respect to the basis (q), for a system (EJ of n + 1 elements of K to be a system of homogeneous coordinates of ~(x) with respect to the basis (it), it is necessary and sufficient that there exist A # 0 in K such that t=o n In particular, if e, = yiit with y, # 0 (0 < i < n), then Ei = &y, where p # 0. 7. PROJECTIVE LINEAR VARIETIES Let W be a vector subspace of a vector space V; the canonical image of W - (0) in the projective space P(V) derived from V is called a projcctioe linear variety (or simply a linear varicty when no confusion is to be feared) ; as the equivalence relation A(W) on W - (0) is induced by the relation A(V), the projective linear variety the image of W - (0) in P(V) can be identified with the projective space P(W) derived from W and hence we may speak of the dimension of such a variety. In a projective space P(V), the canonical image of a hyperplane (with the origin removed) of V is a linear variety called a projective hyperplane (or simply a hyperplane) ; if P(V) is of finite dimension the hyperplanes in P(V) are the linear varieties of dimension n - 1. Every proposition concerning vector subspaces of a vector space goes over to a proposition concerning projective linear varieties. For example, if a projective space P(V) is of finite dimension n and (et)oGiGn is a basis of V, every linear variety L c P(V) of dimension r can be defined by a system of n - r homogeneous linear equations between the homogeneous coordinates E,, (0 < i < n) of a point of P(V) with respect to the basis (et), the left hand sides of (4) being linearly independent forms on V. In particular, a projective hyperplane is defined by a single homogeneous linear equation with coefficients not all zero. Conversely, the points of P(V) satisfying an arbitrary system of homogeneous linear equations with respect to the E,{ form a linear variety L; if the system considered consists ofk < n + 1 equations, L is of dimension >n - k. Every intersection of linear varieties of P(V) is a linear variety; for every subset A of P(V), there exists a smallest linear variety L containing A; it is called the linear variety generated by A and A is called a generating system of L. IfW is the vector subspace of V generated by ;'(A), then L = P(W). If L and M are any two linear varieties in P(V) and N the variety generated by L u M, then (8 7, no. 3, Corollary 3 to Proposition 4) dim L + dim M = dim(L n M) + dim N. In particular, if P(V) is finite -dimensional and dim L + dim M 2 dim P(V), it follows from (5) that L n M is non-empty. Let (x,), (y,) be two families of points in the vector space V with the same indexing set, such that y, = A,xl, where A; # 0 for all t. If the family (x,) is free, so is (y,) and conversely; then it is said that the family of points ~(x,) of P(V) is projectively free (or simply free). It amounts to the same to say that for every index K, the point x(x~) does not belong to the linear variety gener - ated by the ~(x,) for L # K. A family of points ofP(V) which is not projectively free is called projectively related (or simply related). For a family (x,) of points of V - (0) to be such that the family (~(x,)) is projectively free and generates P(V), it is necessary and sufficient that (x,) be a basis of V. If P(V) is of dimension n the number of elements in such a family is therefore n + 1. Note that giving such a family (x(x,)) in P(V) does not determine (even to within a left factor) the homogeneous coordi - nates of a given point of P(V) with respect to a basis (yJ of V such that ~(y,) = ~(x,) for all L (cf. no. 6). (5) 8. PROJECTIVE COMPLETION OF AN AFFINE SPACE Let V be a (left) vector space over a field K and consider the vector space IC, x V over K; the projective space P(Ks x V) is called the projective space canonically associated with the vector space V. If V is of dimension n, P(K, x V) is of the same dimension n. Consider in K, x V the affine hyper- plane V, = (1) x V, whose direction (no. 3) is the subspace Vo = (0) x V; if a line (passing through 0) of K, x V is not contained in V,, it contains a Point (a, x) with a # 0 and x E V, hence it contains also the point x) = (1, a-'x) of V,; the converse is immediate and it is seen that there is a one-to-one correspondence between the points of V, and the 333 1 I 332 I1 LINEAR ALGEBRA lines (passing through 0) of K, x V not contained in V,, each of the latter meeting V1 in one and only one point. It follows that the mapping x ++ +(x) = x(1, x) is an injection (called canonical) of V into the projective space P(K, x V) ; V is often identified with its image under this injection. The complement of +(V) in P(K, x V) is the projective hyperplane P(V,) called the hyperplane at inJinity of P(K, x V) (or of V, by an abuse of language) ; its points are also called the “points at infinity” of P(K, x V) (or of v). If (a,) is a basis of V and in K, x V the basis is taken consisting of the ele- ments e, = (0, a,) and the element e, = (1,0), the points at infinity in p(Ks x V) are those whose homogeneous coordinate of index w is 0. Let M be an affine linear variety in V (no. 3) and D its direction; the canoni - cal image $(M) of M in P(K, x V) is contained in the canonical image = x(M,) of the vector subspace M, of K, x V generated by the affine variety MI = (1) x M of K, x V. More precisely, if (a,) is an affinely free system of M generating M, the elements (1, a,) form a basis of M, and therefore a is just the projective linear variety generated by +(M); if M is finite- dimensional, % has the same dimension as M. The complement of +(M) in id is the intersection of % and the hyperplane at infinity and is equal to the canonical image x(M,), where Mo = (0) x D. Conversely, let N be a projective linear variety not contained in the hyper - plane at infinity and let R = x (N); R n V1 is an affine linear variety of K x V of the form {I) x M, where M is an affine linear variety of V and it is immediately seen that N is the affine linear variety generated by There is therefore a one-to-one correspondence between the affine linear varieties of V and the projective linear varieties of P(Ks x V) not contained in the hyperplane at infinity; for two affine linear varieties of V to be parallel, it is necessary and sufficient that the projective linear varieties which they generate have the same intersection with the hyperplane at infinity (which is sometimes expressed by saying that the two affine linear varieties in ques- tion have the same points at infinity). 9. EXTENSION OF RATIONAL FUNCTIONS If the results of no. 8 are applied to the vector space V = K, of dimension 1, it is seen that there exists a canonical injection + of K, into the projective line P,(K) = P(K, x K,); for all E K, +(4) is the point with homogeneous coordinates (1, E) with respect to the canonical basis (4 1, no. 11) of K, x K,. The complement of +(K) in P,(K) consists of the single point with homo - geneous coordinates (0, 1) with respect to the above basis; it is called the “point at infinity”, P,(K) is also called the projective field associated with K and denoted by K, the point at infinity in R being denoted by 00. *Consider in particular the case where K is a commutative field and let -1 +(W 334 PROJECTIVE LINEAR MAPPINGS 3 9.10 feK(X) be a rational function in one indeterminate over K (IV, $4); if f # 0, there is a unique expression f = ap/q, where a E K* and p and q are two relatively prime monic polynomials (VII, 3 1); let m and n be their respective degrees and let r = sup(m, n). We write PlV, X) = TY(X/T), Ql(T.9 X) = T‘q(X/T); p1 and q1 are two homogeneous polynomials of degree r over K such that p(X) = pl(l, X), q(X) = ql(l, x). Hence, for every element 4 E K which is not a zero of q(X), f (4) = a@(<)/q(E) is defined and we may write f(4) = aP,(L 4)/q1(1, 4) = aP,(A, q/ql(A, At) for all A # 0 in K. Consider then the mapping (?, 4) kl(% 41, P1(?, 4)) of K2 into itself; it is compatible with the equivalence relation A(K2) and therefore defines, when passing to the quotients, a mappingj’ of lX into itself which coincides with 5 ++ f (4) at the points where this rational function is defined; it is said, by an abuse of language, that j’ is the canonical extension of f to R. For example, iff = 1/X, thenf(0) = co andf(co) = 0; if f = (ax + b)/(cX + d) with ad - bc # 0, then f( -d/c) = co, ~(co) = a/c if c # 0, ~(co) = co if c = 0. Iff = aoXn + . - + a, is a polynomial of degree n > 0, then J(.o) = a** 10. PROJECTIVE LINEAR MAPPINGS Let V, V’ be two left vector spaces over a field K, f a linear mapping of V of a line (passing through 0) in V not contained in N is a line (passing through 0) in V’; hence, on passing to the quotients, f defines a mapping g of P(V) - P(N) into P(V’). Such a mapping is called a projective linear nzupping (or, simply, a projective mapping) ; although it is defined on B(V) - P(N) and not on P(V) (when N # {0}), we shall say by an abuse of language that g is a projective mapping of P(V) into P(V’). The projective linear variety P(N), where g is not defined, is called the centre of g. Note that, when g is defined on the whole of P(V) (that is when N = {0}), g is an injection of P(V) into P(V‘). When bases (ah)heL, (bJWeM are given in V and V’ respectively, a projec - tive mapping of P(V) into P(V’) maps a point of P(V) with homogeneous 335 into V‘ and N = f -1 (0) its kernel. It is immediate that the image under f LINEAR ALGEBRA I1 59.11 PROJECTIVE SPACE STRUCTURE 11. PROJECTIVE SPACE STRUCTURE Given a set E and a field K, a (left) projective space structure on E with respect to the field K is defined by giving a non-empty set @ of bijections of subsets of the projective space P(KY) onto E satisfying the following axioms: (EP,) The set of dejnition of every mapping f E @ is a linear variety of P( Ky’) . (EP,,) For every ordered pair of elementsf, g of 0 dejned respectively on the linear varieties P(V) and P(w), the bijection h = f of P(V) onto P(W) is a pro- jective mapping. (EP,,,) Conversely, iff E @ is dejned on the linear variety P(V) and h is a bijective projective mapping of P(V) onto a linear variety P(W) c P(KF’), then f 0 h-1 E @. Let E be a set, (V,),,, a family of vector spaces over K and suppose given for each A E L a bijection f, of P(V,) onto E such that, for every ordered pair of indices A, p, f, of, is aprojective mapping of P(V,) onto P(V,). Then we can define on E a projective space structure with respect to K as follows: let (el),eI be a basis of a space V, and write a, =f,(n(e,)); let b, be the element of index a, in the canonical basis of KiE’ (4 1, no. 11). The relation L # K implies b, # b, because of the hypothesis that f, is bijective; hence the b, form a basis of a vector subspace W, of KF’ and there therefore exists a bijective projective mapping h of P(W,) onto P(V,) such that h(n(b,)) = x(e,) for all t E I. If @ is taken to be the set of all bijective projective mappings fA o h o g-l, where g runs through the set of all bijective projective mappings P(W) c P(KY’), it is immediately verified that @ satisfies axioms (EP,), (EPI,) and (EP,,,). It is moreover immediate that @ depends neither on the choice of index A E L, nor on the choice of basis (e,) in V,, nor on the choice of h. In particular (taking L to consist of a single element), every projective space P(V) derived from a vector space V (no. 5, Definition 4) thus has a well determined “projective space structure” in the sense of the definition given in this no. Hence any set with a projective space structure can be called a projective space. With the same notation, a linear variety in a projective space E is a subset M of E such that, for at least one bijectionfE @ defined on P(V) c P(Ky’), f (M) is a linear variety in P(V) in the sense of no. 7 (this property then holds for all f E @). It follows from the above that every linear variety in a projective space has canonically a projective space structure. is a linear variety of dimension n (it suffices that this hold for one mapping -1 -1 A projective space E is said to be of dimension n if, for all f E @, f -11 (E) f E a)). coordinates t, (AEL) to a point of P(V’) with a system of homogeneous coordinates q, (p E M) of the form The centre of g is the linear variety defined by the equations If C is the centre of g and M is a linear variety of P(V), the image under g of M - (M n C) is a linear variety of P(V’) denoted (by an abuse of language) by g(M). Then (7) dimg(M) + dim(M n C) + 1 = dim M (9 7, no. 4, formula (12)). If M’ is a linear variety of P(v’), il(~f) u c is a linear variety of P(V) and (8) dirn(gl(Mf) u C) = dim C + dim(M’ (7 g(P(V))) + 1. -1 It is said, by an abuse of language, that g (M‘) u C is the inverse image of M’ under g. As the values taken by a linear mapping on a basis (e,) of V can be chosen arbitrarily in V‘, it is seen that there exists a projective linear mapping of P(V) into P(V’) taking arbitrary values at the points x(e,). But (even when g is everywhere defined) giving g(x(?,)) does not determine g uniquely (Exer- The composition of two projective mappings which are bijections is a projective mapping; so is the inverse mapping of such a bijection. The bijec- tive projective mappings of a projective space P(V) onto itself thus form a group, called the projective group of P(V) and denoted by PGL(V) ; we write PGL,(K) or PGL(n, K) instead of PGL(KC). 2 cise 10). Remark. In a projective space P(V) over a field K, let H = P(W) be a hyperplane. There exists a bijective linear mapping f of V onto K, x W such that f (W) = W; let g be the projective mapping obtained from f by passing to the quotients. It has been seen (no. 8) that the complement of P(W) in P(Ks x W) can be identified with an affine space whose translation space is W. When P(V) is identified with P(Ks x W) by means of g, it is said that H has been taken as hyperplane at injnity in P(V) ; the complement of H in P(V) is then identified with an affine space whose translation space is W. 336 337 LINEAR ALGEBRA 3 10.2 MATRICES OVER A COMMUTATIVE GROUP A subfamily M' = (mrK)(L,K)E~xL of a matrix M = (mLK)(r,K)EIxK, whose indexing set is the product of a subset J of I and a subset L of K, is called a submatrix of the matrix M; it is said to be obtained by suppressing in M the rows of index t $ J and the columns of index K # L; conversely, M is said to be obtained by bordering M' with the rows of index t $ J and the columns of index K # L. DEFINITION 2. The transpose of a matrix M = (mrK)(L,IOEIxK, denoted by tM, is the matrix (m:L)(K,L)EKxI over H given by m:, = m,, for all (t, K) E K x L. It follows from this definition that the transpose of a matrix of type (I, K) is a matrix of type (K, I) and that (1) t(tM) = M. 2. MATRICES OVER A COMMUTATIVE GROUP Let G be a commutative group (written additively). The sct of matrices over G, with the given indexing sets I, K, has a Commutative group structure since it is the set of mappings from I x K to G; this group is written additively, so that if M = (mLK) and M' = (miK) are two of its elements, then M + M' = (mLK + miK); the identity element of this group is therefore the matrix all of whose elemcnts are zero (called the zero matrix). Clearly (2) t(M + M') = tM + tM'. 3 10. MATRICES 1. DEFINITION OF MATRICES DEFINITION 1. Let I, K, H be three sets; a matrix oftype (I, K) with elements in H (or a matrix oftype (I, K) over H) is any family M = (mrK)(L,K)EI xK of elements of H whose indexing set is the product I x K. For all t E I, the family (mr,JKEK is called the row of M of index t; for all K E K, theJamily (m,K)LEI is called the column of M of index K. If I (resp. K) is finite, M is said to be a matrix with a finite number of rows (resp. columns). The set of matrices of type (I, K) over H is idcntified with the product HI K. The names "row" and "column " arise from the fact that, in the casc where I and K are intervals (l,p), (1, q) of N, the elements of the matrix are envisaged as set out in a rectangular array with p rows (arranged horizontally) and q columns (arranged vertically) : mll m12 -.. mm (m21. m,2 . . ..* .m2j mp1 mp2 0. * mPq When p and q are explicit integers sufficiently small for it to be prac- ticable, it is a convention that the above array is a symbol effectively denot - ing the matrix in question; this notation enables us to dispense with the use of indices, it being understood that the indices of an element are deter - mined by its place in the array; for example, when we speak of the matrix (; : ;) we mean the matrix (mi,)lcla2,1Cjc3 such that mll = a, m12 = b, m13 = c, m21 = d, m22 = e, m23 =f. Instead of matrix of type ((1, p), (1, q)), we also say matrix of type (p, q), or matrix with p rows and q columns, if no confusion arises; the set of matrices of type (p, q) over H is sometimes denoted by MPSq(H). Every matrix over H for which one of the indexing sets I, K is empty is identical with the empty family of elements of 13; it is also called the empty matrix. When I = {io} (resp. K = {/lo}) is a set consisting of a singlc element, M is called a row matrix (resp. column matrix) and the row (resp. column) index can then be suppressed in the notation; when I and K are both sets with one element, a matrix of type (I, K) is often identified with the unique element in this matrix. 338 The sum of two matrices is thus only defined if the indexing sets of the rows and the columns are the same for the two matrices. Let H', H" be two sets, G a commutative group (written additively) and f: (h', h") H h'h" a mapping from H' x H" to G. Given two matrices M' = (m' ik ) (l,k)EIXK> M" = (dl)(k,l)EKxL over H' and H" respectively such that the indexing set K of the columns of M' isjinite and equal to the indexing set of the rows of M", the product OfM and .M' viaf, denoted by M'M or f (M, M"), is the matrix (3) over G. The above definition supposes that the indexing set of the columns of M' is equal to the indexing set of the rows of M"; in particular the product WM' has no meaning if I # L. In formula (3) the elements of the same 339 I1 LINEAR ALGEBRA row of M' figure multiplied on the right by the elements of the same column of M"; the multiplication is said to be made "rows by columns ". LetfO be the mapping (h", h') ++ h'h" of H" x H' to G; it follows immedi - ately from the definitions that t(M'M") = tM".tM' (4) where the product on the left (resp. right) hand side is calculated via f (resp. via f O). When H' and H" are themselves commutative groups (written additively) andjis Z-bilinear (4 3, no. l), the distributivity formulae (M' + N')M" = M'M " + N'M " (5) { M(M" + N") = M'M " + M"" are immediately verified, the indexing sets being such that the sums and pro- ducts appearing are defined. Now let HI, H2, H,, H12, H,, and H be commutative groups (written addi- tively), f,, : H, x H, -+ H12, f23 : H, x H, + H23 mappings and j3:H12 x H3+H, ji:H, x H23-tH Z-bilinear mappings; suppose further that, for all x, E Hi (i = 1,2,3) f3(f12('1, 'Z), '3) =fl(xl,f23(x2~ '3)) (which may also be written as above (xlx2)x, = x1(x2x3)); then, if M' = (m;J, M" = (m3, M" = (m:) are matrices over H,, H,, H3 respectively, (6) (M'M")M" = M'(M"M") when the products on the two sides (calculated respectively via flz, f3, f23 and fl) are defined; for by virtue of the hypotheses made. The two sides of (6) are also denoted by M'M"M". Analogous conventions are made for products of more than three factors. Remark. The above formulae extend to a more general situation. To be pre- cise : G,, where each G,, is a commutative group written additively; then the sum M + M' may be defined when, for each ordered pair (t, K), mrr E G,, and m:, E GLK. (a) Suppose H = (L, K)q MATRICES OVER A RING 3 10.3 (b) Let I, K, L be three sets with K assumed finite and let H' = u Hik, H" = u HiI, H = u Hi, be three sets; suppose that each Hi, is a commutative group written additively and for each triple (i, k, I) let (i, k)E 1 x K (k. I) E K x L (1. I) E I x L be a mapping. Then if M' = (m&)({,k)ErxK, M" = (m~l)(k,l)EKxLarematrices such that mlk E and mi, E Hil for all i, k, 1 we can define the product MM via thef,,,. We leave to the reader the task of writing down and proving the formulae analogous to (4), (5) and (6). 3. MATRICES OVER A RING The most important matrices in Mathematics are matrices over a ring A. The set A' of matrices over A corresponding to indexing sets I, K then has canonically an (A, A)-bimodule structure (4 1, no. 14). For every ordered pair (i, k) E I x K, let E,k be the matrix (ajI) such that atk = 1 and a,, = 0 for (j, I) # (i, k); the E,k are called the matrix units in the set of matrices A' K; if I and K are finite, they form the canonical basis of this set for its left or right A-module structure (4 1, no. 11). Clearly 'Eik = Eki. Unless otherwise mentioned, the product M'M " of two matrices over A (assumed to be defined) will always be understood to be relative to the multiplication (x, y) H xy in A (or, as is also said, will be "calculated in A"). Then we have (no. 2) the associativity and distributivity formulae (7) (XU) z = X( YZ) X(Y + 2) = XY + xz { (X + Y)Z = xz + YZ for three matrices X, Y, Z over A, whenever the sums and products appear - ing in these formulae are defined. In particular, if Eik (resp. ELl, E;) are the matrix units in AIxK (resp. AKxL, ,IxL) respectively, with I = (l,p), K = (1, q), L = (1, r), we obtain ;he formulae (9) EikE,!, = 0 if k # j EikEL, = E:. Let Ao be the opposite ring of A and let a * b ( = ba) denote the product of a and b in AO; then, for two matrices X, Y over A whose product is defined, "(XY) = fY * tx (10) 34 1 340 LINEAR ALGEBRA I1 3 10.4 member that the index i is a row index, to adjoin to it a column index taking only one value and write the matrix M(x) as (xlo). We now consider two (left or right) A-modules E and F with bases (ei)iEI and (fk)ksK respectively; let (f:) be the family of coordinate form corresponding to ( fk). For a linear mapping u of E into F, we shall define the matrix of u with respect to the bases (el), (fk) in each of the following cases: MATRICES AND LINEAR MAPPINGS (D) E and F are right A-modules, u is A-linear. (G) E and F are left A-modules, u is A-linear. In what follows, we shall attach the letter (D) (resp. (G)) to formulae applying to right (resp. left) modules. DEFINITION 3. In each of the above two cases, the matrix of u with respect to the bases (el), (fk) is the matrix M(u) = (Ukl)(k, f)EK such that (14) ukl = f? (u(el)) which is written respectively as where on the right hand side tY and tX are considered as matrices with ele- ments in AO; when A is commutative, then (1 1) ‘(XU) = tY.tX PROPOSITION 1. Let A, B be two rings and M = (mik)O,k)EIxE and M’ = (m;k)(i,k)EIXK two matrices with Jnite indexing sets over an (A, B)-bimodule G. Suppose that for every matrix unit L = (a,)iEI with one row and elements in A and every matrix unit C = (b,) with one column and elements in B, L . M. C = L . M . C (the products being calculated via the external laws of the (A, B)-module G) ; then M = M’. If L is taken to be the matrix unit (a,) with a, = 1, a, = 0 for s # i, and C the matrix unit (6,) with bk = 1, b, = 0 for t # k, the products L. M.C and L . M’ . c are matrices with a single element respectively equal to m,k and mik. Let A, B be two rings and o:A -+ B a homomorphism. For every matrix M = (mJ over A, we shall denote by o(M) the matrix (o(mLK)) over B; clearly o(aM) = o(a)o(M), o(Ma) = o(M)o(a) for a E A, also O(~M) = t(o(M)) and o(M + M’) = o(M) + o(M’) (12) o(MM’) = o(M)o(M’) when the operations considered are defined, the products on the left and right hand sides of (12) being calculated in A and B respectively. When o is denoted by x H xu, we write Mu instead of o(M). Consider in particular an anti-endomorphism o of A, that is a homomorphism of A to the opposite ring AO, or a mapping of A into itself such that for all a, a‘ in A; then, for two matrices M, M‘ over A whose product MM’ is defined, (13) where the products on the two sides are calculated in A; this follows immedi - ately from (10) and (12). o(a + a’) = .(a) + o(a’), o(aa‘) = o(a’)o(a) o(MM‘) = t(o(tM’) . o(’M)) 4. MATRICES AND LINEAR MAPPINGS Let A be a ring and E a (right or left) A-module admitting a basis (el)iEI. For every element x E E, the matrix of x with respect to the basis (e,), denoted by M(x) or x (or sometimes simply x when no confusion can arise), is the column matrix consisting of the components x1 (i E I) of x with respect to (e,) (3 1, no. 11); in calculations it will sometimes be convenient, in order to re- 342 The column of M(u) of index i is therefore equal to M(u(e,)). Clearly if u, v are two linear mappings of E into I: and M(u), M(v) their matrices with respect to the same bases, then M(u + v) = M(u) + M(v) (15) (16) M(V) = YMb) and for every element y of the centre I’ of A. In other words, once the bases (el), ( fk) are fixed, the mapping u H M (u) is a I’-module isomorphism of Hom,(E, F) onto a subset of the set AK I, equal to AK I if K isfinite. PROPOSITION 2. Suppose I and K are finite. For every element x E E, the matrix M(u(x)) with respect to the basis (fk) is given by the formula (17 D) (17 G) M(U(X)) = M(u) .M(x) tM(u(x)) = tM(x) .tM(U). we verify for example (17 GI. Let x = xioei, u(x) = F ykofk with 343 LINEAR ALGEBRA I1 4 10.4 Remarks. (2) Let E arid F be two left A-modules with bases (e,)IEI and (fk)kEK respectively. For every A -linear mapping u:E ---f F, by (14 G), u(et) = Ukffk; these relations can also be interpreted by saying that the column matrix (u(ei)161) with elements in F is equal to the product 'M(u) . (fk), where (fk)kEK is considered as a cohnn matrix with elements in F and the product is calculated for the mapping A x F -+ F defining the law of action on the A-module F (no. 2). (3) Let A, B be two commutative rings and u: A --f B a ring homomorphism. In the notation of Proposition 3, (el @ 1) and (fk @ 1) are respective bases of E(B) = E @A B and F(B) = F @A B (5 5, no. 1, Proposition 4); moreover, if (e:) and (f:) are respectively the dual bases of (e,) and (fk), then (e: @ 1) and (f: @ 1) are respectively the dual bases of (el @ 1) and (fk @ 1) (4 5, no. 4). For every A-linear mapping u: E + F, let M(u) and M(u @ 1) be the matrix of u with respect to (e,) and (fk) and the matrix of the B-linear mapping u @ 1 with respect to (el @ 1) and (fk @ 1). It follows from 4 5, no. 4, formula (20) that M(u @ 1) = a(M(u)). MATRICES AND LINEAR MAPPINGS y,, = 4 x&k,. In order to bring the two indices i along side one another, we consider the transpose matrices tM(x) = (x&), where xAi = xio and t'(u) = (u6), where u;k = Uk{; then yk0 = of index k of the matrix with one row 'M(x) .'M(u), whence (17 G). x&u;, and the right hand side is the element When A is commutative, (17 G) reduces to (17 D) by formula (4) of no. 2. COROLLARY. Let E, F, G be three right (resp. left) modules over a ring A, (ei)lEI, ( fk)kEK, (gJIEL respectivejnite bases of E, F, G, u:E + F, v:F + G two linear mappings, M(u) the matrix of u relative to the bases (e,), (fk), M(v) the matrix of u relative to the bases ( fk), (gl) and M (u o u) the matrix .f u 0 u relative to the bases (ei), then (18 D) (18 GI M(u 0 u) = M(v)M(u) 'M(v 0 u) = 'M(u)'M(v). We prove for example ( 18 G) . For all x E E, by ( 1 7 G) : 'M(x) .'M(v 0 U) = 'M(u(u(x))) = 'M(u(x)) .'M(v) = 'M(x) .'M(u) .'M(v) by associativity; the corollary then follows from no. 3, Proposition 1 since the matrix tM(x) with one row is arbitrary. Remark (1). Formula (1 7 D) can be considered as a special case of (18 D). For there corresponds canonically to every x E E the linear mapping e,:Ad-+ E mapping every a E A to xa (5 2, no. 1). It is immediate that the matrix M(8,) with respect to the basis 1 of Ad and the basis (e,) of E is just the matrix M(x) ; similarly M(Ou(x)) = M(u(x)) and formula (17 D) can therefore be considered as a translation of the relation eu(x) = u ex. PRO POSITION 3. Let E, F be two right (resp. left) A-modules and (ei)iEI, (fk)kEK jinite bases ofE and F respectively. For every linear mapping u of E into F, let M(u) be the matrix ofu with respect to the bases (e,) and ( f,). Then the matrix of% : F* 3 E* with respect to the dual bases (f E) and (e:) is equal to 'M(u). E is canonically identified with its bidual E** and (e,) with the dual basis of (e:) ; then (supposing for example that E and F are right modules) <"(f EL el> = u:, u(e,)>, whence the proposition. 344 Consider a system of ajnite number of right scalar linear equations in a jinite number of unknowns with a,,, xi, 6, in A. Let (el)lsI, ( fk)kEK be the canonical bases of E = A: and F = A:; the system (19) is equivalent to the equation u(x) = b, where x = 7 e,x,, b = fkb, and u:E -+ F is the linear mapping such that the matrix M(u) with respect to the bases (e,) and (fk) are equal to A = (aki)tk,i)EKuL. This matrix is called the matrix of the system of linear equations (19). Recall (5 2, no. 8, Remarks 2 and 3), that, writing c, = 5 fkak,, the system (19) is equivalent to the unique vector equation (20) 7 clxl = 6, and as ct is the column of index i in the matrix A, we see that to say that the system (19) admits a solution amounts to saying that the matrix b = (bkO) with one column is a linear combination of the columns of the matrix A. I We leave to the reader the task of formulating the analogous definitions and remarks for systems of left linear equations. 345 LINEAR ALGEBRA I1 4 10.6 Suppose now that I and K are finite and further that each of the E, (resp. F,) admits aJnite basis. It amounts to the same to say that E (resp. F) admits a basis (er)rER (resp. and that R (resp. S) admits a partition (RJiE1 (resp. (Sk)keK) such that for all ~EI (resp. k EK), (er)tER, is a basis of E, (resp. ( fJsESk is a basis of F,). Then, if X = M(u) is the matrix of u with respect to the bases (er),.eR and (fs)ses, the matrix Xki = M(uki) with respect to the bases (er)r ER, and sk is just the submatrix of X obtained by suppress- ing the rows of index s $ sk and the columns of index r $ R,. Thus we define a one-to-one correspondence MATRIX OF A SEMI-LINEAR MAPPING (24) (Xki)(k, OEK x I between the set of matrices of type (S, R) with elements in A and the set of matrices ofmatrices (xk{)(J& I of type K x I, where each x,, is a matrix over A oftype (S,, R,). Suppose that further G admits a finite basis (gJtET and that T = (TJIEL is a partition of T such that, for each 1 E L, (g,),,,, is a basis of GI; let Y = M(u) be the matrix of u with respect to the bases (jJSEs and (gJtsT, Ylk = M(qk) that of ulk with respect to the bases (jJSESk and (gt)teTI, Z = M(w) the matrix of w = u 0 u with respect to the bases (er)rER and (gJtET and Z,, = M(w,,) that of wl, with respect to the bases (er)rERf and (gJtETI; then it follows from (23) that the submatrices Z,, of 2 = YX are given by (25) zli = ylkxkL in other words, the one-to -one correspondence (24) transform products into products when all the products in question are defined (products of matrices of matrices being defined in the sense of no. 2, Remark) ; when the submat- rices Z,, of the product YX are calculated thus, this product is said to be carried out “in blocks”. This name arises from the fact that, when I = (I,#) and K = (1, q), the table representing the matrix X is envisaged as divided into “blocks” forming an “array of matrices” XI, XI, * * - XI, 7; ..a; X,, Xq, . . . Xq, which is considered as a symbol dcnoting X when p and q are specific integers sufficiently small for this to be practicablc. 6. MATRIX OF A SEMI -LINEAR MAPPING Let A, B be two rings, o:A+B a homomorphism of A into B, E a right (resp. left) A-module with basis (eJiGI and F a right (resp. left) B-modulewith 5. BLOCK PRODUCTS The definitions of no. 4 can be generalized as follows. Let E be a (right or left) A-module, the direct sum of a family (Ei)i,I of submodules. For all x E E, let x = 2 xi with x, E E, for all i E I; we shall say that the column matrix M(x) = (xi)iEI with elements in E is the matrix of x with respect to the decomposi - tion Let F be another A-module (E and F being both right A-modules or both left A-modules) and suppose that F is the direct sum of a family (F,),,, of submodules. For all u E Hom(E, F) and all xi E E,, let u(xi) = 2 u ki (2) with ukl(x,) E F, for all k E K; then uki E Hom(E,, F,) ; we shall say that the matrix M(u) = (uki)(k,i)eKxI of type (K, I) with elements in the set H the sum of the Hom(E,, Fk) is the matrix of u with respect to the decompositions (E,) and (F,) of E and F as direct sum. With these definitions, it is obvious that if u, u are two A-linear mappings of E into F then, for matrices with respect to the same decompositions as direct sums LEI of E as a direct sum. (21) M(u + u) = M(u) + M(u), M(yu) = yM(u) for every element y of the centre of A (no. 2, Remark). Moreover, the definition of the u,, shows that, if K is finite, we can write (22) M(u(x)) = M(u) .M(x) where M(u(x)) is the matrix of ~(x) with respect to the decomposition (F,), the product on the right hand side of (22) being calculated for the mappings (t, z) H t(z) of Hom(E,, F,) x E, into Fk (no. 2, Remark). of submodules, so that there corresponds to every A-linear mapping v:F --> G a matrix M(u) = (vlk) with respect to the decompositions (FJ and (GJ. If I, K and L are jinite, then Let G be a third A-module, the direct sum of a family (GI) I (23) M(v 0 u) = M(u) .M(u) where the left hand side is the matrix (w,,) of w = v 0 u with respect to the decomposition (E,) and (GI) and the product on the left hand side is calcu - lated for the mappings (t, s) ++ to s of Hom(F,, GI) x Hom(E,, F,) into Hom(E,, G,) (no. 2, Remark). This is just formula (32) of 3 1, no. 8, ex- pressed in terms of matrices. Finally, if I and K are assumed to be finite, E* (resp. F*) is canonically identified with the direct sum of the modules ET (resp. F,*) (9 2, no. 6, Proposi - tion 10). Then it is immediately verified that the matrix of ,u with respect to the decompositions (F;) and (ET) is just (tUki)(k.L)eK I. 5 316 347 I1 LINEAR ALGEBRA basis ( fk)kEK. Let u:E + F be a semi-linear mapping relative to a and u(el) = JKfkuki (resp. u(ei) = ksK 1) Ukifk , where the Uki are therefore elements of B; by definition, the matrix M(u) = (uk[) of type K x I is also called the matrix of u with respect to the bases (el) and ( fk). By the same calculation as in Proposition 2 of no. 4, it is immediately verified that for all x E E, if I and K arejnite, (26 D) (26 G) M(u(x)) = M(u) *o(M(x)) tM(u(x)) = a(tM(x)) .tM(u)). (resp. Let C be a third ring, 7 :B --f C a homomorphism, G a right (resp. left) C-module with basis (g,),,, and v a semi -linear mapping of F into G relative to r; if M(v) is the matrix of u with respect to (fk) and (gl) and M(v o u) the matrix of v o u relative to (e,) and (g,), then, if I, K and L are finite, (27 D) (27 G) M(v O ~).7(4M(X))) = M(v(u(x))) M(v 0 u) = M(v). r(M(u)) tM(v 0 u) = r(tM(u)) .tM(u)). (resp. To show for example (27 D), note that for all x E E, by (26 D), = M(v). T(M(U(4)) = M(v) ..r(M(u)). 7(4M(x))), whence (27 D) by Proposition 1 of no. 3. Suppose finally that 5:A + B is an isomorphism; then recall that h:F* -+ E* is a semi -linear mapping relative to 0-l (9 2, no. 5); when I and K are finite, the matrix t~ with respect to the dual bases (f:) and (ef) is given by (28) M(tu) = c-'(tM(u)) for, by definition, supposing for example that E and F are right modules, ('~(f:), eiy = (~;r", u(ei)) when ts is denoted by x H xu. Remark. Let A be a ring and a an anti-endomorphism of A (no. 3); consider the two following situations: (GD) E is a left A-module, F a right A-module and u a Z-linear mapping of E into F such that u(ax) = u(x)o(a) for aE A, XEE; in other words, u is a semi-linear mapping relative to D of the right Ao-module E into the right A-module F. (DG) E is a right A-module, F a left A-module and u a Z-linear mapping of E into F such that u(xa) = c(a)u(x) for a E A, x E E; in other words, u is a semi-linear mapping relative to c of the left Ao-module E into the left A-module F. SQUARE MATRICES 5 10.7 In the two cases, the matrix M(u) of u relative to bases of E and F has its elements in A; if these bases are finite, then, for all x E E, we have the respective formulae (17 GD) (17 DG) tM(u(x)) = o(tM(x)) .tM(u), the products on the two sides being calculated in A. This follows immedi - ately from (26 D) and (26 G) respectively. M(u(x)) = M(u) .o(M(x)) 7. SQUARE MATRICES DEFINITION 4. A matrix whose rows and columns have the same indexing set is called a square matrix. A square matrix with n rows and n columns is called a matrix of order n. Remark. It should be noted that a matrix for which the indexing sets of the rows and columns have the same cardinal but are not identical, must not be considered as a square matrix; in particular, the product of two such matrices over a ring is not deJined. Clearly addition and multiplication of square matrices over A with a finite set as indexing set of the rows and columns, define on the set of these matri - ces a ring structure because of formulae (7), (8) and (9) (no. 3); the matrix (st,), where sij is the Kronecker index (for i E I, j E I), is the unit element of this ring and is denoted by I,, or 1, when I has n elements. When I = (1, n), the ring of matrices thus defined is denoted simply by M,(A); the group of invertible elements of M,(A) is denoted by GL,(A) or GL(n, A). For a square matrix U = (a,?) of order n over A to be right (resp. left) invertible, it is necessary and sufficient that, for every system (bJIGiG,, of i elements ofA, the system oFn equations in n unknowns j=1 5 aijxj = b, (1 < i < n) have one solution (xi) in A. Let I be a finite indexing set, A a ring and E a right (resp. left) A -module with basis (eJtpI. For every endomorphism u of E, the matrix M(u) of u with respect to the two bases identical with (e,) is a square matrix; more briefly, it is called the matrix of u with respect to the basis (ei). Suppose that I = (1, n). The mapping u H M(u) (resp. zi - tM(u)) is an isomorphism of the ring End,(E) onto M,(A) (resp. onto the opposite ring of M,(A), as follows from formulae (18 D) (resp. 18 G)) (no. 4). The invertible 349 348 I1 LINEAR ALGEBRA SQUARE MATRICES In particular, for two diagonal matrices of order n, 9 10.7 elements of the ring M,(A) called invertible matrices, correspond under the mapping u H M(u) (resp. u ++ tM(u)) to the automorphisms of E; the group GL(n, A) is therefore canonically identified with the group GL(Ai). If u is an automorphism of E, its contragredient ii is an automorphism of the left (resp. right) A-module E*, such that u' = (%-l = "u-') (3 2, no. 5, Definition 6); if M(6) is the matrix of u' with respect to the dual basis (e:), then, by virtue of Proposition 3 (no. 4), (29) M(ii) = (tM(21))-1 = tM(u-1). For every invertible matrix X, it therefore follows that "X-1) = (")-I; this matrix is also denoted by tX-l and called the contragredient of the matrix X. Let (I be an automorphism of the ring A; for every semi-linear mapping u:E -+ E relative to c, the matrix M(u) of this mapping with respect to a basis (e,) of E is also a square matrix. It follows immediately from (27 D) (no. 6) that, if u is bijective, then M(u-1) = (D-l(M(U)))-l. Let E be an A-module which is the direct sum of a finite family (EJtE1 of submodules; for every endomorphism u of E, the matrix M(u) = (uk,) of u with respect to the two decompositions of E identical with (E,) (no. 5) is a square matrix of linear mappings. In order that u(E,) c E, for all i E I, it is necessary and sufficient that ukt = 0 for k # i. When 1 = (1, n), the rela- tions u(E,) c E, + Et+, + ... + E, (1 < i < n) are equivalent to the relations ukt = 0 for k < i. Examples ofsquare matrices. I. Diagonal matrices. In a square matrix M = (mJct,K)EIXI, the elements both of whose indices are equal are called diagonal elements and the family (mll)LEI is called the diagonal of M; a square matrix M = (mlK) over a ring, whose elements other than the diagonal elements are zero, is called a diagonal matrix. For every family of elements of a ring A, the diagonal matrix (qK) such that mrl = at for all t E I is denoted by diag(a,),.I (or diag(a,, a2, . . . , a,) when I = (1, n)). In the set M,(A) of square matrices of order n over A, the unit matrix I, is a diagonal matrix and also every mul - tiple aI, = Ina of this matrix by a scalar a (the diagonal matrix (called scalar) all of whose diagonal elements are equal to a). For every family (dJlGt4, of elements of A and every matrix X = (xi,) of type (n, q) (resp. (p, n)) over A, writing D = diag(d,), (30) XD = (xt,dj). diag(a,) + diag(b,) = diag(a, + b,) diag(a,) . diag(b,) = diag(a,b,). The diagonal matrices therefore form a subring of M,(A) isomorphic to the product ring A"; the scalar matrices form a subring isomorphic to A. 11. Permutation matrices; monomial matrices. Let x be any permutation of a finite set I and let (e,),,, be the canonical basis of the A-module E = A:; there exists one and only one endomorphism u, of E such that, for all i E I, u,(et) = en(,! (3 1, no. 11, Corollary 3 to Proposition 17). For all i E I, the column of index i in the matrix M(u,) with respect to the basis (ei) has all its elements zero except the one in the row of index x(i), which is equal to 1. By an abuse of language, M(u,) is called the matrix of the permutation x. It is immediate that for any two permutations 0, T of I, u,, = u, o u, and that for the identity permutation E, u, is the identity; the mapping x H M (u,) is therefore an isomorphism of the symmetric group 6, onto the group of permuta - tion matrices. (31) Each row and each column of a permutation matrix contains only a single element #O. A finite square matrix R over a non-zero ring A, with this property, is called a monomial matrix; let ri be the unique element # 0 in the column of R of index i and let n(i) be the index of the row where this element is; clearly n is a permutation of the indexing set I and R = M(u,)D, where D = diag(ri). 111. Triangular matrices. In the ring M,(A) of square matrices of order n over a ring A, any matrix (a,,) such that a,, = 0 for i > j (resp. i < j) is called an upper (resp. lower) triangular matrix; it is also said that such a matrix has only mos below (resp. above) its diagonal. It is immediately established that the upper (resp. lower) triangular matrices form a subring S (resp. T) of M,(A), S n T being obviously the ring of diagonal matrices. The set S' (resp. T') of matrices in S (resp. T) whose diagonal elements are invertible is a multiplicative group of matrices called the upper (resp. lower) total triangular group, this follows immediately from 3 1, no. 11, Remark 5. The set s1 (resp. TI) of matrices in S (resp. T) whose diagonal elements are all equal to 1 is a subgroup of the above group, called the upper (resp. lower) strict triangular group, and every matrix M E S' (resp. M E T') whose diagonal is (di)? my be written as M = DM, = MiD, where D = diag(d,) and MI and MI matrices belonging to S, (resp. Tl). IV. Diagonal and triangular matrices of matrices. Let (Ik)lCkGp be a par- tition of the finite set I; every square matrix over a ring A with indexing set I can be written in the form of a square matrix of matrices corresponding 35 1 350 I1 LINEAR ALGEBRA to the same partition (Ik) of the indexing set of the rows and the indexing set of the columns (no. 5) CHANGE OF BASES 3 10.8 (32) ‘I LP1 XP2 . .* x,, ...... \I where each Xkk is a square matrix with Ik as indexing set of the rows and columns. With this notation, (32) will be called a diagonal (resp. upper triangular, resp. lower triangular) matrix of matrices if all the matrices Xij such that i # j (resp. i > j, resp. i < j) are zero. The interpretation of endomorphisms u whose matrix is a diagonal, resp. triangular, matrix of matrices has been seen earlier, by considering the corresponding matrix M(u) of linear mappings. The lower triangular (resp. upper triangular, diagonal) matrices of matrices for a given partition (Ik) of I form subrings of the ring of matrices A’”’. In particular, the ring of diagonal matrices of matrices relative to the partition (I,) is isomorphic to the product k=l fi End,@,). 8. CHANGE OF BASES PROPOSITION 4. Let E be a right A-module with Jinite basis (e,) , ,, of n elements. For a family of n elements el = 2 ejaji (1 < i < n) to be a basis of E, it is necessary and suficient that the square matrix P = (aj,) of order n be invertible. P is just the matrix, with respect to the basis (e,) of the endomorphism u of E defined by u(e,) = e; (1 < i < n). Now, for u to be an automorphism of E, it is necessary and sufficient that (u(e,)) be a basis of E (4 1, no. 11, Corollary 3 to Proposition 17) ; whence the proposition. The invertible matrix P is called the matrix of passage from the basis (e,) to the basis (e;). It can also be interpreted as the matrix of the identity mapping 1, with respect to the bases (e;) and (e,) (in that order); then clearly the matrix of passage from the basis (e;) to the basis (e,) is the inverse P-l of P. PROPOSITION 5. Let (e,), (e;) be two bases of n elements of E and P the matrix of passage from (e,) to (e;). If (e:) and (el*) are the respective dual bases of (ei) and (el), the matrix of passage from (e:) to (e;*) is the contragredient tP-l of P. The transpose of the identity mapping 1, is the identity mapping I,* ; by Proposition 3, no. 4, the matrix of 1,. with respect to the bases (e;*) and (eT) (in that order) is the transpose of the matrix of 1, with respect to the bases (e,) and (e;) (in that order), that is the transpose of P-’. PROPOSITION 6. Let E and F be two right A-modules, (e,) and (ei) two bases of E n y=l with n elements, (5) and (f,’) two bases of F with m elements, P the matrix of passage from (e,) to (e;) and Q the matrix ofpassage from ( fj) to (j,’). For every linear mapping u ofE into F, let M(u) be the matrix ofu with respect to the bases (e,) and (A) and M’(u) the matrix ofu with respect to the bases (e;) and (f(); then M’(u) = Q-’M(u)P. (33 D) We may write u = 1, 0 u 0 1,. Formula (33) follows immediately from no. 4, Corollary to Proposition 2 when the matrix of 1, is taken with respect to (e;) and (e,), that of u with respect to (e,) and (A) and that of 1, with respect to (A) and (f2. COROLLARY 1. If u is an endomorphism of E and M(ri) and M‘(u) its matrices with respect to the bases (e,) and (e;) respectively, then (34 D) M’(u) = P-lM(u)P. COROLLARY 2. If M(x) and M‘(x) are the matrices with one column of the same element x E E with respect to the bases (e,) and (e;) respectively, then (35 D) M(x) = P.M‘(x). This is a special case of Proposition 6, applied to the mapping 0,:a H xu of A, to E (no. 4, Remark 1). Formula (35) is equivalent to for the elements xi and xi of the matrices M(x) and M’(x) respectively. Formulae (36 D) are called formulae of change of coordinates. Observe that they express the components of x relative to the “old” basis (e,) as functions of the components of x relative to the “new” basis (e;) and the elements of P, that is the components of the “new” basis relative to the “old” basis. Remarks. (1) We now start with a 1eJt A-module E with two bases (e,), (ei), each with n elements; if we write e; = 5 ajiei, P = (aji) is also called the matrix ofpassing from (e,) to (e;) ; it is also the matrix of the automorphism of E such that u(e,) = el, with respect to the basis (e,) and also the matrix of 1, with respect to the bases (el) and (e,) in that order. The above results then hold with only the following modifications: formulae (33 D) to (36 D) are respectively replaced by j=1 (33 G) (34 G) (35 G) tM’(u) = tP.’M(u) .‘&-I tM’(u) = ‘P.’M(u) .tP-l ‘M(u) = tM’(~) .tP. 352 353 LINEAR ALGEBRA I1 EQUIVALENT MATRICES ; SIMILAR MATRICES 3 10.9 Clearly the relation “X and X’ are equivalent” is an equivalence relation (Set Theory, 11,s 6, no. 1) on the set Am” of matrices of type (m, n) over A, which justifies the terminology. With this definition, Proposition 6 of no. 8 can be stated by saying that when the bases are changed in two right A-modules E, F (with finite bases), the matrix of a linear mapping u: E -+ F with respect to the new bases is equivalent to the matrix of u with respect to the old bases. Conversely, if relation (40) holds and u : A: -+ A! is a linear mapping whose matrix is Xwith respect to the respective canonical bases (e,) and (f,) of A: and A:, then X’ is the matrix of u with respect to the bases (el) and (f;) such that Q is the matrix of passage from (e,) to (pi) and P-l the matrix of passage from (fr) to (36 G) j=l (2) Under the hypotheses of Proposition 4, consider an element x* E E* ; as the matrix of passage from (e:) to (el*) is tP-l (Proposition 5), for the matrices M(x*) and M’(x*) of x* with respect to these two bases respectively, tM(x*) = tM’(x*) . P -1 or also (37 D) tM’(x*) = tM(x*) .P which is equivalent to the system of equations (38 D) xi* = j=1 C ..Fa,, (1 < i < n) for the elements (x:) and (xi*) of the matrices M(x*) and M’(x*). The corresponding formulae for a left A-module E are (37 G) M’(x*) = tP.M(X*) (38 GI j=1 n (3) Let A, B be two rings, o:A -+ B a homomorphism of A into B, E a right (resp. left) A-module, (e,), (e;) two bases with n elements of E, F a right (resp. left) B-module, (fr), (f;) two bases with m elements of F and P (resp. Q) the matrix of passage from (e,) to (e;) (resp. from (fr) to (f;)). For every semi-linear mapping u: E -+ F, relative to Q, let M (u) be the matrix of u with respect to (e,) and (5) and M’(u) its matrix with respect to (e;) and (f;). Then (39 D) M’(u) = Q-lM(u)o(P) (resp. (39 G) (27 D) and (27 G) (no. 6). tM’(u) = CS(~P) .tM(~) .“-’). The proof is the same as that for (33 D) and (33 G), this time using formulae 9. EQUIVALENT MATRICES; SIMILAR MATRICES DEFINITION 5. Two matrices X, X’ with m rows and n columns over a ring are called equivalent ;f there exists an invertible square matrix P .f order m and an invert - ible square matrix Q of order n such that (40) X’ = PXQ. 354 Examples of equivalent matrices. (1) Two matrices X = (xi,) and X’ = (x&) with m rows and n columns “dzxer only in the order of their rows” if there exists a permutation u of the interval (1, m) of N, such that for every ordered pair of indices (i, j), xi, = xu(,),, (we also say that X‘ is obtained by per- forming the permutation u-l on the rows of X). The matrices X and X’ are then equivalent, for X’ = PX, where P is the matrix of the permutation u-l (cf. no. 7, Example 11). Similarly X and X’ are said to dzffer only in the order of their columns if there exists a permutation T of (1, n) such that x;, = for every ordered pair of indices (i,j), X and X’ are also equivalent, for x’ = XQ where Q is the matrix of the permutation ‘F. Note that in the above notation P is the matrix of passage from a basis (fr)l<,C,,, to the basis (fu-i(,,)16,6m and Q the matrix of passage from a basis 1 < 16 n to the basis (eT(1)) 1 6 1 C ?Lo (2) Let j, k be distinct elements of (1, n) and let a E A. Suppose that for 1 < i < m, xi, = xij + xika and xi, = xl, for j # 1 and 1 < i < m; x’ is said to be derived from X by adding to the column of X of index j the column of index k multiplied on the right by a. In this case X and X’ are also equivalent: for if Q = I, + aEk, (an invertible triangular matrix, as seen in no. 7), then X’ = XQ. Similarly, let h, i be two distinct elements of (1, m) and a an element of A; if X‘ is derived from X by adding to the row of X of index i the row of index h multiplied on the left by a, X and x’ are equivalent, for X’ = PX, where P = I,,, + aE,,. (3) Finally, if, for a given index j, xi, = X,,C for 1 < i < m, where c is invertible and xi, = xll for 1 < i < m and 1 # j, X and X’ are equivalent; for X‘ = XQ, where Q is the matrix diag(ak) with a, = c, ak = 1 for k # j. Then X’ is said to be derived from X by multiplying the column of X of indexj on the right by a. Similarly, if X’ is derived from X‘ by multiplying the row of X of index i on the left by an invertible element c E A, X‘ and X are equiva - lent, for X’ = PX where P is the matrix diag(b,) with bl = c, bh = 1 for h # i. 355 T I1 LINEAR ALGEBRA DEFINITION 6. Two square matrices X, X’ of order n over a ring A are called similar if there exists an invertible square matrix P of order n such that X’ = PXP-1 Clearly the relation “X and X‘ are similar” is an equivalence relation on M,(A) meaning that X and X’ are transformed into one another by an inner automorphism of this ring. With this definition, Corollary 1 to Proposition 6 of no. 8 can be stated by saying that when the basis of an A-module E (with a finite basis) is changed, the matrix of an endomorphism u with respect to the new basis is similar to the matrix of u with respect to the old basis. (41) Remarks. (1) Two square matrices which difrer only in the order of their rows (or the order of their columns) are equivalent, but in general not similar. A matrix similar to a square matrix X = (xu) can be obtained by performing the same permutation a-l on the rows and columns, that is by considering the matrix X‘ = (xi,), where xi, = x,(,),~(~) for every ordered pair of indices; for if X is the matrix of an endomorphism u of A: with respect to a basis (el)16lcn, X’ is the matrix of u with respect to the basis (2) Let X and X’ be two square matrices of order n which can be written (%(iJ 1 c i < R. in the form of diagonal matrices of square matrices (no. 7, Example IV) : TENSOR PRODUCT OF MATRICES OVER A COMMUTATIVE RING 9 10.10 finite bases (e,,JheL, (fJUaM, (u,)~~~; let A = (aph) be the matrix of C$ with respect to (eh) and (u,), B = (b,,) that oft) with respect to (f,) and (u,). For every ordered pair (A, p) E L x M = N, let ghu = eh @f,; for every ordered pair (p, c) E R x S = T let w,, = u, @ v,; the ghn then form a basis of E @ F and the woo a basis of U @ V (3 3, no. 6, Corollary 2 to Proposition 7). The tensorproduct of A by B, denoted by A @ B, is the matrix X = (xTV)(=, V)ET xN whose elements are given by (42) ~(o, a), (A, u) = ao*bau- Then A @ B is the matrix of + @ t) with respect to the bases (ghJ and (w,,). By definition (4 3, no. 2, formula (3)) (+ €4 t))(g*u) = (+ @ t))kh @fu) = +(ed @ t)(f,) = 2 aohbuu(Uo @ 0,) = 2 aohbauwpa* Definition (42) of the elements of A @ B shows that this matrix corresponds and also the matrix The fact that (+, t)) +> + @ $ is a C-bilinear mapping and formula (9) bijectively with the matrix of matrices (aoAB)(,, (Abau)(a, u) E s x M (no. 5) * of§ 3, no. 5, can be expressed by the identities 1 +B2) =A @Bi +A @B2 B = A1 @ B + A2 @ B (A1 + 4) (43) (“ (cA) @ B = A @ (cB) = c(A @ B) for c E C (44) (45) (A1 63 Bd(A2 @ B2) = (442) @ (BlB2) when the operations appearing are defined. The transpose of a tensor product of matrices is given by (46) t(A @ B) = (tA) @ (‘B). If A and B are invertible square matrices over C, A @ B is invertible and (A @ B)-’ = (A-l) @ (B-l). Let (ei)heL be another basis of E and (fJuEM another basis of F; if P is the matrix of passage from the basis (eh) to the basis (e;) and Q the matrix of passage from the basis (f,) to the basis (fJ, the matrix of passage from the basis (eh @f,) to the basis (ei @fJ is P @ &. If A’ is equivalent (resp. similar) to A and B’ equivalent (resp. similar) to B, then A‘ @ B’ is equivalent (resp. similar) to A @ B. The definition of tensor product of matrices can be generalized in an (47) i corresponding to the same partition of the indexing set (1, n) for X and X‘. If, for 1 < i < p, XI and X,’ are equivalent (resp. similar), then X and X‘ are equivalent (resp. similar): for, if X( = P,X,Q, for 1 < i < p, then X’ = PXQ where as follows from calculating the “block” product (no. 6). Moreover, if Q1 = P<l for all i, then Q = P-l. 10. TENSOR PRODUCT OF MATRICES OVER A COMMUTATIVE RING Let C be a commutative ring, E, F, U, V four C-modules and C$:E -+ U, $:F -+ V two C-linear mappings. Suppose that E, F, U, V have respectively 356 357 I1 LINEAR ALGEBRA obvious way to an arbitrary finite number of matrices over C; in particular we have the associativity formula Q 10.12 MATRICES OVER A FIELD the caye where n > 2. Taking X = E,,, Y = El, with i # li, we obtain g(E,,) = 0; then taking X = E,,, Y = El, with i # j, we find g(E,,) = g(EjJ) ; the proposition follows immediately since the E,, form a basis of M,(C). 12. MATRICES OVER A FIELD The finite matrices with m rows and n columns over a field K are in one-to-one correspondence with the linear mappings of the right vector space E = K: into the right vector space KZ when the matrices of these mappings are taken with respect to the canonical bases of E and F. By definition, the rank of such a matrix X is the rank of the linear mapping u : E -+ F corresponding to it; as this number is by definition the dimension of the subspace u(E) of F, it amounts to the same (identifying the columns of X with the images under u of the canonical basis of E) to give the following definition: DEFINITION 7. Given a matrix X with m rows and n columns over a jield K, the dimension of the subspace of KZ generated by the n columns of X is called the rank of X with respect to K and denoted by rg(X). It can also be said that the rank of X is the maximum number of linearly inde- pendent columns of X (as elements of K:). Obviously rg(X) 6 inf(m, n); for every submatrix Y of X, rg(Y) 6 rg(X). If E and F are two finite-dimensional vector spaces over K and u a linear mapping of E into F, the rank of the matrix M(u) with respect to any two bases is equal to the rank of u. PROPOSITION 9. If the elements of a matrix X with m rows and n columns belong to a subjield KO of ajeld K, the rank of X with respect to KO is equal to the rank of X with respect to K. Let F, be the right vector K O-space generated by the canonical basis of the right vector K -space E = K:; by hypothesis the columns of X belong to E,. Let V, (resp. V) be the vector sub-K O-space of F, (resp. the vector sub-K- space of E) generated by these columns. Then V = V, K (3 8, no. 2, Proposition 2) and hence dim, V = dim,, V,. PROPOSITION 10. The rank of a matrix X over a jield K is equal to the rank of its tranrpose 'X over the oppositejeld KO. In the notation introduced before Definition 7, the rank of u is equal to that of% (Q 7, no. 5, Proposition 10) and the proposition therefore follows from no. 4, Proposition 3. It is thus seen that the rank of X can also be defined as the maximum number of linearly independent rows of X (considering them as elements of the left vector K-space K,"). The square matrices of order n over a field K correspond bijectively with the endomorphisms of E = K: and form a ring isomorphic to the ring 359 I for every partition (I1, 12) of the finite indexing set I. 11. TRACE OF A MATRIX Let C be a commutative ring; for every square matrix X = (xij) over C corres- ponding to the finite indexing set I, the trace of X is the element (49) Tr(X) = 1EI xil. u of E, (50) Tr(u) = Tr(M(u)) M(u) being the matrix u with respect to the basis (e,) ; this follows immediately from 3 4, no. 3, formula (17), when this formula is applied to the endomor- phism x H (x, e;)e, (where 6:) is the dual basis of (el)) ; from this we pass to the general case by linearity. Formula (49) shows that Let E be a C-module admitting a finite basis (el),EI; for every endomorphism (51) for every basis (e,) of E (cf. Q 4, no. 3, formula (1 7)). then (52) Tr(XY) = Tr(YX) as follows from the above and Proposition 3 of Q 4, no. 3; (52) can also be obtained directly, for if X = (xlf), Y = (y,,) (1 < i 6 m, 1 < j 6 n), then If X is a matrix of type (m, n) over C and Y a matrix of type (n, m) over C, (53) Tr(XY) = c 1. I XifYjl by (49). The latter formula proves moreover: PROPOSITION 7. Let C be a commutative ring and for every matrix P E M,(C) let fp be the linear form XH Tr(PX) on M,(C); the mapping PH fp is a C-linear bijection ofMn(C) onto its dual. PROPOSITION 8. If g is a linear form on the C-module M,(C) such that g(XY) = g(YX) for all matrices X, Y in M,(C), there exists one and only one scalar c E C such that g(X) = c .Tr(X) for every matrix X E M,(C). Since the proposition is trivial for n = 1, attention may be confined to 358 11 LINEAR ALGEBRA End,(E) (no. 7) ; corresponding to the automorphisms of E are the invertible square matrices. PROPOSITION 11. Let X be a square matrix of order n over a jeld K. The following properties are equivalent : (a) Xis invertible in M,(K). (b) X is right invertible in M,(K). (c) Xis lgt invertible in M,(K). (d) X is of rank n. This is just a translation of 8 7, no. 4, Corollary to Proposition 9. PROPOSITION 12. For a system ofm linear equations in n unknowns f: al,xj = b, (1 < i < m) over ajield K to have at least one solution, it is necessary and suflcient that the matrix A = (a,,) ofthe system and the matrix B, obtained 6y bordering A with an (n + 1)-th column equal to (b,), be matrices of the same rank. It has been seen (no. 4) that the existence of a solution of (54) is equivalent to the fact that the column (6,) is a linear combination of the columns of A and the proposition therefore follows from 5 7, no. 3, Corollary 4 to Proposition 4. Note that the condition of Proposition 12 is always fulfilled when m = n and A is invertible, that is of rank n (Proposition 11). If x and b then denote the matrices with one column (xi) and (b,) respectively, system (54) is equiva - lent to A. x = b and its unique solution is x = A - l. b. 13. EQUIVALENCE OF MATRICES OVER A FIELD PROPOSITION 13. Let E, F be two jnite-dimensional vector spaces over a Jield K. Ifu:E -+ F is a linear mapping of rank r, there exist bases of E and F such that, with respect to these bases, (54) 1=1 (55) Every matrix of type (m, n) over K and of rank r is equivalent to a matrix of the form The second assertion is trivially equivalent to the first. To show the latter, let dim E = n, dim F = m. The kernel N = u (0) is of dimension n - r (4 7, no. 4, formula (11)); let V be a supplementary subspace of N in E and (e,) , ,, a basis of N. Then the u(ej) (1 < j < r) form a basis of u(E) ; hence there exists a basis ( f,)lcjQm of F such thatf, = u(e,) for 1 < j < r (3 7, no. 1, Theorem 2) (55). -1 , ,, a basis of E such that (e,) , Cr is a basis of V and (e,)r + 360 EQUIVALENCE OF MATRICES OVER A FIELD 8 10.13 and clearly with respect to the bases (e,) and (A) the matrix M(u) is given by (55). COROLLARY . For two matrices over a jeld, of type (m, ti), to be equivalent, it is I necessary and suflcient that they have the same rank. We shall now recover Proposition 13 by another more explicit method. For every ring A, every h E A, every integer m > 1 and every ordered pair of distinct integers i, j in [ 1, m), we write 1 I I (56) Bl,(h) = 1, + AEl, an invertible matrix of order m by no. 8. Lemma 1. Let X = (&,) be a matrix of type (m, n) over a ring A. Suppose that m 3 2 and that there exists an element Ell in thejrst column of X which is invertible in A. Then there exist two invertible square matrices P E M,(A), Q E M,,(A) and a of the form B,,(h) of order m (resp. n) and I I 1, I I matrix Y of type (m - 1, n - 1) over A such that P (resp. Q) is aproduct of matrices I 1 /1 0 ... o\ (57) The matrix B,,(A)X is obtained by adding to the row of X of index i the row of indexj multiplied on the left by A (no. 9, Example 2) ; if Ell is invertible, then there exists h E A such that, for the matrix X' = Bl,(h)X = (ELl), til = 1 ; multiplying X' on the left by suitably chosen matrices Bkl(pk) of order m (for 1 6 k 6 m), a matrix X" = (ELI) is obtained such that = 1, EE1 = 0 for k # 1. Then the matrix obtained is multiplied successively on the right by suit- able matrices Bl,(v,) of order n (2 < j < n) and a matrix is obtained of the form (57). PROPOSITION 14. Let X be a matrix of type (m, n) over a jeld K. If X is of rank r, there exist two invertible square matrices P E M,(K), Q E M,,(K) such that P (resp. &) is a product ofmatrices oforder m (resp. n) ofthe form B,,(h) and , I ~ 11 0 ... 0 0 ... o\ (58) 0 1 ... 0 0 ... I 0 0 ... 0 0 ... 36 1 I1 LINEAR ALGEBRA (a matrix (.ill) all of rrhsc trrmr are zero exccpt the ql, jor 1 < i < r, with *q,t = 1 jor 1 < i < r - 1, q,, = 8, # 0). Ifr # m or r # n, zt may also be assumed that 6, = 1. The proposition is obvious if X = 0; suppose tliercfore X # 0. If m = 72 = 1 the proposition is obvious (with P = I,,,, Q = I,,, 6, # 0 arbitrary). If n = 1, m 2 2, we can apply Lemma 1 (since X # 0), which gives the desired form (58) with r = I, sr = 1. We argue by induction on n > 1; there exists an element 4,j # 0 in X; ifj = 1, Lemma 1 can be applied and reduces the problem to the case where X has the form (57). The induction hypothesis then applies to Y and there are therefore invertible matrices P’ E Mm-i(K), Q ‘ E Mn- i(K) which are products of matrices of the form B,,(h) of order m - 1 (resp. n - l), such that P’YQ’ is of the form (58). But, if B,,(A) belongs for example to M, - , (K), then 9 11.1 GRADED COMMUTATIVE GROUPS formula (58) then follows from the formula for block products writing p= (’ O) and Q = (I O) 0 P’ 0 Q’ If finallyj # 1, it would be sufficient to consider the matrix XBjl(l) to reduce it to the above case. Proposition 14 recovers Proposition 13 immediately. COROLLARY 1. IfX is an invertible square matrix of order n over ajield K, there exist three invertible matrices P, Q, D oJ order n such that X = PDQ, P and Q being products of matrices of the form B,,(A) and D a diagonal matrix of the form D = diag(1, 1,. . ., 6), where 8 # 0 (cf. Exercise 13). COROLLARY 2. For every jield K, the group of invertible matrices GL(n, K) is generated by the permutation matrices (no. 7, Example 2), the diagonal matrices diag(a, 1, . . ., 1) (a # 0 in K) and the matrices B12(h) (A E K). It has been seen (no. 9) that the right (resp. left) product of a matrix by the matrix of a suitable transposition exchanges any two columns (resp. rows). Then the matrix diag(1,. . ., 1, a) is equal to the product of diag(a, 1,. . . , 1) and permutation matrices and every matrix Bi,(A) is equal to the product of BI2(~) and permutation matrices, whence the corollary. I I Remarks. (1) In Chapter 111, we shall see that, if m = n = r and K is comm - utative, then, for all choices of P and Q satisfying the conditions of Proposition 14, the element 6, is always the same and equal to the determinant of X (111, 5 8, no. 6). (2) The argument of Proposition 14, slightly modified, shows that there is a permutation matrix R such that (with the same conditions on P) Ir N pxR= (0 0) if rn = n = r does not hold, and PXR = diag(1,. . ., 1, 6) otherwise. Observe also that the method of proof gives an explicit determina - tion of the matrices P, Q, R when Xis given explicitly. tj 11. GRADED MODULES AND RINGS From no. 2 of thisparagraph onwards, A will denote a commutative monoid (I, 3 2, no. l), written additively, with an identity element denoted by 0. 1. GRADED COMMUTATIVE GROUPS We are going to translate into another language the definitions concerning direct sums (8 1, no. 8). DEFTNITION 1. Given a commutative group G written additively and a set A, a gradua- tionof type A on G is a family (Gh)AGL ofsubgroups ofG, ofwhich G is the direct sum. The set G, with the structure dejined by its group law and its graduation, is called a graded (commutative) group of type A. A is called the set ofdegrees of G. An element x E G is called homogeneous if it belongs to one of the Gh, homogeneous of degree A if x E G,. The element 0 is therefore homogeneous of all degrees; but if x # 0 is homogeneous, it belongs to only one of the Gh; the index A such that x E G, is then called the degree ofx (or sometimes the weight of x) and is sometimes denoted by deg(x). Everyy E G my be written uniquely as a sum 2 y, of homogeneous elements With&. E G,,; !h. is called the homogeneous component of degree A (or simply the component ofdegree A) ofy. When the word “weight” is used instead of “degree”, the adjective Examples. (1) Given any commutative monoid A (with identity element 0) and a Commutative group G, a graduation (G;,)hEA is defined on G by taking Go = G and G;, = (0) for A # 0; this graduation is called trivial. h cc homogeneous” is replaced by “isobarie”. 362 363 LINEAR ALGEBRA I1 (2) Let A, A‘ be two sets and p a mapping of A into A’. Let (G,)AEA be a graduation of type A on a commutative group G; for p E A‘, let GL be the sum of the G, such that p(A) = p; clearly (GL),EA. is a graduation of type A‘ on G, said to be derived from (G,) by means of the mapping p. When A is a commutative group written additively and p the mapping A ++ - of A onto itself, (GL) is called the opposite graduation of (G,). (3) If A = A, x A, is a product of two sets, a graduation of type A is called a bigraduation of types A,, A,. For all A EA,, let GI, = ,g2 G,, and, for all p €A2, let GI = @ G,,; clearly (GI,),EA, is a graduation of type A, and (GE),EA2 a graduation of type A, on G; these graduations are called the partial graduations derived from the bigraduation (G,,). Note that G,, = GE, n GL; conversely, if (GE,),eAl and (G;),EA2 are two graduations on G such that G is the direct sum of the G,, = GE, n GE, these subgroups form a bigraduation of types A,, A, on G, of which (GL) and (G;) are the partial graduations. We leave to the reader the task of generalizing this to the case where A is a finite product of sets. (4) Let A, be a commutative monoid written additively, with identity element denoted by 0; let I be any set and A,) = A denote the submonoid of the product A’, consisting of the families (A,) I of finite support. Let p : A + A, be the surjective (codiagonal) homomorphism of A into A, defined by p((h,)) = 2 A,. From every graduation of type A a graduation of type A, is derived by means of p (Example 2) ; it is called the total graduation associated with the given “multigraduation” of type A. ?.€A1 LEI The definitions and examples of this no. extend immediately to the case where G is a group which is not necessarily commutatiue; it is simply necessary to replace everywhere the notion of direct sum by that of “restricted sum” (8 1, no. 6, Remark). Note that in this case the Gh are normal subgroups of G and that for 1 # every element of G, is permutable with every element of G,. 2. GRADED RINGS AND MODULES DEFINITION 2. Given a ring A and a graduation (A,) oftype A on the additive group A, this graduation is said to be compatible with the ring structure on A ;f (1) A,A, c A, , for all A, p in A. The ring A with this graduation is then called a graded ring oftype A. PROPOSITION 1. If every element of A is cancellable and (A,) is agraduation @‘type A compatible zoith the structure of a ring A, A, is a subring of A (and in particular 1 €Ao). GRADED RINGS AND MODULES 3 11.2 As A,A, c A, by definition, it suffices to prove that 1 E A,. Let 1 = 2 be the decomposition of 1 into its homogeneous components. If x E A,, then x = x. 1 = zA xe,; comparing the components of degree p, (since p + A = p implies A = 0) x = xe,. Since this relation is true for every homogeneous element of A, it is true for all x E A; in particular 1 = 1 .e, = e, E A,. ,€Ae, DEFINITION 3. Let A be a graded ring Of type A, (A,) its graduation and M a le$ (resp. right) A-module; a graduation (MA) of type A on the additive group M is compatible with the A-module structure on M ;f for all A, p in A. The module M with this graduation is then called a L$t (resp. right) graded module of type A over the graded ring A. When the elements of A are cancellable, it follows from (2) and Proposi - Clearly if A is a graded ring of type A, the left A -module A, (resp. the right tion 1 that the MA are A,-modules. A-module Ad) is graded of type A. Examples. (1) On any ring A the trivial graduation of type A is compatible with the ring structure. If A is graded by the trivial graduation, for a gradua - tion (MA) of type A on an A-module M to be compatible with the A-module structure, it is necessary and sufficient that the M, be submodules of M. (2) Let A be a graded ring of type A, M a graded A-module of type A and p a homomorphism of A into a commutative monoid A‘ whose identity element is denoted by 0. Then A is a graded ring of type A’ and M a graded module of type A’ for the graduations of type A’ derived from p and the graduations of type A on A and M by the procedure of no. 1, Example 1 : this follows immedi - ately from the relation p(A + p) = p(A) + p(p). In particular, if A = A, x A, is a product of two commutative monoids, the projections pr, and pr, are homomorphisms and the corresponding graduations are just the partial graduatiom derived from the graduations of type A (no. 1, Example 3) ; these partial graduations are thus compatible with the ring struc - ture of A and the module structure of M. Similarly, if A = At) (where A, is a commutative monoid with identity element denoted by 0), the total graduation (no. 1, Example 4) of type A, de- rived from the graduation of type A on A (resp. M) by means of the codiagonal homomorphism is compatible with the ring structure on A (resp. with the module structure on M). (3) Let A be a graded ring of type A, M a graded A-module of type A and ho an element of A; for all A E A, let ME, = and let M‘ be the Z-module @ ME,. As A,ML c M, , + ,,, = MI, ,, M’ is an A-module and the ME, form h€A 364 3 65 11 LINEAR ALGEBRA on M' a graduation of type A compatible with the A-module structure of M'; the graded A-module M' of type A thus defined is said to be obtained by sh;fting by A. the graduation of M and it is denoted by M(Ao). When A is a group, the underlying A-module of the graded A-module M' is identified with M. "(4) Let B be a commutative ring. The polynomial ring B[X] in one in- determinate is graded of type N by the subgroups BX" (n > 0) (cf. 111, § 2, GRADED SUBMODULES 3 11.3 $isms can be taken as the morphism of the species of graded ring structure of type A (Set Theory, IV, 3 2, no. 1). Similarly, if u : M --f M' and u' : M' 4 M" are two graded homomorphisms of graded A-modules of type A, of respective degrees 8 and a', u' o u : M -+ M" is a graded homomorphism of degree 8 + 8'. If 8 admits an inverse - 8 in A and u: M 4 M' is a bijective graded homomorphism of degree 8, the inverse mapping u': M' -+ M is a bijective graded homomorphism of degree - 8. It follows as above that the graded homomorphism of degree 0 can be taken as the moqhisms of the species of graded A-module of type A. But a bijective graded homomorphism u: M 4 N of degree #O is not a graded A-module isomor - phism if M and N are non-zero and the elements of A are cancellable. Examples. (6) If M is a graded A-module and M(Ao) is a graded A-module ob- tained by shifting (no. 2, Example 3), the Z-linear mapping of M(Ao) into M which coincides with the canonical injection on each M,,,, is a graded homo - morphism of degree A, (which is bijective when A is a group). (7) If a is a homogeneous element of degree 8 belonging to the centre of A, the homothety x - ax of any graded A-module M is a graded homomorphism of degree 6. Remark (3) A graded A -module M is called a graded free A-module if there exists a basis (m,),Gx of M consisting of homogeneous elements. Suppose it is and A is a commutative group; let A, be the degree of m, and consider for each t the shifted A-module A( - A,) (no. 2, Example 3) ; if e, denotes the element 1 ofA considered as an element of degree A, in A( - A'), the A-linear mapping u: 0 A( - A,) --f M Assuming always that A is a commutative group, now let N be a graded A-module, (nJt a system of homogeneous generators of N and suppose that n, is ofdegree p,. Then the A-linear mapping v: 2 A( - pt) --f N such that u(e,) = 17, for all L is a surjcctive graded A-module homomorphism of degree 0. If N is a finitely generated graded A -module, there is always a finite system of homogeneous generators of N and hence there is a surjective homomorphism of the above type with Ijnite. such that u(e,) = m, for all L, is a graded A-module isomorphism. tEI - no. 9 and IV).* *(5) Let B be a commutative ring, E a B-module, Q a quadratic form on E and C(Q) the Clifford algebra of Q (cf. IX, 3 9). The sub-B-modules C+ (Q) and C- (Q) form on C(Q) a graduation of type 2/22 compatible with the ring structure on C(Q)., Remarks. (1) The graduations most often used are of type Z or of type Z"; when we speak of graded (resp. bigraded, trigraded, etc.) modules and rings without mentioning the type, it is understood that we mean graduations of type Z (resp. Za, Z3, etc.) ; a graded ring (resp. module) of type N is also called a graded ring (resp. module) with positive degrees. (2) The graded Z-modules of type A, when Z has the trivial graduation, are just the graded commutative groups (whose set of degrees is a commutative monoid) of Definition 1 (no. 1). DEFINITION 4. Let A, A' be two graded rings of the same type A and (A,), (A;) their respective graduations. A ring homomorphism h : A --f A' is called graded ;fh(A,) c A; for all A E A. Let M, M' be two graded modules of type A over a graded ring of type A. Let u : M -+ M' be an A-homomorphism and 6 an element of A; u is called graded Of degree 6 ifu(MA) c MA+6 for all A E A. Let A be a graded ring of type A, A' a graded ring of type A' and p: A -+ A' a homomorphism. A ring homomorphism h : A + A' is called graded if h is a graded homomorphism of graded rings of type A' when A is given the gradua - tion of type A' derived from its graduation of type A by means of p (no. 1, Example 2) ; this therefore means that h(A,) c A;(*) for all h E A. An A-homomorphism u: M -+ M' is called graded if there exists S E A such that u is graded of degree 8. If u # 0 and every element of A is cancellable, the degree S of u is then determined uniquely. If h: A + A', h' : A' --f A" are two graded homomorphisms of graded rings of type A, so is h' o h: A --f A"; for a mapping h: A --f A' to be a graded ring isomorphism, it is necessary and sufficient that h be bijective and that h and the inverse mapping h' be graded homomorphisms; it also suffices for this that h be a bijective graded homomorphism. Thus it is seen that graded homomor - 3- GRADED SUBMODULES ~OPOSITION 2. Let A be a graded ring oftype A, M a graded A-module oftype A, (MA) its graduation and N a sub -A-module of M. The following properties are equiva - lent : (a) N is the sum ofthefamily (N n MA),,,. (b) The homogeneous components Of every element of N belong to N. I I (c) N is generated by homogeneous elements. 366 367 LINEAR ALGEBRA I1 GRADED SUBMODULES 3 11.3 Every element of N can be written uniquely as a sum of elements of the M, and hence it is immediate that (a) and (b) are equivalent and that (a) implies (c). We show that (c) implies (b). Then let (xJLE1 be a family of homogeneous generators #O of N and let 8(t) be the degree of x,. Every element of N can be written as a,x, with at E A; if at, is the component of at of degree A, the con- clusion follows from the relation Remark (I) In the above notation, the relation a,%, = 0 is therefore equivalent to the system of relations + z= ar,px, = 0. When A is a group, these rela - tions can be written 2 a,, h-B(r)~, = 0. When a submodule N of M has the equivalent properties stated in Proposi - tion 2, clearly the N n M, form a graduation compatible with the A-module structure of N, called the graduation induced by that on M; N with this gradua - tion is called a graded submodule of M. COROLLARY 1. IfN is a graded submodule ofM and (xJ is a generating system ofN, the homogeneous components of the x, form a generating system of N. COROLLARY 2. IfN is afinitely generated submodule of M, N admits afinite generating system consisting of homogeneous elements. It suffices to apply Corollary 1 noting that an element of M has only a finite number of homogeneous components # 0. A graded submodule of A, (resp. Ad) is called a graded lgt (resp. right) ideal of the graded ring A. For every subring B of A(B n A,) (B n A,) c B n A, ,; if B is a graded sub-Z-module of A, the graduation induced on B by that on A is therefore compatible with the ring structure on B; B is then called a graded subring of A. Clearly if N (resp. B) is a graded sub-A-module of M (resp. a graded subring of A), the canonical injection N --f M (resp. B + A) is a graded module homomorphism of degree 0 (resp. a graded ring homomorphism). If N is a graded submodule of a graded A-module M and (M,),EA the graduation of M, the submodules (M, + N)/N of M/N form a graduation compatible with the structure of this quotient module. For, if N, = MA n N, (MA + N)/N is identified with MJN, and it follows from Proposition 2 and 8 1, no. 6, formula (26) that M/N is their direct sum. Moreover, and hence A,((M, + N)/N) c (M,+, + N)/N, which establishes our asser- 368 A,(M, + N) = A,M, + N = M,,, + N I tion. The graduation ((M, + N) /N) , A is called the quotient graduation of that on M by N and the quotient module M/N with this graduation is called the graded quotient module of M by the graded submodule N; the canonical homo- morphism M -+ M/N is a graded homomorphism of degree 0 for this gradua- tion. If b is a graded two-sided ideal of A, the quotient graduation on A/b is com- patible with the ring structure on A/b; the ring A/b with this graduation is called the quotient graded ring of A by 6; the canonical homomorphism A + A/b is a homomorphism of graded rings for this graduation. PROPOSITION 3. Let A be a graded ring oftype A, M, N two graded A-modules oftype A and u : M + N a graded A-homomorphism of degree 8. Then : (i) Im(u) is a graded submodule of N. (ii) If 8 is a regular element of A, Ker(u) is a graded submodule of M. (iii) If8 = 0, the bijection M/Ker(u) + Im(u) canonically associated with u is an isomorphism of graded modules. Assertion (i) follows immediately from the definitions and Proposition 2(c). x, is its decomposition If x is an element of M such that u(x) = 0 and x = into homogeneous components (where x,, is of degree A), then ; u(x,) = U(X) = 0 and u(x,) is of degree A + 8; if 8 is regular the relation A + 8 = p + 8 implies h = p, hence the u(x,) are the homogeneous components of u(x) and necessarily u(xh) = 0 for all A E A, which proves (ii). The bijection v: M/Ker(u) + Im(u) canonically associated with u is then a graded homomorphism of degree 8, as follows from the definition of the quotient graduation; whence (iii) when 8 = 0. COROLLARY . Let A, B be two graded rings of type A and u : A + B a graded homo - morphism of graded rings. Then Im(u) is a graded submodule of B, Ker(u) a graded two-sided ideal of A and the bijection A/Ker(u) + Im(u) canonically associated with u is an isomorphism of graded rings. It suffices to apply Proposition 3 to u considered as a homomorphism of de- Pee 0 of graded Z-modules. ~OPOSITION 4. Let A be a graded ring oftype A and M a graded A-module of type A. (i) Every sum and every intersection of graded submodules of M is a graded sub- module. (ii) If x is a homogeneous element of M of degree p which is cancellable in A, the (%) If all the elements of A are cancellable, the annihilator .f a graded submodule annihilator of x is a graded left ideal of A. of M is a graded two -sided ideal of A. 369 I1 LINEAR ALGEBRA If (N,) is a family of graded submodules of M, property (c) of Proposition 2 shows that the sum of the N, is generated by homogeneous elements and property (b) of Proposition 2 proves that the homogeneous components of every element of (;7 N, belongs to To prove (ii), it suffices to note that Ann(x) is the kernel of the homo - morphism a - ax of the A-module A, into M and that this homomorphism is graded of degree p; the conclusion follows from Proposition 3(ii). Finally (iii) is a consequence of (i) and (ii) for the annihilator of a graded submodule N of M is the intersection of the annihilators of the homogeneous elements of N, by virtue of Proposition 2. Remark 2. Let M be a graded A-module and E a submodule of M; it follows from Proposition 4(i) that there exists a largest graded submodule N' of M contained in E and a smallest graded submodule N" of M containing E; N' is the set of x E E all of whose homogeneous components belong to E and N" is the submodule of M generated by the homogeneous components of a generating system of E. PROPOSITION 5. Let A be a graded ring of type A. If every element ofA is cancellable, then, for every homogeneous element a E A, the centralizer of a in A (I, S 1, no. 5) is a graded subring of A. N,; whence (i). Suppose that a is of degree 8; let b = b, be an element permutable with a, b, being the homogeneous component of b of degree A for all A E A. Then by hypothesis 7 (ab, - b,a) = 0 and ab, - b,a is homogeneous of degree A + 6; as 6 is cancellable, it follows that ab, = b,a for all A, which proves our assertion. COROLLARY . If every element of A is cancellable, the centralizer of the graded subring B of A (and in particular the centre of A) is a graded subring of A. It is the intersection of the centralizers of the homogeneous elements of B. Remark (3) A direct system (A,) I$*,) of graded rings of type A (resp. a direct system (Ma) fBa) of graded A, -modules of type A) is a direct system of rings (resp. A,- modules) such that each A, (resp. Ma) is graded of type A and each (resp. fOa) is a homomorphism of graded rings (resp. an A,-homomorphism of degree 0 of graded modules). If (resp. (Mk)AsA) be the graduation of %, (resp. Ma) and we write A = lim A,, Ah = lim A: (resp. M = lirn Ma, MA = lim Mk), -2 + -+ -2 it follows from 5 6, no. 2, Proposition 5 that (Ah) (resp. (MA)) is a graduation 4 11.4 CASE OF AN ORDERED GROUP OF DEGREES 370 of A (resp. hf) and it follows from I, 4 10, nos. 3 and 4 that this graduation is compatible with the ring structure on A (resp. the A-module structure on M). The gradcd ring A (resp. graded A-module M) is called the direct limit of the direct system of graded rings (A,, c$~,) (resp. graded modules (Ma, foa)). If 4,: A, --f A (resp. fa: Ma -+ M) is the canonical mapping, 4, (resp. fa) is a homomorphism of graded rings (resp. a homomorphism of degree 0 of graded &-modules). 4. CASE OF AN ORDERED GROUP OF DEGREES An order structure (denoted by <) on a commutative group A written additively is said to be compatible with the group structure if, for all p E A, the relation < p implies A + p < p + p. The group A with this order structure is then called an ordered group. We shall study these groups in detail in VI, 5 1 ; here we restrict ourselves to the remark that in such a group the relation A > 0 implies A + p > p for all p, for it implies A + p 2 p by definition and the relation 6 + p = p is equivalent to E = 0. Let A be an ordered commutative group, A a graded ring of type A and (Ah) its graduation and suppose that the relation A,, # (0) implies A 2 0; then it follows from the definitions that 3, = 2 A, is a graded two -sided ideal ofA, by virtue of the remark made above. h>O PROPOSITION 6. Let A be an ordered commutative group, A a graded ring of type A, (Ah) its graduation, M a graded A-module of type A and (MA) its graduation. Suppose that the relation A, # (0) implies A 2 0 and that there exists A, such that MA, # (0) and MA = {O}for A < A,. Then, $3, = zo A,,, 8,M # M. Let x be a non-zero element of Mho; suppose that x E 8,M. Then x = 2 six,, where the a, are homogeneous elements #0 of 3, and the xi homogeneous elements #O of M with deg(x) = deg(a,) + deg(x,) for all i (no. 2). But, as thesis. COROLLARY 1. With the hypotheses on A and A of Proposition 6, $ M is afinitely generated graded A-module such that 3,M = M, then M = (0). Suppose M # (0). Let A, be a minimal element of the set of degrees of a finite generating system of M consisting of homogeneous elements #O; then the hypotheses of Proposition 6 would be fulfilled, which implies a coptradic- tion. COROLLARY 2. With the hypotheses on A and A 0s Proposition 6, let M be a4nitely \ deg(ai) > 0, A, = deg(a,) + deg(x,) > deg(x,), which contradicts the hypo - 371 CASE OF AN ORDERED GROUP OF DEGREES 8 11.4 I1 LINEAR ALGEBRA generatedgraded A-module and K a graded submodule of M such that N + 3,M = M; then N = M. M/N is a finitely generated graded A-module and the hypothesis implies that 3,. (M/N) = M/N; hence M/N = 0. COROLLARY 3. With the hypotheses on A and A of Proposition 6, let u : M -+ N be a graded homomorphism of graded right A-modules, where N is assumed to be Jinitely generated. If the homomorphisnt @ 1 : M @A (A/~o) + N @A (A/~o) is surjective, then u is surjective. u(M) is a graded submodule of N and the (A/3,)-module (N/u(M)) @A is isomorphic to (N @A (A/o,))/Im(u @ 1) (4 3, no. 6, Proposition 6). The hypothesis therefore implies (N/u(M)) @A (A/3,) = 0 and hence N = u(M) by Corollary 1. Remark. It follows from the proof of Corollary 1 that Corollaries 1 and 2 (resp. Corollary 3) are still valid when, instead of assuming that M (resp. N) is finitely generated, the following hypothcsis is made: there exists a subset A+ of A satisfying the following conditions: (1) for lea+, M, = {0} (resp. N, = (0)); (2) every non-empty subset ofA+ has a least elrment. This will be the case if A = Z and M (resp. N) is a graded module with positive degrees. PROPOSITION 7. Suppose that A = Z. With the hyjotheses on A and M of Proposition 6, consider the graded A,-module N = M/3,M and suppose the following conditions hold: (i) each ofthe N, considered as an A,-module admits a basis (ylh)leIh; (ii) the canonical homomorphism 3, @A M --f M is injective. Then M is a graded free A-module (no. 2, Remark 3) and, to be precise, ifx,, is an element of M, whose image in N, is y,,, the family (x,,)(~, (where I is the set the sum of the I,) is a basis of M. We know (no. 2, Remark 3) that there is a graded free A -module L (of graduation (L,)) and a surjective homomorphism p: L --f M of degree 0 such that #(el,) = x,, for all (t, ,) E I being a basis of L consisting of homogeneous elements et, E L,). It follows from the above Remark that @ is surjective. Consider the graded A-module R = Ker(p) and note that R, = (0) for A < A, by definition; we need to prove that R = (0) and by Proposition 6 372 it will suffice to show that 3,R = R. Consider the commutative diagram (5 3, no. 6, Proposition 5) 1 @j I @P 30 @R + 3, @ L + 30 @ M ---+ 0 'I O4R------+L-------tM----tO f P wherej is the canonical injection, a, b, c deriving from the canonical injection so + A (4 3, no. 4, Proposition 4) ; it must be shown that a is surjective. Note that, as L is free, b is injective (4 3, no. 7, Corollary 6 to Proposition 7) and c is injective by hypothesis. Then let t be an element of R and t its class in R/3,R; then there is an exact sequence (4 3, no. 6, Proposition 5 and Corollary 2 to Proposition 6) R/3,R & L/3,L -+ M/3,M - 0 wherej and derive fromj and p when passing to the quotients and ji is by hypothesis a bijection; then.j(i) = 0, in other wordsj(t) E 3,L. Then there is anelementz~3,@Lsuchthatj(t) = b(z);asp(b(z)) =O,c((l @p)(z)) =O and, as c is injective, (1 @ p) (z) = 0. In other words, z is the image of an ele- mentt'E 3, @ Runder 1 @jandthenj(a(t')) = b(z) =j(t);asjisinjective, this implies t = a(t ') . I We shall show later (Commutative Algebra, 11, 9 3, no. 2, Proposition 5) how this proposition can be extended to non -graded modules. Lemma 1. For a commutative group A to be such that there exists on A a total ordering compatible with the group structure of A, it is necessary and suficient that A be torsion-frue. If there exists such an order structure on A and if A > 0, then h + p > 0 for all p 2 0 and in particular, by induction on the integer n > 0, n. A > 0, which proves that A is torsion -free (since every element # 0 of A is either > 0 or < 0). Conversely, if A is torsion -free, A is a sub-Z-module of a vector Q-space (9 7, no. 10, Corollary 1 to Proposition 26) which may be assumed of the form Q"); if I is given a well-ordering (Set Theory, 111, 9 2, no. 3, Theorem 1) and Q its usual ordering, the set Q(I) with the lexicographical ordering is totally ordered (Set Theory, 111, 4 2, no. 6) ; it is immediate that this ordering is com- patible with the additive group structure of Q"). PROPOSITION 8. Let A be a torsion-j?ree commutative group and A a graded ring of type A- If the product in A of two homogeneous elements # 0 is #O, the ring A has no divisor Let A be given a total ordering compatible with its group structure (Lemma 373 of 0. LINEAR ALGEBRA 11 1) and let x = hEA~,, c y = hEA 2 y, be two non-zero elements of A (x, and y, being homogeneous of degree A for all h E A) ; let a (resp. p) be the greatest of the elements A E A such that x, # 0 (resp. y, # 0) ; it is immediate that if A # a or p # p, either x,y, = 0 or deg(x,y,) < a + p; the homogeneous component of xy of degree a + p is therefore x,yp, which is non-zero by hypothesis; whence xy # 0. 5. GRADED TENSOR PRODUCT OF GRADED MODULES Let A be a commutative monoid with its identity element denoted by 0, A a graded ring of type A and M (resp. N) a graded right (resp. left) A-module of type A. Let (A,) (resp. (M,), (N,)) be the graduation of A (resp. M, N) ; the tensor product M BZ N of the Z-modules M and N is the direct sum of the MA @z N, (3 3, no. 7, Proposition 7) and hence the latter form a bigradua- tion of types A, A on this Z-module. Consider on M QZ N the total graduation of type A associated with this bigraduation (no. 1, Example 4); it consists of the sub-Z-modules PA = (M, BZ N,). It is known that the Z-module M @A N is the quotient of M BZ N by the sub-Z-module Qgenerated by the elements (xu) @ y - x @ (ay), where x E M, y E N and a E A (3 3, no. 1) ; if, for all A E A, xA, y,, a, are the homogeneous components of degree A of x, y, a respectively, clearly (xu) @ y - x @ (zy) is the sum of the homogeneous ele - ments (xAuv) @ y, - x, @ (avy,), in other words Qis a graded sub-Z-module of M gZ N (no. 3, Proposition 2) and the quotient M @AN = (M @zN)/Q therefore has canonically a graded Z -module structure of type A (no. 3). More - over (no. 3, Proposition 5), the centre C of A is a graded subring of A; the graduation which we have just defined on M @A N is compatible with its module structure over the graded ring C. For M mZ N has canonically two C-module structures, for which respectively c(x @ y) = (xc) @ y and (x @ y)c = x @ (cy) for x E M, y EN, c E C (3 3, no. 3); if x E M,, y EN,, c E C n A,, the two ele - ments c(x @ y) and (x @ y)c belong to (M gZ N),+,+, and their difference belongs to Q and hence their common image in M BAN belongs to (M N)A+u+v, which establishes our assertion. When we speak of M @AN as a graded C-module, we always mean with the structure thus defined, unless otherwise mentioned. Note that (M @AN), can be defined as the additive group of M @*N generated by the x, @ y,, where x, EM,, y, EN, and Let M’ (resp. N’) be another graded right (resp. left) A-module and u : M --f M’, v : N -+ N’ graded homomorphisms of respective degrees a and p. Then it follows immediately from the above remark that u @ u is a graded (C-module) homomorphism of degree a + p. ,+,=A p+v=A. GRADED MODULES OF GRADED HOMOMORPHISMS 81 1.6 When A is conimutativc, a graduation (compatible with thc A-module structure) is similarly defined on the tensor product of any finite number of graded A-modules; it is moreover immediate that the associativity isomor - phisms such as (M @ N) @ P 4 M @ (N @ P) (3 3, no. 8, Proposition 8) are isomorphisms of graded modules. Remark. When A has the trivial graduation (no. 1, Example I), (M N)A is then simply the direct sum of the sub-Y-modules M, @A N, of M @A N such that p + v = A. Let M (resp. N) be a graded right (resp. left) A-module of type A, P a graded Z-module of type A and letf be a Z-bilinear mapping of M x N into p satisfying condition (1) of 3 3, no. 1, and such moreover that f(xA, y,) for ‘A Yu N,, A> p in A* Thenf(x, y) = g(x @I y), where g: M @A N -+ P is a Z-linear mapping (4 3, no. 1, Proposition 1) and it follows from the above condition that g is a graded Z-module homomorphism of degree 0. Let B be another graded ring of type A and p: A -+ B a homomorphism of graded rings (no. 2) ; then p* (Bd) is a graded right A-module of type A. If E is a graded left A-module of type A and p*(B,) @A E is given the graded Z- module structure of type A defined above, the canonical left B-module struc- ture (3 5, no. 1) is compatible with the graduation of EC3, = = p*(Bd) @AE. The graded B-module thus obtained is said to be obtained by extending the ring of scalars to B by means of p and when we speak of E(=) or p* (E) as a graded B-module, we always mean this structure, unless otherwise mentioned. 6. GRADED MODULES OF GRADED HOMOMORPHISMS We suppose in this no. that the monoid A is a group. Let A be a graded ring of type A and M, N two graded left (for example) A-modules of type A. Let H, denote the additive group of graded homomor@hisms of degree A of M into N (no. 2); in the additive group HomA(M, N) of all homomorphisms of M into N (with the non-graded A-module structures) the sum (for A E A) of the H, is direct. For, if there is a relation 7 u, = 0 with u, E H, for all A, it follows that %(xu) = 0 for all p and all x, E M,. As the elements of A are cancellable, %(xu) is the homogeneous component of u,(x,) of degree A + p; hence %(xu) = 0 for every ordered pair (p, A) and every x, E M,, which implies UA 0 for all h E A. We shall denote (in this paragraph) by Homgr,(M, N) the additive group of graded A-module homomor@hism of M into N. Let C be the 375 I additive subgroup of HomA(M, N) the sum of the HA and we shall call it the 374 I1 LINEAR ALGEBRA 41 1.6 We sometimes consider on the graded dual M*Er the graduation derived from the above with the aid of the isomorphism A H - A of A (no. 1, Example 2) so that the homogeneous elements of degree A in M*g' are the graded linear forms of degree - A on M (when A has the trivial graduation, these are the zero linear forms of the M, of index p # A). Then, if u: M -+ N is a graded homomorphism of degree 6, u' becomes a graded homomorphism of degree -6. GRADED MODULES OF GRADED HOMOMORPHISMS centre of A, which is a graded subring (no. 3, Corollary to Proposition 5) ; for the canonical C -module structure on Hom,(M, N) (4 1, no. 14, Remark l), Homgr,(M, N) is a submodule and the graduation (HA) is compatible with the C- module structure: for, if c, E C n A,, x, E N, and uA E H,, then by definition (cvu,)(xu) = cy.u,(x,) c Nh+,+, and hence c,uh E HA+,. Let M' and N' be two graded left A -modules of type A and u': M' -+ M, u': N -+ N' graded homomorphisms of respective degrees a and p. Then it is immediate that Hom(u', u') : w I-+ u' o w o u' maps Homgr,(M, N) into Homgr,(M', N') and that its restriction to Homgr,(M, N) is a graded homo - morphism into Homgr,(M', N') ofdegree a + p. In particular Homgr,(M, M) is a graded subring of End,(M), which is de- noted by Endgr,( M) . Remark. If M and N are graded left A-modules, Homgr,(M, N) is in general distinct from Hom,(M, N). However these two sets are equal when M is a finitely generated A-module. For M is then generated by a finite number of homogeneous elements x, (1 < i < n); let d(i) bc the degree of x,; let u E HomA(M, N) and for all A E A let zl, , denote the homogeneous component of u(x0 of degree A + d(i). We show that there cxists a homomorphism u,: M -+ N such that u,(x,) = z,, for all i. It sufices to prove that if; a,x, = 0 with a, E A for 1 < i 6 n, then 2 a,z,, , = 0 for all A E A (4 1, no. 7, Remark). It can be assumed that each a, is homogeneous of degree d'(i) such that d(i) + d'(i) = p for all i (no. 3, Remark 1); then 2 a,u(x,) = 0; taking the homogeneous component of degree A + p on the left-hand side, we obtain a,z,, , = 0, whence the existence of the homomorphism u,; clearly moreover uA is graded of degree A. Finally, u, = 0 except for a finite number of values of A, and u = 2 u, by definition, which proves our assertion. In particular, Homgr,(A,, M) = HomA(A,, M) for every graded left A- module M; moreover HomA(A,, M) has a graded le$ A-module structure (and not just a graded C-module structure), and it is immediate that with this structure the canonical mapping of M into HomA(A,, M) (3 1, no. 14, Remark 2) is a graded A-module isomorphism. Similarly, Homgr,(M, A,) has a graded right A -module structure (and not only a graded C-module structure) ; it is called the graded dual of the graded A-module M and is denoted by M*gr, or simply M* when no confusion results. If u: M + N is a graded homomorphism of degree 8, it follows from the above that the restriction to N*g' of tu = Hom(u, IAs) is a graded homomorphism of the graded dual N*g' into the graded dual M*g', of degree S, called the graded transpose of u. f A Suppose A is commutative and graded of type A and let M, N, P, Q be four graded A-modules of type A. Then there are canonical graded homomorphisms of degree 0 HomgrA(M, HomgrA(N, P)) -+ Homgr,(M &IA N, P) (3) (4) HomgrA(M, N, @A HomgrA(M, @A p, (5) HomgrA(M, ') @A HomgrA(N, Q) --f HomgrA(M @A N, @A Q) (the tensor products being given the graduations defined in no. 5) obtained by restricting the canonical homomorphisms defined in 3 4, nos. 1, 2 and 4; for, if u: M -t Homgr,(N, P) is graded of degree S, then, for all x E MA, u(x) is a graded homomorphism N-tP of degree 6 + h and hence, for YEN,, U(X) (y) E P, if u: M @, N -+ P corresponds canonically to u, it is then seen that u is a graded homomorphisms of degree 6, whence our assertion con - cerning (3) ; moreover it is seen that this homomorphism is bijective. The argu- ment is similar for (4) and (5). If in particular P = Q = A in (5), then there is a canonical graded homo - morphism of degree 0 M*gr N*gr -+ (M @A N)*gr. (6) 376 377 APPENDIX APPENDIX PSEUDOMODULES 1. ADJUNCTION OF A UNIT ELEMENT TO A PSEUDO -RING Let A be a pseudo-ring (I, 5 8, no. 1). On the set A' = Z x A we define the following laws of composition : { (m, a)(n, b) = (mn, mb + nu + ab). It is immediately verified that A' with these two laws of composition is a ring in which the element (1, 0) is the uiiit element. The set (0) x A is a two- sided ideal of A' and L: x w (0, x) is an isomorphism of the pseudo-ring A onto the sub-pseudo-ring (0) x A by means of which A and (0) x A are identified. A' is called the ring derived from the pseudo-ring A by adjoining a unit element. (m, a) + (n, b) = (m + n, a + b) (1) If A already has an identity element E, the element e = (0, E) of A' is an idempotent belonging to the centre of A' and such that A = eA' = A'e. Then (eA', (1 -e)A) is a direct decomposition (I, 4 8, no. 11) of A' and the ring (1 - e)A' is isomorphic to Z. moreover, by restricting the set of operators of this module structure to (0) x A (identified with A), we obtain on E the pseudomodule structure given initially. For a subset M of E to be a subgroup with operators of the pseudomodule E (in which case the induced structure is obviously also a left pseudomodule structure over A), it is necessary and sufficient that M be a submodule of the associated A'-module E and this sub-A'-module is associated with the pseudo- module M. Moreover, the quotient A-module E/M is then associated with the quotient group with operators E/M, which is obviously a pseudomodule over A. groups with operators are identical with the A'-linear mappings E -+ F of the A'-modules associated respectively with the pseudomodules E and F. If (EL),EI is a family of pseudomodules over A, the groups with operators E and @ E, are pseudomodules over A and the associated A-modules are respectively the product and direct sum of the associated A'-modules E,. There are analogous results for inverse and direct limits of pseudomodules. The theory of pseudomodules over A can thus be entirely reduced to that of A'- modules. 4 If E, F are two pseudomodules over A, the homomorphisms E --f F of LEI &€I 2. PSEUDOMODULES Given a pseudo-ring A with or without a unit element, a lejl pseudomodule over A is a commutative group E (written additively) admitting A as set of operators and satisfying axioms (MI), (MII) and (MIII) of 3 I, no. 1, Definition I. Right pseudomodules over A are defined similarly. Let A' be the ring obtained by adjoining a unit element to A. If E is a left pseudomodule over A, a Zejlt A'-module structure on E is associated with it by writing, for all x E E and every element (n, u) E A', (2) (n, a).x = nx -I- ax. Axioms (MI) to (MIv) of 5 1, no. 1, Definition 1 are immediately verified; 378 379 EXERCISES EXERCISES 91 1. Let E, F be two A-modules and u: E -+ F a linear mapping. Let M be a submodule of E, N a submodule of F, i: M + E the canonical injection and q: F +F/N the canonical surjection. Show that u(M) = Im(u o i) and -1 u (N) = Ker(q o u). 2. Let E be a left A-module; for every left ideal a of A let aE denote the sum (a) Show that for every family ofA-modules, a. @ E, is canonically of the ax where x E E, which is therefore a submodule of E. ?.EL identified with @ aE,. (b) If a is a finitely generated left ideal of A, show that for every family (EA)h,L of A-modules, a. n E, is canonically identified with n aE,. Give an example where a is not finitely generated and the above property fails to hold (take A commutative and all the E, equal to A). *(c) Let K be a commutative field, A the ring K[X, Y] of polynomials in two indeterminates over K, m = AX and n = AY. If E is the quotient A- module of A2 by the submodule of A2 generated by xe, - Ye, (where (el, e,) is the canonical basis of A2), show that (m n n)E # (mE) n (nE)., heL ,EL h€L 3. Let E be an A-module. (a) Let (LJUEM be a family of right directed ordered sets and, for all p E M, let (FAU)AELu be an increasing family of submodules of E. Show that (6) Take A to be a field and E the A-module AN; let G be the submodule A“) of E; on the other hand, for every finite subset H of N, let F, be the sub- 380 module n M, of E, where M, = A for n $ H and M, = 0 for n E H. Show that the family (FH), where H runs throughout the set Q, of finite subsets of N is right directed and that nsN 4. Let E, F,, F, be three A-modules and f,: F, -+ E, f2: F, -+ E two A- linear mappings. The jbre product of F, and F, relative to fl and f2 is the submodule of the product F, x F, consisting of the ordered pairs (x,, x,) such that fl(xl) = f2(x2); it is denoted by F, x (a) For every ordered pair of A-linear mappings u1 : G -+ F,, u2 : G -+ F, such that f, o u, = f, o u,, there exists one and only one A-linear mapping u: G -+ F, x F, such that p, 0 u = u,, p, 0 u = u,, where p1 and p2 denote the restrictions to F, x F, of the projections pr, and pr,. (6) Let E’, F;, F& be three A-modules, f;: F; -+ E’, fi: F& -+ E’ A-linear mappings and F; x E, Fa the fibre product relative to these mappings. For every system of A-linear mappings u,: F, -+ F;, u,: F, -+ F&, w: E + E’ such that f; o u, = w 0 f,, fi o u2 = w 0 f,, let u be the A-linear mapping of F, x F, into F; x E, F& corresponding to the two linear mappings u, 0 p1 and u2 0 p,. Show that if u, and u, are injective, so is v; give an example where u1 and u2 are surjective but u is not surjective (take E’ = (0)). If w is injective and u1 and u2 surjective, show that u is surjective. F,. 5. Let E, F,, F, be three A-modules and f,: E -+ F,,f,: E -+ F, two linear mappings. The amalgamated sum of F, and F, relative to f, and f, is the quotient module of F, x F, by the submodule the image of E under the mapping z ++ (fl(z), -f,(z)); it is denoted by Fl OE F2. (a) For every ordered pair of A-linear mappings u1 : F, -+ G, u, : F, -+ G such that u, 0 f, = u2 0 f,, show that there exists one and only one A -linear mapping u: F, BE F, --f G such that u oj, = u,, u oj, = u,, where j, and j, denote the respective compositions of the canonical mapping Fl F2 --f Fl BE F2 with the canonical injections F, -+ F, x F,, F, --f F, x F,. (6) Let E’, F;, FL be three A-modules, fl: E’ --f F;, fl: E’ -+ FL two A- linear mappings and 5‘; GE, Fh the amalgamated sum relative to these map- pings. For every system of A-linear mappings u,: Fi -+ F,, u2: F& -+ F,, w: E‘ + E such that u1 0.f; = f, 0 w, u, 0 fi = f, 0 w, let u be the A-linear mapping of F; BE, F& into F, OE F, corresponding to the two linear mappings jl 0 u1 andj, 0 u,. Show that if u1 and u2 are surjective, so is u; give an example where v, and u2 are injective but u is not injective. Show that if w is surjective and v1 and v, injective, then v is injective. 38 1 I1 LINEAR ALGEBRA 6. Given an A-module E, let y(E) denote the least of the cardinals of the (u) If 0 -+ E -+ F + G -+ 0 is an exact sequence of A-modules, then generating systems of E. Y(G) G Y(F) < Y(E) + Y(G)- Give an example ofa product F = E x G such that y(E) = y(F) = y(G) = 1 (take E and G to be quotients of Z). (6) IfEis thesumofafamily (F,),,Lofsubmodules, then y(E) < zLy(F,). (c) If M and N are submodules of a module E, show that SUP(Y(M), Y" G Y(M n N) + Y(M + N). 7. (u) Let A be a ring and E an (A,A)-bimodule; on the product B = A x E a ring structure is defined by setting (a, X)(U', x') = (ad, ax' + xu'); E (canonically identified with (0) x E) is then a two-sided ideal of B such that Ea = {O}. (b) By suitably choosing A and E, show that E can be a left B-module with y(E) (Exercise 6) an arbitrary cardinal, although y(B,) = 1. *8. Let A be a commutative ring, B = A[X,,],,,, a polynomial ring over A in an infinity of indeterminates and C the quotient ring of B by the ideal generated by the polynomials (X, - X,)X, for i 2 3. If El and E2 are the classes in C of X, and X, respectively, show that the intersection in C of the monogenous ideals CE, and CE, is not a finitely generated ideal., 9. Show that in an A-module E, if (Fn)ndl is a strictly increasing sequence of finitely generated submodules, the submodule F of E the union of the F, is not finitely generated. Deduce that if E is an A-module which is not finitely generated, there exists a submodule F of E such that y(F) = x, (=Card(N)). 10. Let E be a Z-module the direct sum of two submodules M, N respec - tively isomorphic to 2/22 and 2/32; show that there exists no other decom- position of E as a direct sum of two non-zero submodules. 11. Let A be a ring admitting a field of left fractions (I, fj 9, Exercise 15). Show that in the A-module A,, a submodule distinct from A, and (0} admits no supplementary submodule (cf. I, 3 9, Exercise 16). 12. Let G be a module and E, F two submodules such that E c F. (u) If F is a direct factor of G, F/E is a direct factor of G/E. If also E is a (b) If E is a direct factor of G, then E is a direct factor of F. If also F/E is a direct factor of F, E is a direct factor of G. direct factor of G/E, then F is a direct factor in G. EXERCISES (c) Give an example of two submodules M, N of the Z-module E = Z2 such that M and N are direct factors of E but M + N is not a direct factor of E. (d) Let p be a prime number and U, the submodule of the Z-module Q/Z consisting of the classes mod Z of rational numbers of the form k/pn (k E Z, n E N). Let E be the product Z-module M x N, where M and N are both isomorphic to Up, M and N being canonically identified with submodules of E. Consider the endomorphism u: (x, y) H (x, y + px) of E; show that u is bi- jective. If M' = u(M), show that M and M' are both direct factors of E but that M n M' is not a direct factor of E (use (b), showing that the Z-module Up ha no direct factor other than itself and (0)). 7 13. Let E be an A-module and B the ring EndA E. Show that for an element 11 E B the following three conditions are equivalent: (1) Bu is a direct factor of the left B-module B,; (2) uB is a direct factor of the right B-module B,; (3) Ker(u) and Im(u) are direct factors of the A-module E. (Observe that every left ideal which is a direct factor of B, is of the form Bp, where p is a projector in E; cf. I, 3 8, Exercise 12 and use 11,s 1, no. 9, Corollaries 1 and 2 to Proposi - tion 15.) 14. Let L be a well-ordered set, E an A-module and (E,),,, an increasing family of submodules of E such that: (1) there exists a A E L such that E = (0); (2) E is the union of the E, for A E L; (3) if A E L is such that the set of p < A admits a greatest element A', E, is the direct sum E,. and a submodule F,; (4) ifh E L is the least upper bound of the set of p < A, then E, is the union of the E, for p < A. Under these conditions, show that E is the direct sum of the FA. (Prove by transfinite induction that each E, is the direct sum of the F, such that p < 1.) fi 15. Let c be an infinite cardinal. (u) Let I be a set and Rja, b{ a relation between elements of I such that for all a E I the set of E I for which Ria, p{ holds has cardinal < c. Show that there exists a well-ordered set L and an increasing family (I,),,L of subsets of 1 such that: (1) there exists a A E L such that I, = o ; (2) I is the union of the 1, for h E L; (3) if h E L is such that the set of p < A has a greatest element A', card(& - 1,') < c; (4) if A E L is the least upper bound of the set of p < A, 1, is the union of the I, for p < A; (5) if a E I, and Rla, p{ holds, then p E I,. (Note first that for all a E I the set of @ E I for which there exists an integer n and a sequence (Y,)~~,~,, of elements of I such that y1 = a, y,, = p and %,, y,+,{ holds for 1 < j < n - 1, has cardinal < c. Then construct the I, by transfinite induction.) (b) Let E be an A-module the direct sum of a family (Ma) a , I of submodules such that y(M,) < c for all a E I (cf. Exercise 6). Letfbe an endomorphism of E. Show that there exists a well-ordered set L and an increasing family (1~)h.L of subsets of I satisfying properties (I), (2), (3) and (4) of (a) and such 382 383 I1 LINEAR ALGEBRA EXERCISES moreover that if we write E, = @ Ma, then f (E,) c E,. (Apply (a) to the relation Ria, pi: “there exists x E Ma such that the component off(x) in M, is #O”.) For all heL such that the set of p < A has a greatest element h’, write FA = Ma. Show that E is the direct sum of the F, (apply Exercise 14). (c) With the hypotheses and notation of (b), suppose further that f is a projector and write P = f (E). Let P, = P n E,; show that if A E L is such that the set of p. < A has a greatest element A‘, P,, is a direct factor of P, (cf. Exer- cise 12), that a supplementary submodule PL of PA, in P, is isomorphic to a direct factor of F, and that y(PL) < c. Show that P is the direct sum of the P; (apply Exercise 14). (d) Deduce from (c) that if an A-module E is the direct sum of a family (Ma),,, of submodules such that y(M,) 6 c, then every direct factor P of E is also the direct sum of a family (Nh)h,L of submodules such that y(N,) < c. 16. Let A be a ring such that there exists an A-module M with a generating system of n elements but containing a free system of n + 1 elements. (a) Show that there exists in A: a free system of n + 1 elements. (b) Deduce that there exists in M an inznite free system (construct such a system by induction using (a)). (c) Let C be a ring, E the free C -module C:”, (en) its canonical basis and A the endomorphism ring of E. Let u1 and u2 denote the endomorphisms of E defined by the conditions ul(e2,,) = en, ul(e2n+l) = 0, ~~(e~,,+~) = en, u2(e2,,) = 0 for all n 2 0. Show that u1 and u2 form a basis of the A-module A, and deduce that in the A-module A, there exist infinite free systems. * (d) Let A be the tensor algebra of a vector space E of dimension 2 2 over a commutative field (cf. 111, 3 5, no. 1). Show that in the A-module A, there exist infinite free systems (observe that two linearly independent vectors in E are linearly independent in A, and use (b)). On the other hand, for every integer n > 0, every basis of the A-module A: has n elements (cf. 3 7, no. 2, Proposition 3). * aEIh @ CL E Ih-I*. 17. Let A be a ring. (a) Show that if the A-module A: is isomorphic to no A: for m > n, then, (b) Show that if the A-module A: contains no free system of n + 1 elements, 18. Let A be a ring for which there exists an integer p with the following of p elements of A, there exists a family of elements of A, one at least of which is not a right divisor of 0, such for allp < n, A: is isomorphic to no A: for q > p. the A-module A: contains no free system ofp + 1 elements for p < n. property: for every family (6,) that 2 ciat = 0. (a) Show, by induction on n, that for every family (xJ ofp” elements of A:, there exists a family (cj) ofp” elements of A, one at least of which is not a right divisor of 0, such that 7 cjxj = 0. contains no free system of n + 1 elements. (b) Deduce from (a) and Exercise 16 that for all n > 0, the A-module A: 19. Let A be a non-zero ring with no divisors of zero. (a) Show that for n > 1, the A-module A: is not monogenous. (b) Given an integer n 2 1, for the A-module A: to contain no free system ofn + 1 elements, it is necessary that A admit a field of left fractions (I, 3 9, Exercise 15) ; conversely, if this condition is satisfied, A: contains a free system of n + 1 elements for no integer n > 0. (Use Exercises 17 and 18 and I, 3 9, Exercise 16.) (c) Show that if every left ideal of A is monogenous, A admits a field of left fractions (use (a) and I, tj 9, Exercise 16). 20. (a) Let A be a ring admitting a field of left fractions (I, 4 9, Exercise 15). Let E be an A-module; show that, if in E (xJ1 is a free system and y, z two elements such that xl,. . ., x,, y on the one hand and xl,. . ., x,, z on the other are two related systems, then x2, . . . , xT, y, z is a related system. (6) Let A be the quotient ring Zj6Z and let (el, e2) be the canonical basis of A2; show that if a = 2e1 + 3e2, a and el form a related system, as do a and e2, although e, and e2 form a free system. 21. Let A be a ring, 6, c two elements of A which are not right divisors of 0, M the A-module AIAbc and N the submodule AcjAbc of M. For N to be a direct factor in M, it is necessary and sufficient that there exist two elements X, y ofA such that xb + cy = 1. (If E is the class of 1 in M, show that a supple - mentary subspace of N in M is generated by an element of the form (1 - yc)~ whose annihilator is the ideal Ac.) 22. If an A-module E admits a basis whose indexing set is I and one of the two sets A, I is infinite, show that E is equipotent to A x I. 23. (u) Let M,N be two subsets of an A-module E and rn and n their annihilators; show that the annihilator of M n N contains rn + n and give an example where it is distinct from m + n. (b) In a product module n E,, the annihilator of a subset F is the inter- section of the annihilators of its projections. (c) In a free A -module, the annihilator of an element # 0 contains only left divisors of 0 in A; in particular, if A is a ring with no divisors of 0, every element #o of a free module is free. t 3 84. 385 I1 LINEAR ALGEBRA 24. Let E be an A-module and M, N two submodules of E. The transporter of M into N, denoted by N: M, is the set of a E A such that aM c N; it is a two- sided ideal of A, equal to Ann( (M + N)/N). (a) Let A be a commutative ring, M an A-module, N a sub-modulc of M and x an element of M; show that N n Ax = (N:Ax)x. (b) Let A be a commutative ring and a, b two ideals of A. Show that a:b 5) a and that (a: b)/a is an A-module isomorphic to Hom(A/b, A/a). 25. Let E, F be two A -modules and u: E +F a linear mapping. Show that the mapping (x, y) H (x, y - ~(x)) of the product module E x F to itself is an automorphism of E x F. Deduce that if there exists a linear mapping U: F -+ E and an a E E such that u(u(a)) = a, there exists an automorphism w of E x F such that w(a, 0) = (0, ~(a)). 26. Let E be a free A -module with a basis containing at least two elements. (a) Show that every semi -linear mapping of E into itself (relative to an automorphism of A) which permutes with all the automorphisms of E, is necessarily a homothety x ++ ax (a E A). (b) Deduce from (a) that the centre Of End,(E) is the ring of central homo- theties, isomorphic to the centre of A. (c) Deduce from (u) that the centre of the group GL(E) is the group of invertible central homotheties of E, which is isomorphic to the multiplicative group of invertible elements of the centre of A. 27. Let A be a commutative ring. Show that if an ideal 3 of A is a free A-module, then 3 is a monogenous A -module (in other words a principal ideal). Give an example showing that the proposition does not extend to the left ideals in a non-commutative ring (Exercise 16). EXERCISES §2 1. (a) If A is a ring with no divisors of zero, show that every element # 0 of a (b) Give an example of a projective A -module whose annihilator is non-zero projective A -module is free. (cf. I, 3 8, no. 11). (c) Show that the Z-module Q is not a projective Z -module. *(See Commu- tat;uk Algebra, 11, 4 5, Exercise 11 for an example of a finitely generated pro - jective module over an integral domain, which is not free.) * 2. Show that every projective A -module is the direct sum of a family of projective submodules each of which admits a countable number of generators (“Kap!ansky’s Theorem”; apply Exercise 15 of 4 1, to the case where E is a free module). 3. Let c be a cardinal and P a projective A -module such that y(P) < c (3 1, Exercise 6). Show that if L is a free A -module with an infinite basis of cardinal 2 c, P @ L is isomorphic to L. (Observe that there exists a free A- module M, whose basis has cardinal < c, isomorphic to the direct sum of P and an A-module Q. Note that hl‘”) and 1’ @ M“) are isomorphic.) 7 4. (a) Let E, E’ be two A -modules, N a submodule of E and N’ a sub- module of E’ such that E is projective and E/N and E’/N’ are isomorphic. Show that there exists an exact sequence o + N It, EON‘ --L E! + 0. (Observe that there exists a homomorphism$ E -+ E’ such thatf(N) c N’, which gives when passing to the quotients an isomorphism E/N --f E’/N’; define u and u in an analogous way to that used for the exact sequence (29) of 4 1, no. 7.) (b) Deduce from (u) that if E‘ is projective, E @ N’ and E’ @ N are iso- morphic. (c) Let 0 --f Em -+ -+. . . --f E, --f Eo + 0 be an exact sequence of A-modules such that E,, El, . . ., are projective. Show that the A- modules @ Em-2h and @ &-2h+ are isomorphic (with the convention that E, = (0) for i < 0). 5. Let u be a linear mapping from an A-module E to an A-module F and v = % its transpose. For every submodule N’ of F*, the submodule of E ortho - gonal to u(N’) is il(N), where N is the submodule of F orthogonal to N’. 6. (a) If E is an A-module, every injective endomorphism u of E is not a left divisor of zero in the ring End,(E). Conversely, if for every sub -A-module F # (0) of E there exists an endomorphism u # 0 of E such that u(E) c F, an element which is not a left divisor of zero in End,(E) is an injective endo - morphism. The above condition is satisfied if there exists a linear form x‘ E E* and an x E E such that (x, x’) is invertible in A and in particular if E is free. (b) Every endomorphism #O of the Z-module Qis bijective, although there exists no linear form #O on Q. (c) Let Up be the Z-module defined in 3 1, Exercise 12(d). Show that the endomorphism x -px of Up is not injective and is not a left divisor of zero in 7. (a) If u is a surjective endomorphism of an A-module E, u is not a right divisor of zero in End,(E). Conversely, if for every submodule F # E of E there exists a linear form x* E E* which is zero on F and surjective, every ele- ment of End,(E) which is not a right divisor of zero is a surjective endo- morphism. (b) If E is the Z-module defined in Exercise 6(c), show that every endo- morphism #O of E is surjective, although there exists no linear form $0 on E. h30 h>O Endz(U,) - 386 387 I1 LINEAR ALGEBRA (c) Show that if E is a free Z-module #O, there exist non-surjective endo- morphisms of E which are not right divisors of 0 in End,(E). 8. (u) Let E be a free A-module, F, G two A-modules and u: E -+ G, v: F -+ G two linear mappings. Show that if u(E) c v(F), there exists a linear mapping w: E -+ F such that u = v o w. (6) Give an example of two non-zero endomorphisms :I, u of the Z-module E defined in Exercise 6(c) such that there exists no endomorphism w of E for which u = u 0 w (although u(E) = v(E) = E). (c) Give an example of two endomorphisms u, u of the Z-module Z such that u (0) = u (0) = {0}, but such that there exists no endomorphism w of Z for which u = w 0 u (cf. 5 7, Exercise 14). 9. Given an A-module E, for every submodule M of E (resp. every sub - module M' of E*), let M" (resp. M") denote the orthogonal of M in E* (resp. the orthogonal of M' in E). Consider the following four properties : -1 -1 (A) The canonical homomorphism cE: E --f E** is bijective. (B) For every submodule M of E, the canonical homomorphism (C) For every submodule M of E, Moo = M. (D) For every ordered pair of submodules M, N of E, E*/Mo -+ M* is bijective. (M n N)" = M" + No. (u) Show that condition (B) implies (D) (prove that for x* E (M n N)", there exists y* E E* such that (x + y, y*) = (x, x*) for x E M and y E N). (6) Let A be a ring with no divisors of zero, but having no field of left fractions *(for example the tensor algebra of a vector space of dimension > 1, cf. III,§ 5, Exercise 5)*. Ifwe take E = A,, condition (A) is satisfied, but none of conditions (B), (C), (D) (for an example where (B), (C), (D) hold, but not (A), see 9 7, no. 5, Theorem 6). (c) Take A to be a product ring K', where K is a commutative field and I an infinite set. Show that the A-module E = A satisfies conditions (A) and (B), but not (C) (observe that the annihilators of the ideals of A are all of the form KJ, where J c I). (d) Suppose that for two submodules M, N of E such that N c M and N # M, the dual (M/N)* is non-zero; then show that condition (B) implies (e) The kernel of cE is the orthogonal (E*)" of E* in E. Give an example where neither E* nor (E*)" is 0 (consider a module containing an element whose annihilator contains an element which is not a divisor of 0). (C) (f) Let M be a direct factor of E; show that Moo = M + (E*)". 10. Give an example of an A-linear mapping u: E -+ F such that 'U is bi- jective but u is neither injective nor surjective (cf. Exercise 6(6) and (c)). 388 EXERCISES Deduce an example where yo E F is orthogonal to the kernel of tu but where the equation u(x) = yo has no solution. 7 11. Let A be a ring and I an A-module. I is called injective if, for every exact sequence E' -+ E + E" of A-linear mappings, the sequence Hom(E", I) -+ Hom(E, I) -+ Hom(E', I) is exact. (u) Show that the following properties are equivalent : (E) I is injective. (@) For every exact sequence 0 -+ E' -+ E -+ E" -+ 0 of A-linear mappings, the sequence 0 + Hom(E", I) --f Hom(E, I) -+ Hom(E', I) -+ 0 is exact. mapping of N into I can be extended to an A-linear mapping of M into I. exists an element b E I such thatf(u) = ub for all a E a. (y) For every A-module M and every submodule N of M, every A -linear (8) For every left ideal a of A and every A-linear mapping8 a -+ I, there (?J I is a direct factor of every A-module containing it. (0) For every A-module E which is the sum of I and a monogenous sub- module, I is a direct factor of E. (To prove that (6) implies (y), show that property (8) implies that if E is an A-module and F a submodule of E such that E/F is monogenous, then every linear mapping of F into I can be extended to a linear mapping of E into I. Then use Zorn's Lemma. To prove that (0) implies (a), consider the A-module A, x I = M, the submodule N of M consisting of the elements (a, -f(u)) for u E a, and apply (0) to the quotient module M/N.) (b) For a Z-module E to be injective, it is necessary and sufficient that for all XE E and every integer n # 0, there exists ~EE such that ny = x. In particular, the Z-modules Q and Q/Z are injective. 12. (u) For a product hEL n E, ofA-modules to be injective (Exercise 1 l), it is necessary and suficient that each of the E, be injective. (b) Let K be a commutative field, L an infinite set and A the product ring KL. Show that A is an injective A-module (use (u)) but that the ideal a ofA the direct sum of the factors of KL (canonically identified with ideals of A) is not an injective A-module. 7 13. (u) Let A be a ring and I an injective A-module (Exercise 11) such that for every non-zero monogenous A-module E, there exists a non-zero A-homomorphism of E into I. Let M be an A-module, N a submodule of M and m the set of u E Hom,(M, I) such that u(x) = 0 in N. Show that for all y @ N, there exists a u E such that u(y) # 0. 389 EXERCISES I1 LINEAR ALGEBRA (6) Let A be a ring. For cvery (A, A) -bimodule T and every left (resp. right) A-module M, Hom,(M,T) has canonically a right (resp. left) A- module structure deriving from that on T, and there is a canonical A-homo- morphism c~,~: M -+ Hom,(Hom,(M, T), T) such that c~,~(x) is the A-homomorphism u H u(x). Show that if I is an (A, A)-bimodule which, as a left A-module, satisfies the conditions of (a), then the canonical homo - morphism cM, I is injective for every left A-module M. (c) Suppose that I is an (A,A)-bimodule which is injective as a left A- module and a right A-module and that for every (left or right) monogenous A-module E # 0, there exists a non-zero A-homomorphism of E into I. Show that under these conditions, if P is a left (resp. right) projective A-module, Hom,(P, I) is a right (resp. left) injective A-module. (Suppose that P is a left projective A-module. Observe that for every submodule N’ of a rigid A-module N, HornA(”, I) is isomorphic to a quotient module of Hom,(N, I) and apply (6) to the right module N and the left module P.) (d) Let C be a commutative ring, A a C-algebra and I an injective C-module such that for every monogenous C-module E # {0} there exists a non-zero C-homomorphism of E into I. For every A-module M, Hom,(M, I) is an A-module; show that for every projective A-module P, Hom,(P, I) is an in- jective A-module (same argument as in (c)). 14. Let A be a ring. Show that for every A-module M there exists an in- jective A-module I such that M is isomorphic to a submodule of I. (Apply Exercise 13(d) with C = Z, using Exercise 11 (b) and the fact that every A- module is the quotient of a free A -module.) 7 15. An A-module homomorphism u: E --f F is called essential if it is in- jective and if, for every submodule P # {0} of F, ul(P) # (0); it suffices that this condition holds for every momogenous submodule P # {0} of F. If E is a submodule of F, F is called an essential extension of E if the canonical injection E -+ F is essential. (a) Let u: E --f F, v: F -+ G be two A-homomorphisms of A-modules. If u and v are essential, so is v o u. Conversely, if v o u is essential and u is injective, then u and u are essential. be a family of submodules of F containing E and whose union is F. Show that if each of the F, is an essential extension of E, so is F. (c) Let (uJLpL be a family of essential homomorphisms u,: E, -+ F,. Show that the homomorphism 0 u,: @ E,+ @ F, is essential. (Prove it first when L has two elements, then use (b).) (d) The Z-module Q is an essential extension of Z but the product Q” is not an essential extension of ZN. (e) Let E be a submodule of a module F. Show that there exists a submodule (b) Let E be a submodule of an A-module F and let heL hsL hsL 390 Q of F such that Q n E = {0} and that the restriction to E of the canonical homomorphism F 3 F/Q is essential. 16. A submodule E of an A-module F is called irreducible with respect to F (or in F) if E # F and E is not the intersection of two submodules of F distinct from E. (a) For an A-module F to be an essential extension of each of its submodules #(O} it is necessary and sufficient that {0} be irreducible with respect to F. (b) Let be a family of submodules of an A-module F. (EJheL is called a reduced irreducible decomposition of {0} in F if each of the F, is irreducible in F, ifthe intersection of the E, reduces to 0 and if none of the E, contains the intersection of the E, for p # A. When this is so, show that the canonical homo - morphism of F into (F/E,) is essential (if El, = wQ, E,, observe that the image Ff, of El, in F/E, is non-zero and that the image of F in @ (F/E,) contains @ Ff,; then apply Exercise 15(c)). Conversely, if is a family of sub- modules ofF irreducible in F and the canonical homomorphism F --f ,FL (F/E,) is essential, the family (E,) is a reduced irreducible decomposition of {0} in F. 7 17. (a) Let M, N, M’, N’ be four submodules of a module E such that h€L hoL M n N = M‘ n N’ = P. Show that M = (M + (N n M’)) n (M + (N n N’)). Deduce that if M is irreducible in E, then necessarily P = M’ n N or P = N’nN. (b) Let (Ni)l <,<,, be a finite family of submodules of E irreducible in E and M its intersection. Show that if (P,) , <,,, is a finite family of submodules of E such that M = 9 P,, then for every index i such that 1 6 i < n there exists an index +(i) such that 1 < +(i) < m and such that, if we write NI = 9, N,, then M = Pmca n N; (use (a)). Deduce that if none of the P, contains the intersection of the P, of index #j, then m < n (successively replace the N, by suitable P,, using the above result). Conclude that two reduced irreducible decompositions of M in E, one of which is finite, necessarily have the same number of terms (cf. Exercise 23). 7 18. (a) For an A-module I to be injective, show that it is necessary and sufficient that I admit no essential extension distinct from itself (show that this condition implies condition (6) of Exercise 11 (a), arguing as for the proof that (0) implies (6) in that exercise). (b) Let I be an injective A-module. For a submoduIe E of I to be injective, it is necessary and sufficient that E admit no essential extension distinct from 39 1 XI LINEAR ALGEBRA itself and contained in I (using Exercise 15(e)), show that this condition implies that E is a direct factor of I). (c) Let I be an injective A-module, E a submodule of I and h: E --f F an essential homomorphism. Show that there exists an injective homomorphism j: F 3 I such thatj 0 h is the canonical injection of E into I. (d) Let I be an A-module and E a submodule of I. Show that the following properties are equivalent : (a) I is injective and is an essential extension of E. (p) I is injective and for every injective homomorphism j of E into an iniective A-module J, there exists an injective homomorphism of I into J I extending j. (y) I is injective and is the smallest injective submodule of I containing E. (8) I is an essential extension of E and for every essential homomorphism \-I - A: E -+ F, there exists an injective homomorphism j: F -+ I such that j o h is the canonical injection of E into I. (Use (c) and (a).) When these equivalent conditions hold, I is called an injective envelope of E; I is then also an injective envelope of every submodule of I containing E. (e) Let I be an injective A-module and E a submodule of I. Show that every maximal element of the set of essential extensions of E contained in I is an injective envelope of E (use (6)). In particular, every A-module admits an in- jective envelope (cf. Exercise 14). (f) If I, I' are two injective envelopes of the same A-module E, show that there exists an isomorphism of I onto I' leaving invariant the elements of E. 7 19. (a) Let E, F be two A-modules and M an injective submodule of E @ F. Let I be an injective envelope of M n E in M and J a supplementary submodule of I in M. Show that the restriction to I (resp. J) of the canonical projection of E @ F onto E (resp. F) is injective (compose with the projection of I into E an extension E -+ I of the injection M n E -+ I). (b) Let E be an A-module and M a submodule of E, maximal in the set of injective submodules of E; M admits in E a supplementary submodule N which contains no injective submodule #{O). Show that for every injective submodule P of E, the image of P under the projection of E onto M (under the decomposition of E as a direct sum M @ N) is an injective envelope of P n M in M (use (a)). If M' is another submodule of E which is maximal in the set of injective submodules of E, show that there exists an automorphism of E trans - forming M into M' and leaving invariant the elements of N. 7 20. (a) Let A be a ring admitting a field of left fractions K (I, fj 9, Exercise 15). Show that K, considered as a left A-module, is an injective envelope of A,. (Apply criterion (a) of Exercise 18(d) and criterion (6) of Exercise 11 (a) noting that iff: ct 3 K is an A-linear mapping of a left ideal a of A into K, the 392 EXERCISES element x-'f(x) for x E a, x # 0 (the inverse being taken in K) does not depend on x E a.) (b) Let Up be the sub-Z-module of Q/Z consisting of the canonical images of the rational numbers of the form k/p", where p is a given prime number, k E Z and n E; N (11, fj 1, Exercise 12(d)). Show that U, is an injective envelope of each of the Z-modules p-"ZjZ isomorphic to Z[p"Z. Show that there exist automorphisms of Up distinct from the identity automorphism leaving in - variant the elements of a submodule p-"Z/Z. 7 21. (a) Let (Ej),EJ be a finite family of A-modules and for alljE J let I, be an injective envelope of E,; show that @ I, is an injective envelope of @ E,. toJ (6) An A-module M is called indecomposable if it is distinct from (0) and it is not the direct sum of two submodules distinct from (0). Show that if I is an injective A-module, the following properties are equivalent: JEJ (a) (0) is an irreducible submodule in I. (p) I is indecomposable. (y) I is an injective envelope of each of its submodules #O. (Use (a) and Exercise 18 (e).) (6) I is isomorphic to the injective envelope I(A/q), where q is an irreducible left ideal of A. Moreover, when this is so, for all x # 0 in I, Ann(x) is an irreducible left ideal of A and I is isomorphic to I(A/Ann(x)). Deduce that for the injective envelope of an A-module E to be indecom - posable, it is necessary and sufficient that (0) be an irreducible submodule of E. (c) Give an example of an indecomposable Z-module E such that its injec - tive envelope I(E) is decomposable (cf. VII, fj 3, Exercise 5). (d) Let E be an A-module, I an injective envelope of E and I = ,TL I, a decomposition of I as a direct sum of afinite family of indecomposable injective submodules; for all A E L let J - @ I, and let N, = JA n E. Show that (N,),EL is a reduced irreducible decomposition of (0) in E (Exercise 16 (b)) and that I, is an injective envelope of E/NA. Conversely, every reduced irreducible decomposition of (0) in E can be obtained by the above procedure uniquely up to isomorphism (use Exercise 16 (b)). - Ufh 22. Let I be an injective A-module. If I is indecomposable, every injective endomorphism of I is an automorphism of I. Deduce that, for I to be indecom - posable, it is necessary and sufficient that the non-invertible elements of the ring End(1) form a two-sided ideal of that ring. 393 LINEAR ALGEBRA I1 7 23. Let M be an A-module, the direct sum of a family (finite or other - wise) (Mh)APL of submodules such that for all A E L the non-invertible ele - ments of End(M,) form a two-sided ideal of this ring (which implies that M, is indecomposable) . (a) Letf, g be two endomorphisms of M such thatf + g = 1,. Show that for every finite sequence (A,) of distinct indices in L there exists a family of submodules N, (1 < k < s) of M such that for every k one of the endo- morphismsf, g restricted to Mhk is an isomorphism of this submodule onto N, and that M is the direct sum of the Nk (1 < k < s) and the M, for A distinct from As. (Reduce it to the case s = 1 ; if xA is the canonical projection of M onto MA corresponding to the decomposition as a direct sum of M, and @ M,, note that either xA of or x, 0 g, restricted to MA, is an automorphism of MA.) (b) Let f be an idempotent endomorphism of M. Show that there exists at least one index A E L such that the restriction off to MA is an isomorphism of MA ontof(MA); moreoverf(MA) is a direct factor of M. (If g = 1, -f, note that there exists a finite sequence of indices such that the inter- section of Ker(g) and the direct sum of the M,, is #(O) and use (a).) (c) Deduce from (b) that every indecomposable direct factor of M is iso - morphic to one of the MA. (d) Let (NJKEK be a family of indecomposable submodules of M of which M is the direct sum; every N, is therefore isomorphic to an MA and vice versa, by virtue of (c) . Let T be the set of classes of indecomposable submodules of M (under the relation of isomorphism) such that an MA (or an N,) be- longs to one of these classes. For all ~ET, let R(t) (resp. S(t)) be the set of A E L (resp. K E K) such that MA E t (resp. N, E t). Show that, for all t E T, Card(R(t)) = Card(S(t)). (In the notation of (a), let J(K) be the set )\EL such that the restriction of x, to N, is an isomorphism of N, onto MA; show that J(K) is finite and that the J(K) cover R(t) when K runs through S(t). Deduce that Card(S(t)) < Card(R(t)) when R(t) is infinite. When R(t) is finite, show with the aid of (a) that M is the direct sum of the MA for A $ R(t) and a subfamily of (NK)KES(t) of cardinal equal to that of R(t).) (e) Deduce from (d) and Exercises 22 and 21 (d) that if (E,),,L and (Ek)KEK are two reduced irreducible decompositions of (0) in a module F, L and K are equipotent. 24. Let M be an A-module, I its injective envelope (Exercise 18), a a two-sided ideal of A and Q the submodule of I consisting of the t E I such that at = (0); Q has a natural (A/ a)-module structure. Let N be the sub- module of M consisting of the x E M such that ax = (0); if N is considered as an (A/a)-module, show that Q is isomorphic to an injective envelope of N. 25. (a) For an A-module Q to be injective, it is sufficient that, for every k uich EXERCISES projectiue A-module P and every submodule P' of P, every linear mapping of P' into Q can be extended to a linear mapping of P into Q (use the fact that every A-module is a quotient of a projective A -module). (b) For an A-module P to be projective, it is sufficient that, for every injective A-module Qand every quotient module Q" of Q,every homomorphism of P into Q" be of the form C+ 0 u, where I$ : Q-+ Q" is the canonical homomor - phism and u is a homomorphism of P into Q (use the fact that every A- module is a submodule of an injective A-module). 26. For every Z -module G and every integer n > 0, let ,G denote the ker- nel of the endomorphism x - nx of G. If 0 -+ G' -+ G -+ G" -+ 0 is an exact sequence, define a canonical homomorphism d : ,G" -+ G'/nG' such that the sequence d 0 --+ ,,G' --+ ,G" G'/nG' -+ G/nG -+ G"/nG" --+ 0 is exact. If ,G and G/nG are finite, we write then show that nG', nG", G'InG', G"/nG" are finite and that h,(G) = Card(G/nG) - Card(,G); h,(G) = h,(G') + h,(G"). such thatf((x,, yl), . . . , (x,, y,)) = g r ,z xi B yi 1 . (Note that if 394 395 LINEAR ALGEBRA I1 EXERCISES the difference combination with integer coefficients of elements of the form (2) of no. 1 .) (xi,yt) - 2 j (x;,yI) in the Z-module Z(ExF) is a linear *2. Consider the field C of complex numbers as a vector space over the field R of real numbers. (a) Show that the canonical mapping C C is not injective. (6) Show that the two C-module structures on C BR C arising from each C -+ C \, of the factors are distinct*. Show that if F is finitely generated, the canonical mapping 3. (a) Let (Eh)hof. be a family of right A-modules and F a left A-module. (n h€L Eh) @AF (Eh @A F, is surjective. (b) Give an example of a commutative ring A and an ideal rn of A such that the canonical mapping AN @A (A/rn) -+ (A/m)N is not injective (cf. 5 1 , Exercise 2 (b)). (c) Give an example of a commutative ring A such that the canonical mapping AN @A AN -+ AN is not surjective (observe that for given n, the , image of N under a mapping of the form rnw t=1 c u,(rn)u,(n), where the ui and u, belong to AN, generate a finitely generated ideal of A). (d) Deduce from (b) and (c) an example where the canonical homomorph - ism (22) (no. 7) is neither injective nor surjective. 4. Let E be a right A-module and F a free left A-module. Show that if x is a free element of E and y an element #O of F, then x @ y # 0; if also A is commutative and y is a free element of F, then x @ y is free in the A-module I I I I @AF* §4 1. Let A, B be two rings, E a left A-module, F a right A-module and G an (A, B)-bimodule; Hom,(F, G) then has a left A-module structure and HornA@, G) a right B-module structure. Show that the Z-modules HomA(E, Hom,(F, G)) and Hom,(F, HomA(E, G)) are both canonically isomorphic to the Z-module of Z-bilinear mappings f of E x F into G such that f(ax,y) = mf (x,y) and f(x,yp) =f(x,y)p for a EA, p E B, x E E, y E F. 2. Let A be the ring 2/42, rn the ideal 22/42 of A and E the A-module , I I 396 A/m. Show that the canonical mapping E* @A E -+HornA(& E) is neither injective nor surjective. The same is true of the canonical mapping E @A E -+ HOmA(E*, E) and the canonical mapping E* @A E* -+ (E @A E)*. 3. Show that under the conditions of no. 2, if E is assumed to be a finitely generated left A-module and F a projective right B-module, the canonical homomorphism (7) (no. 2) is bijective. 4. Given an example where the canonical homomorphism (21) (no. 4) is neither injective nor surjective, although F, = C, El = C and E, and F, are finitely generated C-modules (cf. Exercise 2). 5. Let A, B be two rings, E a right A-module, F a left A-module and G a (u) Show that there exists one and only one Z-linear mapping (B, A)-bimodule. ?:E @A F -+ HOmB(HOmA(E, G), G @A F) which maps the tensor product x @ y, where x E E and y E F, to the mapping V,,~:U H u(x) @ y of HomA(E, G) into G @A F. When B = A and G = Jd, the homomorphism reduces to the canonical homomorphism (1 5) (no. 2). (6) Show that if F is a projective A-module and E and G are such that for x # 0 in E there exists u E HomA(E, G) such that u(x) # 0, then the homo - morphism q is injective. 7 6. Let A, B be two rings, E a left A-module, F an (A, B)-bimodule and (u) Show that there exists one and only one Z-linear mapping G a right B-module. a:Hom,(F, G) @A E -+ Hom,(Hom,(E, F), G) such that for x E E and u E Hom,(F, G), cr(u ($3 x) is the mapping u H u(U(X)) of HomA(E, F) into G. (b) If E is a finitely generated projective A-module, show that CT is bijec- tive. (c) Suppose that G is an injective B-module (3 2, Exercise 11) and that E is the cokernel of an A-linear mapping A: -+ At. Show that cr is bijective (start from the exact sequence A: -+A: -+ E -+ 0 and use (b), the definition of inkcfive modules and Theorem 1 of 9 2, no. 1, and Proposition 5 of 9 3, no. 6). 7. Show that, if El, E,, F,, F, are four modules over a commutative ring 397 I1 LINEAR ALGEBRA c, the canonical homomorphism (21) (no. 4) is composed of the homomor- nhisms EXERCISES where the first two homomorphisms come from the canonical homomorphism (7) (no. 2) and the last is the isomorphism of no. 1, Proposition 1. 8. Let E, F be two finitely generated projective modules over a commuta - tive ring C. Show that if u is an endomorphism of E and u an endomorphism of F then Tr(u €3 v) = Tr(u)Tr(v). 9. Let A be a ring, C its centre, E a right A-module, F a left A-module and E*, F* the respective duals of I.: and F. An additive mapping f of E* mC F* into A is called doubly linear iff (uw) = uf (w) and f (wu) =f(w)a for all w E E* BC F* and all a E A. Show that there exists one and only one C-linear mapping 4 of E @A F into the C-module L of doubly linear map- pings of E* BC F* into A such that for all x E E, y E F, x* E E*, y* E F*. If E and F are finitely generated projec- tive modules, 4 is bijective. (4(x @Y))(X* BYY*) = (x*,x)(Y,Y*> §5 1. Give an example of a homomorphism p :A -+ B of commutative rings and two A-modules E, F such that the canonical homomorphism (17) (no. 3) is neither injective nor surjective (cf. fj 4, Exercise 2). 2. Give an example of a free A-module and a ring B such that the homo - morphism (20) (no. 4) is not injective (§ 3, Exercise 3). 3. Give an example of a monogenous A -module E and a ring B such that the homomorphism (20) (no. 4) is not surjective (cf. fj 4, Exercise 2). 4. Let p:A --f B be a ring homomorphism. For every A-module E, show that the diagram E E** (E**)(B, is commutative. 5. Let K be a field, A the product ring KN, a the ideal K") of A and B the ring A/a. Show that a is a projective A-module which is not finitely generated, although a(B) = (0). 6. With the rings A and B defined as in fj 1, Exercise 7, let M be an A- module; show that for M to be finitely generated (resp. free, projective), it is necessary and sufficient that M RAB be finitely generated (resp. free, pro- jective). 7. Let p:A -+ B be a ring homomorphism. For every left B-module F, define a canonical B-module homomorphism F-t F(P*(F)) F(E) + F(P*(W))) -+ F(E) P*F) -+ P*(F(P*(F))) -+ P*(F) and show that if E is a left A-module, the composite mappings are the identity mappings. §6 1. Let (En,fnm) be the inverse system of Z-modules whose indexing set is N, such that En = 2 for all n and, for n < m, fnm is the mapping x ++ 3m-nx. For all n, let u, be the canonical mapping En -+ 2/22. Show that (un) is an inverse system of surjective linear mappings but that lim u, is not surjective. 2. Let (Ea,fpa) be a direct system of A-modules. Show that the A-module lim E, is canonically isomorphic to the quotient of the direct sum @ E, = F by the submodule N generated by the elements of the formjl,( foa(xa)) - ja(xa), for every ordered pair (a, p) such that cc < and all xu E E,, the j,: E, -+ F being the canonical injections. 3. Let (Fn,fmn) be the direct system of Z-modules such that F, is equal to up (3 1, Exercise 12 (d)) for all n 2 0 and fmn is the endomorphism * ++P"-"x of U, for n < m. Show that lim F, = (0) although the fmn are t 3 sujective. ---f 7 4. Let (Fa,&,) be a direct system of A-modules. (a) For every A-module E, define a canonical Z-module homomorphism ~:lim Hom,(E, Fa) -+ HOmA(E, lim Fa). where E is not injective (cf Exercise 3, taking E = lim Fa). 4 4 (6) Show that if E is finitely generated, E is injective. Give an example --+ 399 398 I1 LINEAR ALGEBRA (c) Show that if E is finitely generated and thefpa are injective, E is surjec - tive. Give an example where E = lim 4 F, is free, thefpa are injective and & EXERCISES is not surjective. (d) Show that if E is projective and finitely generated, E is bijective. *(el Let A be an integral domain and a an ideal of A which is not finitely \-I - generated. If (a,) is the family of finitely generated ideals contained in a, show that E = A/a is canonically isomorphic to the direct limit of the F, = A/ a,, but that the homomorphism E is not surjective., §7 1. Show that if E is a vector space #{O} over a field K with an infinity of elements, the set of generating systems of E is not inductive for the order relation I> (form a decreasing sequence (S,) of generating systems of E such that the intersection of the S, is empty). 7 2. Let K be a field, L a subfield of K, I a set and (xt)l,t,m a free family of vectors in K: such that the coordinates of each of the xi belong to L. Let V be the vector subspace of Kf generated by the x,; show that there exists a finite subset J of I, with m elements, such that for every vector z = (<a)apI of V, all the <, belong to the subfield of K generated by L and the m elements <,-, where p E J. (By applying Corollary 3 of no. 5, to the space Lt, show that it can be reduced to the case where there exist m indices pt E I (1 < i < m) such that P‘P,(X,) = h)* T[ 3. (a) Let G be a group and M an infinite subset of G. Show that if G’ is the subgroup of G generated by M, then Card(G’) = Card(M). (b) Let K be a field and M an infinite subset of K. Show that if K’ is the subfield of K generated by M, then Card(K’) = Card(M). (Consider two sequences (A,),ao, (P,JnbO of subsets of K such that A, = M, P, is the multiplicative subgroup of K* generated by A, n K* and A,,, the additive subgroup of K generated by P,; then apply (a).) (c) Let K be a field and I an infinite set; show that Card(K) < dim(Kt). (Reduce it to the case I = N and argue by reductio ad absurdum. Let B be a basis of K: such that Card(B) < Card(K) and let L be the subfield of K generated by the coordinates of all the elements of B; then Card(L) < Card(K). Form a sequence (En),, , of elements of K such that for all n, 5, does not belong to the subfield of K generated by L and the Et of index i < n; then apply Exer- cise 2 to obtain a contradiction, by considering the point x = (5,) of KF.) (d) Deduce from (c) that for every field K and every infinite set I, dirn(K:) = (Card(K))Card(l) (“Erdos-Kaplansky theorem”). 4QO 4. Give an example of an infinite sequence (F,) of vector subspaces of a vector space E such that codim(f) F,) > 2 codim(F,). (Take codim(F,) = I for all n and use Exercise 3 (d).) 5. Let (HJhsL be a family of hyperplanes (passing through 0) of a vector space E over a field K, which is a covering of E. (a) Show that, if K is finite, then Card(L) >/ 1 + Card(K) and, if K is infinite, Card(L) 2 KO = Card(N). (Prove by induction on r that if K has at least r elements, then E cannot be the union of r hyperplanes.) Show by examples that these inequalities cannot be improved without an additional hypothesis (when K is infinite, consider the space E = K:N)). (6) If E is finite-dimensional, show that Card(L) >, 1 + Card(K). (Argue by induction on dim(E).) T[ 6. Let S be a non-empty finite subset of a vector K-space E, not contain- ing 0. Suppose that there exists an integer k > 1 with the following property: (*) for all x E S, there is a partition of S - {x} into h free subsets with h Q k. For such a partition (Nt)l,l,h and every sequence (rJ1, jSm of integers of (1, h) a subspace F,,,,. ,.,m of E is defined by induction on m as follows: F, is the subspace generated by N,,, F,, , . , the subspace generated by the intersection of F,, ...I p-l and NTp. Suppose that: (**) for all x E S and every partition (Nt)l,i,h of S - {x} into at most k free subsets, there exists at least one sequence (r,)l ,fSm such that x # Fr,,, . . .?,,,a (a) Let n be the least of the integers m >/ 1 for all possible choices of x E S and let (NJ,,,,, and (r,),,,,,,, satisfy conditions (*) and (**). Show that of necessity n = 1. (Argue by reductio ad absurdum by assuming x $ F,,,, , . , ,, and 11 > 1. Then necessarily x E F,. For simplicity write E, = Frlr,,. , ,, for 1 6 P d n and x = a,yt with at E K and yt E N,,; there is at least one y, not belonging to En-,, denote it by y; then write N = Nrn - {y}, N; = Nt for 1 < i d h and i # r,, Ni, = N u {x}, so that (Ni),,,,, is a partition of s - {y} consisting of free subsets. Define EL = E and for 1 < p < n, Eb as the subspace generated by the intersection of EL- and Nip. Now y En- ; let q be the least integer such that y$E,; show that Ek Q E,. Let s be the 1-t integer such that Ej Q: E,; show that r, = r,; deduce finally a contra- diction from the relations y E E,, Ej- , c E,-, and Ej Q E,.) (b) Conclude from (a) that with the given hypotheses there exists a par- tition of S into h free subsets for an integer h < k. 7 7. Let S be a non-empty finite subset of a vector K -space E, not contain - ing 0. Suppose that there exists an integer k > 1 such that for every subset 40 1 I1 LINEAR ALGEBRA T of S, Card(T) < k.rg(T). Show that there exists a partition of S into h free subsets for some integer h < k. (Prove that S satisfies condition (**) of Exercise 6, arguing by induction on Card(S) and taking amongst all the subspaces F,,...,,,, one of those with the smallest possible dimension; if G is such a subspace andj an index for which Card(G n N,) is the least possible, Drove that EXERCISES k.Card(G n N,) 6 Card(G n (S - {XI)) 6 Card(G n S) < k.Card(G n N,).) 8, Let E be a vector space. Show that every endomorphism of E which is not a right divisor of 0 in End@) is surjective (3 2, Exercise 7). 9. Let K be a field. (a) In the vector space E = K:z), let (en)neZ denote the canonical basis; let u be the automorphism of E defined by the relations u(e,) = en+, for all n E Z. If u = 1, - u, show that u is an injective endomorphism of E, but that dim(Coker(u)) = 1. (b) In the vector space E = KiN), let (en)neN denote the canonical basis; let v be the endomorphism of E defined by v(eo) = e,, u(e,) = en-l + en for n 3 1. Show that u is an automorphism of E and that, if u = 1, - u, u is a surjective endomorphism of E such that dim(Ker(u)) = 1. (c) Let I be a non-empty set. Show that every automorphism u of the vector space E = Kg’ can be written as a difference u = u - w of two automorph- isms of E, except when K has only two elements and I consists of a single element. (Note that it suffices to prove that one particular automorphism u is of the desired form; when K is a field with two elements, consider separately the cases where Card(1) = 2 and Card(1) = 3.) 7 10. Let E be a vector space over a field K. Show that every endomorphism u of E is an automorphism or can be written as u = u - w, where u and w are automorphisms of E. (Observe that u can always be replaced by s1 0 u 0 s2, where s1 and s2 are two automorphisms of E. Distinguish two cases, according to whether dim(Ker(u)) < dim(Coker(u)) or dim(Ker(u)) > dim(Coker(u)) ; when rg(u) is finite, then the first case always holds. When dim(Ker (u)) < dim(Coker(u)), it can be reduced to the case where Ker(u) n Im(u) = (0); if dim(Ker(u)) = dim(Coker(u)), it can also be assumed that Im(u) + Ker(u) = E and then apply Exercise 9 (c). If dim(Ker(u)) < dim(Coker(u)), there is a supplementary subspace in E of Ker(u) of the form W @ Im(u) with dim(W) = dim(Im(u)) = dim(E) ; apply the remark at the beginning with s,(u(Im(u))) = Im(u) and use Exer- cise 9 (u) and (c). When dim(Ker(u)) > dim(Coker(u)). it can be reduced to the case where Ker(u) c Im(u); this time take sl(Ker(u)) = -1 u (Ker(u)) and use the results of the preceding cases and Exercise 9.) mapping. Show that for every vector subspace V of F, 11. Let E, F be two vector spaces over a field K and u:E + F a linear dim(: (V)) = dim(V n Im(u)) + dim(Ker(u)). 12. Let E, F be two vector spaces and u, u two linear mappings of E (a) Show that the following inequalities hold : into F. rg(u) 6 rg(u + u) + rg(u) rg(u + u) 6 inf(dim(E), dim(F), rg(u) + rg(u)). If E and F are finite-dimensional, show that rg(u + u) can take any integer value satisfying the inequalities Irg(u) - rg(u)[ 6 rg(u + u) 6 inf(dim(E), dim(F), rg(u) + rg(u)). dim(Ker(u + u)) < dim(Ker(u) n Ker(v)) + dim(Im(u) n Im(v)). dim(Coker(u + u)) < dim(Coker(u)) + rg(u). u(Ker(u)) c Im(u + u) and dim(u(V)) 6 rg(u).) (b) Show that (c) Show that (If V is a supplementary subspace of Ker(v) in E, note that 13. Let E, F, G be three vector spaces and u: E -+ F, u: F + G two linear (a) Show that there exists a decomposition of E as a direct sum of Ker(u) mappings. and two subspaces M, N such that Ker(u 0 u) = M @ Ker(u) and Im(u 0 u) = v(u(N)). if Ker(u) and Ker(v) are finite-dimensional, show that dim(Ker(u 0 u)) can take every integer value satisfying the above inequalities. rg(u) = rg(u 0 u) + dim(Im(u) n Ker(v)) (6) dim(Ker(u)) 6 dim(Ker(u 0 u)) < dim(Ker(u)) + dim(Ker(u)); (c) The following equalities hold: ~ ,, rg(u) = rg(v 0 u) + codim,(Im(u) + Ker(u)). If F is finite-dimensional, show that rg(v o u) can take any integer value satisfying the inequality sup(0, rg(.) + rg(u) - n) < rg(u 0 u) < inf(rg(u), rg(u)). 14. Let E, F, G be three vector spaces and u:E 4 F, w:E 4 G two linear mappings. Show that if-i(O) c -1 w(O), there exists a linear mapping u of F into G such that w = u o u (cf. 3 2, Exercise 8). 402 403 I1 LINEAR ALGEBRA 15. Let E, F be two vector spaces over a field K and u:E -+ F a linear EXERCISES mapping. (a) The following conditions are equivalent: (1) Ker(u) is finite -dimen - sional; (2) there exists a linear mapping v ofF into E such that v o u = 1, + w, where w is an endomorphism of E of finite rank; (3) for every vector space G over K and every linear mapping f:G -+ E, the relation rg(u of) < +a implies rg(f) < +a. (b\ The following conditions are equivalent: (1) Coker(u) is finite-dimen - - \-I sional; (2) there exists a linear mapping v of F into E such that u o v = 1, + w, where w is an endomorphism of F of finite rank; (3) for every vector space G over K and every linear mapping g : F -+ G, the relation rg( g o u) < + m implies rg(g) < +a. 16. Let E, F be two vector spaces over K and u:E -+ F a linear mapping. u is said to be ofjnite index if Ker(u) and Coker(u) are finite-dimensional and the number d(u) = dim(Ker(u)) - dim(Coker(u)) is then called the index of u. Let G be a third vector space over K and u: F -+ G a linear mapping. Show ~~~ ~ that if two of the three lineAar mappings u, v, u 0 u are of finite index, so is the third and d(v 0 u) = d(u) + d(v) (use Exercise 13 (u)). 17. Let E, F, G, H be four vector spaces over K and u : E -+ F, v : F + G, w : G -+ H three linear mappings. Show that rg(v o u) + rg(w o v) < rg(u) + rg(w o v o u). 18. Let E, F be two vector spaces over a field K and u, v two linear mappings of E into F. For there to exist an automorphismfof E and an automorphism g of F such that v = g o u of, it is necessary and sufficient that rg(u) = rg(u), dim(Ker(u)) = dim(Ker(v)) and dim(Coker(u)) = dim(Coker(v)). 19. Let E be a vector space over a field K. (u) Show that every mappingfof E into E which is permutable with every automorphism of E is of the form x H ar, where cc E K (show first that for all x E E there exists p(x) E K such that f(x) = p(x)x, using the fact that f commutes with every automorphism of E leaving x invariant). (b) Let g be a mapping of E x E into E such that, for every automorphism u of E, g(u(x), u(y)) = u(g(x,y)) for all x, y in E. Show that there exist two elements d~, p of K such that g(x,y) = ar + @y on the set of ordered pairs (x, y) of linearly independent elements of E; moreover there exists a mapping (pofK x KintoKsuchthatg(Ax, px) = +(A, p)xforall XEE (same method). If further g(u(x), u(y)) = u(g(x,y)) for every endomorphism u of E, then g(x,y) = ar + @y for all x, y in E. Generalize to mappings of E" into E. 20. Let E be a vector space. (u) Show that if M and N are two vector subspaces of E and M', N' the orthogonals of M and N respectively in E*, the orthogonal of M n N is M' + N' (cf. 5 2, Exercise 9 (u)). (6) If E is infinite-dimensional, show that there exist hyperplanes H' of E* such that the subspace of E orthogonal to H' reduces to 0 (consider a hyperplane containing the coordinate forms corresponding to a basis of E) . (G) Show that if E is infinite -dimensional there exists an infinite family (v,) of subspaces of E such that, if Vl is the subspace of E* orthogonal to V,, the subspace of E* orthogonal to (d) Deduce from (b) that if E is infinite-dimensional, there exists a decompo - sition V' 0 W' of E* as a direct sum, W' being finite -dimensional, for which the sum V + W of subspaces V, W of E orthogonal respectively to V' and W' is distinct from E. (e) For a subspace of E to be finite -dimensional, it is necessary and sufficient that its orthogonal in E* be of finite codimension. V, is distinct from 2 V:. 2 1, With the notation of Theorem 8 of no. 6, let u denote the linear mapping x n ((x, x:)) of E into K: and yo = (q,). K(dI) is identified with a subspace of the dual of Kf under the canonical mapping of Ky) into its bidual; then show that if N' is the kernel of {u, the condition of Theorem 3 (no. 5) expresses the fact that yo is orthogonal to the intersection of N' and KY). When I is infinite and the system (21) (no. 5) is of finite rank, show that this intersection is dis- tinct from N' (note that N' is then of finite codimension). 22. Let E and F be two vector spaces over a field K and u a linear mapping of E into F. If V is a vector subspace of E and V' the orthogonal of V in E*, show that the dual of u(V) is isomorphic to h(F*)/(V' n %(F*)). If W' is a subspace of F* such that *u(W) is finite-dimensional and W is the orthogonal of W' in F, *u(W') is isomorphic to the dual of the space u(E)/(W n u(E)). 23. Show that for a linear mapping u of a vector space E into a vector space F to be such that rg(*u) = rg(u), it is necessary and sufficient that rg(u) be finite (cf. Exercise 3 (d)). 24. Let E be a vector space of finite dimension n > 1 over a commutative field K; show that, unless n = 2 and K is a field with two elements, there dsts no isomorphism (p of E onto E* depending only on the vector space structure of E. (Note that if (p is such an isomorphism, then necessarily (x, +(y)) = (u(x), +(u(y))) for x, y in E and for every automorphism u of E (Set Theory, IV, 5 1, no. 5).) 25. (u) Let K be a field and L, M two sets; there are canonical inclusions (Kt)(') c (K:MM')L c KtxM. Show that for two of these vector spaces to be equal, it is necessary and sufficient that one of the sets L, M be finite. 404 405 I1 LINEAR ALGEBRA (6) Let be a family of right vector spaces over K and (F,JltEM a family of left vector spaces over K. For the canonical mapping (26) (no. 7) to be bijective, it is necessary and sufficient that one of the vector spaces EXERCISES n Ex, n F, be finite-dimensional (use (u)). h€L ,EM 26. (u) Let E, F be two left vector spaces over a field K; for the canonical homomorphism E* @, F -+ Hom,(E, F) to be bijective, it is necessary and sufficient that one of the spaces E, F be finite -dimensional. (6) Let E be a right vector space and F a left vector space over a field K. For the canonical homomorphism E @, F -+ Hom,(E*, F) to be bijective, it is necessary and sufficient that E be finite-dimensional (use Exercise 3 (d)). 27. (u) Under the hypotheses of Proposition 16 (i) (no. 7), for the canonical homomorphism (27) (no. 7) to be bijective, it is necessary and sufficient that E or F be finite -dimensional (note that the image under (27) (no. 7) ofan element of Hom,(E, G) BL F maps E to a subspace of G @L F contained in a subspace of the form G BL F’, where F’ is a finite-dimensional subspace of F). (6) Under the hypotheses of Proposition 16 (ii) (no. 7), for the canonical homomorphism (28) (no. 7) to be bijective, it is necessary and sufficient that one ofthe ordered pairs (El, E,), (El, Fl), (E,, F,) consist of finite-dimensional vector spaces (use (u) and Exercise 7 of 5 4). 28. Let p : K -+ A be an injective homomorphism of a field K into a ring A and let E be a left vector space over K. For the canonical mapping (E*)(A, + (Eo,)* to be bijective, it is necessary and sufficient that E be finite -dimensional or that A, considered as a right vector K-space, be finite- dimensional (use Exercise 25 (6)). When one of these two cases holds, there exists a canonical bijection (E**)(,, -+ (E,,)**. 29. Let A be an integral domain, K its field of fractions, E an A-module (u) Show that every linear form on E is zero on T(E), so that the duals (6) Show that the canonical mapping (E*)(,, -+ (E,,,)* is injective and (c) Give an example of a torsion-free A-module E such that E* = {0} and 7 30. Let A be an integral domain and E, F two A-modules. Show that if x is a free element of E and y a free element of F, then x @ y # 0 in E @A (Reduce it to the case where E and F are torsion-free, then to the case where E and F are finitely generated and use Exercise 29 (6) to show that there is an A-bilinear mappingfof E x F into K such thatf(x, y) # 0.) 406 and T(E) its torsion module. E* and (E/T(E)) * are canonically isomorphic. that it is bijective when E is finitely generated. EW # {Oh *31. Let KO be a commutative field, A the polynomial ring Ko[X, Y] in two indeterminates over KO, which is an integral domain, and E the ideal AX + AY of A. Show that in the tensor product E mAE the element x @Y - Y @X is #O and that XY(X BY - Y @X) = 0 (consider the A-bilinear mappings of E x E into the quotient module A/E)., 32. Extend the results of no. 10 to the case where A is a non-commutative ring admitting ajeld of left fractions K (I, 5 9, Exercise 15). (Note that if Et (1 < i < n) are elements of K, there exists an a # 0 in A such that all the a& belong to A.) 33. Let A be an integral domain. An A-module E is called divisible if for all x E E and all a # 0 in A, there exists y E E such that ocy = x. (a) Let E, F be two A-modules. Show that if E is divisible, so is E @A F. If E is divisible and F is torsion -free, HomA(E, F) is torsion-free. If E is a torsion A-module and F is divisible, then E I; = (0). If E is a torsion A- module and F is torsion-free, then HomA(E, F) = (0). (b) Show that every injective A-module (5 2, Exercise 11) is divisible and conversely that every divisible torsion -free A-module is injective (use criterion (6) of 2, Exercise 11). 34. Let A be an integral domain, P a projective A-module, P‘ a projective submodule of P and j: P‘ -+ P the canonical injection. Show that for every torsion -free A -module E, the homomorphism j @ 1 : P’ @A E -+ P @A E is injective (reduce it to the case where E is finitely generated and embed E in a free A-module) . T[ 35. (u) Let K be a field and E a (K, K)-bimodule. Suppose that the dimensions of E, as a left and right vector space over K, are equal to the same finite number n. Show that there exists a family (e,) of n elements of E, which is a basis for each of the two vector space structures on E. (Note that if (6j)1Sj,, is a family of m < n elements of E, which is free for each of the two vector space structures on E and V is the left vector subspace and W the right vector subspace generated by (6,), either V + W # E, or V n CW and W n CV are non-empty; in the latter case, if y E V n CW, z E W n CV, Y + z forms with the 6, a family of m + 1 elements which is free for each of the two vector space structures on E.) (b) Let F be a subbimodule of E, whose two vector space structures over K have the same dimension p < n. Show that there exists a family (eJ1 ,t4n of elements of E which is a basis of E for each of its vector space structures and such that (e,),,,,, is a basis of F for each of its two vector space structures (same method). (C) Let (6,)1~j~,,-1 be a family of n - 1 elements of E, which is free for each of the two vector space structures on E. Let V (resp. W) be the hyper - Plane generated by the (6,) in E considered as a left (resp. right) vector space. 407 I1 LINEAR ALGEBRA Show that if V C W (resp. W C V), then necessarily V = W (note that if a 4 W, the set of A E K such that ha E W is a left ideal). *36. (u) Let KO be a commutative field and K = Ko(X) the field of rational functions in one indeterminate over KO (Chapter IV). A (K, K)- bimodule structure is defined on K as follows: the product t.u of an element E K by a left operator t E K is the rational function t(X)u(X) ; the product u.t of u by a right operator t E K is the rational function u(X)t(X2); the left (resp. right) vector K-space structure thus defined on K is then l-dimen- sional (resp. 2-dimensional) . Deduce examples of (K, K)-bimodules such that dimensions of the two vector space structures of such a bimodule are arbitrary integers. (b) Deduce from (u) an example of a (K, K)-bimodule E whose two vector space structures have the same dimension and such that there exists a sub- bimodule F of E whose two vector space structures do not have the same dimension. * 737. (a) Let E be a left vector space (finite-dimensional or otherwise) and A, (1 < i < n) vector subspaces of E. Suppose that, for every sequence (uJ1 G,Cn of points of E such that a, E A, for all i, the vector subspace of E generated by the a, is of dimension < m (m an integer < n). Then prove that there exists a subspace W of E of dimension h < m, containing h + (n - m) of the subspaces A,. (Argue by induction on n. Prove first that (for fixed n) attention may be confined to the case where m < n and dim(A,) < m for all i; it may also be assumed that A, # (0) for all i and that the A, are not all of dimension 1. Then argue (for fixed n) by induction on d = z, dim(A,). If for example dim(A,) 2 2, consider a 1-dimensional subspace B, of A,, of dimension 1 and apply the induction hypothesis to A,, . . . , A,-,, B,; con- clude that there exists a number k such that 1 < k < m and a k-dimensional subspace U of E containing k of the A,, for example A,, . . . , A,; replacing, if need be, k by an integer k’ such that 1 < k’ < k, show also that it can be assumed that there exist vectors 6, E A, for 1 < i < k such that b,, . . . , bk form a basis of U; then project onto a supplementary subspace of U in E and use the induction hypothesis.) (6) Let K be a finite field and let E be an N-dimensional vector space over K. Let Ui (1 < i < n) be vector subspaces of E such that, for every subset H of (1, n), the intersection of the U1 such that i E H has dimension <N - Card(H). Show that the union of the U, cannot be equal to E (argue by duality using (u)). 38. Let K be a commutative field of characteristic 0 and E a finite-dimen- sional vector space over K. Let PI, . . . , prn be projectors of E onto vector sub- spaces of E such that 1, = p1 + pz + . . . + prn. Show that E is the direct rn EXERCISES sum of the p,(E) and that the pi are pairwise permutable (take the traces of the two sides). 39. Let K be a field, L a subfield of K and (K,)aEI a left directed family of subfields of K such that L = a91 K,. Let E be a vector space over K and (u,),~,~,,, a finite family of elements of E; show that if the family (u,) is free over L, there exists an index t( such that (u,) is free over K,. (Argue by induc - tion on m - r, where r is the rank of the family (ai) over K.) §8 1. Let K be a field, K’ a subfield of K and V a non-zero right vector space over K. (0) Let V’ be a K‘-structure on V. Show that K’ is equal to the set of p E K such that, for all x E V’, xp E V’. (6) Let I?’ be the group of automorphisms of K leaving invariant the ele- ments of K’. Show that if there exists no element of K not belonging to K and invariant under all the automorphisms 0 E I?’, there exists no element of v not belonging to V’ and invariant under every bijective dimorphism of V (relative to an automorphism of K) which leaves invariant all the elements of V’. (c) Conversely, let E be a subset of V and G the group of bijective dimor - phisms of V leaving invariant all the elements of E; for all u E G, let be the corresponding automorphism of K and let I? be the group of automorphisms of K the image of G under the homomorphism u H oU. Suppose that G con- tains no automorphism of V distinct from the identity automorphism and that there exists no element of V not belonging to E and invariant under all the dimorphisms u E G. Show that if K’ is the subfield of elements of K in- variant under all the automorphism 0 E I?, E is a K‘-structure on V. (Prove first that E is an additive subgroup of V containing a basis of V; let K” be the set of elements p E K such that the relation x E E implies xp E E. Show that the elements of K” are invariant under all Q E I? and that K” is a subfield of K; deduce that E is a K”-structure on V.) 2. Let K be a field, K’ a subfield of K, V a right vector space over K, v’ a K’-structure on V and R’ the canonical image of the dual V‘* in the dual v* ofV (no. 4). For R to be a generating system of V*, it is necessary and sufficient that V bc finite-dimensional or that K be a finite-dimensional right vector K’-space (use Exercise 25 of 3 7). 3. Let K be a field, L a subfield of K and let KL denote the field K con- sidered as a left vector space over L; E = EndL(KL) has canonically a (K, K)- bhodule structure. The dual (KL)* is contained in E and is a (K, L)-sub- bhodule of E. When L is contained in the centre of K, the two vector L-space 409 408 LINEAR ALGEBRA I1 EXERCISES structures on E, obtained by restricting to L the field of scalars of the two vector K-space structures on E, are identical. Suppose henceforth that L is contained in the centre of K and that the dimension of K, is finite and equal to n. (a) Show that (KL)* is 1-dimensional under its left vector space structure Over K (calculate the dimension of (K,)* over L in two ways). (6) Show that E is an n-dimensional left vector space over K (same method). (c) Let F be a finite-dimensional left vector space over K and F[,, the corresponding vector space over L. If u, # 0 is a linear form on K,, show that the mapping x' H u, 0 x' of the dual F* of F (considered as a vector space over L) onto the dual (FIL1)* of F,,] is bijective (use (a)). 7 4. Let K be a field of finite rank over its centre Z and L a subfield of K containing Z. (a) Show that the left and right vector space structures of K with respect to L have the same dimension. (6) Show that properties (a), (b), (c) of Exercise 3 also hold in this case (if u, is a linear form #O on L, and v, a linear form #O on K,, note that E. (u, 0 u,) = u, o ({no) for E E K, that {. (u, 0 u,) runs through the dual (Kz)* when E runs through K and use Exercise 3 (c)). 5. Let KO be a subfield of a field K such that K is of dimension 2 as a right vector space over KO. Let E be a left vector space over K. (a) Let E, be a subset of E which is a vector space over KO and let V be the largest vector sub -K-space of E,; if W, is a vector sub-KO-space of E, supplementary to V, show that the vector sub-K-space W of E generated by Wo is such that V n W = (0) (note that if an element p E K does not belong to KO, the relation px E E, for an x E E, implies x E V). (6) Let EL be another vector sub-K O-space of E and V' the largest vector sub-K-space of E& For there to exist a K-automorphism of E mapping Eo to Eb, it is necessary and sufficient that V and V' have the same dimension with respect to K, that the codimensions of V in E, and V' in EA (with respect to KO) be equal and that the codimensions of E, and EL in E (with respect to KO) be equal (use (a)). 39 *l. In an affine space E over a field K, a quadruple (a, 6, c, d) of points of E is called a parallelogram if 6 - a = c - d, in which case (a, d, c, 6) is also a parallelogram. Show that if K is of characteristic #2 the midpoints of the ordered pairs (a, c) and (b, d) are then equal; what can be said when K is of characteristic 2 ?* *2. Let E be an affine space over a field of characteristic 22 and a, b, C, d any four points of E. Show that if x, y, z, t are the respective midpoints of the ordered pairs (a, 6), (6, c), (c, d), (d, a), the quadruple (x, y, z, t) is a parallelo - gram (Exercise I).* *3. Let K be a commutative field of characteristic #2, E an affine plane Over K and a, 6, c, d four points of E any three of which do not lie on a straight line. Let D,, denote the line passing through two distinct points x, y of E. Suppose that the lines Dab and D,, have a common point e and that D,, and Dbc have a common pointf. Show that the midpoints of the three ordered pairs (a, c), (b, d), (e,f) lie on a straight line. What becomes of this property when Dab and Dcd are parallel or when Dad and Dbc are parallel? Consider the case where K is a field with 3 elements., *4. Let K be a field whose characteristic is different from 2 and 3, E an affine space over K, a, 6, c three points of E not on a straight line and a', 6', c' the respective midpoints of the ordered pairs (6, c), (c, a) and (a, 6). Show that (in the notation of Exercise 3) the lines D,,, Dbb, and D,,, pass through the barycentre of the three points a, 6, c. What becomes of this property when K is of characteristic 2 or of characteristic 3 ? Generalize to a system of n affinely independent points. * 5. For a non-empty subset V of an affine space E over a field K with at least three elements to be a linear variety, it is necessary and sufficient that for every ordered pair (x, y) of distinct points of V the line D,, passing through x and y be entirely contained in V. If K has two elements, for V to be a linear variety, it is necessary and sufficient that the barycentre of any three points of V belong to V. 6. (a) Let E be an affine space of dimension 22 over a field K. For an affine mapping u of E into itself to transform every line of E into a parallel line, it is necessary and sufficient that the linear mapping v associated with u be a homothety t H yt of ratio y # 0 belonging to the centre of K. If y = 1, u is a translation; show that if y # 1, there exists one and only one point a E E such that ~(a) = a. If a is taken as origin of E, u is then identified with a central homothety for the vector space structure thus determined on E; u is called a central homothety of the affine space E of centre a and ratio y. (b) Let ul, u2 be two affine mappings of E into E each of which is either a translation or a central homothety. Show that u1 o u2 is a translation or a central homothety of E; if ul, u2 and u1 0 u2 are all three central homotheties show that their centres lie on a straight line. What can be said when u1 and u2 are central homotheties and u1 0 u2 is a translation? (c) Show that the set of translations and central homotheties is a normal subgroup H of the affine group of E and that H/T is isomorphic to the i i 410 41 1 LINEAR ALGEBRA I1 multiplicative group of the centre of K: show that H can only be commu - tative if H = T, in other words if the centre of K has only two elements. 7. Let E (resp. E') be an affine space of finite dimension n 2 2 over field K with at least 3 elements (resp. over a field K') and let u be an injective mapping of E into E' mapping any three points in a straight line in E to three points in a straight line and such that the affine linear variety generated by u(E) in E' is equal to E'. (a) Show that u maps every system of affinely independent points of E to a system of affinely independent points (use Exercise 5). (b) Let D,, D, be two parallel lines in E and D;, DL the lines of E' con- taining respectively u(D1) and u(D,). Show that Di and DL are in the same plane; if further u is surjective, show that Di and DL are parallel (in the con- trary case, show that there would be 3 points not in a straight line in E whose images under u would be in a straight line) (cf. Exercise 17). (c) Suppose henceforth that if D, and D, are parallel lines in E, the lines of E' containing respectively u(D,) and u(D2) are parallel. Show that if an origin a in E and the origin a' = .(a) in E' are taken, there exists an iso- morphism a of K onto a subfield K, of K' such that if E is considered as a vector space over K and E' as a vector space over K,, u is an injective semi-linear mapping (relative to 0) of E into E' (3 1, no. 13). (Consider first the case n = 2; given a basis (el, e,) of E, show that for any two elements a, p of K, the points (a + P)e, and (@)el of E can be constructed starting with the points 0, el, e,, ore,, pel by constructing parallels to given lines and intersections of given lines; deduce that u(Ae,) = h%(el), where a is an isomorphism of K onto a subfield of K', then show that also u(Ae,) = A'u(e,) by considering the line joining the points Ae, and Ae,. Pass finally from here to the case where n is arbitrary, arguing by induction on n.) If u is bijective, show that K, = K'. assuming also that u maps every system of affinely independent points of E to a system of affinely independent points of E'. (d) Extend the result of (c) to the case where K is a field with 2 elements I 8. Let E be a left affine space over a field K and T its translation space. (a) For a mapping f of E into a left vector space L over K to be affine, it is necessary and sufficient that f(t + f(At + 4 -f@) = Af(t + 4 -f(x)) -fb) =f(t + Y) -f (Y) for all A E K, t E T, x, y in E. Let x H [XI be the canonical injection of E into the vector space KY) of formal linear combinations of elements of E and let N be the subspace of K:E) generated by the elements EXERCISES [t + XI - [XI - [t + Yl + [!/I [At + XI - [XI - ACt + XI + A[%] for A E K, t E T, x, y in E; finally, let V be the quotient space of Ky) by N and + the mapping of E into V which associates with every x E E the class of [XI modulo N. Then is an affine mapping and for every affine mapping f of E into a left vector space L over K, there exists one and only one linear map- ping g of V into L such that f = g 0 +. (b) Let +o: T --f V be the linear mapping associated with +, so that +(t + x) - +(x) = +o(t). Show that + (and therefore also +o) is injective (consider the mapping x H x - a of E into T for some a E E) ; for every family (A,),GI of elements of K with finite support and every family (x,),~~ of elements of E, if C A, = o Deduce that +,(T) is a hyperplane passing through 0 in V and +(E) a hyper - plane parallel to +o (T). 9. Let K be a finite field with q elements and V an n-dimensional vector space over K. (a) Show that the set of sequences (x,, x,, . . ., xm) of m < n vectors of V forming a free system has cardinal equal to (qn - l)(q" - q). . . (gn - qrn-1) (argue by induction on n). varieties in an n-dimensional projective space over K is equal to (b) Deduce from (a) that the cardinal of the set of m-dimensional linear (p+1 - l)(q"+1 - q). . . (q"+1 - q") . (q"+l - 1)(q"+1 - 4). . .(q"+l - q") 10. In an n-dimensional projective space P(V) over a field K, a projective basis is a set S of n + 2 points any two of which form a projectively free system. IfS = andS' = areanytwoprojective basesofP(V), show that there exists a transformation f E PGL(V) such that f (a,) = a; for 0 d i d I? + 1. For this transformation to be unique, it is necessary and sufficient that K be commutative. (Reduce it to the case where a[ = a, for all i and note that it is always possible to write (in the notation of no. 6) a, = ~(b,), where (bi)lGiGn+l is a basis of V and bo = b, + b, +.a - + bn+,.) *Give an example where K is a field of quaternions and there exists an infinite set T of Points of P(V) any n + 1 of which form a projectively free system and a trans- fomationf E PGL(V), distinct from the identity, leaving invariant all the points of T., 11. Let V be a 2-dimensional vector space over a field K and a, 6, c, d four 413 412 I1 LINEAR ALGEBRA distinct points of the projective line P(V). The cross-ratio of the quadruple (a, b,c, d), denoted by [I; !I, is the set of elements E K such that there exist two vectors u, v in V for which (in the notation of no. 6) a = x(u), b = ~(u), = x(~ + v), d = X(U + kv). This definition extends immediately to any quadruple of distinct points of a set with a projective line structure (no. 11). (a) Show that [I; !] is the set of conjugates of an element # 1 of the multi - plicative group K* and conversely that, when a, b, c, are distinct points of p(v) and p the set of conjugates of an element # 1 of K*, there exists a point d E p(V) such that [i !] = p. For d to be unique, it is necessary and sufficient that p consist of a single point. (6) Show that EXERCISES [; 3 = [” dc a] = [a dc 61-l and (denoting by p-l (resp. 1 - p) the set of conjugates Ak-lh-l = (hSA-l)-l (resp. 1 - 1Eh-l = A(1 - S)h-l), where (c) Let (a, b, c, d), (a’, b‘, c‘, d’) be two quadruples of distinct points of P(V). For there to exist a bijective semi-linear mapping of V onto itself such that the bijective mappingfof P(V) onto itself, obtained when passing to the quotients, satisfies the conditions f(a) = a‘, f(b) = b‘, f(c) = c’, f(d) = d‘, it is necessary and sufficient that there exist an automorphism Q of K such that is an element of p). d‘ c’ For there to exist a transformationfof the projective group PGL(V) satisfying the above conditions, it is necessary and sufficient that d‘ c’ 12. Let P(V) be a (left) projective space of finite dimension n over a field K. Show that there exists on the set of projective hyperplanes of P(V) an n- dimensional (right) projective space structure over K canonically isomorphic to that of P(V*) (V* being the dual of V). If M is a linear variety of dimension r < n in P(V), derive an (n - r - 1)-dimensional projective space structure on the set of projective hyperplanes containing M. In particular if M is of dimension n - 2, the cross-ratio of a quadruple (Hl, H,, H,, H4) of distinct hyperplanes containing M can be defined. Show that if D c P(V) is a line not meeting M and a, is the intersection of D with Hi (1 < i 6 4), then *13. In a projective plane P(V) over a field K of characteristic 22, let a, b, c, d be four points forming a projective basis (Exercise 10) ; denoting by D,, the line passing through two distinct points x, y of P(V), let e,f, g be the points of intersection of the lines Dab and Dcd, D,, and Dbd, Dad and D,, respectively; let h be the point of intersection of D,, and Def; show that [i i] = { - l} (“theorem ofthe complete quadrilateral”; reduce it to the case where Dad is the line at infinity of an affine plane). What is the corresponding result when K is of characteristic 2?, 14. In a projective plane P(V) over a field K with at least three elements, let D, D’ be two distinct lines. In order that, for any distinct points a, 6, c of D, a‘, b‘, c‘ of D’, the intersection points r of Dab, and D,,,, q of D,,, and D,,., j~ of DbC, and Deb, lie on a straight line, it is necessary and sufficient that K be commutative (“Pappus’s theorem”; reduce it to the case where q and r are on the line at infinity of an affine plane). Apply this theorem to the projective space of lines of P(V) (Exercise 12). 15. In a projective plane over a field K with at least three elements, let s, a, b, c, a’, b‘, c‘ be seven distinct points such that {s, a, b, c} and {s, a’, b‘, c‘} are projective bases (Exercise 10) and the lines D,,, D,,, D,, pass respectively through a‘, b’, c’. Show that the intersection points r of D,, and D,,,,, p of D,, and Dbrc,, q ofD,, and D,,,, lie on a straight line (“Desargues’s theorem”; method analogous to that of Exercise 14). 16. (a) Let E = P(V) and E’ = P(V’) be two projective spaces of the same dimension n 2 2 over two fields K, K respectively and let u be a bijective mapping of E onto E’, mapping any three points on a straight line to three points on a straight line. Show that there exist an isomorphism Q of K onto K’ and a bijective semi-linear mapping v of V onto V‘ (relative to Q) such that u is the mapping obtained from v when passing to the quotients (‘‘jiundamental theorem ofprojective geometry”; use Exercise 7). Suppose also that V’ = V and K 1s commutative; for u to be a projective mapping, it is necessary and sufficient that also [ U(4 44 1 = [: for every quadruple (a, b, c, d) of distinct points 44 u(c) on a straight iine‘ i’i P(v~. - 414 415 EXERCISES I1 LINEAR ALGEBRA (b) Let p be an integer such that 1 < p < n - 1. Show that the first con - clusion of (u) holds when the image under u of every p-dimensional projective linear variety is contained in a p-dimensional projective linear variety. 17. Let V be a vector space of finite dimension n over a field K, (e,) ,, a basis of V, K' a subfield of K and V' the n-dimensional vector space over K' generated by the e,. Give an example of an injective mapping of V' into V, mapping any three points on a straight line of the affine space V' to three points on a straight line in the affine space V, but not necessarily mapping two parallel lines to sets contained in two parallel lines. (Embed V in the projective space E canonically associated with it and consider a projective transformation u of E into itself such that the inverse image under u of the hyperplane at infinity is distinct from this hyperplane and contains no point of V'; for example K can be taken to be infinite and K' finite.) 7 18. Let E = P(V) be a projective plane over a field K and u a bijective mapping of E onto itself, arising when passing to the quotients from a bijective semi-linear mapping v of V onto itself, relative to an automorphism CT of K. (a) Show that the four following properties are equivalent: (a) for all x E E, x, ~(x) and u2(x) lie on a straight line; (p) every line of E contains a point invariant under u; (y) through every point of E there passes a line invariant under u; (6) for every line D of E, the three line D, u(D) and u2(D) have a common point. (Show first that (u) and (y) are equivalent; deduce by duality (Exercise 12) that (p) and (8) are equivalent; prove finally that (y) implies (p) and deduce by duality that (p) implies (y) .) (b) Suppose that u has the properties stated in (a). Show that if there exists in E a line D invariant under u and containing only a single point invariant under u, u arises from a transvection v of V when passing to the quotients (5 10, Exercise 11). (Show that every line invariant under u contains a; by con- sidering a line not passing through a, show that there exists a line Do passing through a and containing at least two points invariant under u; conclude that all the points of Do are necessarily invariant under u.) (c) Suppose that u has the properties stated in (u). Show that if there exists in E a line E invariant under u and containing only two points invariant under u, u arises from a dilatation v of V when passing to the quotients (3 10, Exercise 11). (If a, b are the two points of D invariant under u, show that every line invariant under u passes through a or through b; then note that there exist at least two other points c, d distinct from a, b and invariant under u and therefore the line D,, passes through a or b; conclude by proving that all the points of D,, are invariant under u.) (d) Suppose that u has the properties stated in (u) and that every line of E invariant under u contains at least three distinct points invariant under U; then there exists in E a projective basis (Exercise 10) each point of which is invariant under u; conclude that there exists a basis (eJl st of V such that u arises from a semi-linear mapping v of V into itself such that v(ei) = e, for 1 < i < 3, when passing to the quotients. The set of points of E invariant under u is then the projective plane P(V,), where V, is the vector space over the field KO of invariants of (T, generated by el, e2, e3. (e) Suppose henceforth that u satisfies the conditions of (u) and (d) and that neither u nor u2 is the identity. Show that there exists y E K such that yo = y and (1) (F" - tY = Y(F" - t) for all E E K (use conditions (a) of (u) and the existence of 1: E K such that # c). Show that y # - 1 and that, for all E E K such that # 5, (2) (1 + Y)%Y = Yt(1 + Y) I (apply (1) replacing 5 by t2) ; extend (2) to all E E K by noting that t = 9 - <, where q' # q and # 1:. Conclude that y # 1, then deduce from (1) and (2) that (3) F" = (1 + Y)S(l + y1-l for all E K and that y + y-' belongs to the centre of I(. Obtain the converse. *Give an example where y does not belong to the centre of K, but where y +- y-l is in this centre., *19. Let K be a commutative field, l? the projective field obtained by ad- joining to K a point at infinity (no. 9) and f and g two elements of K(X). Show that, writing h(X) = f (g(X)), iff; g, are the canonical extensions off, g, h to K, then = 30 i.* 9 10 1. (a) Let E be a right A-module. On the additive group E' (with r 2 1) an external law of composition is defined with M,(A) as set of operators, denoting by x.P, for every element x = of E' and every square matrix P = (atj) E M,(A), the element y = (y,) of E' such that 2 yi = j=lxprt (1 < i < r). This external law defines with the additive law on E' a right M,(A)-module structure on E'; by restricting the ring of operators ofthe ring of scalar matrices I.a, the product A-module structure on E' is recovered. Show that for the A-module E to have a generating system of r elements, it is necessary and sufficient that the M,(A)-module E* be monogenous. (b) Let (xi)lsrsn be a generating system of E and (y,)1sj4m a family of elements of E (resp. a generating system of E). Let z, z', z" be three elements OfEm+n such that zi = x, for 1 < i < n, z,+~ = 0 for 1 <j < m, 2: = 2, for 416 417 I1 LINEAR ALGEBRA 1 < i < n, zk+, = yi for 1 <j < n, zr = 0 for 1 < i < n, z;+, = yj for 1 <j < n. Show that there exist two invertible matrices P, Q of M,+,(A) such that z' = z.P (resp. z" = z.Q). (c) If A is commutative, up : x H x. P is an endomorphism of the A-module Er and the mapping P H up is a homomorphism of the ring M,(A) into the ring End,(E'). If E is a faithful A-module this homomorphism is injective. 2. (a) Let X be a square matrix over a ring A, which can be written as an upper triangular matrix (Xij) of matrices (1 < i < p, 1 < j < p). Show that if each of the square matrices XI, (1 < i < p) is invertible, so is X and X-l can be written as an upper triangular matrix ( Ylj) (1 < i < p, 1 < j < p) corresponding to the same partition of the indexing set. When A is a field, prove that this sufficient condition for X to be invertible is also necessary. (with a EA, b EA) which is invertible without either a or b being invertible in A and whose inverse (:: ::) is such that b' # 0 and c' # 0 (take A to be the endomorphism ring of an infinite-dimensional vector space). (b) Give an example of a ring A and a matrix (i 3. Let A be the quotient ring Z/3OZ. Show that in the matrix (; ; -3 over the ring A, the two rows are linearly independent but any two columns are linearly dependent. 4. Let A be a ring, C its centre, B the ring of matrices M,(A), A the additive subgroup of A generated by the elements a@ - pa for a E A, p E A, and D the additive subgroup of B generated by the matrices XY - YX for X E B, Y E B; A and D are C-modules. For every matrix X = (?&) E B, let 0(X) be the element c El, of A/A, where, for all a E A, Cr denotes the class of a mod. A. Show that 0 is a surjective homomorphism of B onto A/A, whose kernel is equal to D, SO that B/D is isomorphic to A/A as a C-module. (Observe that D contains the matrices aE,, and the matrices a(Ei, - E,,) for a E A and i # j.) n L=l 7 5. Let A be a ring, L, M, N any three indexing sets U = (axw) a matrix of ALxM and V = (bwv) a matrix of AMXN; if, for every ordered pair (A, v) E L x N, the family (axwbllv) (p E M) has jinite support, the element cAV = aAwb,, is defined and it is also said that the matrix (cAv) is the product UV of U by V. When the products UV' and UV" are defined, so is U( V' + V") EXERCISES and UV' + UV" = U(V' + V"); is the converse true? Give an example of three infinite matrices, U, V, W such that the products UV, VW, U( VW) and (UV) W are defined but U(VW) # (UV) W (take U to be a matrix with one row all of whose elements are equal to 1, W its transpose and V a matrix all of whose elements are 0, 1 or - 1 and which has only a finite number of elements #O in each row and each column. Determine by induction the elements of V such that UV = 0 but VW' has a single element #O.) 6. Let X be a matrix with m rows and n columns over a field K; show that the rank rg(X) is equal to the greatest of the ranks of the submatrices of X with an equal number of rows and columns. (Let rg(X) = r; if a,, . . . , a, are Y columns of X forming a free system in K! and a basis of KT is formed with these r vectors and m - r vectors of the canonical basis, show that the com- ponents of a,, . . ., a, over the r other vectors of the canonical basis form a matrix of rank r.) 7. Let X be a matrix with m rows and n columns over a field K; if r is the rank of X, show that the rank of a submatrix with m rows and s columns, obtained by suppressing n - s columns of X, is 2 r + s - n. 8. Let X = (alj) he a matrix with m rows and n columns over a field K. For X to be of rank 1, it is necessary and sufficient that there exist in K a family (A,) of n elements not all zero such that ai, = A,pj for every ordered pair of indices. of m elements not all zero and a family (pj) 9. Let X, Y be two matrices with m rows and n columns over a field K; if there exist two square matrices P, P, of order m and two square matrices Q, Q, of order n such that Y = PXQ and X = PIXQ 1, show that X and Y are equivalent. 10. Let X, X', Y, Y' be four square matrices of order n over a ring A, such that X is invertible. For there to exist two invertible square matrices P, Q of order n such that X' = PXQ and Y' = PYQ, it is necessary and sufficient that X' be invertible and that the matrices YX-l and Y'X'-l be similar. 11. Let E be a right vector space over a field K, of dimension 2 1, and H a hyperplane of E. Every endomorphism u of E leaving invariant each of the elements of H gives, on passing to the quotients, an endomorphism of the 1- dimensional quotient space E/H, an endomorphism which is therefore of the form i ++ kp(k), where p(i) E K is such that p(.+A) = A-lp(.+)A for A E K*. An automorphism of E leaving invariant the elements of H is called a transvection of hy@erplane H if the corresponding automorphism of E/H is the identity and a dilatation ofhyperplane H otherwise; when u is a dilatation, the set of elements F(2) for i E E/H which is a class of conjugate elements in the multiplicative group K*, is called the class of the dilatation u. t 418 419 LINEAR ALGEBRA I1 EXERCISES (a) Show that for every dilatation there exists one and only one supple - mentary line of H, invariant under the dilatation. -1 (b) Let + be a linear form on E such that H = 4 (0) ; show that for every transvection u of hyperplane H there exists a unique vector a E H such that U(x) = x + a+(.) for all x E E. Let r(E, H) be the subgroup of GL(E) con- sisting of the automorphisms leaving invariant each element of H; show that the transvections of hyperplane H form a normal commutative subgroup @(E, H) of r(E, H) isomorphic to the additive group H; the quotient group r(E, H)/O(E, H) is isomorphic to the multiplicative group K*. For r(E, H) to be commutative, it is necessary and sufficient that one of the two following conditions be satisfied: (a) K is a field with two elements; (p) K is commutative and dimE = 1. (c) Suppose that E is finite-dimensional; show that for every transvection u there exists a basis of E such that the matrix of u with respect to this basis has all its diagonal elements equal to 1 and at most one other element #O. (d) Show that the centralizer (I, 3 5, no. 3) of the group O(E, H) in the group GL(E) is the composition Z(E) O(E, H) = O(E, H)Z(E) of O(E, H) and the centre Z(E) of GL(E) (cf. 3 1, Exercise 26). The only automorphisms belonging to this centralizer and leaving invariant at least one element #O of E are the transvections of the group O(E, H). If K has at least 3 elements, the centralizer of r(E, H) in GL(E) is equal to Z(E). (e) Show that the normalizers (I, $5, no. 3) of O(E,H) and I'(E,H) in GL(E) are both equal to the subgroup consisting of the automorphisms leaving H invariant. 7 12. Let E be a right vector space of dimension 2 1 over a field K. Let F(E) denote the normal subgroup of GL(E) consisting of the automorphisms u such that the kernel of 1, - u is ofjnite codimension (hence F(E) = GL(E) if E is finite -dimensional). Let SL(E) denote the normal subgroup of GL(E) generated by all the transvections (Exercise 11) ; it is contained in F(E). (a) If dim E 2 2, show that for every ordered pair of non-zero vectors x, y of E, there exists a transvection or product of two transvections, which maps x toy (in other words SL(E) operates transitively on E - {O}). (b) Let V, W be two hyperplanes of E, fo = x, + V a class mod. V distinct from V and j, = yo + W a class mod. W distinct from W. Show that if dim E 3 2, there exists a transvection or product of two transvections, which maps V to W and i, to yo (consider first the case where V and W are distinct). (c) If dim E 2 2, show that any two transvections distinct from the identity \I are conjugate in the group F(E). (d) If dimE 2 3, show that any two transvections distinct from the identity are conjugate in the group SL(E) (reduce it with the aid of (6) to the case where the hyperplanes of the two transvections are identical, then use (a)). (e) Suppose that dim E = 2. For any two transvections to be conjugate in SL(E), it is necessary and sufficient that the subgroup Qof K* generated by the squares ofelements of K* be identical with K*. (If u is a transvection distinct from the identity, a a vector of E which is not invariant under u and b = ~(a) - a, show that, for every transvection u' conjugate to u in SL(E), u'(a) - a = ah + bp, with p = 0 or p E Q; for this use the fact that in a group G every product sts-lt is a product of squares.) (f) Suppose that dim E 2 2. For two dilatations u, u' to be such that there exists a u E SL(E) such that u' = uuu-l, it is necessary and sufficient that the classes (Exercise 11) of u and u' be the same (use (6)). Deduce that if the class of a dilatation is contained in the commutator group of K*, this dilatation belongs to the group SL(E) (observe that (uuu-~)u-~ = ~(uu-~u-')). 7 13. Let E be a right vector space of dimension 2 2 and H, a hyperplane of E. (a) Show that every automorphism u belonging to F(E) (Exercise 12) is the product of an automorphism of SL(E) and possibly a dilatation of the hyper- plane H, (proceed by induction on the codimension of the kernel of 1, - u, using Exercises 12(a) and 12(f)). (b) Show that SL(E) contains the commutator group of F(E) and is identical with this group except when E is a space of dimension 2 over the field with 2 elements. (To show that SL(E) contains the commutator group of F(E), use (u) and Exercise 12(f). To see that SL(E) is contained in the commutator group of F(E), except in the case indicated, show that for every homomorphism of F(E) into a commutative group, the image of a transvection is the identity element, using Exercises 11 (6) and 12(c).) (c) Show that SL(E) is equal to its commutator group except when dim E = 2 and K has 2 or 3 elements. (If dim E 2 3, note that every trans- vection u can be written as UWU-~W-~, where u is a transvection and w E SL(E), using Exercise 12(d). If dim E = 2, note first that every automorphism u whose matrix with respect to a basis of E is of the form is a product of transvections, arguing as in (a) ; then consider the commutator uuu-lu-l, where u is the transvection whose matrix with respect to the same basis is (i 7 14. Let E be a vector space of dimension 22 over a field K. (a) Show that the projective group PGL(E) is canonically isomorphic to the quotient of the linear group GL(E) by its centre (isomorphic to the multiplica - tive group of the centre of K, cf. 3 1, Exercise 26). 420 42 1 I1 LINEAR ALGEBRA (b) Let PSL(E) denote the canonical image of the group SL(E) in PGL(E) ; it is a normal subgroup of PGL(E), containing the commutator group of PGL(E) when E is finite-dimensional. Show that PSL(E) is a doubly transitive (cf. T, fj 5, Exercise 14) group of permutations of P(E). (c) Let a # 0 be an element of E and let e = .(a) EP(E) (in the notation of 9: 9, no. 6). Show that the transvections of the form x H x + a+(x) (where 4 E E* is such that +(a) = 0) constitute a commutative subgroup of SL(E); if re is the image of this subgroup in PSL(E) , re is a normal subgroup of the subgroup Qe of PSL(E) leaving e invariant. Show also that the union of the subgroups conjugate to re in PSL(E) generates PSL(E). (d) Let C be a primitive group (I, 5 5, Exercise 13) of permutations of a set F which is equal to its commutator group. Suppose that there exists an element c E F such that the subgroup (Dc ofZ leaving c invariant contains a commutative normal subgroup rC such that the union of the subgroups conjugate to rC in 2 generates Z. Show that under these conditions X is simple. (Let A be a normal subgroup of X distinct from the identity; using I, fj 5, Exercise 17, show that 2 = A. (Dc = A. rc and, using the fact that rc is commutative, deduce that every commutator in 2 is contained in A.) (e) Deduce from (c), (d) and Exercise 13(c) that the group PSL(E) is simple except when dim E = 2 and K has 2 or 3 elements. *(f) If dim E = n + 1 and K is a field with y elements, show that the order of PGL(E) is (p+l - 1 ) (Y n+l - 9). . .(@+I - qn-yqn (use (a), Exercise 9(a) of fj 9, and the fact that K is necessarily Commutative (V, fj 11, Exercise 14 and VIII, fj 11, no. 1, Theorem l)).* (9) Show that if dim E = 2 and K is a field with 2 (resp. 3) elements, PSL(E) is isomorphic to the symmetric group 6, (resp. the alternating group U4). (Considering PSL(E) as a group of permutations of P(E), note that it contains every transposition (resp. every cycle (a b c) and usef).) (h) Show that, except in the cases considered in (g), every normal subgroup ofSL(E) distinct from SL(E) is contained in the centre of SL(E) (use (d) and the fact that every transvection is a commutator in SL(E) (Exercise 13(c))). 7 15. Let A be a ring; for every integer n 2 1, An is canonically identified with the submodule of A") consisting of the elements whose coordinates of index > n are 0 and A; denotes the complementary submodule consisting of the elements whose coordinates of index <n are 0. Every endomorphism u E GL,(A) is identified with the automorphism of A") whose restriction to A" is u and whose restriction to A; is the identity, so that GL,(A) is identified with a subgroup of GL(A(") ; let F denote the subgroup of GL(A(") the union of the GL,(A), in other words the subgroup leaving invariant the elements of the canonical basis of A") except for a finite number of them (cf. Exercise 12). EXERCISES Let T, be the subgroup of GL,(A) generated by the B,,(A) for 0 < i < n, 0 < j < n, i # j, A E A (no. 13) and let T be the subgroup of F the union of the T,. (u) Show that for n 2 3, T, is equal to its commutator group. (b) Let a, b be two invertible elements of A. Show that (aba ib - where P, Q, P', Q' belong to T,. Deduce that the matrix belongs to T,. (c) Deduce from (6) that the commutator group of GL,(A) is contained in T,, (replace A by M,(A) in (6)). Conclude (using (a)) that the commutator group of F is equal to T. 16. (u) Let U be a square matrix of order n over the field Q all of whose non-diagonal elements are equal to the same rational number r > 0 and whose diagonal (dl, . . . , d,) is such that d, > r for all i and dl > r except perhaps for one value of i. Show that U is invertible (if x = is a solution of the equation U. x = 0, show that the x1 all have the same sign and deduce that they are all zero). (6) Let E be a finite set with m elements denoted by a, (1 < i < m) and (Aj)lGrcn a family of n subsets of E, distinct from one another; suppose that Card(A, n A,) = r 2 1 for every ordered pair (i,j) of distinct elements of (1, n). Show that necessarily n < m. (Let V = (c,,) be the matrixof type (m, n) .. such that cij = 1 when a, €Aj, cl, = 0 otherwise. Apply (a) to the matrix ' V. V of order n.) 7 17. Let K be a field (commutative or otherwise), E an n-dimensional right vector space over K and u an endomorphism of E. (a) Show that if n > 1 and u is not a central homothety, there exists a basis of E such that the matrix of u with respect to this basis is of the form (E~,) with aij = 0 for j > i + 1 and 2 (b) Show that if n > 2, there exists a basis of E such that the matrix of u With respect to this basis is of the form (q,) with air = 0 forj > i + 1 and n-1 = 1 (argue by induction on n). cql+l = 0. (Argue by induction on n using (a), noting that it amounts to the same to prove the proposition for u or for u + yl,, where y belongs to the centre of K, and studying separately the case where u is of rank 1 or 2.) (c) Let A = (q,) be a matrix with the properties described in (b). Also let s be the matrix of order n (E,,) such that qj = 0 except when i = j + 1, in which 422 423 I1 LINEAR ALGEBRA I APPENDIX case E!+~, = 1. Show that there exists a matrix B = (PIj) of order n such that A = a,,, + (BS - SB) with A E K. (Take B such that Pll = 0 for 1 < i 6 n and Pij = 0 forj > i + 2.) (d) Suppose that K is commutative. Deduce from (c) that every endomor- phism u of E such that Tr(u) = 0 can be written as uw - LOU, where u and w are endomorphisms of E. ~ 1 § 11 1. Let A be a commutative monoid, written additively, with its identity element denoted by 0. Let Z(A) be the algebra of A over Z (111, 9 2, no. 6), that is a commutative ring, which is a free Z-module with a basis (Xu)oeA and the multiplication (aX")(bX') = abXU+'. For every Z-module, we write E(A) = E & Z(A), every element of E(A) hence being written uniquely in the form 2 z, @ Xu, with z, E E, the family (2,) having finite support. For every Z -module homomorphism f: E -+ E', let f (A) denote the Z(A)-module homomorphism f @ 1 : E(A) -+ E"A). Similarly, if A' is another additive monoid with its identity element denoted by 0 and if ci: A -+ A' is a monoid homomorphism, a(E) denotes the homomorphism E(A) --f E(A') such that a(E) (2, @ X") = Z, @ Xa(O). (E(A))(A') is identified with E(A A') by associating with ozA, (zA z,,,, 8 Xu) @ Xa' the element ,,,, 2 z,,,, @ X(a*a'). Finally, let E(E) denote the homomorphism E(A) -+ E mapping 2 z, @ X" to 2 2,. (u) Let E be a Z-module, (E,),€A a graduation on E of type A and, for all Q E A, let p, be the projector E -+ E, corresponding to the decomposition of E as the direct sum of the E,. For all x E E, let +E(x) = c p,(x) @I Xu. Show that the homomorphism +E: E -+ E(A) thus defined has the following proper - ties : I oeA aeA (1) $E = I 6(E) $E = +$) +E, I I (2) where 6: A + A x A is the diagonal mapping. Show that a bijection of the set of graduations on E of type A is thus defined on to the set of homomorphisms of E into ECA) satisfying conditions (1) and (2). $E+F to be graded of degree 0, it is necessary and sufficient that +F 0 f = f (A) o +E. For a submodule E' of E to be graded, it is necessary and sufficient that +E(E') c E'CA). (c) Let A be a ring; ACA) = A & Z(A) is given a ring structure such that j (b) Let E, F be two graded Z-modules of type A. For a homomorphism I I (a @I Xa)(b @ XT) = (ab) @ Xu+'. Let (A,),,,beagraduationoftheadditive I 424 group A; for this graduation to be compatible with the ring structure on A and 1 E A,,, it is necessary and sufficient that +*: A --f A(A) be a ring homomorphism. State and prove an analogous result for graded A-modules. APPENDIX 1. Let A be a pseudo-ring and M a left pseudomodule over A such that for all z # 0 in M, z E Az. Show that if x, y are two elements of M such that Ann(%) c Ann(y) in A, there exists an endomorphism u of M such that u(x) = y (prove that the relation bx = ax for a, b in A implies by = uy). 425 CHAPTER I11 Tensor Algebras, Exterior Algebras, Symmetric Algebras Recall the exbonential notation introduced in Chapter I, of which rue shall make frequent use (I, fj 7, no. 8) : mapping a: L -+ N of Jinite support we shall write Let (xh)hGL be a family of pairwise permutable elements of a ring A; for every If p is another mapping of L into N offinite support, a + (3 denotes the mapping A++a(A) + B(V of L into N; with this law of composition the set N(L) of mappings of L into N of Jinite support is the free commutative monoid derived from L and pxD = XUfD, For all aEN(L), we write la1 = 2 a(A) EN; then (a + PI = la1 + IpI; 1.1 is called the order ofthe “multiindex” a. For all A E L, let 6, denote the element Such that ah(),) = 1, 6,(p) = 0 for p # A (Kronecker index); the 6h for A E L are the only elements ofN(L) of order 1. N(L) is given the ordering induced by the product Ordering on NL, so that the relation a < p iS equivalent to “a(A) i p(A) for all 1 E v’; then the multiindex A H P(A) - a(A) is denoted by p - a, so that it the unique multiindex such that a + (p - a) = p. For all a E N(L), there are only a f;nite number ofmultiindices p < a; the 8, are the minimal elements ofthe set NCL’ - (0); the relation a < p implies la( < IpI and ifboth a < p and la1 = IpI, then a = p. Finally, we write a! = n (a(h)) !, which is meaningful since O! = 1. From 3 4 to 3 8 inclusive, A denotes a commutative ring and, unless otherwise stated, the algebras considered are assumed to be associative and unital and the algebra homo- moqhh are assumed to be unital. hEL h€L 42 7 Ill TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 8 1. ALGEBRAS 1. DEFINITION OF AN ALGEBRA DEFINITION 1. Let A be a commutative ring. An algebra over A (or an A-algebra, or simply an algebra, when no confusion is to be feared) is a set E with a structure deJined by giving the following: (1) an A-module structure on E; (2) an A-bilinear mapfkg (11, 5 3, no. 5) ofE x E into E. The A-bilinear mapping of E x E into E occurring in this definition is called the multiplication of the algebra E; it is usually denoted by (x, y) H x.y, or simply (x, y) H xy. Let (aJioI and be two families of elements of A, offinite support (I, 3 2, no. 1). Then, for all families (xi)I6I and (Y,),~ of elementsof E, the general distributivity formula (I, 4 3, no. 4) holds; in particular (2) (ax) y = x(ay) = a(xy) for a E A, x E E and y E E. The bilinear mapping (x, y) H yx of E x E into E and the A-module structure on E define an A-algebra structure on E, called opposite to the given algebra structure. The set E with this new structure is called the opposite algebra to the algebra E; it is often denoted by EO. The A-algebra E is called commutative if it is identical with its opposite, in other words if multiplication in E is commutative. An isomorphism of E onto Eo is also called an anfi- automorphism of the algebra E. Wh'en multiplication in the algebra E is associative, E is called an associative A-algebra. When multiplication in E admits an identity element (necessarily unique (I, 3 2, no. l)), this element is called the unit element of E and E is called a unital algebra. Examples. (1) Every commutative ring A can be considered as an (associative and commutative) A -algebra. (2) Let E be a pseudo-ring (I, 3 8, no. 1). Multiplication on E and the unique Z-module structure on E define on E an associative Z-algebra struc- ture. (3) Let F be a set and A a commutative ring. The set AF of all mappings of F into A, with the product ring structure (I, 8 8, no. 10) and the product 3 1.2 SUBALGEBRAS. IDEALS. QUOTIENT ALGEBRAS &module structure (11, 3 1, no. 5) is an associative and commutative A- algebra. (4) Let E be an A-algebra; the internal laws (x, y) HXY + yx and (x, y) H xy - yx define (with the A-module structure on E) two A -algebra structures on E, which are not in general associative; the first law (%Y) HXY + YX is always commutative. DEFINITION 2. Given two algebras E, E' over a commutative ring A, a homomorphism of E into E' is a mapping f: E -+ E' such that (1) f is an A-module homomorphism; (2) f(xy) = f(x)f(y) forallxEEandyczE. Clearly the composition of two A -algebra homomorphisms is an A-algebra homomorphism. Every bijective algebra homomorphism is an isomorphism. Therefore A -algebra homomorphisms may be taken as rnorphisms of the species of A-algebra structure (Set Theory, IV, 2, no. 1). We shall always suppose in what follows that this choice of morphisms has been made. If E, E' are two A -algebras, let HomA-alg(E, E') denote the set of A-algebra homo - morphisms of E into E'. Let E, E' be two algebras each with a unit element. A homomorphism of E into E' mapping the unit element of E to the unit element of E' is called a unital homomorphism (or unital algebra morphism). - 2. SUBALGEBRAS. IDEALS. QUOTIENT ALGEBRAS Let A be a commutative ring and E an A-algebra. If F is a sub-A-module of W which is stable under the multiplication on E, the restriction to F x F of the multiplication of E defines (with the A-module structure on F) an A-algebra structure on F. F, with this structure, is called a subalgebra of the A-algebra E. Every intersection of subalgebras of E is a subalgebra of E. For every family (xJIoI of elements of E, the intersection of the subalgebras of E containing all the xi is called the subalgebra of E generated by the family (xi)iEI and (xl)IGI is called a generating system (or generatingfamily) of this sub - algebra. If u:E -+ E' is an A-algebra homomorphism, the image u(F) of every subalgebra F of E is a subalgebra of E'. Let E be an associative algebra. For every subset M of E, the set M' of elements of E which are permutable with all the elements of M is a subalgebra of E called the centralizer subalgebra of M in E (I, 3 1, no. 5). The centralizer M" of M' in E is also called the bicentralizer of M; clearly M c M". It follows that M' is contained in its bicentralizer M", which is just the centralizer of M"; but the relation M c M" implies M" c M', so that M' = M" (cf. Set Theory, 111, 3 1, no. 7, Proposition 2). If F is a subalgebra of E, the celzflz of F is the intersection F n F' of F and its centralizer F' in E. Note that if 428 429 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS F is commutative, then F c F’ and hence F’ =I F”; the bicentralizer F” of F is in this case the centre of F. For certain non-associative algebras (for example Lie algebras) the notions of centralizer of a subalgebra and centre are defined differently (Lie Groups and Lie algebras, I, A subset a of an A-algebra E is called a l~t ideal (resp. right ideal) of E when a is a sub-A-module of E and the relations x E a, y E E imply yx E a (resp. xy E a). It amounts to the same to say that a is a left ideal of E or a right ideal of the opposite algebra EO. A two-sided ideal of E is a subset a of E which is both a left ideal and a right ideal. When E is associative and admits a unit element e, then, for a E A and x E E, ax = (ae)x = x(ae) by virtue of (2) (no. 1) and hence the (right, left, two-sided) ideals of the ring E (I, 5 8, no. 6) are identical with the (right, left, two-sided) ideals of the algebra E. Every sum and every intersection of left (resp. right, two-sided) ideals of the algebra E is a left (resp. right, two-sided) ideal. The intersection of the left (resp. right, resp. two-sided) ideals containing a subset X of E is called the left (resp. right, resp. two-sided) ideal of E generated by X. Let b be a two-sided ideal of an A-algebra E. If x = x‘ (mod. 6) and y = y‘ (mod. b), then and and hence xy = x’y‘ (mod. 6). Hence an internal law can be defined on the quotient A-module E/b, which is the quotient of the multiplication law (x, y) H xy of E by the equivalence relation x = x‘ (mod. b) (I, 4 1, no. 6). It is immediately verified that this quotient law is an A-bilinear mapping of (E/b) x (E/b) into E/b; it therefore defines with the A-module structure on E/b an A-algebra structure on E/b. E/b, with this algebra structure, is called the quotient algebra of the algebra E by the two-sided ideal b. The canonical mapping p: E -+ E/b is an algebra homomorphism. Let E, E’ be two A-algebras and u:E -+ E’ an algebra homomorphism. The image u(E) is a subalgebra of E’ and the kernel b =-i(O) is a two-sided ideal of E; further, in the canonical decomposition of u: v is an algebra isomorphism. More generally, all the results of Chapter I, 9 8, no. 9 are still valid (and also their proofs) when the word “ring” is replaced by “algebra”. Let A be a commutative ring and E an A-algebra. On the set E = A x E, we define the following laws of composition : 1, no. 6). x(y - y’) E b (x - x’) y’ E b E & E/b A u(E) E’ (A,%) + (P,Y) = (A + P, x + Y) VP, 4 = OP, h). (A, X>(P,Y) = (AP, XY + P* + AY) 430 I DIAGRAMS EXPRESSING ASSOCIATIVITY AND COMMUTATIVITY 5 1.3 I It is immediately verified that E, with these laws of composition, is an algebra over A and (1, 0) is a unit element of this algebra. The set (0) x E is a two-sided ideal of E and x I+ (0, x) is an isomorphism of the algebra E onto the subalgebra (0) x E, by means of which E and (0) x E are identi - fied. is called the algebra derived from E by adjoining a unit element; it is associa - tive (resp. commutative) if and only if E is. I ! m:E @AE-+E 3. DIAGRAMS EXPRESSING ASSOCIATIVITY AND COMMUTATIYITY Let A be a commutative ring and E an A-module; being given a bilinear map- ping of E x E into E is equivalent to being given an A-linear mapping: (11, $3, no. 5). An A-algebra structure on E is therefore defined by giving an A-module structure on E and an A-linear mapping of E @A E into E. Let E’ be another A-algebra and m’ : E’ @A E’ -+ E‘ the A-linear mapping defining the multiplication of E’. A mapping f: E -+ E’ is an A-algebra homomorphism if and only iff is a mapping rendering commutative the diagram E@AE + f @f E’BAE‘ -I E 7 E’ f For an A-algebra E to be associative, it is necessary and sufficient (taking account of the associativity of tensor products, cf. 11, 5 3, no. 8) that the diagram of A-linear mappings EBAE E 1- m be commutative. Similarly, for the A-algebra E to be commutative, it is neces- sary and sufficient that the diagram of A-linear mappings be commutative, where a denotes the canonical A-linear mapping defined by 43 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS b(x @ y) = y @ x for x E E, y E E (11, $3, no. 1, Corollary 2 to Proposition For all c E E, let qc denote the A-linear mapping of A into E defined by the condition qc(l) = c. For c to be a unit element of E, it is necessary and sufficient that the two diagrams 1). be Commutative (i and i‘ denoting the canonical isomorphisms (11, $3, (4 (Be) = (.PI8 = a(Pe) ; hence q is an A-algebra homomorphism. Observe that the A-module structure on E can be defined using q, for (3) ax=q(a).x foraEA,xEE (where, on the right hand side, multiplication is in E). The image of the homomorphism q is a subalgebra of E whose elements commute with all those of E. The kernel of the homomorphism q is the annihilator of the element e of the A-module E; by (3), it is also the annihilator of the A-module E (11, 5 1, no. 12). When the algebra E is iinital and associative, q is a ring homomorphism. Conversely, let p: A -+ B be a ring homomorphism such that the image p(A) 2 is contained in the centre of B, assuming also that the ring A is commutative; then an A-algebra structure is defined on B which is associative and unital, by writing (cf. (3)) )a = p(h).x for AEA,xEE. 4. PRODUCTS OF ALGEBRAS Let (EJtE1 be a family of algebras over the same commutative ring A. It is immediately verified that on the product set E = E,, the product A- module structure (11, § 1, no. 5) and the multiplication (4) ((xi), (Yi)) H (~JJ RESTRICTION AND EXTENSION OF SCALARS $ 1.5 define an A-algebra structure; with this structure, the set E is called the product algebra of the family of algebras (EJiEI. When all the algebras E, are associative (resp. commutative, resp. unital), so is their product. Moreover, all the properties stated in I, $ 8, no. 10, extend without modification to arbitrary products of algebras. 5. RESTRICTION AND EXTENSION OF SCALARS kt A, and A be two commutative rings and p :A, --+ A a ring homomorphism. If E is an A-algebra, we denote (conforming with 11, 0 1, no. 13) by p,(E) the A,-module defined by addition on E and the external law A.x = p(A)x for all A E A, and all x E E. Multiplication in E and the A,-module structure on p,(E) define an A,- algebra structure on p,(E). When A, is a subring of A and p the canonical injection, the algebra p,(E) is said to be obtained from E by restricting the ring A of scalars to A,. By an abuse of language, this is also sometimes said when p is arbitrary. Let F be an A,-algebra. An A,-algebra homomorphism F + p,(E) is called a semi-homomorphism (relative to p) or a p-homomorphism of F into the A- algebra E; it is also called an A,-homomorphism if no confusion arises. If E, E’ are two A-algebras, every A-algebra homomorphism E --+ E’ is also an A,,-algebra homomorphism p* (E) --+ p* (E’). Consider now two commutative rings A and B and a ring homomorphism p:A --f B. For every A -module E, the B-module p*(E) = E @A B, obtained from E by extending the ring A of scalars to B, has been defined (11, 3 5, no. 1). If E is also an A-algebra, we shall define on p*(E) a B-algebra structure. For this, observe that (E @A B) @, (E @A B) is canonically isomorphic to (E@.,E) BAB (11, $5, no. 1, Proposition 3). If m:EBAE-tE defines the multiplication on E, the mapping m @I 1,: (E @A E) @A B -+ E @A B is therefore canonically identified with a B-linear mapping m’:p*(E) @B P*(E) --f p*(E) which defines the desired B-algebra structure on p* (E) . Hence (5) (x @ P) (x’ €3 P’) = (xx’) @ (PP’) for *, X‘ in E, P and P’ in B. The B-algebra p*(E) is said to be derived from the A-algebra E by extending the ring A of scalars to B (by means of p). It is also denoted by E<B, or E B. When E is associative (resp. commutative, resP. ~nital), so is p* (E) . ~~Pos~~oN 1. For every A-algebra E, the canonical mapping +E:~ H x @ 1 of E into Eo, is an A-homomorphk .f algebras. Moreover, for every B-algebra F 432 433 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS and every A-homomorphism f :E --f F, there exists one and only one B-homomorphism J:E(B) 3 F such thatf(x @ 1) = f(x) for all x E E. The first assertion follows immediately from the definition of multiplica - tion in E(B), which gives (x @ 1) (x' @ 1) = (xx') @ 1 for x E E and x' E E. The existence and uniqueness of the B-linear mappingfof Eo, into F satisfying the relationf(x @ 1) =f(x) for all x E E follow from 11, 3 5, no. 1, Proposi - tion 1; here it all amounts to verifying thatf(yy') =J'(y)f(y') for y and y' in E(B); as the elements of the form x @ 1 (with x E E) generate the B-module E(B), attention may be confined to the case where y = x @ 1, y' = x' @ 1 with x E E, x' E E; as yy' = (xx') @ 1, the relation f(yy') = f(y)J(y') then follows from f (xx') = f(x) f (x'). It can also be said that f Hfis a canonical bijection (6) HomA-alg.(E, p*(F)) --f HomB-alg.(p*(E), F)' The ordered pair consisting of Eo, and +E is therefore a solution of the universal mapping problem (Set Theory, IV, 4 3, no. 1) where C is the species of B-algebra structure and the a-mappings the A-homomorphisms from E to a B-algebra. COROLLARY . Let E, E' be two A-algebras; for every A-homomorphism of algebras u : E -+ E', u @ 1 is the unique B-homomorphism of algebras v : E @A B -+ E' @A B rendering commutative the diagram E 5 E@AB Let C be a third commutative ring and o:B -+ C a ring homomorphism; it is immediate that the canonical C-homomorphism o*(p*(E)) -+ (00 P)*(E) mapping (x@ 1) @ 1 to .x@ 1 for all x EE (11, 9 5, no. 1, Proposition 2) is an algebra isomorphism. 6. INVERSE AND DIRECT LIMITS OF ALGEBRAS Let I be a preordered set and (A,, +ij) an inverse system of commutative rings with I as indexing set. Let (Ei,J,) be an inverse system of A,-modules with 1 as indexing set (11, 9 6, no. 1) and suppose further that each Ef has an Ai- algebra structure and that, for i < j, fti is an A,-homomorphism of algebras (relative to +fj) (no. 5). Let A = lim t Ai and E = t lim Ei, which has an A- 4 1.6 module structure, the inverse limit of the structure of the A,-modules Ei (11, 5 6, no. 1) ; it is immediately verified that the law of composition on E, con- sidered as the inverse limit of the Ei considered as magmas under multiplication (I, 3 10, no. 1)) with the A-module structure on E, defines on E an A-algebra structure; (E,,&) is called an inverse system of A,-algebras and the A-algebra E is called its inverse limit. If fi : E 4 E,, +f : A -+ A, are the canonical mappings, fi is an A-homomorphism of algebras (relative to +,). If the E, are associative (resp. commutative), so is E; if each E, admits a unit element e, andJ;,(ej) = e, for i < j, e = (e,) is a unit element of the algebra E. Let (E;,Ji,) be another inverse system of A,-algebras and for all i let u,:Et -+ E; be an A,-algebra homomorphism, these mappings forming an inverse system; then u = lirn ut is an A-algebra homomorphism. Suppose now that all the A, are equal to the same commutative ring A and the +,, to Id,, so that E = lim E, is an A-algebra. Let F be an A-algebra (US is an inverse system of mappings; then u = lirn ui is a homomorphism of the algebra F into the algebra E. Conversely, for every A-algebra homo- morphism v:F + E, the family of vi =fi 0 u is an inverse system of A-algebra homomorphisms such that v = lirn v,. As moreover, writing& = Hom( IF&), INVERSE AND DIRECT LIMITS OF ALGEBRAS t and, for all i E I, let u,: F --f E, t be an A-algebra homomorphism such that (HomA,,,.(F, Ei),&) is clearly t an inverse system of sets, it is seen that the ~ ..._ above remarks can also be expressed by saying that the 'canonical mapping v H (J; 0 v) is a bijection IF : HomA-alg. (F, lim E,) --f lim HomA-,lg. (F, E,). f- +-- Moreover, for every A -algebra homomorphism w : F -+ F', the zZt = Hom(w, l~,) :HOmA-alg.(F', E,) --f HOmA-alg.(F, E,) form an inverse system of mappings and the diagram IF' HOmA-,,,. (F', lim E,) ---+ lim HOmA-,l,. (F', Ei) lirn di - 1- IF c 1- Hom(m, IS) HOmA-,,,. (F, lim c Ei) ---+ lim HOmA-,l,. (F, E,) is commutative. Suppose now that I is right directed. Consider a direct system of commu - tative rings (Af, +,J and a direct system (Ef,At) of Ai-modules, with I as indexing set; suppose that each E, has an At-algebra structure and that, for i 4 j, fr, is an A,-homomorphism of algebras (relative to +j,) (no. 5). Let A = lirn 4, E = lim E,; E has an A-module structure, the direct limit of _. - * the structures of the A,-modules E, (11, 3 6, no. 2); moreover, the law of 434 435 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS composition on E considered as the direct limit of the E,, considered as magmas under multiplication (I, 5 10, no. 3), with the A-module structure on E, defines an A-algebra structure on E; (Ei,fj,) is called a direct system of A,- algebras and the A-algebra E is called its direct limit. IfJ;:E, -+ E, +,:A, + A are the canonical mappings, fi is an A,-homomorphism of algebras (relative to 4,). If the E, are associative (resp. commutative), so is E; if each E, admits a unit element e, andJ;.,(e,) = ej for i < j, E admits a unit element e such that fi(e,) = e for all i E I. Let @:,A;) be another direct system of A,-algebras and for all i let u,: E, 3 E; be an A,-algebra homomorphism, these mappings forming a direct system; then u = lirn u, is an A-algebra homomorphism. --+ Suppose now that all the rings A, are equal to the same ring A and the to IdA, so that E = lim E, is an A-algebra. Let F be an A-algebra and for all i let u,:E, -+ F an A-algebra homomorphism such that (u,) is a direct system of mappings; then u = lim u, is a homomorphism of the algebra E into the algebra F. Conversely, for every A-algebra homomorphism v: E -+ F, the family of v, = v 05 is a direct system of A-algebra homomorphisms such that u = lim u,. As moreover, writingfij = Horn(& IF), (HOmA-alg.(Et, F),Jj) is clearly an inverse system of sets, it is seen that the above remarks can also be expressed by saying that the canonical mapping u H (v 05) is a bijection --f --f I 4 dF:HOmA-,1,,(lim 3 El, F) -+ lfiHOmA-alg.(Et, F). Further, for every A-algebra homomorphism w : F --f F’, the Wi = Hom(l,,, w) :HOmA-alg.(Et, F) -+ HomA-alp.(Ei, F’) form an inverse system of mappings and the diagram dF E,, F) --+ lim HomA-alg.(Ei, F) dw t HOmA-,l,.(% E,, F’) __f lim HomA-alg.(Ei, F’) is commutative. 7. BASES OF AN ALGEBRA. MULTIPLICATION TABLE By definition, a busis of an A-algebra E is a basis of E for its A-module struc - ture. Let (a,),EI be a basis of E; there exists a unique family (yb)({, j, k)EI XIXI of elements of the ring A such that for every ordered pair (i,j) E I x I, the set of k E I such that y: # 0 isfinite and (7) fj 1.7 The yFj are called the constants ofstructure of the algebra E with respect to the basis (a,) and relations (7) constitute the mult$lication table of the algebra E (relative to the basis (a,)). BASES OF AN ALGEBRA. MULTIPLICATION TABLE Relations (7) can be imagined written down by setting out the right hand sides of these relations in a square table I )...Ia,I... it being understood that the element appearing in the row of index i and the column of index j is equal to the product a,q. Conversely, given an A-module E, a basis (aJIEI of E and a family (y:,) of elements of A such that for every ordered pair (i,y) E I x I the set of k E I such that y: # 0 is finite, then there is on E one and only one A-algebra struc- ture under which relations (7) hold, since the A-module E @A E is free and admits as basis (a, @ a,)(i, j)EI (cf. 11, 5 3, no. 7, Corollary 2 to Proposition 7) * Let E be an A-algebra and (at)tEI a generating system of the A-module E (for example a basis). For E to be associative, it is necessary and sufficient that the a, satisfjr the associativity relations The mapping (x, y, z) H (xy)z - x(yz) is an A-trilinear mapping E x Ex E+E and hence defines an A-linear mapping E BAE gAE--+E; if the latter mapping is zero for all the elements a, @ aj @ ak, which form a generating system of the A-module E @A E @A E, it is identically zero. Similarly, for E to be commutative, it is necessary and sufficient that the a, satisfy the commutativity relations (9) (a,a,)ak = a,(ajak) for all i, j, k (8) a,a, = a,a, for all i,j. The proof is analogous, this time considering the A-bilinear mapping (x, !/) I-+ xy - yx. Finally, for an element e E E to be a unit element, it is necessary and sufficient that the a, satisfy the relations a, = eat = ate for all i, 43 7 436 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS as is seen this time by considering the A-linear mappings XHX - xe and x H x - ex. When (~1,)~~~ is a basis of E and (yf,) the corresponding family of con- stants of structure, relations (8) are equivalent to the relations for all i, j, k, s. Similarly relations (9) are equivalent to yf, = y;, for all i, j, k. Let (a,), I be a basis of the A-algebra E ; if p : A -+ B is a ring homomorphism, (a, @ 1) is a basis of the B-algebra p*(E) = E @A B (11, 5 5, no. 1, Proposi - tion 4). If (y:j) is the family of constants of structure relative to the basis (ai), the family (p(yFj)) is the family of constants of structure of p*(E) relative to the basis (ai @ 1). $2. EXAMPLES OF ALGEBRAS Throughout this paragraph, A denotes a commutative ring. 1. ENDOMORPHISM ALGEBRAS Let B be an associative A-algebra with a unit element denoted by 1 and let M be a right B-module. We know that the ring E = End,(M) also has a module structure over the centre of B. Now the image of the homomorphism h:a H a. 1 of A into B is contained in the centre of B (5 1, no. 1) ; hence h gives E an A-module structure. Further, for a E A and5 g in E, a(f 0 9) = f 0 (ag) = (af) O g; hence multiplication in E and the A-module structure on E define an associa - tive A-algebra structure on E; the identity mapping of M is a unit element of this algebra. 2. MATRIX ELEMENTS Let B be a unital associative A-algebra and M,(B) the set of square matrice $ order n over B (11,s 10, no. 7). Then Mn(B) has an A-module structure defined by a. (bij) = (abij) (a E A, bij E B, 1 < i < n, 1 < j < n); this structure and matrix multiplication define a unital associative A-algebra structure on M,(B) . The canonical bijection of M,(B) onto End,(B:) (11, 3 10, no. 7) is an A- algebra isomorphism. 438 QUADRATIC ALGEBRAS 5 2.3 When B = A, the A-algebra Mn(A) admits a canonical basis (Eij) consisting of the matrix units (11, 5 10, no. 3) ; the corresponding multiplication table is (1) EijEhk = Sj,Eik. The unit element I, is equal to 2 EH. n t=1 3. QUADRATIC ALGEBRAS Let a, p be two elements of A and (el, e2) the canonical basis of A2. The quadratic algebra of type (a, p) over A is the A-module A2 with the algebra structure defined by the multiplication table (8 1, no. 7) (2) e; = el, e1e2 = e2e1 = e,, ei = ae, + pe,. An A-algebra E isomorphic to a quadratic algebra is also called a quadratic algebra. It amounts to the same to say that E admits a basis of two elements one of which is the unit element. It can be shown that every unital A-algebra which admits a basis of two elements is a quadratic algebra (Exercise 1). If a basis (el, e,) of an A-algebra has multiplication table (2), it is called a basis of ope (a, p). By an abuse of language, a quadratic algebra is said to be of type (a, p) when it has a basis of type (a, p). PROPOSITION 1. A quadratic algebra E is associative and commutative. The fact that E is commutative follows from the equation e1e2 = e2e1 in (2); similarly, to verify associativity, it suffices to see that x(yz) = (y)z when X, y, z are each equal to el or e2. Now, this relation is obvious if at least one of the elements x, y, z is equal to el ; it is also true for x = y = z = e, since E is commutative; whence the proposition. Let e denote the unit element of a quadratic algebra E and let (e, i) be a basis of E of type (a, p) ; every other basis of E containing e is therefore of the form (e,j) with j = ye + Si (11, 5 7, no. 2, Corollary to Proposition 3); moreover, for (e,j) to be a basis of E, it is necessary and sufficient that S be invertible in A; the condition is obviously sufficient; conversely, if i is the canonical image of i in EIAe, i and 3 = Si must each form a basis of E/Ae, whence the necessity of the condition. Then ja = (yz + aS2)e + (2yS + p8')i = (aS2 - y2 - pyS)e + (2y + p8)j; thus it is seen that E is of type (as2 - y2 - PY%2Y + PS) (3) 4.39 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS for all invertible 8 E A and all y E A. In particular, if E is of type (x, 2p’), i is also of type (x + p’,, 0) as is seen by taking y = - p’ and 6 = 1. PROPOSITION 2. Let E be a quadratic A-algebra and e its unit element. For all u E E, let T(u) be the trace of the endomorphism m,:x H ux of the free A-module E (11, 5 4, no. 3). Then the mapping s deJined by s(u) = T(u) .e - u is an automorphism ofthe algebra E and s2(u) = u for all u E E. Let (e, i) be a basis of E of type (a, p); then T(e) = 2, when s(e) = e, and T(i) = p, whence s(i) = pe - i. Hence (e, s(i)) is a basis of E, whose type is given by (3) with y = p and 6 = - 1, which again gives (a, (3) ; it follows that s is an automorphism of the algebra E. As mS(,,) = sm,,s-l, the endomorphisms mu and mS(,,) of the A-module E have the same trace (11, 3 4, no. 3, Proposition 3), whence s2(u) = T(u).e - s(u) = T(u).e - (T(u).e - u) = u CAYLEY ALGEBRAS 5 2.4 for all u E E. The automorphism s is called conjugate of u. If u = Ee + qi, with 4, q in A, (4) T(u)e = u (5) u.s(u) = (42 conjugation of the A-algebra E and s(u) the then s(u) = (4 + Pq)e - qi, whence + s(u) = (24 + Bde + PSq - q2)e = N(u)e where we have written N(u) = t2 + psq - aq2. The elements T(u) and N(u) (or, when A and Ae are canonically identified, the elements T(u)e and N(u)e) are called respectively the trace and norm of u. When p = 0, the above formulae are simplified to (6) s(k + qi) = 4e - qi, T(4e + qi) = 24, N(<e + qi) = E2 - a$. Clearly T is a linear form on E *and N is a quadratic form on E (IX, 3 3, no. 4),. As E is commutative and associative, it follows from (5) that (7) N(w) = N(u))N(v). For u to be invertible in E, it is necessary and sufficient that N(u) be invertible in A. For, as N(e) = 1, the necessity of the condition follows from (7), writing u = u-l. Conversely, if N(u) is invertible in A, it follows from (5) that u is invertible and that (8) u-1 = (N(u)) -ls(u)- *It can be proved that N(u) is the determinant (3 8, no. 1) of the endomorphism m,, (cf. 9 g no. 3, Example l).* The following proposition gives the structure of quadratic algebras over a commutative fisld: PROPOSITION 3. Let E be a quadratic A-algebra of type (a, p). (i) IfA is ajeld and contains no element 7: such that C2 = a + p7:, E is a (com- mutative) jeld (cf. V, 5 3). (ii) If the ring A contains an element 7: such that c2 = a + p7: and p - 27: is invertible (resp. zero), E is isomorphic to A x A (resp. is of type (0, 0)). We prove (i). Let 6, q be two elements of A and u = te + qi. If q # 0 and we write 0 = -Eq-l, then N(u) = q2(02 - - a) by (5), whence N(u) # 0 by virtue of the hypothesis on A; if q = 0, then N(u) = k2. In any case, if u # 0, then N(u) # 0, hence N(u) is invertible in A and therefore u is invertible in E. We now prove (ii). The canonical basis (el, e,) of the algebra A x A is of type (0, 1). We have seen (formula (3)) that E is of type (as2 - y2 - BY& 2Y + PSI for all ycA and all 6 invertible in A. If p - 27: is invertible, take 8 = (p - 2c)-l and y = -<(p - 2iJ-l; then 2y + pa = 1 and aa2 - y2 - py8 = S2(a - 7:’ + p7:) = 0; thus E is of type (0, 1) and hence isomorphic to A x A. If p - 21: = 0, it has already been remarked that E is of type (a + c2, 0) and hence of type (0,O) since a + A quadratic A-algebra of type (0, 0) is also called an algebra of dual numbers over A. = 2ca - pl: = 0. I 4. CAYLEY ALGEBRAS I DEFINITION 1. A Cayley algebra over A is an ordered pair (E, s), where E is an algebra over A with a unit element e and s is an antiautornorphism of E such that u + s(u) EAe and u.s(u) EAe for all u E E. s is called the conjugation of the Cayley algebra (E, s) and s(u) the conjugate of u. The condition u + s(u) E Ae implies that u and s(u) are permutable. We write (9) T(u) = u + ~(u) (10) N(u) = U.S(U) = S(U).U and these elements of the subalgebra Ae are called respectively the Cayley trace and norm of u. i I ! I 440 441 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The ordered pair consisting of a quadratic algebra E and its conjugation s (which is an antiautomorphism since E is commutative) (no. 3) is a Cayley algebra. Let (E,s) be a Cayley algebra; as s(e) = e, s(u + s(u)) = u + s(u), in other words s(u) + s2(u) = u + s(u) or also ?(u) = u (1 1) SO that s2 is the identity mapping of E. It follows that (12) T(s(u)) = T(u), N(s(u)) = N(u). Finally, the relation (u - u)(u - ~(u)) = 0 gives (13) u2 - T(u).u + N(u) = 0 for all u E E. PROPOSITION 4. Let E be an A-algebra and s and s’ antiautomorphisms of E such that (E, s) and (E, s’) are Cayley algebras. If E admits a basis containing the unit element e, then s’ = S. Clearly s’(u) = s(u) = u for all u E Ae. If T, N (resp. T’, N’) are the trace and norm functions for (E, s) (resp. (E, s‘)), it follows from (13) that (T(u) - T‘(u)).u - (N(u) - N’(u)) = 0. Let B be a basis of E containing e and u an element of B distinct from e; then T(u) - T’(u) = 0, whence s(u) = s’(u). As s and s‘ coincide on B, they are equal. In what follows, we shall write ii = s(u), so that u + ii = T(u), {- u + v = ii + i7, UU = iiu = N(u), E = u - - au = aii, UV = V.ii (14) for u, v in E, a E A; moreover (15) T(e) = 2e, N(e) = e. From the formula T(uv) = uv + UV = uv + V.C = uv + (T(v) - v)(T(u) - u), we deduce that (16) whence, exchanging u and V, (17) T(vu) = T(uv). 442 uv + uu = T(u)v + T(v)u + (T(uv) - T(u)T(v)) 5 2.5 On the other hand, N(u + v) = (u + v)(ii + 5) = N(u) + N(v) + T(u?), whence CONSTRUCTION OF CAYLEY ALGEBRAS. QUATERNIONS (18) T(vii) = T(uF) = N(u + V) - N(u) - N(v). , Now, (16) applied with u replaced by ii gives T(uF) = T(u)T(u) + iiv + vii - T(u)v - T(v)ii = T(u)T(v) - uv - V.U; whence (19) T(vU) = T(u5) = N(u + V) - N(u) - N(v) = T(u)T(v) - T(uv). Finally, clearly for all a E A, (20) N(au) = a2N(u) ; in particular N(2u) = 4N(u), so that formula (19) gives (21) (T(u))~ - T(u2) = 2N(u). Clearly T is a linear form on the (&)-module E. As (u, ZJ) H T(vi) is a bilinear form on this module, *it follows from (18) and (20) that N is a quadratic form (cf. IX, 5 3, no. 4).* 5. CONSTRUCTION OF CAYLEY ALGEBRAS. QUATERNIONS Let (E, s) be a Cayley algebra over A, for which we shall use the notation of no. 4, and let y E A. Let F be the algebra over A whose underlying module is E x E and whose multiplication is defined by (x, y)(x’, y‘) = (xx’ + yyy, YE‘ + y’x); (22) clearly (e, 0) is unit element of F and E x (0) is a subalgebra of F isomorphic to E; we shall identfy it with E in what follows, so that x E E is identified with (x, 0) and in particular e is identified with the unit element of F. Let t be the permutation of F defined by (23) t((X,Y)) = (2, -Y> EE). PROPOSITION 5. (i) The orderedpair (F, t) is a Cayley algebra over A. and norm T, and N, of F are given by the formulae (ii) Let j = (0, e), so that (x, y) = xe -I- yj for x E E, y E E. The Cayley trace (24) TF(Xe + Yj) = T(X), NF(Xe + yj) = N(x) - yN(y). (iii) For F to be associative, it is necessary and sujicient that E be associative and commutative. 443 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS To prove both (i) and (ii), it therefore suffices to show that t is an antiauto- morphism of F. Clearly t is an A-linear bijection. On the other hand, t ((x, y) . (x', y')) = t ((xx' + yy'y, yx' + y'x)) = (F'X + yjy', -yx - y'x) = @', -Y') (2, -y) = t ((x', y'))t ((x, Y)) and hence t is an antiautomorphism. It remains to prove (iii). As E is identified with a subalgebra of F, E may be assumed to be associative. Let u = (x,y), u' = (x',y'), u" = (x",y") be elements of F. Then (uu')u" = ((xx' + yg"" + yj"(3x' + y'x), (YX' + y'x)?" + y"(xx' + y3'y)) ._ .. u(u'u") = (x(x'x" + yj"y') + (y(x"j' + JC'j")y, y(X"X' + yy'y") + (y'jz" + y".').). Examining these formulae shows that the commutativity of E implies the associativity of F. Conversely, if F is associative, formulae (27) applied with y = y' = 0, X" = 0 and y" = e give (0, x'x) = (0, xx'), that is x'x = xx' for all x, x' in E; thus E is then commutative. Note also that, in the above notation, for x, y in E, (28) yj =jU, x(yj) = (yx)j, (xj)y = (4.i (.j)(yj) = 3xe (29) j2 = e. The Cayley algebra (F, t) is called the Cayley extension of (E, s) deJined by y. Examples. (1) If we take E = A (and hence s = I*), the algebra F is a quadratic A-algebra with basis (e, j) where j2 = ye. (2) Take E to be a quadratic algebra of type (a, p), so that the underlying module of E is Aa, with multiplication table (2) (no. 3) for the canonical basis. Take s to be conjugation of E (no. 3, Proposition 2). Then, for all y E A, the Cayley extension F of (E, s) defined by y is called the quaternion algebra Of tyke (a, p, y), which is associative by no. 3, Proposition 1 and Proposition 5 above; its underlying module is A4 and, if (e, i,j, k) denotes the canonical basis of A4, the corresponding multiplication table is given by i2 = ae + pi, ;j = k, ji = pj - k, jz = ye, ki = -aj, kj = yi, k2 = -aye. ik = aj + pk, jk = Bye - yi, 2.5 Further, for u = pe + ti + yj + qk (with p, E, y, Z: in A), we have (writing I CONSTRUCTION OF CAYLEY ALGEBRAS. QUATERNIONS ii instead of t (u) and identifying A with Ae) : u = (p + Pt)e - Si - yj - Ck TF(U) = 2p f p< NF(u) = p2 + Ppt - at2 - y(r2 + PrC - a<'). I Formulae (30) follow from (28) and (29) and formulae (31) from (23) and (24), taking account of the formulae for the quadratic algebra E. Then, for u, v in F, NF(uv) = NF(U)NF(U) I (32) - for NF(uv) = UU.UU = uv(z7.U) = u(vC)U = (uii)(vV) by virtue of the associa - tivity and the fact that NF(u) belongs to the centre of F. An A-algebra isomorphic to a quaternion algebra is also called a quater- nion algebra; if a basis of such an algebra has multiplication table (30), it is called a basis of type (a, p, y). By an abuse of language, a quaternion algebra is said to be of type (a, p, y) when it has a basis of type (a, p, y). I When (3 = 0, formulae (30) and (31) simplify to ;j = k, jt = -k, j2 = ye, jk = -yi, ik = aj, aye, k2 = - kj = yi, (33) and ii = pe - ti - yj- Ck NF(u) = p2 - at2 - yq2 + ayc2. Then (a, p, y) is replaced throughout by (a, y) in the above expressions. It is immediate that the quaternion algebras of types (a, y) and (y, a) are iSomorphic. Note that formulae (32) show that F is not commutative when -1 # 1 in A. *Taking A to be the field R of real numbers and a = y = - 1, p = 0, the corresponding algebra F is called the algebra of Hamiltonian quaternions and is denoted by H. If u = pe + ti + yj + ck (p, E, y, < in R) is an element #O in H, the formula uii = iiu = p2 + t2 + q2 + <2 (cf. (34)) shows that N(u) # 0 in R, so that u admits an inverse u-l = N(u) -lii in H and that H is therefore a non-commutative jeld. (3) If E is taken to be a quaternion algebra (cf. Example 2), the Cayley extension of E defined by an element 8 E A is in general non-associative TF(U) = 2P (34) 1 444 445 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (proposition 5); it is called an octonion algebra over A (cf. Appendix, no. 3). 6. ALGEBRA OF A MAGMA, A MONOID, A GROUP Recall that a magma is a set with a law of composition (I, 3 1, no. 1). Let S be a magma written multiplicatively and let E = A(s) be the A-module of formal linear combinations of elements of S (11,s 1, no. 11) ; we know that a canonical mapping s - e, is defined of S into A") such that the family (e,),,s is a basis (called canonical) of A(s), every element of ACS) being then written uniquely in the form zs a,es, where (a,) is a family of elements of A of finite support. Then an A-algebra structure is defined on E by taking as multiplication table of the canonical basis eset = esf. The algebra E thus defined is called the algebra of the magma S over A. If x = 2 sses and y = 2 qses are two elements of E, then (35) sss 8ES xy = SES c ( tu=s c t;t?,)e,. When S is a monoid (resp. group), E is called the algebra of the monoid (resp. group) S over A; it is then an associative algebra (3 1, no. 7); similarly, when S is a commutative monoid, its algebra is associative and commutative. Finally if the magma S admits an identity element u, e,, is unit element of the algebra E; as the element e, is free, A is then identified with the sub- algebra Ae, of E. When A # {0}, S is sometimes identified with its image under the injection SH~,, so that an element of E is written as 2 ass; but this identification is not possible (without causing confusion) when S is written additively. Then es is also often written instead of e,. 8E8 Let B be another commutative ring and p :A + B a ring homomorphism; consider the algebras E = A(s) and E' = B(s) of the same magma S over A and B and let (eJSSs and (e$sSs be their respective canonical bases. The algebra BCS) is canonically identified, under the A-linear mapping j such that j(e, @ 1) = el for all s E S, with the algebra A(s) @A B obtained from A'" by extending the ring of scalars to B (11, 9 1, no. 11 , Corollary 3 to Proposi - tion 17). PROPOSITION 6. Let S be a magma, F an A -algebra and f a homomorphism of 4 2.6 S into F with only its multiplicative structure. Then there exists one and only one A- algebra homomorphism f : A(s) -+ F rendering commutative the diagram f S+F ALGEBRA OF A MAGMA, A MONOID, A GROUP (36) A(s) - F f (where the vertical arrow on the l~t is the canonical mapping s H e,). Let f:A(s) -+ F be the unique A-module homomorphism such that J(eJ = f(s) (11,s I, no. 1 I, Corollary 3 to Proposition 17) ; it suffices to verify that fis an algebra homomorphism and for this it suffices to prove that f(e,e,) = f(es)f(ef), which follows immediately from the definition and the hypothesis f (st) = f (s) f (t). Proposition 6 expresses the fact that the ordrred pair consisting of A@) and the canonical mapping s H e, is a solution of the universal mapping problem (Set Theory, IV, 5 3, no. 1) where C is the species of A-algebra structure and the a-mappings the homomorphisms of S into an A-algebra with only its multiplicative law. COROLLARY . Let S, S' be two ma<gmas and g:S + S' a homomorphism. Then there exists one and only one A-algebra homomorphism u : A(s) +- A(s') rendering commuta - tive the diagram s S' I I A'S) J. -----f A'S') J- U (where the vertical arrows are the canonical mappings). It suffices to apply Proposition 6 taking f to be the composite mapping s 4 S' -+ A(s'). In particular, if T is a stable subset of the magma S (I, 1, no. 4), the set of elements 2 ases of A(s) is a subalgebra of A(s) canonically isomorphic to the Example. Let V be an A-module and S a monoid which operates on V on the left; this means (I, 3 5, no. 1) that there is given a mapping (s, x) ++ s.x of S into V such that s.(x + y) = s.x + s.y, s. (ax) = a(s.x) and s. (t.x) = (st) .X for S, t in S, x, y in V and a E A and, denoting by e the identity element of S, 4.47 algebra A(T) SET and sometimes identified with the latter. 44.6 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS e.x = x for x E V. Writing f (s) (x) = s. x, f is a homomorphism of S into the algebra End,(V) (with only its multiplicative law), mapping the identity element e to the unit element 1,. Applying Proposition 6, an A-algebra homo - morphismf:A'S) -+ End,(V) is obtained, which gives the underlying group of v a lejl module structure over A('). This allows us to reduce the study of commutative groups with operators to that of modules. For let M be a commutative group with operators written additively, all of whose external laws are written multiplicatively. Let R be the sum set (Set Theory, 11, § 4, no. 8) of the domains of operators of the various external laws on M, cach of these domains being canonically identified with a subset of R. Let Mo(i2) be the free monoid (I 5 7, no. 2) constructed on R; a law of action (s, x) t-+ s.x is defined on M with Mo(0) as domain of operators, by induction on the length of the word s in Mo(R); if s is of length 0, it is the empty word e and we write e.x = x for all x E M. If x is of length n > 1, it can be written uniquely as s = tu, where u is of length n - 1 and t of length 1, so that t~ R; we then write s.x = t.(u.x). For any two words s, s' in Mo(R), the relation s. (s'.x) = (IS') .x is verified by induction on the length of s. Then applying the method described above, a left Z(M"(n))-module structure is obtained on M and it is verified without difficulty that the usual notions in the theory of groups with operators (stable subgroups, homomorphisms) are the same for commutative groups with operators and the modules thus associated with them. 7. FREE ALGEBRAS DEFINITION 2. Let I be a set; let M(1) (resp. Mo(I), resp. N")) denote the free magma (resp. free monoid, resp. free commutative monoid) derived from I. The algebra ofM(1) (resp. Mo(I), resp. N")) over A is called the free algebra (resp.free associa - tive algebra, resp. free commutative associative algebra (or, by an abuse of language, free commutative algebra)) of the set I over the ring A. We shall denote the free algebra (resp. free associative algebra, resp. free commutative algebra) of I over A by LibA(1) (resp. Libas,(I), resp. Libasc,(I)). By composing the canonical mapping of I into M(1) (resp. Mo(I), resp. "I)) with the canonical mapping of M(1) (resp. Mo(I), resp. N(I)) into LibA(1) (resp. Libas,(I), resp. Libasc,(I)), a canonical mapping is obtained of I into Lib,(I) (resp. Libas,(I), resp. Libasc,(I)), which is injective if A # (0). We shall denote the image of an element i E I under this canonical mapping by X, and we shall say that Xi is the indeterminate of index i of Lib,(I) (resp. Libas,(I), resp. LibascA(1)). As Mo(1) and N(I) each have an identity element, Libas,(I) and Libasc,(I) 448 FREE ALGEBRAS 4 2.7 are unital associative algebras and further Libasc,(I) is commutative. If e is the unit element of Libas,(I) (resp. Libasc,(I)), the mapping a +> ue is an isomorphism ofA onto a subring ofthe centre of Libas,(I) (resp. Libasc,(I)), which is identified with A (no. 1). PROPOSITION 7. Let I be a set and F an algebra (resp. unital associative algebra, resp. unital commutative associative algebra) over A. For every mapping f: I -+ F, there exists one and only one homomorphism (resp. unital homomorphism) J of Lib,(I) (resp. Libas,(I), resp. Libasc,(I)) into F such thatf(X,) = f (i) for all iEI. Let F, be the magma (resp. monoid) obtained by giving the set F its multi- plicative law of composition. There is one and only one homomorphism (resp. unital homomorphism) g of M(1) (resp. Mo(I), resp. N")) into F, such that g(i) = f(i) for all iE I (I, 5 7, nos. 1, 2 and 7); Proposition 7 then follows from no. 6, Proposition 6. Remarks. (1) We shall later define an isomorphism of Libas,(I) onto the tensor algebra of the free module A") (5 5, no. 5) and also an isomorphism of Libasc,(I) onto the symmetric algebra of A") (5 6, no. 6). (2) Let p be a unital homomorphism of A into a commutative ring B. As has been seen (5 2, no. 6), an isomorphism cr is derived from p of Lib,(I) (resp. Libas,(I), resp. Libasc,(I)) onto the algebra (LibA(I))(B) (resp. (Libas*(I))(,), resp. (LibaSCA(I))(B)) obtained by extending the scalars to B by means of p; if Xf, Xf are the indeterminates of index i corresponding respectively to A and B, then cr(X:) = Xf @ 1. (3) Let J be a subset of I; we know that M(J) is identified with a stable subset of the magma M(1) and hence (no. 6) Lib,(J) is canonically identi - fied with a subalgebra of Lib,(I), generated by the Xi such that i E J; it is said that only the indeterminates of indices belonging to J occur in an element of Lib,( J). The definition given in no. 6 of the algebra of a magma shows that Lib,(I) is the union of the directed family of subalgebras Lib,(J) when J runs through the set ofjnite subsets of I. There are analogous results for LibaSA(1) and Libasc,(I). (4) With element s of M(1) (resp. Mo(I), resp. N(I)) is associated its length I($), which is an integer 2 1 (resp. 20) such that l(ss') = l(s) + l(s') (I, 5 7, nos. 1, 2 and 7). If e, is the element of Lib,(I) (resp. Libas,(I), resp. LibascA(I)) corresponding to s, the total degree (or simply the degree of an element x = 7 a,e, # 0 of Lib,(I) (resp. Libas,(A), resp. Libasc,(I)) is the greatest of the numbers I (s) when s runs through the (non-empty by hypo- thesis) set of elements such that a, # 0. For example, if i, j, k are three dis- tinct elements of I, the element (Xi(XjX,J)Xi - (X&) (X,X,) is an elemerlt fO of total degree 4 in LibA(I). 449 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 8. DEFINITION OF AN ALGEBRA BY GENERATORS AND RELATIONS Let F be an algebra over A and (xJiEI a family of elements of F. By no. 7, Proposition 7, there exists a unique homornorphismf: LibA(1) 4 F such that f(x,) = xi for all i E I. For f to be ~- surjective, it is necessary and sufficient that -. be a generating system of F’. If u E Lib,(I), f(U) is called the element of F derived from U by substituting the elements xi for the indeterminates Xi, or also the value of U for the values xi of the indeterminates Xi; it is usually denoted by U((X~)[€~); in particular U((XJiEI) = U. If A is a homomorphism of F into an algebra F‘ over A, then A(U ((Xi) i E I)) = u ((A(%) 1 i €1) Consider in particular the case where F = Lib,(J), where J is another set; for every family (Hi),,, of elements of Lib,(J) and every family (Y;),,~ of elements of an A-algebra F’, (37) E I)) ((Y;)j E J) = ((Y;)~E J)) 1 E 1). In the above notation, every element U of Lib,(I) such that U( (xJteI) = 0, or also such thatf(U) = 0, is called a relator of the family (xi)ieI in F. The two-sided ideal Ker( f) consisting of these elements is called the ideal of relators Let (Rj)jEj be a family of elements of LibA(I). ((xt)tEI, (Rj)jEJ) is called a presentation of the algebra F if (xi)iEI is a generating system of F and the two- sided ideal of Lib,(I) generated by the R, is equal to the ideal of relators of the family (xi),EI; the xi are called the generators and the R, the relators of the of (Xi) * presentation. Consider now any set I and a family (Rj)jEJ of elements of LibA(I). The quotient algebra E of Lib,(I) by the two-sided ideal generated by the family (R,) is called the universal a1,gebra deJined by the generating system I related by the \- -,/ - family of relators (R,),E J. Clearly, if xi denotes the image of Xi in E, is a presentation of E. Moreover, if (xi)iEI is a family of elements of an algebra F and R,((xi){.3 = 0 for all j E J, there exists a unique homomorphism g:E --f F such that g(gi) = xi for all i E I; for ((xi)teI, (Rj)jEJ) to be a presen- tation of F, it is necessary and sufficient that g be bijective. These remarks justify the following abuse of language: instead of saying ((xJiEI, (R,),€ J) is a presentation of F”, it is also said that “F is the algebra generated by the generators xi subject to the relations Rj((xOiE1) = 0”. When the Rj are of the form P, - Qj, it is also said that “F is the algebra generated by the xt subject to the relations pi( (xi)) = Qj( (xi))”. Let H be a set; we shall say that an element S of Lib,(H) is a universal << 450 DEFINITION OF AN ALGEBRA BY GENERATORS AND RELATIONS 9 2.8 of elements relator for an A-algebra F is S ( (xJII H) = 0 for every family (xJh of F with H as indexing set. Examples. (1) Take H = {1,2, 3); the algebras which admit (x1x2)x3 - x1(x2x3) as universal relator are the associative algebras. The algebras which admit x,x, - X,X, as universal relator are the commutative algebras. *The algebras which admit the universal relators X,X, and (x1x2)x3 + (x2x3)x1 + (x3x1)x2 as universal relator are the Lie algebras., Let I be a set; let a family (S,),,, of elements of Lib,(H) be given and consider the set T of elements of Lib,(I) of the form sk((uh))hEH), where k runs through K and, for each k, (U,),,, runs through the set of families of elements of Lib,(I) with H as indexing set; consider a family (R,),, with T as set of elements. Then let F be the universal algebra defined by the generat - ing system I related by the family (Rj),EJ and let u:Lib,(I) --f F be the canonical homomorphism, so that Ker(u) is generated by the elements sk((Uh)hEH) for all k E K and all families (Uh)h,H of elements of LibA(1); clearly each of the S, (k E K) is a universal relator for F. Now let F’ be an algebra admitting a generating system (xJieI, for which each of the S, is a universal relator, and let u’:Lib,(I) -+ F’ be the homomorphism such that u’(X,) = xi for all i E I; clearly Ker(u) c Ker(u’) and hence u‘ can be writ- ten uniquely in the form u’ = h 0 u, where h:F -+ F’ is a homomorphism such that h(X,) = xi for all i E I. For this reason F is called the universal algebra &&d by the generating system I, corresponding to the family of universal relators (SJlceIC. By an abuse of language, F is sometimes called the universal algebra generated by 1 and subject to the identities sk((z&)) = 0 for every family (uJhsH of elements of F. Example 2. Let L’ be the universal algebra generated by I and subject to the identities (uu)w - U(VW) = 0 for every family of three elements of L’ and let L” be the algebra obtained by adjoining a unit element to L’; then there exists a unique unital isomorphism g of L” onto Libas,(I) such that g(xi) = Xi for all i E I. For clearly L” is associative and the existence of the homomorphism g follows from the definition of L‘ and the remarks preceding it; but then clearly L” satisfies the universal property (no. 7, Proposition 7) which characterizes Libas,(I), whence the conclusion. Considerations analogous to the above can be applied to associative alge - bras (resp. commutative associative algebras), taking account of the foiiow- hg remarks. When the context gives sufficient indication that the algebras 45 1 I 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS considered are unital algebras, by an abuse of language, a family of elements of an algebra F, such that the subalgebra generated by the xi (i E I) and the unit element is equal to F, is often called a generating system of F. Then let F be a unital associative algebra over A and let (xi)iEI be a family of elements of F; by virtue of no. 7, Proposition 7, there exists a unique unital homomorphism f:Libas,(I) +F such that f(X,) = xi for all iE1; if u E Libas,(I), f (U) is also called the element of F derived from U by substituting the elements xi for the indeterminates Xt and it is also denoted by U((X~)~~~). Then the notions of relator, presentation and universal relator go over immediately to associative algebras; it suffices simply to replace Lib,(I) throughout by Libas,(I). The universal unital associative algebra dejined by the generating system I related by the family ofrelators (Rj),EJ is the quotient algebra of Libas,(I) by the two-sided ideal generated by the family (R,) . The universal unital associa- tive algebra generated by the generating system I, corresponding to a family of universal relators is defined similarly. We leave to the reader the task of stating the analogous definitions relative to commutative associative algebras with Libasc,(I) substituted for Libas,(I). Example 3. Let L‘ be the universal unital associative algebra generated by I and subject to the identities uv - vu = 0 for every family of two elements of L’. It is seen, as in Example 2, that L‘ is canonically isomorphic to Libasc,(I). 0 2.9 2,3,. . . elements. Note that, by virtue of the conventions made above, A[X] and A[Y] are for example (distinct) subalgebras of A[X, Y, Z] if A # (0). POLYNOMIAL ALGEBRAS The elements xv == n xy, iEI where v runs through “I) form a basis of the polynomial algebra A[(XJiE1]. These elements are called monomials in the indeterminates Xi and the number 1.1 = &v(i) is called the degree (or total degree) of the monomial Xv. The unique monomial of degree 0 is the unit element of A[(Xi)iEI]; it is often identified with the unit element 1 of A. Every polynomial u of A[(Xi)iE,] Can be written uniquely as u = v E C N(I) a,xv with a,EA; the elements a,, zero except for a finite number of indices v E N(I), are called the coeficients of u; the elements a,XV are called the terms of u (the element a,XV often being called the “term in X””) ; in particular the term a,Xo (identified with a,, E A) is called the constant term of u. If J is a subset of I, u belongs to A[(X,), EJ] if and only if a, = 0 for v $ N(J). It follows that A[(Xi)iEI] is the union of the subalgebras A[(XJiEJ], where J runs through the set of finite subsets of I. If a, = 0 for IvI > n, u is said to be a polynomial of degree < n. When a, = 0, (by an abuse of language) u is said to contain no term in Xv; in particular, when a, = 0, u is said to be a polynomial with no constant term. For every non-zero polynomial u = 2 a,Xv, the degree (or total degree) of u is the greatest of the integers IvI of the multiindices v such that a, # 0. Let F be a unital associative A-algebra and let (xJiEI be a family of ele- ments of F, which are pairwise permutable. The subalgebra F’ of F generated by the x, and the unit element is commutative (4 1, no. 7), which allows us to define substituting the x, for the Xi in the polynomial u E A[(Xi)iE,] (although F is not necessarily commutative): u((xi)iEI) is an element of F‘ and there - fore of F and h: u-+u((~~)~~~) is a homomorphism of A[(X,),.,] into F. The elements ofthe kernel ofh are the relators of the family (xi) in A[(Xi)iEI], also called Polynomial relators (with coefficients in A) between the x,. The image of the homomorphism h is the subalgebra F’, also denoted by A[(X~)~~,] (even when F is not commutative) ; if a is the ideal of polynomial relators between the xi, then there is an exact sequence of A-modules 0 - a - A[(X,),.,I - h A[(x,j,.,l --+ 0. - 9. POLYNOMIAL ALGEBRAS Let B be a unital commutative associative A-algebra and let (xi)iaI be a family of elements of B; the subalgebra of B generated by the xi (i E I) and the unit element is denoted by A[(x~)~~~]~ or simply A[(X,)~,~] when no confusion can arise. For every set I, the algebra Libasc,(I) is therefore equal to A[(Xi)iE,] (also denoted by AIXiliEI); the latter notation, which has the advantage of indicating the notation chosen to denote the indeterminates, is the one we shall generally use in the rest of this Treatise. The elements of A[(Xi)!,,] are called polynomials in the indeterminates Xi (i E I) with coe$cients in A; it is a convention that, when it is said “let A[(XJiG1] be a polynomial algebra”, the Xi are always understood to be the indeterminates. For every subset J of I, the use of the above notation amounts to identifying Libasc,(J) with the algebra of Libasc,(I) generated by the Xi of index i E J and the unit element (cf. no. 7, Remark (3)). For I = (1 , 2, . . . , n), we write AIX1, Xz, . . . , X,] instead of A[(XJiEI]. If I and I’ are two equipotent sets, the algebras Libasc,(I) and LibaSCA(1’) are isomorphic. A[X] is often used to denote the polynomial algebra corres- ponding to an unspecified indexing set with a single element, X denoting the unique indeterminate; similarly, A[X, Y], A[X, Y, Z], . . . are used to de- note the polynomial algebras corresponding to unspecified indexing sets with 452 POSITION 8. Let A[(XJiSI] be a polynomial algebra, J a subset of I and K the cwfllement of J in I. Writing A’ = A[(X,),,,] and denoting by XL (k E K) the 453 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS indeterminates in the polynomial algebra Libasc,, (K) = A [(X;) K], there exists a unique ring isomorphism of A'[(Xl,)ksK] onto A[(XI)iEI] which coincides with the identity on A' and maps Xl, to XI, for all k E K. Clearly A[(X,),,,] is an A-algebra generated by the XI, for k E K. On the other hand, as a polynomial relator between the X, (k E K) with coefficients in A' can be written uniquely as 2 h,( (X,) je J)Xv where v runs through a finite subset of NCK) and where the h, are elements of A[(X,),,J], the h, must be polynomial relators between the X, with coefficients in A and hence are all zero, which proves the proposition. The isomorphism described in Proposition 8 is often used to identify the elements of A[(X,),,,] with polynomials with coefficients in A' = A[(X,),E J]. Ifu is an element #O of A[(Xi)i,I], its total degree considered as an element of A'[(Xk)kEK] is also called its degree with respect to the Xi of index i E K. Remark. Let I and J be two sets and (Pj),. a family of elements of Z[(Xi)tsI] ; if Q is an element of Z[(Xj),EJ] such that Q((P,)jsJ) = 0, then, for every family (b,),,, of pairwise permutable elements of a ring B, Q((PJ((~JLEI))JEJ) = 0. The relations which hold of the form Q((Pj)jEJ) = 0 are sometimes called polynomial identities. For example (X, + X2)2 - x: - xi - 2X1X2 = 0 xl; - x; - (X, - x2)(xl;-1 + x;-2x2 + . *. + Xp) = 0 P 3- - xy-1 + x;-2x2 + . . * + X;-' with Q= Y; - Y2, PI = XI + Xz, P, = X; + X: + 2X1X2 with Q= Yl - Y2Y3, Pi = X; - X:, Pz = XI - Xz, are polynomial identities. 10. TOTAL ALGEBRA OF A MONOID The algebra of a monoid S over A is (as an A-module) the submodule of the product AS consisting of the families (a,),Es of finite support; the multiplica- tion in this algebra is defined by the relations (aS)(Ps) = (y,), where, for all s E s, (38) ys = t==, 2 atPu T I 2.1 1 (cf. no. 6, formula (35)). The sum on the right hand side of (38) is mean - ingful because (a,) and (P,) are families of finite support and so therefore is the double family (at(ju)(l,U)ESYS. But the right hand side of (38) is also meaningful for arbitrary elements (a,), p,) of AS when the monoid S satisfies the following condition: (D) For all s E S, there exists only ajnite number oforderedpairs (t, u) in S x S such that tu = S. Suppose then that S satisfies condition (D) ; a multiplication law can then be defined on the product A-module AS by formula (38). It is immediate that the multiplication thus defined on AS is A-bilinear; also it is associative, since, for a, p, y in AS, FORMAL POWER SERIES OVER A COMMUTATIVE RING This multiplication and the A-module structure on AS therefore define on As a unital associative algebra structure over A; we shall say that the set AS, with this structure, is the total algebra of the monoid S over A. It is immediate that the algebra A") of the monoid S over A (also called the restricted algebra of S when necessary to avoid confusion) is a subalgebra of the total algebra of S over A (and is identical with the latter when S is finite). By an abuse Of language, every element (c,),, of the total algebra of S over A is also denoted by the same notation 2 E,e, (or even 2 E,.s) as the elements of the restricted algebra of S; of course the summation symbol appearing in this notation corresponds to no algebraic operation because it is taken over an infinity of terms #O in general. With this notation, multiplication in the total algebra of S is also given by formula (35) of no. 6. If S is commutative, so is its total algebra AS. If T is a submonoid of S, the total algebra AT of the monoid is canonically identified with a subalgebra of the total algebra of S. If p: A + B is a ring homomorphism, the canonical extension ps: AS -+ BS is an A-homomorphism of the total algebra of S Over A into the total algebra of S over B, which extends the canonical homo - morphism A'S) + B'S). SEQ 11. FORMAL POWER SERIES OVER A COMMUTATIVE RING For every set I, the additive monoid "I) satisfies condition (D) of no. 10; for, if s = (nJiEI with ni = 0 except for the indices i in a finite subset H of I, the relation s = t + u with t = (pi)iEI and u = (qJtEI is equivalent to PI -!- pi = n, for all i; but this impliesp, = q, = 0 for i 4 H andp, < n,, qr 6 n, for i E H; there are therefore n (n, + 1) ordered pairs (t, u) in N(I) such that t+tl=s. i€H 454 455 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS We can therefore consider the total algebra of the monoid N") over A, which contains the (restricted) algebra A[Xi], of this monoid. It is a unital commuta - tive associative algebra called the algebra of formal power series in the indeterminates xi (i E I) with coe@cients in A and denoted by A[[Xl]]lsI; its elements are called formal power series in the indeterminates X, (i E I) with coefficients in A. Such an element (C(~)~E~(I) is also denoted, following the convention made in no. 10, by .,& avXv; the a, are the co@cients of the formal power series and the a,XV its terms; a polynomial in the X, is therefore a formal power series with only a finite number of terms # 0. Clearly an algebra isomorphism A[[X,]]iEI, -+ A[[X1]IiEI2 is canonically derived from every bijection CS: I, -+ I, by mapping the formal power series z) cqnO. a X:' to the formal power series Let J be a subset of I; the algebra A[[Xi]ItEJ can be identified with a sub- algebra of A[[X,]ItEI consisting of the formal power series 2 cqn,).u X:' where a(,,) = 0 for every element (n,) EN(') such that n, # 0 for at least one index i E I - J. Further, if K = I - J, A[[X,]IiEI is canonically identified with (A[[X,]],EJ) [[Xk]IkEKJ by identifying the formal power series c .n X:' with the formal power series (mk) c Ij(mk). a Xp, where q,,,). n iEI1 X::,). (n,) t e 1 commutative monoid written additively whose identity element is denoted by 0. 1. GRADED ALGEBRAS DEFINITION 1. Let A be a commutative monoid, A a graded ring oftype A (11,s 11, no. 2) , (AJheA its graduation and E an A -algebra. A graduation (EJheA of type A 4 3.1 Further, if w(u) # w(v), then necessarily u + v # 0 and the two sides of GRADED ALGEBRAS (40) are equal. with y(pj, = a(,,') for the sequence (ni) such that n, = Pi for i E J and n, = m, for iEK. Given a formal power series u = 2 CI,X", the terms xVXv such that IvI = p are called the terms in u of total degree p. The formal power series up whose terms of total degree p are those of u and whose other terms are zero, is called the homogenouspart of u of degree p; when I isjnite, up is a polynomial for all p; uo is identified with an element of A (also called the constant term of u). If u and u are two formal power series and w = uv, then (39) wp = 2 U,vp-r r=O for every integerp 2 0. For every formal power series u # 0, the least integerp 2 0 such that up # 0 is called the total order (or simply the order) of u. If this order is denoted by ~(u) and u and u are two formal power series #O, then (40) (41) w(uv) 2 W(.) + w(v) if uv # 0. 456 w(u + v) 2 inf(w(u), w(v)) if u + v # 0 I Note that the order of 0 is not dejned. By an abuse of language, it is a convention to say that " f is a formal power series of order >p (resp. >p)" if the homogeneous part off of degree n is zero for all n < p (resp. n < p) ; 0 is therefore a "formal power series of order >p" for every integer p 0. Let J be a subset of I and let A[[X,]]iEI be identified as above with B[[Xk]]ke~, where K = I -J and B = A[[X,]],EJ; corresponding to the above definitions applied to B[[Xk]lkEK there are new definitions for the formal power series u EA[[X~]],.~; a term a(,,').rI X:' is said to be ofdegreep in the Xt of index i E K if ,z ni = p and the formal power series Of B[[Xk]]k,, with the same terms of degree p as u and the others zero is called the homogenous part gdegreep in the Xi of index i E K. If u # 0, the order wK(u) with respect to the Xi ofindex i E K is the smallest ofthe integersp 2 0 such that the homogeneous part of u of degree p in the Xi of index i E K is # 0. Inequalities (40) and (41) still hold when w is replaced by wK. tEI $3. GRADED ALGEBRAS is compatible both with the A-module andwith the ring structure on E, in other words: ;f, for all A, p in A, % A-algebra E, with this graduation, is then called a graded algebra of type A over the graded ring A. When the graduation on A is trivial (that is (11,s 1 1, no. 1) A, = A, A, = (0) for h # 0), condition (1) means that the E, are sub-A-modules of E. This leads to the definition of the notion of graded algebra of type A over a non-graded 45 7 1. T I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS commutative ring A: A is given the trivial graduation of type A and the above definition is applied. When we consider graded A-algebras E with a unit element e, it will always be understood that e is Of degree 0 (cf. Exercise 1). It follows that if an invertible element x E E is homogeneous and of degree p, its inverse x-l is homogeneous and of degree -p: it suffices to decompose x-l as a sum of homogeneous elements in the relations x-lx = xx-l = e. Let E and E' be two graded algebras of type A over a graded ring A of type A. An A-algebra homomorphism u: E -+ E' is called a graded algebra homo - morphism if u(Eh) c Ei for all A E A (where (E,) and (EL) denote the respective graduations of E and E') ; where E and E' are associative and unital and u is unital, this condition means that u is a graded ring homomorphism (11, $ 11, no. 2). Let E be a graded A-algebra of type N. E is identified with a graded A-alge- bra of type Z by writing En = (0) for n < 0. Remark. Definition 1 can also be interpreted by saying that E is a graded A- module and that the A-linear mapping m:E@,E+E defining the multiplication on E (4 1, no. 3), is homogeneous of degree 0 when E @A E is given its graduation of type A (11, TO define a graded A-algebra structure of type A on the graded ring A, with E as underlying graded A-module, therefore amounts to defining for each ordered pair (A, p) of elements of A a Z-bilinear mapping 11, no. 5). mh,:E, E,-tEh+lA such that for every triple of indices (A, p, v) and for a E A,, x E E,, y E E,, Examples. (1) Let B be a graded ring of type A; if B is given its canonical Z- algebra structure (3 1, no. 1, Example 2), B is a graded A-algebra (Z being given the trivial graduation). (2) Let A be a graded commutative ring of type A and M a graded A- module of type A. Suppose that all the elements of the monoid A are cancellable, which allows (11,s 11, no. 6) us to define on Homgr,(M, M) = Endgr,(M) a graded A-module structure of type A; as this graduation is compatible with the ring structure on Endgr,(M) (11,s 11, no. 6), it defines a unital graded A-algebra structure on the A-algebra Endgr,(M). (3) Algebra of a magma. Let S be a magma and +: S -+ A a homomorphism. For all h E A, we write S, = 4 (A) ; then ShS, c S,,,. Let A be a graded commutative ring of type A and (Ah),EA its graduation; we shall define a graded A-algebra structure on the algebra E = ACS) of the magma S ($2, a*mWv(x,!/) = mh+W.v(ax,Y) = mfi,h+v(x, -1 I GRADED SUBALGEBRAS, GRADED IDEALS OF A GRADED ALGEBRA $ 3.2 no. 6). To this end, let E, denote the additive subgroup of E generated by the elements of the form a. s such that M E A,, s E S, and p + v = A. As the S, are pairwise disjoint, E is the direct sum of the A,S, and hence also the direct therefore define on E the desired graded A-algebra structure. If S admits an identity element e, it may also be supposed that +(e) = 0. A particular case is the one where the graduation of the ring A is trivial; then E, is the sub-& module of E generated by S,. More particularly, if we take S = N(I), A = N and + the mapping such that +((n,)) = 2 n,, the ring A having the trivial graduation, a graduation is thus obtained on the polynomial algebra A[X& for which the degree of a homogeneous polynomial # 0 is the total degree defined in $ 2, no. 9 (cf. $ 6, no. 6). We now take S to be the free monoid Mo(B) of a set B (I, $ 7, no. 2) and 4 the homomorphism Mo(B) + N which associates with each word its length. Thus a graded A-algebra structure is obtained on the free associative algebra of the set B (5 2, no. 7; cf. 3 5, no. 5). ~ I sum of the E, and it is immediate that the E, satisfy conditions (1) and (2) and lE1 I 2. GRADED SUBALGEBRAS, GRADED IDEALS OF A GRADED ALGEBRA Let E be a graded algebra of type A over a graded ring A of type A. If F is a sub-A-algebra of E which is a graded sub-A-module, then the graduation (F,) on F is compatible with its A-algebra structure, since F, = F n E,; in this case F is called a graded subalgebra of E and the canonical injection F -+ E is a graded algebra homomorphism. Similarly, if a is a left (resp. right) ideal of E which is a graded sub-A-module, then EhaW c ah+, (resp. a,E, c ah+W), since ah = a n E,; then a is called a graded ideal of the algebra E. If b is a graded two-sided ideal of E the quotient graduation on the module E/b is compatible with the algebra structure on E/b and the canonical homomorphism E+ E/b is a graded algebra homomorphism. If u: E -+ E' is a graded algebra homomorphism, Im(u) is a graded sub- algebra of E', Ker(u) is a graded two-sided ideal of E and the bijection E/Ker(u) -+ Im(u) canonically associated with u is a graded algebra iso- morphism. PROPOSITION 1. Let A be a graded commutative ring oftype A, E a graded A-algebra oftype A and S a set ofhomogeneous elements of E. Then the sub-A-algebra (resp. le$t ideal, right ideal, two-sided ideal) generated by S is a graded subalgebra (resp. graded ideal). The subalgebra of E generated by S is the sub-A-module generated by the finite products of elements of S, which are homogeneous; similarly, the left bp. right) ideal generated by S is the sub-A-module generated by the ele- ments of the form u1(u2(. . . (uns)). . .) (resp. ( . . . ((su,)u,- . .)u2)u1) where 458 459 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS s E S and the u, E E are homogeneous (n arbitrary) and these products are homogeneous, whence in this case the conclusion by virtue of 11, 9 11, no. 3, Proposition 2) ; finally the two-sided ideal generated by S is the union of the sequence (3n)n21, where 3, is the left ideal generated by S and 3,, (resp. ~,,+1) the right (resp. left) ideal generated by 32n-1 (resp. 3,,,), which com- pletes the proof. 3. DIRECT LIMITS OF GRADED ALGEBRAS Let (A,, +pa) be a directed direct system of graded commutative rings of type A (11,s 11, no. 3, Remark 3) and for each u let E, be a graded A,-algebra of type A; for u < let fpa: E, 3 ED be an A,-homomorphism of graded algebras and suppose that fya =fvD ofD, for CL < p < y; then we shall call (E,,fDa) a directed direct system of graded algebras of type A over the directed direct system (4, +Da) ofgraded commutative rings of type A. Then we know (11,s 11, no. 3) that E = lim E, has canonically a graded module structure of type A over the graded ring A = lim A, and a multiplication such that E'EN c Eh + (where (Eh) denotes the graduation on E) ; then this multiplication and the graded A-module structure on E define on E a graded A-algebra structure oftype A; E, with this structure, is called the direct limit of the direct system (E,, foa) of graded algebras. The canonical homomorphisms E, -+ E are taken A,-homo - morphisms of graded algebras. Moreover, if F is a graded A-algebra of type A and (u,) a direct system of A,-homomorphisms u,: E, -+ F, u = lim 3 u, is an A-homomorphism of graded algebras. + 3 TENSOR PRODUCT OF A FINITE FAMILY OF ALGEBRAS 8 4.1 $4. TENSOR PRODUCTS OF ALGEBRAS From 4 to 8 inclusive, A denotes a Commutative ring and, unless otherwise mentioned, the algebras considered are assumed to be associative and unital and the algebra homo- morphisms are assumed to be unital. 1. TENSOR PRODUCT OF A FINITE FAMILY OF ALGEBRAS A always denotes a commutative ring with unit element. Let be a jinite family of A-algebras and let E = 9 E, be the tensor product A-module of the A-modules E, (11,s 3, no. 9). We shall define an A-algebra structure on E. Let mi: E, @A E, -+ E, be the A-linear mapping defining the multiplication on E, (3 1, no. 3). Consider the A-linear mapping m' = @ m,: (8 (Ei @A E,) -+ E, = E; i€I ,€I 460 I the composite mapping where 7 is the associativity isomorphism (11, 3 3, no. 9) is an A-linear mapping m: E @A E -+ E; we shall see that m defines an (associative and unital) algebra structure on E. For, on explicitly performing the multiplication defined by m, we obtain the formula for xi,yi in E, and i E I. It is therefore seen already, by linearity, that if ei is the unit element of E,, e = @ ei is unit element of E. On the other hand, the associativity of each of the E, Is1 implies the relation whence, by linearity, the relation x(yz) = (.xy)z for all x, y, z in E. DEFINITION 1. Given a family (E,),,, of algebras over A, the tensor product of this family, denoted by @ E, (or, when I is the interval (1, n) of N, by giving the tensor product ofthe A-modules E, the multiplication deJined by (1). El @A E, @ . . . @A En, or ,€I simply El @ E, @ . . . En) is the algebra obtained Relation (1) shows that the tensor product 2 EP of the opposite algebras to the E, is the opposite algebra to @ E,; in particular, if the Ei are commutative, so is (8 E,. indexing set I. For each i E I, let&: E, -+ F, be an A-algebra homomorphism. Then the A-linear mapping ,€I Let (€1 (E,)lEI and (FJiEI be two families of A-algebras with the same finite i is an A-algebra homomorphism, as follows from (1). For every partition (I,),,, of I, the associativity isomorphisms I (11, 3 3, no. 9) are also algebra isomorphisms, as follows from (1) and their definitions. 46 1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS UNIVERSAL CHARACTERIZATION OF TENSOR PRODUCTS OF ALGEBRAS 9 4.2 I11 When I is the interval (1, n) of N and all the algebras E, are equal to the Same algebra E, the tensor product algebra (8 tEI E, is also denoted by E@”. We shall restrict our attention in the remainder of this no. to the properties of tensor products of two algebras, leaving to the reader the task of extending them to tensor products of arbitrary finite families. Let E, F be two A -algebras; if a (resp. b) is a left ideal of E (resp. F), the canonical image Im(a @ b) of a @A b in E @A F is a left ideal of E @A F; there are analogous results when “left ideal” is replaced by “right ideal’’ or “two -sided ideal”. Moreover: PROPOSITION 1. Let E, F be two A-algebras and a (resp. 6) a two-sided ideal ofE (resp. F) . Then the canonical A-module isomorphism (E/a) @ (W) -+ (E @ F)/(Im(a @ F) + WE @ 6)) (11, 3 3, no. 6, Corollary 1 to Proposition 6) is an algebra isomorphism. COROLLARY 1. Let E be an A-algebra and a an ideal of A. Then the A-module aE is a two-sided ideal of E and the canonical (A/a)-module isomorphism (A/a) @A E -+ E/aE This follows from (1) and the definition given loc. cit. is an (A/a)-algebra isomorphism. COROLLARY 2. If a, b are two ideals of A, (A/a) @A (A/b) is canonically iso- morphic to A/@ + b) . COROLLARY 3. Let E, F be two A-algebras and a an ideal of A contained in the annihilator of F. Then the (A/a)-algebra E @A F is canonically isomorphic to PROPOSITION 2. Let (EA)hPL and (F,JVEM be two families of A-algebras. The canonical mapping (11, 3 3, no. 7) (E/aE) @A/a F. is an algebra isomorphism. This follows immediately from 11, 3 3, no. 7, Proposition 7 and the definition of multiplication on E @ F. PROPOSITION 3. Let A, B be two commutative rings, p: A+B a ring homomor - phism and E, F two A-algebras. Then the canonical B-module isomorphism p*(E) @B p*(F) + p*(E @A F, (11, 3 5, no. 1, Proposition 3) is a B-algebra isomorphism. 462 PROPOSITION 4. Let A, B be two Commutative rings, p : A -+ B a ring homomorphism, E an A-algebra and F a B-algebra. Then the canonical A-module isomorphism p*(F) @A --f p*(F @El p*(E)) (11, 9 5, no. 2, Proposition 6) is an A-algebra isomorphism. The verifications are trivial on account of 3 1, no. 5. In particular, the A-algebra structure on B @A E, obtained by restricting the ring B of scalars to A, is identical with the structure of the algebra B @A E, the tensor product of the A-algebras B and E. Finally, if (Ai, +,,) is a direct system of commutative rings, (E&) and (F,,gj,) two direct systems of A,-algebras (9 1, no. 6) and A = lim A,, the lim (Ei gA, Fi) -+ (lim E,) @A (lim F,) (11,s 6, no. 3, Proposition 7) is also an A-algebra isomorphism, as follows from the definitions. Examples of tensor products of algebras. (1) Let A be a commutative ring and M, N two A -modules; the canonical mapping canonical A -module isomorphism -+ + + + @A --f @AN) (2) (11, 3 4, no. 4) is an A-algebra homomorphism, as follows from 11, 3 3, no. 2, formula (5). When M or N is a Jinitely generated projective A-module, we know that this homomorphism is bijective (11, 3 4, no. 4, Proposition 4). In particular we recover the definition of the tensor product of two square matrices. (2) Let S, T be two monoids and A(s) and A(T) the algebras of the monoids S and T over the ring A (111, 9 2, no. 6) ; then there is a canonical A -algebra isomorphism A‘S) @A ACT) -+ A‘S x T). (3) The elements e, @ e, (resp. e(s, t), where s runs through S and t runs through T, form a basis ofA(” @A A(T) by virtue of 11,s 3, no. 7, Corollary 2 to Proposi - tion 7 (resp. of AcsxT); the desired isomorphism is obtained by mapping e, 8 e, to e(8,t) and it follows from the definitions that this is indeed an algebra isomorphism. 2. UNIVERSAL CHARACTERIZATION OF TENSOR PRODUCTS OF ALGEBRAS PROPOSITION 5. Let (EJIEI be aJinite family of A-algebras and, for each i E I, let be the unit element of E,. For each i E I, let ui : E, -+ E = 8 E, be the A-linear mapping dejned by ui(xt) = 9 x; with x; = xi and xi = e, forj # i. 463 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (i) The u, are A-algebra isomorphisms; further, for i # j, the elements ui(xl) and u,(x,) are permutable in E for all x, E E, and x, E E, and E is generated by the union of the iubalgebrar u,(E,). (ii) Let F be an A-algebra and, for all i E I, let v,: E, 3 F be an A-algebra homo- morphism, where the vi are such that, for i # j, vi(xi) and vj(xj) are permutable in F for all xi E Ei and x, E E,. Then there exists one and only one A-algebra homomorphism W: E --f F such that v, = w o u, for all i E I. (4) (i) The mapping ui is an algebra homomorphism by definition of the multi - plication on E. If i # j, xi E Ei, x, E E,, then u,(x,) = xk with xi = xi, xk = e, for k # i, u,(x,) = 9 x; with x; = x,, xi = e, for k # j. Clearly x;xE = xlx; for all k E I and hence ul(xi) and u,(x,) commute in E by formula (1) (no. 1) defining the multiplication in E. The last assertion follows from the relation 9 xi = (ii) For each i E I, let xi be an element of E,. The product n tEI v,(x,) is then defined in F independently of any ordering on I since the algebra F is associa - tive and the elements vI(x,) are pairwise permutable. The mapping (xi)iEI -+ u vt(xl) of El into F is obviously A-multilinear and there there - fore exists one and only one A-linear mapping w: E --+ F such that u,(q). (5) Now, the desired A-algebra homomorphism w: E -+ F must satisfy (5), which follows from (4) and the fact that 9 xi = sT1[ u,(x,). This proves the uniqueness of w; it remains to show that the A-linear mapping w defined by (5) is an A-algebra homomorphism and satisfies (4). The fact that w satisfies (4) is obvious: it suffices to apply (5) to the case where x, = e, forj # i and we obtain w(ui(xt)) = v,(x,). Finally, w is an algebra homomorphism, for 464 I MODULES AND MULTIMODULES OVER TENSOR PRODUCTS OF ALGEBRAS 5 4.3 I 1 I since u,(xi) commutes with vj(yj) forj # i; hence which, by linearity, completes the proof. The ordered pair consisting of E and the canonical mapping +: (xi) H 9 xi of n E, into E is a solution of the universal mappingproblem (Set Theory, IV, 3 3, no. 1) where C is the species of A-algebra structure, the morphisms being the A-algebra homomorphisms and the cr-mappings the mappings n u, of n E, into an A-algebra such that the u, are A-algebra homomorphisms and ui(x,) and uj(xj) commute for i # j, for all xi E El and x, E E,. COROLLARY . Let (El),Er, (Fi),EI be two finite families of A-algebras and, for all i E I, let&: E, -+ F, be an algebra homomorphism. Ifu,: E, + @ E,, v,: F, + (8 F, are the canonical homomorphisms, the mapping f = @A (cf. no. 1) is the unique A- algebra homomorphism such that f 0 ui = vi Qfl for all i E I. It suffices to note that the homomorphisms g, = u1 ofl are such that g;(x,) = v,(J(x,)) and gj(xj) = v,(fr(x,)) commute for i # j, x, E E, and x, E E,; then apply Proposition 5. When, in Proposition 5, the algebra F is assumed to be commutative, the hypothesis that vi(xi) and v,(xj) are permutable for i # j is automatically satisfied. Hence, when F is commutative, there is a canonical bijection I i I ~ 1 ~ IEI fEI t (6) HomA-alg ($3 E,, F) + IJ HomA-alg. (~1, F), namely the one which associates with every homomorphism w of 9 E, into F the family of w 0 ut. F is the =me as that of F(E) (9 1, no. 5). Note that if E is a commutative A -algebra, the ring structure of E I ~ I 3. MODULES AND MULTIMODULES OVER TENSOR PRODUCTS OF ALGEBRAS DEFINITION 2. Let E be a (unital) A-algebra. A l$t (resp. right) E-module is a l$t (resp. right) module over the underlying ring of E. unless otherwise mentioned, all the modules and multimodules considered If M is an E-module, the homomorphism 3: A -+ E (3 1, no. 4) then I in this no. are left modules and multimodules. I 465 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS defines on M an A-module structure, said to be underlying the E-module struc - ture on M; for a EA, s E E, x E M, (7) a(sx) = S(M) = (ccr)x, so that for all s E E, the homothety h,: x H sx of M is an endomorphism of the underlying A-module structure. Conversely, being given an E-module struc - ture on M is equivalent to being given an A-module structure on M and an A-algebra homomorphism s H h, of E into End,(M). DEFINITION 3. Let E and F be two (unital) A-algebras and M a set with an E-module structure and an F-module structure. M is called a (left) bimodule over the algebras E and F ;f: (1) M is a bimodule over the underlying rings of E and F (11, 5 1, no. 14); (2) the two A-module structures underlying the E -module and F-module structures on The latter condition says that if e and e‘ are the unit elements of E and F M are identical. respectively, then (8) then M is used to denote the common value of the two sides. It can also be said that being given on M a bimodule structure over E and F is equivalent to being given an A-module structure on M and also two A-algebra homomorphisms s H h: of E into End,(M) and t H hi of F into End,(M) such that h:hi = hih: for all s E E and t E F. Consequently (no. 2, Proposi - tion 5) an A-algebra homomorphism u H h, of E @, F into End,(M) is ca- nonically derived such that hSmt = hihi = hrhj for s E E and t E F. In other words, an (E @A F)-module structure is thus defined on M, which is said to be associated with the given bimodule structure over E and F and under which (Ee)x = (ae’)x for a E A, x E M; (s @ t) .x = s(tx) = t(sx) for s E E, t E F and x E M. The given E-module and F-module structures on M can be derived from this (E F)-module structure by restrictions of the ring of scalars, corresponding to the two canonical homomorphisms E -+ E @, F and F -+ E @, F. Conversely, if an (E @, F)-module structure is given on M, by means of the canonical homomorphisms E + E @, F and F -+ E @, F an E-module structure and an F-module structure on M and it is immediate that M is a bimodule over the algebras E and F with these two structures and that the given (E @A F)-module structure is associated with this bimodule structure. Thus a one-to-one correspondence has been established between (E @A F)- module and bimodules over the algebras E and F. Clearly every sub -bimodule of M is a submodule for the associated (E @A F)-module structure and con- MODULES AND MULTIMODULES OVER TENSOR PRODUCTS OF ALGEBRAS $4.3 versely. There are analogous results for quotients, products, direct sums and inverse and direct limits. Finally, if M’ is another bimodule over the algebras E and F and$ M + M’ is a bimodule homomorphism,fis also an (E @, F)- module homomorphism and conversely. There are obviously corresponding statements for right bimodule structures, or when for example there is a left E-module structure and a right F -module structure; in this case we speak of an (E, F)-bimodule and being given such a structure amounts to being given a left bimodule structure over E and F”. Examples. (1) Let B be an A-algebra; the ring B has canonically a (B, B)- bimodule structure (11, 4 1, no. 14, ExarnyYe 1) and, if e is the unit element of B, then (ae)x = x(ae) = ax for all x E B and all t( E A; B can therefore be con - sidered as a left bimodule over the algebras B and B” (opposite to B) ; there is therefore associated with the (B, B)-bimodule structure on B a (B @A B”)- module structure such that, for b, x and 6’ in B, (6 @ b’).~ = bxb’ (9) the right-hand side being the product in the ring B. (2) Let E and F be two A-algebras, e, e’ their respective unit elements, M an E-module and N an F-module; these module structures define on M a bi- module structure over the rings A and E and on N a bimoduIe structure over the rings A and F; from these is therefore derived a bimodule structure over the rings E and F on the tensor product M 8, N, defined by x. (m @ n) = (x.m) @ n, y. (m @ n) = m @ (y.n) for x E E, y E F, m E M, n E N (11,s 3, no. 4) ; it is also seen that conditions (8) hold and hence the above bimodule structure is associated with an (E @A F)- module structure on M @, N, such that (10) (x @Y).(m 634 = (x.m) @ (y.n) for x E E, y E F, m E M, n E N. In particular, taking M = E,, E, @,N has canonically an (E @, F)- module structure; on the other hand, E 8, N is canonically identified with E @A (F, gF N) = (E @A F) gF N, where E F is considered as having its right F-module structure defined by the canonical homomorphism 7.1: F -+ E @A F; for x, x’ in E, y E F, n EN, x’ @ n is thus identified with (x’ €4 e’) @ n and (x @ y) . (x’ @ n’) = (xx‘) @ (y.n) with ((xx’) @ y) @ n. The (E @A F)-module E, @, N is thus identified with the (E 8, F)-module derived from N by extending the scalars to E @, F by means of the homo - morphism u (11, § 5, no. 1). The canonical mapping n ++ e @ n of N into E, @A N is identified with the canonical mapping n - (e @ e’) @ n of N into (E @A F) gF N; this is known to be an F-homomorphism. 466 467 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS With the same notation, let P be a right (E @A F)-module; then there is a canonid Z-module isomorphism (11) where on the right-hand side P is considered as a right F-module by means of the canonical homomorphism v. For P is canonically identified with P @EBaF (E @A F) and (E @A F) BF N with E @A (F gFN) and hence with E @A N, which establishes the stated isomorphism (11,s 3, no. 8, Proposi - tion 8 and IT, 3 3, no. 4, proposition 4). P @EgjAF (Es @AN) +P @FN All the above extends to multimodules (11, 3 1, no. 14). 4. TENSOR PRODUCT OF ALGEBRAS OVER A FIELD Let K be a commutative jield and E, F two algebras over K whose respec - tive unit elements e, e‘ are non-zero. Then the homomorphisms yjE: K -+ E and qF: K --f F (4 1, no. 3) are injections which allow us to identify K with a subfield of E (resp. F). The canonical homomorphisms u: E --f E @K F and v: F + E BK F, defined by U(X) = x @ e’ and v(y) = e @ y are injective (XI, 5 7, no. 9, Proposition 19) and allow us to identify E and F with subalgebras of E I& F, both having as unit element the unit element e @ e’ of E BK F. In E BK F, E n F = K (11, fj 7, no. 9, Proposition 19). If E’ and F‘ are subalgebras of E and F respectively, the canonical homo - morphism E‘ @& F’ + E @K F is injective and allows us to identify E’ @K F’ with the subalgebra of E BK F generated by E’ u F’ (11,s 7, no. 7, Proposition 14). PROPOSITION 6. Let E, F be two non-zero algebras over a commutative jield, K, C (resp. D) a subalgebra ofE (resp. F) and C’ (resp. D’) the centralizer of C in E (resp. D in F). Then the centralizer of C BK D in E gK F is C’ BK D’. It all reduces to verifying that an element z = x, @ yf of the centralizer of C gK D (x, E F, y, E F) belongs to C‘ @K D’; we know that C’ @K D’ = (C’ gK F) n (E mK D’) (11, 3 7, no. 7, Corollary to Proposition 14). The y, may be assumed to be linearly independent over K; for all x E C, of necessity (x @ e’)z = z(x @ e‘), that is (xxi - x,x) Byt = 0, whence xxf = xp for all i (11, 3, no. 7, Corollary 1 to Proposition 7) ; hence of necessity x, E C’ for all i and therefore z E C’ mK F; it can similarly be shown that z E E BK D’, whence the proposi- tion. COROLLARY . IfZ and Z’ are the respective centres of E and F, the centre of E BK F is z &Z’. 468 4.4 Let E and F be two subalgebras of an algebra G over a commutative field K; suppose that every element of E commutes with every element of F. Then the canonical injections i: E + G, j: F -+ G define a canonical homomorphism h = i @j: E BK F -+ G (no. 2, Proposition 5) such that (i@j)(x@y) =xy forxEE,yEF. DEFINITION 4. Given an algebra G over a commutativejeld K, two subalgebras E, F of G are said to be linearly disjoint over K if they satisjy the following conditions: TENSOR PRODUCT OF ALGEBRAS OVER A FIELD (1) every element of E commutes with every element of F; (2) the canonical homomorphism of E mK F into G is injective. PROPOSITION 7. Let G be an algebra over a commutativeJeld K and E, F two sub- algebras of G such that every element of E commutes with every element of F. For E and F to be linearly disjoint over K, it is necessary and sujkient that there exist a basis of E over K which is a free subset of G for the right F-module structure on G. When this is so : (i) the canonical homomorphism h: E @K F -+ G is an isomorphism of E BK F onto the subalgebra of G generated by E u F; (ii) E n F = K; (iii) every free subset of E (resp. F) over K is a free subset ofG with its right or left F-module (resp. E-module) structure. The condition of the statement is obviously necessary, since every basis of E over K is a basis of E gK F with its right F-module structure (11, 3 3, no. 7, Corollary 1 to Proposition 7). To see that the condition is sufficient, note that the image H of E gK F under h is the set of sums 7 x,y, = y,x, in G, with xi E E and yr E F; if (ah) is a basis of E over K, H is therefore also the submodule of the (right or left) F-module G, generated by (ah). The condition of the state- ment therefore means that there exists a basis (ah) of E which is also a basis of the F-module H; it follows that h is injective. Assertion (iii) follows from the fact that every free subset of E is contained in a basis of E (11, 4 7, no. 1, Theorem 2). COROLLARY 1. For the canonical homomorphism of E BK F into G to be bijective, it is necessary and suficient that there exist a basis ofE over K which is a basis ofthe (right or left) F-module G. COROLLARY 2. Let E, F be two subalgebras of G, ofjinite rank over K and such that every element of E commutes with every element of F. For E and F to be linearly disjoint over K, it is necessary and suficient that the subalgebra H of G generated by E v F be such that (12) [H:K] = [E:K]. [F:K]. This says that the rank over K of the surjective canonical homomorphism 469 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS h: E saying that this homomorphism is bijective (11, 3 7, No. 4, Proposition 9). 5. TENSOR PRODUCT OF AN INFINITE FAMILY OF ALGEBRAS Let A be a commutative ring and an arbitrary family of (unital) A- algebras. For every finite subset J of I, let E, denote the tensor product 8 E, of the algebras E, of index i E J; let e, denote the unit element of E, and e, = @ e, the unit element of E,; IetfJ, , denote the canonical homomorphism E, -+ E, for i E J (no. 2, Proposition 5). If J, J' are two finite subsets of I such that J c J', a homomorphism fJTJ: E, --f EJ, is canonically derived (no. 2, Proposition 5), by the condition fj,J 0 fJ. , = fJ., , for all i EJ. Moreover the uniqueness of fTJ implies that if J, J', J" are three finite subsets of I such that J c J' c J", then fJ-, =fJwJr 0 fJ?J. In other words, (Ej,fJ*J) is a direct system of A-algebras whose indexing set is the right directed set %(I) of finite subsets of I. DEFINITION 5. The direct limit E of the direct system (EJ, fJ,J) is called the tensor product of the family of A-algebras (EJ, If I is finite, E is identified with 9 E,. By an abuse of notation, E is also F 3 H is equal to the rank of E BK F over K, which is equivalent to iEJ denoted by @ E, even if I is infinite. LEI For every finite subset J of I, let fJ denote the canonical homomorphism @ E, -+ % E, (writingf, instead off;,)) ; if e is the unit element of E,, then fJ(eJ) = e for allJ E $(I). It is immediate that if all the algebras EL are com- mutative, so is 2 E,. PROPOSITION 8. (i) The homomorphismsf,: Ei --f E = gI E, are such thatfor two indices i,j such that i # j,ft(xi) and_f;.(xj) commute in E for all xi E E, and x, E E,; further, E is generated by the union ofthe subalgebrasf,(E,). (ii) Let F be an A-algebra and, for all i E I, let ui: Ei --> F be an A-algebra homomorphism such that, for i # j, ui(xi) and uj(xj) commute in F for all xi E E, and xI E E,. Then there exists one and only one A-algebra homomorphism u: E + F such that ui = u ofi for all i E 1. (i) As, for every finite subset J of I,$ = f, o f,.,, the first assertion in (i) follows from no. 2, Proposition 5, taking J containing i andj; the second also follows from no. 2, Proposition 5, taking account of the fact that E is the union of thefJ(EJ) when J runs through $(I). (ii) For every finite subset J of I, it follows from no. 2, Proposition 5 that there exists a unique homomorphism uJ: EJ -+ F such that uJ ofJ,{ = ui t€J 4 4.5 TENSOR PRODUCT OF AN INFINITE FAMILY 017 ALGEHRAS for all i E J; it immediately follows from this uniqueness property that, for J C J', U, = u,, 0 fJ,J; in other words, the U, form a direct system of homo - morphisms. Let u = lim u,: E + F; then by definition u, = u 0 fJ for every follows from these relations and the fact that theA(E,) generate the algebra E. indexing set and, for all i E I, let u, : E, -+ E; be an algebra homomorphism. Then there exists one and only one A-algebra homomorphism u: @ E, -+ @ E; such that, for all E, EI finite subset J of I and --+ in particular ui = 21 of, for all i E I; the uniqueness of u I COROLLARY . Let (EJtEI, (Ei)LeI be two families of A-algebras with the same i E I, the diagram ie1 LEI is comrnutative,S, and f denoting the canonical homomorphisms. It suffices to apply Proposition 8 to the homomorphismfi o ui. The homomorphism u defined in the Corollary to Proposition 8 is denoted by @ u,. If J is any subset of I, Proposition 8 can be applied to the family , of canonical homomorphismsf,: E, + 8 E, = E; a canonical homo- IEI morphism E, -+ E is derived which is also denoted If1 by f, and which, when J is jinite support H. It is immediate that, IS1 if J and J' are two finite subsets of I con- jinite, coincides with the homomorphism denoted thus above. Now let be an element ofn E, such that the family (xi - eJLe1 has taining H, then fJ( ('1) 1 EJ) = fJ'( f J') * The common value of the fJ((xi)LeJ) for the finite subsets J 3 H of I is de- noted by @ xi. PROPOSITION 9. Let (EL), be a family of A-algebras and for each i E I let B, be a basis of EL such that the unit element e, belongs to B,. Let B be the set of elements of the f.m 8 xi, where (xi) runs through the set of elements of rI B, such that the family (x, - ei) hasjnite support. Then B is a basis ofthe algebra @ E, and this basis con- tains the unit element e. is1 LEI I ,€I For every finite subset J of I, let BJ be the basis of E - @ E, the tensor product of the bases BL for i E J (11,s 3, no. 9). It follows immediately from the - LeJ 470 47 1 I' I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS definitions that B is the union of the fJ(BJ) when J runs through $(I) and that fJ',(BJ) c Br when J c J'; hence (Bj) is a direct system of subsets of the Ej and B = lim B,; the conclusion then follows from 11, 3 6, no. 2, Corollary to Proposition 5. The basis B is also called the tensor product of the bases Bt for i E I; when the conditions of Proposition 9 are fulfilled, the canonical homomorphisms fJ:EJ -+ E = @ E, are injective for every subset J of I, for if Bj is the basis of E, the tensor product of the B, for i E J, it is immediately verified that the restriction of fJ to BJ is injective and maps BJ onto a subset of B. 6. COMMUTATION LEMMAS Lemma 1. Let A be a commutative ring, E an A-algebra, (x,) of elements of E, (?,,) suppose that (13) Then 3 ie1 , ,, a jnite sequence ,, a jinite sequence of elements of A and y an element of E ; x,y = A,yx, for 1 < i < n. (14) (xlxz***~n)~ = (~1-**~n-1)(~n~) (XIXZ. . . x,)y = (AlhZ. . . h,)y(x,x,. . . xn)* The lemma being trivial for n = 1, we argue by induction on n 2 2. Now = (21 *. * xn-l)(AnYXn) = An((x1.. *xn-l)~)~n, which, by the induction hypothesis, is equal to A&. . . An-l)y(x, . . . xn-l)xn = (A,. . . h,-lh,) y(x1. . . x,- &), whence the lemma. Lemma 2. Let A be a commutative ring, E an A-algebra and (x,) twojinite sequences ofn elements of E; suppose that for 1 < j < i 6 n, (15) xtyj = hljyfxt with hj E A. Then , and (y,) ,, (16) (x1xz * * xn) (YlYz * * * Yn) = (T[J 'if) (xlul) (xzvz) * . (XnYn). The lemma being trivial for n = 1, we again argue by induction on n for n 2 2. By virtue of Lemma 1, (x1 . . * .,)(Yl . . * Y,) = Xl(X2. . . X,)Yl(YZ. . . Yn) = (Q All) (xlY1) (xz . * . xn) (Y2 * * * Yn) and it then suffices to apply the induction hypothesis to obtain (16). 5 4.6 COMMUTATION LEMMAS For every family A = (Atj) of elements of A, with 1 < j < i < n, and for every permutation a E G,, we write Observe that, when A = Z and A,, = -1 for every ordered pair (i,j) such that 1 < j < i < n, &,(A) is just the signature E, of the permutation 5 (I, $5, no. 7). Lemma 3. Let A be a commutative ring, E an A-algebra, (xi) ,, a jinite sequence of elements of E and suppose that, for every ordered pair (i,j) of integers such that l<j<i<n, xp, = htjx,xt with A,, E A. (18) Then, for every permutation a E B,, (19) ~o(l)xo(~) . * . Xa(n) = E~(A)x~xz * * . xn- The lemma is trivial for n = 1 and n = 2; we proceed by induction on n for n 2 3. If a(.) = n, relation (19) follows from the induction hypothesis. Suppose therefore that a(.) = k, k # n, and let T be the permutation of (1, n) defined by T(i) = i ~(i) = i + 1 T(n) = k. for i < k fork 6 i < n Let x = 7-l 0 a; the permutation x leaves n fixed; now 5 = T o x and therefore, writing y, = yn(,) = x,(~). If i # n and j # n, the relations x(i) > x(j) and a(i) > a(j) are equivalent (since T is a strictly increasing mapping of (1, n - 1) into (1, n). For i # n,j # n and o(i) > a(j), whence, by the induction hypothesis (using the fact that x(n) = n) : Yn(t)Yn(j) = xo(i)xa(j) = Aa(t), a(j)xu(j)xa(i) = ho(i), atj)Yntj)Yn(t) Now 472 473 I11 TENSOR ALGEBRAS, ESTEKIOR ALGEBRAS, SYMMETRIC ALGEBRAS 3 4.7 TENSOR PRODUCT RELATIVE TO COMMiJTATION FACTORS PROPOSITION 10. Let A be a commutative ring and (AJtoI a jinite family of com- mutative monoids written additively; for each i E I let El be a graded A-algebra of type A,. Finally, let (qj) be a system of commutation factors over the A, with values in A. Then there exist a graded A-algebra E of type A = n A, and for each i E I (i) If +, :A, -+ A is the canonical homomorphism, then hi is a graded homomorphism (11, 5 11, no. 2), in other wordr, h,(Er') c E*i("t), where (EF') and (E") denote the respective graduations on E, and E. (ii) If i # j and xi (resp. xj) is a homogeneous element of E, (resp. E,) of degree a, EA, (resp. Pj €Aj), then an algebra homomorphism hi : El --f E, with the following properties: ,€I (25) hi(Xi)hj(xj) = Eij(ai, Pj)hj(xj)hi(xJ- (iii) For every A-algebra F and every system of homomorphismsf,:E, --f F satis- fying the conditions (26) ff (xi)h(xj) = Eij(ai, Pj)h (xj)ff (xi) 9 where i,j, xi, xj, ui, Pj are as in (ii), then there exists one and only one algebra homo - morphism f: E -+ F such that ff = f 0 h, for all i E I. Moreover the underlying A- module of E is the tensor product @ E,. ,€I Consider the A-module E = @ E,; it is identified with the direct sum of the submodules E", where, for each a = (ai) EA, we write E" = @ EY'; the Ea therefore form a graduation of type A on the A-module E. We shall define on E a graded A-algebra structure of type A. For this let I be given a total order - ing; for every ordered pair of elements a = (a,), P = (Pi) of A, we must first define an A-bilinear mapping of Ea x ED into Ea+O, or alternatively an A-linear mapping map of E" @A EP into Ea+O. We shall define map by the condition ,€I ,€I Finally, (20) and (21) give with x,(~) . . . xO(,,) = a.xl . . . x, which completes the proof of Lemma 3. 7. TENSOR PRODUCT OF GRADED ALGEBRAS RELATIVE TO COMMUTA - TION FACTORS DEFINITION 6. Let (Ai) , I be a jinite family of commutative monoids written addi- tively. A system of commutation factors over the A, with values in a commutative ring A is a system of mappings &,,:A, x A, -+A, where i E I, j E I, i # j satisfying the following conditions: (22) (23) (24) + 4, PI) = 4aI, PMaL P,) Ed% P, + Pi) = El,(% Pr)Et,(Qt, P;) Ed% P,)&,i(P,, .I) = 1, for all u,, a; in A,, Pj, P; in Aj. If I is given a total ordering and the A, are groups, a system of commutation factors is defined over the A, by taking for every ordered pair (i, j) such that i < j an arbitrary Z-bilinear mapping of A, x Aj into the (multiplicative) Z-module A* of invertible elements of the ring A and then writing dPj, a,) = (Ed%, P,)) -' for i < j. Note that, since the qj(a,, Pi) are invertible, Eij(0, Pj) = ~ij(at, 0) = 1, by virtue of (22) and (23). Examples. (1) The trivial system of commutation factors consists of the Eij such that Etj(al, Pi) = 1 for all i, j, u, E A,, P, E A,. (2) If we take A = Z and A, = Z for all i E I, a system of commutation factors is obtained by taking &,,(a,, Pi) = (-l)"i% Note that this number depends only on the parities of at and P, and the Eij can therefore be con - sidered as commutation factors when certain of the A, are equal to 2/22 and the others to Z. These two examples are the most frequent cases encountered in applica - tions. 474 for xi E EY', yi E E:', where The right hand side of (27) obviously belongs to Ea+B and the mapping (xl,. . . , x,, yl,. . . , y,) H €(a, P) @ (xiyi) is A-multilinear in the product of the EP' and the EP' (1 < i < n). Then 1.51 it must be proved that the multiplication 475 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS thus defined on E is associative; now, if y = (y,) is a third element of A and 2, E Eil for 1 < i < n, then and it reduces to verifling the identity + P, YMQ, P) = 4a, P + Y)E(P, Y). + P, Y) = €(a, P)E(P, Y) 4% P + Y) = 4% P)E(Q, Y) But the latter follows immediately from the relations themselves immediate consequences of the definition (28) and (22) and (23). If, for all i E I, e, denotes the unit element of E,, we know that e, is homo- geneous of degree 0 (§ 3, no. l), hence e = @ ,€I e, is homogeneous of degree 0 and it follows from (27), (28) and the relations El,(% 0) = 4, P,) = 1 that e is unit element of E, which completes the definition on E of the desired graded A-algebra structure. Then take h,(xi) = xi @I (8 j#i e,; to verify that ht(x,xl) = h,(x,)h,(x;) for xi, xi in Ei, attention may be confined to the case where xt and x; are homogeneous and then this relation follows immediately from (27) and the relations q,(ai, 0) = q,(O, P,) = 1 ; the same relations and (24) prove also that the h, satisfy conditions (i) and (ii) of the statement and that where the right hand side is the product of the ordered sequence (hi(xi))iEI in E with the given total ordering on I (I, 3 1, no. 2) (it suffices to argue by induc - tion on the number of xi (assumed homogeneous) distinct from the e,). It remains to prove condition (iii) ; note that the mapping where the right hand side is the product of the ordered sequence (fi(xi))tEI with TENSOR PRODUCT RELATIVE TO COMMUTATION FACTORS 3 4.7 the given total ordering on I, is A-multilinear. Then there exists one and only one A-linear mapping f: E --f F such that Clearlyf(e) is the unit element of F and f 0 hi =fi; to verify that f is an algebra homomorphism, in other words thatf(x) f (y) =f(xy) for x, y in E, attention may be confined, by linearity, to the case where x = @xi and y = By,, xi (resp. y,) being homogeneous of degree a, (resp. Pi) for all i E I. The relation to be verified then reduces, by (27), to (€1 i€I But, taking account of relations (26), this is a consequence of Lemma 2 of no. 6. Clearly the algebra E and the canonical mappiiig +: 8 E, + E consti- 3, no. I), (€1 tute a solution of the universal mapping problem (Set Theory, IV, where C is the species of A-algebra structure and the a -mappings from n E, to an A -algebra, satisfying conditions (26). t For a fixed total ordering on I, the graded algebra E defined in the proof of Proposition 10 will be called a graded tensor E-product of type A of the family (EJiEI of graded algebras of type A, and will be denoted by '@ E, (if no con- fusion can arise over the ordering on I) ; similarly, the homomorphism f: E -+ F defined in the proof of Proposition 10 will be denoted by "5. The homo - morphisms hi are called canonical. We also write 'Gmn when I = (1, n) and all the E, are equal to the same algebra G. 161 Remarks. (1) We recover the tensor product of algebras defined in no. 1 (with moreover the graduation the tensor product of those of its factors) taking &,,(a,, P,) = 1 for all i,j, tLi and P,. (2) Suppose that all the A, are equal to Z and write ci,(tci, p,) = (- 1)'%; the tensor €-product '8 E, corresponding to this system of commutation type Z and denoted by '9 E, (or E g@IA F for two algebras, or gGQn instead of hG@n). factors is then called the ,€I skew tensor product of the graded algebras E, of 476 i 477 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS COROLLARY 1. In the notation of Proposition 10, suppose further that F is a graded A-algebra of type A and that each fi is a graded algebra homomorphism relative to +,:A, +A; then f = ‘FQ3fi is a graded algebra homomorphism. This follsws immediately from the definition off and the fact that by definition of the 4,. It is therefore seen that (E, +) is also a solution of another universal mapping problem, where this time C is the species of graded A-algebra structure of tybe A, the morphisms being graded algebra homomorphisms of type A and the a-mappings the mappings nf,, where, in addition to conditions (26), it is assumed that f( is a graded algebra homomorphism relative to COROLLARY 2. Let (El)tsI, (FJlEr be two jnite families of A-algebras, with E, and Ff graded oftype A, for all i E I. For each i E I, let g, 1 El + F, be a graded algebra homomorphism of type A,. Then, zf hi: E, -+ ‘8 E, and hi : F, -+ ‘8 IEI F, are the Canonical homomorphisms, there exists one and only one homomorphism of graded A- algebras of type A, g: ‘% El 3 ‘8 F, such that g 0 h, = h; 0 g, for all i E I. Also, if each g, is bijective, so is g. It suffices to apply Corollary 1 to f( = hl 0 g,, noting that conditions (26) then follow from relations (25) applied to the hi. The homomorphism defined in Corollary 2 is also denoted by ‘qgl (if no confusion can arise) ; if, for each i E I, G, is a third graded A-algebra of type A, and gl : F, -+ G, a graded algebra homomorphism, then as follows immediately from (30). In the case of a skew tensor product of graded algebras of type Z, we write ‘$3h instead of for homomorphisms fi: E, --+ F, of graded algebras of type Z; when I = {1,2}, this homomorphism is also denoted byfl “@f,; when I = (1, n) and all the E, (resp. F,) are equal and all the5 equal to the same homomorphismf, we write gf an. Remark. In the proof of Proposition 10, a total ordering on I was used to define an algebra structure on the tensor product ,% E, of the A-modules 3 4.7 TENSOR PR0I)UCT RELATIVE TO COMMUTATION FACTORS E,. If the ordering on I is changed, another multiplicative structure arises on @ E,, but the new algebra thus obtained is canonically isomorphic to the above, since both are solutions of the same universal mapping problem. For example, when I = { 1,2}, the canonical isomorphism of the algebra El &aA E, onto the algebra Ez E@A El maps ~1 C3 XZ to &Z.~(C(, P)xZ @ rl, where x1 is homogeneous of degree u and xZ homogeneous of degree p. Let J be a subset of I and, for each i E J, consider the canonical homomorph - ism hi: El -+ ‘@ E, = E. By virtue of relations (25) a canonical homomorphism h:E’ = ‘@ E, -+ E is derived canonically (by Proposition 10) from these homomorphisms, such that, for all i EJ, hI = h 0 h,, hl being the canonical homomorphism E, -+ E‘. Taking the total ordering on J induced by that chosen on I, we obtain ,€I tEI iSJ where the middle term is the product of the ordered sequence (ht(x,))iEJ and where, in the right hand term, X; = xi for i E J, xi = e, for i $ J. PROPOSITION 1 1. (“associativity” of the tensor E-product). In the notation of Proposition 10, let (JJA be a partition of I and write A; = n Al for all A E L. Let Ei be a graded tensor E-product of type A: of the family (for some total ordering chosen on JJ. On the other hand, for A, p in L and A # p, we write, for iEJh d = (‘1)IEJh> P: = (Pj),EJw, Then (EL,) is a system of commutation factors ouer the A; with values in A and there exists one and only one homomorphism ofgraded algebras oftype A, u : ”@ EL --+ ‘@ El, such that hsL is1 (33) for all (xi) ~n E,, provided that the total ordering is taken on I which induces on each Jh the chosen total ordering, and which is such that, for A < p in L, i E Jh and jEJ,,i <j. The fact that the EL,, form a system of commutation factors is trivial. Let h,, E, -+ EL, hi: EL --+ ”@ El, be the canonical homomorphisms (for h E L, i €Jh) and write hf = hi o hr,a; it will suffice by virtue of the uniqueness of 479 IEI hsL 478 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS the solution of a universal mapping problem, to show that "@ ?.EL E; and the h; satisfy the Conditions of Proposition 10. Now, for all h E L, let fL:E; -+ F be the unique algebra homomorphism such that f; o htPh = fi for all i E Jh. We show that, for h # P, = (at)tEJA, P; = (Pj)jEJ,,, (34) f;(x;)f;(x:) = &;,(a;, P&)f;(x&)fL(x;) for x; E EL (resp. x& E EL) homogeneous of degree u; (resp. (3:) ; it suffices, by linearity, to verify it when x: = 1EJA (8 x,, x: = IEJU (8 xj, xi (resp. x,) being homogeneous of degree (resp. P,) in E, (resp. E,) for i E Jh, j E J,. But this follows from formula (30) which defines the fi and Lemma 3 of no. 6, taking account of hypothesis (26) and definition (32). There is therefore one and only one algebra homomorphism f: "8 ?.EL E; -+ F such that f 0 h; = f for all A E L; whence f 0 hr = fi for all i E I and the uniqueness offis trivial. 8. TENSOR PRODUCT OF GRADED ALGEBRAS OF THE SAME TYPES Assuming the hypotheses of no. 7, Proposition 10 hold, suppose further that all the At are equal to the same commutative monoid A,; we can then consider on the tensor €-product '@ E, the total graduation of type A,, associated with the graduation of type A = A: on this algebra (11, 3 11, no. 1); we shall call '@ E,, with this graduation, a graded tensor E-product of type A, of the family (E,),,, of graded algebras of type A,. Always preserving the notation of Proposition 10 of no. 7, suppose that F is also a graded A-algebra of type A, and that the fi are homomorphisms of graded algebras of type A,. Then3 '8 E, --f F is also a homomorphism of graded algebras of type A,: for it follows from formula (30) (no. 7) that if xi is homogeneous and of degree a, E A,, %xi and fi(xi) are both homogeneous of degree 2 a, E A,. !€I !€I tPI It can therefore be said that '@ El, with the total graduation of type A,, constitutes, together with the canonical mapping +, a solution of a third universal mapping problem, where C is the species of graded A- algebra of tyke A,, the morphisms are homomorphisms of graded algebras of type A, and the a-mappings are the mappings vfi, where, in addition to conditions (26) (of no. 7), it is assumed that eachfi is a homomorphism of graded algebras of type Ao. fEI 480 TENSOR PRODUCT OF GRADED ALGEBRAS OF THE SAME TYPES 4 4.8 For every subset J of I, thecanonical homomorphism '8 E, --f '@ E, (no. 7) is, in fact, a homomorphism of graded algebras of type A,, as follows immedi - ately from the above. PROPOSITION 12. ("associativity" of the tensor &-product of graded algebras of the same types). With the notation of Proposition 10 of no. 7, suppose that all the A, are equal to the same monoid A,; let (JJAEL be a partition of I. With the notation of Proposition 11 of no. 7, suppose that the right hand side of formula (32) (no. 7) dependr only on the sums a: = ,zA u,, pIL = 2 P,, for every ordered pair (h, p) of distinct indices, all a: E A: and all P& E A;; let ~k,(u:, P:) denote the right hand side of (32). Then (E:,,) is a system of commutation factors over the family where A: = A, for all A E L. If E: is the graded tensor €-product of type A, of the family (Ei),€ JA, there exists one and only one isomorphism ofgraded algebras of type A,, w : '';% Ei -+ '8 E,, such that !EJ !€I jEJ,, (35) i.( ?.EL 8 (8 !EJ,%, x,)) = 8n.i !€I provided that total orderings are chosen on the Jh and on I as described in no. 7, Proposition 1 1. By the hypothesis, for y, 6 in Ao, &;(,,(y, 6) = qo,o(y, 6) for some i, E Jh and some j, E J,, as is seen by considering the elements a; = (a,), EJh and p$ = (P,)jEJ,, such that uto = y, a, = 0 for i # i,, = 6, (3, = 0 for3 #j,; it follows immediately that the E:, form a system of commutation factors. The rest of the proof is then analogous to that of Proposition 11 (no. 7) and is left to the reader. Note that the additional hypotheses of Proposition 12 are fulfilled when A, = Z and that (qj) is, either the trivial system of factors (Et,(ai, P,) = 1 for all i, j), or the system of factors defined by ci,(ai, P,) = (- l)"@~; in the latter case, the right hand side of formula (32) is equal to (-l)y. where Remarks. (1) Let I be an inznite indexing set and A, a commutative monoid; let denote the family such that At = A, for all i and suppose given for every ordered pair of distinct indices (i, j) of I a mapping €{,:Ai x A, + A satisfying conditions (22), (23) and (24) (no. 7); this will also be called a system of commutation factors over the family (A,). Consider a family (EJtE1 of graded A-algebras of type A,; for each finite subset J of I, let EJ denote a graded tensor E-product of type A, of the subfamily (EJtEJ (with an arbitrary choice of a total ordering on J). If J, J' are two finite subsets of I such that J c J', a canonical homomorphism of graded algebras of type A,, 48 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS hJ*J:EJ -+ E,*, has been defined above and the uniqueness properties of these homomorphisms show immediately that if J c J' c J" are three finite subsets of I, then hJ-, = h,",, 0 hj'j. Thus there is a direct system (Ej, h,,,) of graded algebras of type A, (3 3, no. 3), whose indexing set is the right directed set $(I) of finite subsets of I. The graded algebra of type A,, the direct limit of this direct system (3 3, no. 3), is called a graded tensor E-product of type A, of the family (EJie1; it is also denoted by '9 E,. When all the A, are equal to z and E,,(a,, P,) = (- 1).lDj, the tensor product '@ ,€I E, is also called the skew tensor product of the family (EJie1 and is denoted by '8 E,. We leave to the reader the task of formulating and proving the proposition which generalizes Proposition 10 of no. 7 to the case where I is infinite, as Proposition 8 of no. 5 generalizes Proposition 5 of no. 2 to the case where I is infinite. Note that the underlying A-module of '8 E, is the same as that underlying the (non-graded) tensor product of the family (E,) , I of non-graded algebras defined in no. 5. (2) Let E be a graded A-algebra of type A, (where A, is a commutative monoid) and p:A -+ B a ring homomorphism; the graduation on p*(E) (11, 3 11, no. 5) is identical with the graduation on the graded tensor product B E, where B has the trivial graduation. 9. ANTICOMMUTATIVE ALGEBRAS AND ALTERNATING ALGEBRAS DEFINITION 7. A graded A-algebra E of type Z is called anticommutative if for all non-zero homogeneous elements x, y of E xy = (- I)deK(X)dee(Y) YX. (36) The algebra E is called alternating if it is anticommutative and also x2 = 0 for every homogeneous element x E E of odd degree. Remarks. (1) Let E+ be the graded subalgebra of E the direct sum of the E,, (n E Z); it follows from Definition 7 that if E is anticommutative, E+ is a subalgebra contained in the centre of E (and hence commutative). (2) Suppose that 2 is not a divisor of 0 in E; then if E is anticommutative E is alternating, since for x E E homogeneous and of odd degree, x2 = -x2 by (36), whence 2x2 = 0 and x2 = 0 by virtue of the hypothesis. (3) We shall study in detail in 3 7 important examples of alternating alge- bras. hma 4. Let E be a graded algebra of type Z and S a set of homogeneous elements #O; the set F of elmnts of E all of whose homogeneous components x # 0 satish relation (36) for ally E S is a graded subalgebra of E. 482 ANTICOMMUTATIVE ALGEBRAS AND ALTERNATING ALGEBRAS 5 4.9 It suffices to note that: (1) if x', x" are two homogeneous elements of the same degree p, y a homogeneous element of degree q and x'y = (- l)pqyx', x"y = (- l)pqyxr', then also (x' + x") y = (- l)P@y(x' + x") ; (2) if x', X" are two homogeneous elements of respective degrees p', p", y a homogeneous element of degree q and x'y = (- l)p'qyx', x"y = (- I)p"qyx", then (x'x") y = (- l)(p'+p")@y(x'x~')~ PROPOSITION 13. Let E be a graded A-algebra of type Z and S a generating system of the algebra E consisting of homogeneous elements #O; for E to be anticommutative (resp. alternating), it is necessary and sujicient that (36) hold for all x E S and y E S (resp. that this condition hold and further that x2 = 0 for all x homogeneous of odd degree belonging to S) . We consider first the case of anticommutative algebras. By Lemma 4, the subalgebra F consisting of the elements all of whose homogeneous com - ponents x # 0 satisfy (36) for all y E S, contains all the elements of S and hence F = E. If now F' is similarly the subalgebra of E consisting of the ele- ments all of whose homogeneous components x # 0 satisfy (36) for every homogeneous element y # 0, it follows from the above that F' contains all the elements of S and hence F' = E, which completes the proof of the proposi - tion in this case. To prove the proposition in the case of alternating algebras, it can be as- sumed that E is already anticommutative; it is then immediate that every homogeneous element of odd degree in E is of the form 7 z,x,, where z, E E+ and xi E S is of odd degree (using the fact that E+ is contained in the centre of E); it follows that (T z,x')' = 7 ztx; + z,z,(x,x, + x,x,) = 0 since X; = 0 by hypothesis and x,xj + xjxi = 0 by (36). PROPOSITION 14. Let E and F be two graded A-algebras of type Z, both anticom- mutative (resp. alternating). Then the skew tensor product E g@IA F (no. 7) is an anticommutative (resp. alternating) algebra. A generating system of E g@IA F consists of the x @ y, where x (resp. y) is a homogeneous element #O of E (resp. F). Consider two such elements x my, x' @ y', with deg(x) = p, deg(y) = q, deg(x') = p', deg(y') = q', so that x @I y is of degreep + q and x' @I y' of degreep' + q'. Then by definition (no. 7, formula (27)) and by virtue of (36), (. 63 y) (x' @ Y') = ( - 1)4P'(XX') @ (yy') (x' @Y')(X BY) = (-l)""(.'X) @ (y'y) (4 0 (YY') = (- 1)PP'+PP'+S4' and the criterion of Proposition 13 shows that E gBA F is anticommutative since pq' + pp' + qq' - qp' = (p + q) (p' + q') (mod. 2). If further E and F 483 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMhtETRIC ALGEBRAS are alternating and p + q is odd, one of the numbers p, q is necessarily odd, hence (x @ Y)~ = & (x2) @ (y2) = 0 and Proposition 13 shows that E g@A F is alternating. COROLLARY . Let E be an anticommutative (resp. alternating) graded A-algebra of type Z. Then for every ring homomorphism p:A+ B, the graded B-algebra p*(E) (no. 8, Remark 2) is anticommutative (resp. alternating). The ring B with the trivial graduation can be considered as an alternating A-algebra and p* (E) = E ggA B, hence Proposition 14 can be applied. Remark. Let E be an anticommutative graded A-algebra of type Z. Then the A-linear mapping of E @A E into E defined by multiplication of E (3 1, no. 3) is a homomorphism of the graded A-algebra E gBA E into E, for in the notation of Proposition 14, in the algebra E, (xy) (x'y') = (- l)QP'(XX') (yy'). 5 5. TENSOR ALGEBRA, TENSORS 1. DEFINITION OF THE TENSOR ALGEBRA OF A MODULE Let A be a commutative ring and M an A-module. For every integer n 2 0, the A-module the tensor product of n modules equal to M (also called the n-th tensor power of M) is denoted by @" M, or MBn, or Tn(M), or Tl(M), or Tens"(M); then T1(M) = M; also we write To(M) = A. The A-module the direct sum 0 Tn(M) is denoted by T(M) or Tens(M). We shall define a graded A-algebra structure of type N on T(M), by defining for every ordered pair of integers p 2 0, q 2 0, an A-linear mapping m,,:TP(M) @A Tq(M) 4 Tp+q(M) (8 3, no. 1, Remark). For p > 0 and q > 0, mpq is the associativity isomorphism (11, 3 3, no. 9) and, when p = 0 (resp. q = 0), mo, is the canonical isomor - phism of A @A Tq(M) onto Tq(M) (resp. m,, is the canonical isomorphism of TP(M) @A A onto TP(M) (11, 3 3, no. 4, Proposition 4). Then, for x, E M, UEA, nZO (x1@ *.* @X,).(.p+l @ **. @xp+q) EXl@ ... @xp @XP+l@ **. @%+q E.(X1@ ... @xp) =u(x1@ **a @x,). (1) { It is immediate that the multiplication thus defined on T(M) is associatiue and admits as unit element the unit element 1 of A = To(M). 484 FUNCTORIAL PROPERTIES OF THE TENSOR ALGEBRA 5 5.2 DEFINITION 1. For every module M over a commutative ring A, the tensor algebra of M, denoted by T(M), or Tens(M), or TA(M), is the algebra @ Tn(M) with the no. 12) (also denoted by +M) is called the canonical injection ofM into T(M). PROPOSITION 1. Let E be a (unital) A-algebra and f :M -+ E an A-linear mapping. There exists one and only one A-algebra homomorphism g:T(M) -+ E such that multiplication d+ed in (1). The canonical injection +:T1(M) nZo -+ T(M) (11, 3 1, f =go+* In other words, (T(M), +) is a solution of the universal mapping problem (Set Theory, IV, 9 3, no. l), where C is the species of A-algebra structure, the a-mappings being the A-linear mappings from the module M to an A-algebra. Observe that here there is no question of a graduation on T(M). of n elements of M, by definition of the product in T(M), x1 @ x2 @I . . . @ x, = +(x1)+(x2) . . . +(xn); then neces - sarily g(xl @I x2 @ . . . @ x,) = f(xl) f (x2) . . . f (x,) for n 2 1 and g(a) = ae (if e is the unit element of E) for a E A, which proves the uniqueness of g. Conversely, note that, for all n > 0, the mapping of Mn into E is A-multilinear; hence there corresponds to it an A-linear mappingg,:T"(M) -+ E such that Also we define the mapping go: To(M) --f E as equal to yjE (4 1, no. 3), in other words go(a) = ae for a E A. Let g be the unique A-linear mapping of T(M) into E whose restriction to Tn(M) is g, (n 2 0); it is immediate that g 0 + = gl = f and it remains to verify that g is an A-algebra homo - morphism. By construction g(1) = e and it suffices by linearity to show that g(uv) = g(u)g(v) for u E Tp(M) and v E Tq (M) (p > 0, q > 0) ; now it follows from formulae (1) and (2) that this relation is true when u =xl @ x2 @I . . @ x, and v E x, +1 @ . * . @ x~+~ (where the xi belong to E). It is therefore true for u E TP(M) and v E T4(M) by linearity. Remark. Suppose that E is a graded A-algebra of type Z, with graduation (En), and suppose also that For every finite family (XI, . . * 3 Xn) -f (x1)f (~2) . * * f (Xn) g(x1 @ ~2 @ * . . @ xn) = f (X1)f (~2) . . * f (Xn) * (2) f(M) = El. (3) Then it follows from (2) that g(Tp(M)) c E, for all p 2 0 and hence g is a graded algebra homomorphism. 2. FUNCTORIAL PROPERTIES OF THE TENSOR ALGEBRA PROPOSITION 2. Let A be a commutative ring, M and N two A-modules and u:M-+N 485 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS an A-linear mapping. There exists one and only one A-algebra homomorphism u':T(M) -+ T(N) such that the diagram ..jl 1.. T(M) 7 T(N) is commutative. Further, u' is a graded algebru homomorphism. to the algebra T(N) and the linear mapping +N o u:M -+ T(N) ; as The existence and uniqueness of u' follow from no. 1, Proposition 1, applied u(M) c T'(N) = N, the fact that u' is a graded algebra homomorphism follows from the Remark of no. 1. The homomorphism u' of Proposition 2 will henceforth be denoted by T(u). If P is an A-module and v:N --f P an A-linear mapping, then (4) T(v 0 U) = T(v) o T(u) for T(u) 0 T(u) is an algebra homomorphism rendering commutative the dia- gram M UDU P T(u) is sometimes called the canonical extension of u to T(M) (which contains M = T1(M)). Note that the restriction T"(u) :T"(hI) + T"(N) is just the linear mapping u@" = u @ u 63 . . - @ u (n times), for T"(U)(X, C3 * * * C3 xn) = u(x1) 63 . * . C3 U(Xn) since T(u) is an algebra homomorphism and T1(u) = u; the restriction To(u) to A is the identity mapping. T"(u) is called the n-th tensor power of U. PROPOSITION 3. If u: M 4 N is a surjective A-linear mapping, the homomorphism T(u) :T(M) -+ T(N) is surjective and its kernel is the two -sided ideal of T(M) generated by the kernel P c M c T(M) ofu. To(u) :To(M) -+ To(N) is bijective and for every integer n > 0, T"(u) :T"(M) -+ T"(N) 486 FUNCTORIAL PROPERTIES OF THE TENSOR ALGEBRA S 5.2 is surjective, as is seen by induction on n using 11, 3 3, no. 6, Proposition 6; the latter proposition also shows, by induction on n, that the kernel 3, of T"(u) is the submodule of T"(M) generated by the products where at least one of the xi belongs to P. This shows that the kernel 3 = of T(u) is the two-sided ideal generated by P in T(M). 3, If u:M -+ N is an injective linear mapping, it is not always true that T(u) is an injective mapping (Exercise 1). However, this is true when u is an injec- tion such that u(M) is a direct factor of N, for then there exists a linear mapping u:N -+ M such that v 0 u is the identity mapping on M and therefore T(u o U) = T(v) o T(u) is the identity mapping on T(M), hence T(u) is injective and its image (iso- morphic to T(M)) is a direct factor of T(N) (11, 4 1, no. 9, Proposition 15). More precisely: PROPOSITION 4. Let N and P be two submodules of an A-module M such that their sum N + P is a direct factor in M and their intersection N n P is a direct factor in N and in P. Then the homomorphisms T(N) -+ T(M), T(P) -+ T(M) and T(N n P) -+ T(M), canonical extensions of the canonical injections, are injective; ;f T(N), T(P) and T(N n P) are identijed with subalgebras of T(M) by means of these homomorphisms, then (5) T(N n P) = T(N) n T(P). By hypothesis, there exist submodules N' c N and P' c P such that N =N'@(NnP),P = P'@(NnP);then N -i- P = N' @P' @ (N n P) and there exists by hypothesis a submodule M' of M such that M=M'@(N+P) =M'@N'@P'@(NnP) = M'@P'@N = M@N'@P. In particular, N + P, N, P and N n P are direct factors in M, which im- plies, as has been seen above, that the canonical homomorphisms T(N + P) -+T(M), T(P) + T(M), T(N) +T(M), T(N n P) + T(M) are injective. The three algebras T(N), T(P) and T(N n P) are thus identi - fied with subalgebras of T(N + P) and the latter with a subalgebra of T(M) ; 487 Ill TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS writing Q = N n P, it remains to show that, if T(Q), T(N' @ Q) and T(P' @ Q) are identified with subalgebras of T(N' @ P' @ Q), then (6) T(N' 0 Q) n T(P' @ Q) = T(Q). N'@Q ---f N'OP'OQ Now, consider the commutative diagram 1 .1 Q- P'OQ where the horizontal arrows are the canonical injections and the vertical arrows the canonical projections. We derive a commutative diagram T(N' @ Q) T(N' @ P' @ Q) EXTENSION OF THE RING OF SCALARS 5 5.3 (7) T(Q) 7 TO'' 0 Q) where r and s are surjective homomorphisms (Proposition 3) and u and v injective homomorphisms. Hence, to prove (6), note that the right hand side is obviously contained in the left; it therefore suffices to verify that if x E T(N' 0 Q) n T(P' @ Q), then x E T(Q). Now the definition of the homomorphism s shows that its restriction to T(P' 0 Q) (identified with a subalgebra of T(N' @ P' @ Q) is the identity mapping; the hypothesis on x therefore implies that s(u(x)) = x. But then also v(r(x)) = x, in other words x belongs to the image of T(Q) in T(P' @ Q), which was to be proved. Remark. Note in particular that the hypotheses of Proposition 4 always hold for arbitrary submodules N, P of M when A is ajeld (11, fj 7, no. 3, Proposi- tion 4). Moreover, if N c P and N # P, then T"(N) # T"(P) for all n 2 1, since if R is a complement of P in N, then T"(P) n T"(R) = {0} by (4) and COROLLARY . Let K be a commutativejeld and M a vector space over K. For every element z E T(M), there exists a smallest vector space N of M such that z E T(N) and N is ofJinite rank over K. It is understood in this statement that for every vector subspace P of M, T(P) is canonically identified with a subalgebra of T(M). Let ZET(M); z can be expressed as a linear combination of elements each of which is a finite product of elements of M = T1(M) ; all the elements of M which occur in these products generate a vector subspace Q of finite rank and z E T(Q). 488 TYR) # (0). I I Let 5 be the (non-empty) set of vector subspaces P of finite rank such that z E T(P). Every decreasing sequence of elements of 5 is stationary, since they are vector spaces of finite rank. Hence 5 has a minimal element N (Set 7heory, 111, $6, no. 5). It remains to verify that every PE~ contains N; now, t E T(P) n T(N) = T(P n N) (Proposition 4) ; in view of the definition of N, this implies N n P = N, that is P 3 N. The subspace N of M is said to be associated with z. 3. EXTENSION OF THE RING OF SCALARS Let A, A' be two commutative rings and p:A --f A' a ring homomorphism. Let M be an A-module, M' an A'-module and u: M --f M' an A-homomorph- ism; as the canonical injection &,l. : M' --f TA,(M') is also an A-homomorphism (by restriction of scalars), so is the composition M & M' % TA,(M'). An A-algebra homomorphism TA(M) --f p*(T,,(M')) is derived (no. Z), also de- noted by T(u) : TA(M) -+ TK(M'), which is the unique A-homomorphism ren - dering commutative the diagram M 2 M' (8) ..I 1*w TA(') TA(M') If o:A'+A" is a commutative ring homomorphism, M" an A"-module and v : M' -+ M" an A'-homomorphism, the above uniqueness property shows that (9) T(u o U) = T(v) o T(u). PROPOSITION 5. Let A, B be two commutative rings, p:A --f B a rin,q homomorphism and M an A-module. The canonical extension +:T,(B @AM) +B @A TA(M) of the B-linear mapping 1, @ C+~,:B @A M -+ B @A TA(M) is an isomorphism of paded B-algebras. Consider the two A -algebra homomorphisms : the canonical injection j :B = To(B @A M) + T(B @A M) and the homomorphism h = T(i) :T(M) -+ T(B @A M) derived (cf. formula (8)) from the canonical A -linear mapping i:M-+B @*M. 489 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS TO(B BAM) is contained in the centre of T(B @JAM), Proposition 5 of 5 4, no. 2 can be applied and an A-algebra homomorphism +':B €3* T(M) -+ T(B €3A M) is obtained such that, for @ E B, xi EM for 1 < i < n, +'(P €3 (XI €3 x2 €3 . * * 63 x,)) = @((I 63 XI) €3 (1 €3 x2) €3 . . . €3 (1 @ X")), which shows immediately that 3' is also a B-algebra homomorphism. It suffices to prove that 3 0 +' and +' 0 + are the identity mappings on B @A T(M) and T(B @AM) respectively. Now, these two algebras are generated by B @A M and clearly $0 4' and +' o + coincide with the identity mapping on B @A M, whence the conclusion. 4. DIRECT LIMIT OF TENSOR ALGEBRAS Let (A,, be a directed direct system of commutative rings and (Ma, fa,) a direct system ofAa-modules (11,s 6, no. 2) ; let A = lirn A, and M = lim --f Ma, which is an A-module. For a < p an A,-algebra homomorphism (no. 3, formula (8)) f;, = T( f,,) : TA,(M,) -+ TA,(M,) is derived canonically from the A,-homomorphism fpa: Ma + M, and it follows from (9) (no. 3) that (TA,(M,),fi,) is a direct system ofA,-algebras. On the other hand let fa: Ma -+ M be the canonical A,-homomorphism ; an A,-algebra homomorphism fi: TA,(M,) -+ TA(M) is derived (no. 3, formula (8)) and it also follows from (9) (no. 3) that the fa' constitute a direct system of A,-homomorphisms. PROPOSITION 6. The A-homomorphism f' = lirn f& : lim TAa(M,) -+ TA(M) is a graded algebra isomorphism. --f 3- To simplify we write E = TA(M) and E' = lirn TAa(Ma) and let + ga:T,(Ma) -+ E' be the canonical A,-homomorphism. Clearly the composite A,-linear map- pings Ma -----f TA,(M,) -% E' form a direct system and there is therefore one and only one A-linear mapping u = lim(g, 0 +Ma) : M -+ E' such that $Ma + u of= = ga 0 +M~ for all a. This mapping itself factorizes uniquely (no. 1, Proposition 1) into M 2 E --f E', where h is an A-algebra homomorphism. It will suffice to prove that h 0 f' = 1,' and f '0 h = lE. To this end note that, for every index a, (no. 3, formula (8)) h Of: +Ma = h 4M Ofa = Of, = ga +Ma TENSOR ALGEBRA OF A DIRECT SUM, A FREE MODULE AND A GRADED MODULE 4 5.5 whence, by the uniqueness assertion of no. 1, Proposition 1, h of& = ga for all a; it follows that (h of ') o g, = g, for all a and hence h 0 f' = lEt by definition of a direct limit. On the other hand, by virtue of no. 3, formula (8), f' Ofa =f' 'gcz +Mu =f& +Ma = +M 'fa, whence again f' o u = C$I~ by definition of a direct limit; we conclude that f' o h o +M = +M and the uniqueness property of no. 1, Proposition 1 gives f' oh = 1,. Proposition 6 can also be shown by observing that, for every integer n 2 1, there is a canonical A-module isomorphism lim Tl,(M,) + Tl(M), 7. It is immedi - ately verified that these isomorphisms are the restrictions off'. as follows by induction on n from 11, $6, no. 3, Proposition -. 5. TENSOR ALGEBRA OF A DIRECT SUM. TENSOR ALGEBRA OF A FREE MODULE. TENSOR ALGEBRA OF A GRADED MODULE Let A be a commutative ring and M = MA the direct sum of a family of A-modules. It follows from 11, 0 3, no. 7, Proposition 7, by induction on n, that T"(M) is the direct sum of the submodules the images of the canonical injections MAl €3 MA, €3 *. * €3 MAn -+ T"(M) = M@" relative to all the sequences (A,) E L". Identifying M,, €3 MA, @I - - @ MAn with this image, it is seen that T(M) is the direct sum ofall the modules @ @ ' * ' @ MAn where n runs through N and, for each n, (A,) runs through L". THEOREM 1. Let A be a commutative ring, M afree A-module and (eA)AsL a basis of M. Then the elements es = e,, @ eAa €3 . . . €3 eAn, where s = (Al, . . . , A,) wu through the set of allfinite sequences of elements ofL and en is used to denote the unit element ofT(M), form a basis ofthe A-module T(M). We first deduce the following consequence: The elements of this basis are obviously homogeneous and the multiplica - tion table is given by eset = e*t where st denotes the sequence of elements of L obtained by juxtaposing the sequences s and t (I, 5 7, no. 2). 490 49 1 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS It is seen that the basis (es) of T(M), with the multiplicative law (lo), is canonically isomorphic to the free monoid of the set L (I, fj 7, no. 2), the iso- morphism being obtained by mapping each word s of this monoid to the element e,. It follows (4 2, no. 7) that the tensor algebra T(M) of a free module M, with a basis of indexing set L, is canonically isomorphic to the free associative algebra of L over A. In particular (3 2, no. 7, Proposition 7), for every mapping f: L --f E from L to an A-algebra E, there exists one and only one A-algebra homomorphismf: T(M) -+ E such thatf(e,) = f (A). Remark. The above results can equally be obtained as a consequence of the universal properties of the free associative algebra of the tensor algebra, using 11, 5 3, no. 7, Corollary 2 to Proposition 7. COROLLARY . If M is a projective A-module, T(M) is a projective A-module. M is a direct factor of a free A-module N (11, 5 2, no. 2, Proposition 4) and hence T(M) is a direct factor of T(N) (no. 2) ; as T(N) is free (Theorem l), this shows that T(M) is projective (11, 3 2, no. 2). PROPOSITION 7. Let A be a commutative monoid, M a graded A-module of type A and (M,),,A its graduation. For every ordered pair (a, n) EA x N, let Ta*"(M) be the (direct) sum of the submodules Ma, @ M,, @ . . . @I Man of T"(M) such that2 a, = a; then (Ta~n(M))(,,n,sAxN is the only graduation of type A x N compatible with the algebra structure on T(M) and inducing on M = T'(M) the given gradation. It has been seen at the beginning of this no. that T(M) is the direct sum of the Ta*"(M) and the fact that this is a graduation compatible with the algebra structure follows immediately from the definitions. If (T'". ") is another graduation of type A x N on T(M) compatible with the algebra structure and such that Ta*l(M) = T'a.l for a E A, it follows immediately from the definitions that, for all n > 1 and all a E A, Ta*"(M) c T'a*"; but since T(M) is also the direct sum of the Tasn(M), this implies that T'a*n = Ta*"(M) (11, 3 1, no. 8, Remark). 6. TENSORS AND TENSOR NOTATION Let A be a commutative ring, M an A-module, M* the dual of M (11, 5 2, no. 3) and I and J two disjointjnite sets; the A-module egJ Et, where Et = M if i E I, E, = M* if i E J, is denoted by T',(M) ; the elements of T:(M) are called tensors of type (I, J) over M. They are called contravariant if J = @, covariant if I = 0 and mixed otherwise. Let 1', I" be two subsets of I and J', J" two subsets of J such that I' U I" = I, 3 5.6 I' n I" = 0, J' u J" = J, J' n J" = 0 ; then there is a canonical associa- tivity isomorphism (11, 3 3, no. 9) TENSORS AND TENSOR NOTATION T:(M) -+ T::(M) @A T:(M). (11) Considering the tensor algebra T(M @ M*), it follows from no. 5 that Tn(M @ M*) is canonically identified with the direct sum of the T:(M) where I runs through the set of subsets of the interval (1, n) of N and J is the complement of I in (1, n). When I = (1,p) and J = (p + 1,p + q) with integers p > 0, q > 0 (where we replace I (resp. J) by ia whenp = 0 (resp. q = 0)), the A-module T:(M) is also denoted by T:(M) ; the A-modules T:(M) and T:(M) are there - fore by definition the A-modules T"(M) and T"(M*) respectively. When I and J are arbitrary finite sets of cardinals p = Card(1) and q = Card(J), we give each of them a total ordering; then there exists an increasing bijec- tion of I (resp. J) onto (1, p) (resp. (p + 1,p + q) and these bijections there- fore define an isomorphism T:(M) -+ T,P(M). When M is a Jinitely generated projective A-module, it follows from 11, 3 2, no. 7, Corollary 4 to Proposition 13 and 11, 3 4, no. 4, Corollary 1 to Proposi- tion 4 that there is a canonical isomorphism (T:(M))* -+ T,J(M). Suppose now that M is ajnilely generated free A-module and let (eJAeL be a basis of M (L therefore being ajnite set). The basis of M* dual to (eA) (11, 5 2, no. 6) is denoted by The bases (e,) and (e*) of M and M* respec- tively define (no. 5) a basis of T:(M), which we give explicitly as follows: given two mappingsf: I -+ L and g: J -+ L, let ey be the clement 8 x, of T:(M) defined by IEIuJ xi = e ,({) if i E I, xi = eg") if i E J. When (5 g) runs through the set of ordered pairs of mappings f: I -+ L and g:J-+L, the e; form a basis of the A-module T:(M), said to be associated with the given basis (eJ of M. For z E T',(M), we can therefore write where the a; are the coordinate forms relative to the basis (4) ; by an abuse of language, the a;(.) are called the coordinates of the tensor z with respect to the basis (eA) of the module M. The a; constitute the dual basis of (ei), in other words they are identified with the elements of the basis of T:(M), associated with (eJ. When I and J are complementary subsets of an interval (1, n) of N, a; (or a:(.)) is denoted as follows: each elementf(i) for i E I is 493 492 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS written as an upper index in the i-th place with a dot in the i-th place for i E J; similarly, g(i) for i E J is written as a lower index in the i -th place with a dot in the i -th place for i E I. For example, for I = {1,4}, J = {2,3}, we write a?;,’! iff(1) = A,f(4) = p, g(2) = V, g(3) = p. Let (iJAsL be another basis of M and P the matrix of passing from (eh) to (ih) (11, 3 10, no. 8). Then the matrix of passing from (8) to (2) (dual basis of (iJ) is the contragredient ‘P-l of P (11, 3 10, no. 8, Proposition 5). It follows (11, 3 10, no. 10) that the matrix of passing from the basis (ey) of Ti(M) to the basis (2;) (wheref (resp. g) runs through the set of mappings of I into L (resp. of J into L)) is the matrix @ Q,, where Q, = P if i E I, Qt = tP-l if i E J. (13) tsIuJ The transpose of this matrix is therefore identified with @ R,, where R, = tP-l if i E I, R, = P if i E J (14) 1EIuJ Suppose now that M is an arbitrary module. Let i E I, j E J and write I‘ = I - {i}, J’ = J - {j}; we shall define a canonical A -linear mapping c;: T:(M) -+ T::(M), called contraction ofthe index i and the indexj. For this, note that the mapping of MI x (M*)J, which associates with every family where x, EM if i E I and xi E M* if i E J, the element €3 (15) ‘j) k€(IuJ)-{t.l) ‘lC of Ty,(M), is A-multilinear ; ci is the corresponding A -linear mapping. Suppose now that M is free and finitely generated and let (e,JAGL be a basis of M. Given two mappingsf: I --f L, g: J -+ L, letf, denote the restriction of f to I’ = I - {i} and gj the restriction of g to J‘ = J - {j}; then by virtue of (12) The expression for the coordinates of 4 (2) as a function of those of t is ob- tained; for every mappingf’ (resp. 9’) of I’ into L (resp. of J’ into L) and every A E L, let (f’, 1) (resp. (g‘, A)) denote the mapping of I into L (resp. of J into L) whose restriction to I’ (resp. J’) isf’ (resp. 9’) and which takes the value A at the element i (resp. j). Then, if the coordinate forms relative to the basis (e$) of T:,(M) are denoted by @;I, (17) 5 5.6 TENSORS AND TENSOR NOTATION Examples oftensors. (1) Let M be afinitely generatedprojective A-module. We know (11, 3 4, no. 2, Corollary to Proposition 2), that there is a canonical A-module isomorphism 8,: M* @A M -+ End,(M) such that 0M(x* @ x) (for x E M, x* E M*) is the endomorphism Y (Y, x*>x. Hence, by means of TI?;(M) (isomorphic to T:(M)) can be identified with the A-module EndA(M). Suppose that M is a free module and let (eh)AEL be a basis of M; then the coordinates of a tensor z E M* @ M relative to the basis (e’ @ eh) of this module are denoted by <;h. As OM(e’ @ eA) is the endo- morphism y ++ (y, eU)eA, the endomorphism u = eM(z) = OM( C <;hew 8 eh) maps y to 2 <;h(y, e’)eh; writing y = eA, we obtain the relation h. u A. u in other words, the matrix ofthe linear rnapbing u = &(t) is that whose element in the row of index p and column of index A is Ci!’. The definition of the trace of u (11, 3 4, no. 3) shows immediately that Tr(oM(2)) = C?(t). Therefore the element to = zL eA @ eh (whose coordinates <;h are zero for h # p and equal to 1 for = p), which is such that eM(tO) = lM, is the image of the element 1 E A = TE(M) under the mapping the transpose of the contraction c:: T:?;(M) +A. (2) Suppose always that M is afinitely generatedprojective A-module; there is a canonical A -module isomorphism p: M* @AM* + (M @AM)* (11, 3 4; no. 4, Corollary 1 to Proposition 2) and a canonical isomorphism (11, 5 4, no. 2, Corollary to Proposition 2) ; also HomA(M @A M, M) is C=wn&ally isomorphic to the A-module P2(M, M; M) of A-bilinear mappings OfM x M into M (11,s 3, no. 9). Composing these isomorphisms, a canonical bmorphism is obtained xM: T:;:2)(M) = M* @ M* @3 M + P2(M, M; M) 8: (M @A M)* @A M + HomA(M @A M, M) 495 494 5 6.1 DEFINITION OF THE SYMMETRIC ALGEBRA OF A MODULE 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS I (23) such that, for x*, y* in M*, z E M, xM(x* @ y* @ z) is the bilinear mapping (u, v) ++ (u, x*)(v, y*)z. The above formulae are such that the summation is Over indices which appear once as a lower index and once as an upper index. Certain authors allow them- selves because of this circumstance to suppress the summation signs. Hence, by means of xM, Ti?!+(M) (isomorphic to Tl(M)) can be identified with the A-module S,(M, M; M). Suppose that M is a free A-module and let (eJhEL be a basis of M; then the coordinates of a tensor z E M* @ M* @ M relative to the basis (eh @ ev @ e,) of this module are denoted by ti;'. The bi- linear mapping xM(t) maps the ordered pair (e,, e,) to VEL C ti;Ye, 6 6. SYMMETRIC ALGEBRAS 1. DEFI"ITI0N OF THE SYMMETRIC ALGEBRA OF A MODULE and therefore the are just the constants of structure of the (in general non- associative) algebra defined on M by the bilinear mapping xM(z), with respect to the basis (en) (5 1, no. 7). Remark 2. Let (e),)A&, (EA)hEL be two bases of M and P the matrix of passage from (eh) to (5,). On account of what was seen in Example 1, the element of P appearing in the row of index A and the column of index p is denoted by a; and the element of the contragredient tP-l appearing in the row of index A and the column of index p is denoted by p!. Then (in the notation introduced above) (19) for all mappings f ': I + L and g': J -+ L. The coordinates ti of a tensor z E T:(M) with respect to the basis (e,) can therefore be expressed in terms of the coordinates c;: of z with respect to the basis (4) using the formulae The matrix P - of passage from the basis (Q to the basis (eh) is the trans - pose of tP-l, so that pt is the element which appears in the column of index A and the row of index p of P-l. The calculation of 4 in terms of the tys' and that of the are therefore made by replacing a: by p! and p! by E^, in the above calculations and exchanging the roles off and f and those of g and g'. Then in terms of the DEFINITION 1. Let A be a commutative ring and M an A-module. The symmetric algebra ofM, denoted by S(M), or Sym(M), or S,(M), is the quotient algebra over A of the tensor algebra T(M) by the two -sided ideal 3' (also denoted by fjh) generated by the elements xy - yx = x @ y - y @ x of T(M), where x and y run through M. Since the ideal 3' is generated by homogeneous elements of degree 2, it is a graded ideal (11, 5 11, no. 3, Proposition 2); we write 3; = 3' n T"(M); the algebra S(M) is then graded by the graduation (called canonical) consisting of the S"(M) = T"(M)/3;. Now Sl, = 3; = (0) and hence So(M) is canonically identified with A and S1(M) with T'(M) = M; we shall always make these identifications in what follows and denote by +' or +h the canonical injection M -+ S(M). PROPOSITION 1. The algebra S(M) is commutative. By definition +'(x)+'(y) = +'(y)+'(x) for x, y in M and, as the elements +I(%), where x runs through M, generate S(M), the conclusion follows from 3 1, no. 7. I PROPOSITION 2. Let E be an A-algebra and f: M --f E an A-linear mapping such that The exists one and only one A-algebra homomorphism g: S(M) + E such that f =go+'. In other words, (S(M), 4') is a solution of the universal mapping problem (Set Theory, IV, 8 3, no. I), where C is the species of A-algebra structure, the a-mappings being the linear mappings of the A -module M to an A-algebra satisfying (1). The uniqueness ofg follows from the fact that +'(M) = M generates S(M). To prove the existence of g, note that by virtue of 3 5, No. 1, Proposition 1 there exists an &algebra homomorphism g,: T(M) + E such thatf = g, o +; 49 7 496 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS all that needs to be proved is that g, is zero on the ideal 3', for then, ifp: T(M) + S(M) = T(M)/3' is the canonical homomorphism, we can write g1 = g o p, where g: S(M) -+ E is an algebra homomorphism, and the con- clusion will follow from the fact that p o + = +'. Now the kernel of g, is a two- sided ideal which, by virtue of (1) and the relation g, 0 + = f, contains the elements x @ y - y @ x for x, y in M. This completes the proof. Remarks. (1) Suppose that E is a graded A-algebra of type Z, with graduation (En), and suppose also that the linear mapping f (assumed to satisfy (1)) is such that Then the relation g(x,x,. . .x,) = f (xl)f(x2). . .f(x,) with the xi E: M shows that g(SP(M)) c Ep for allp >, 0 and henceg is agradedalgebra homomorphism. (2) Every element of S(M) is a sum of products of the form xlx2. . .x,, where the xi belong to M; care should be taken not to confuse such products taken in S(M) with the analogous products taken in T(M). (3) If n! . 1 is invertible in A, the A-module S"(M) is generated by the ele- ments of the form x", where x E M; this follows from the above remark and I, 3 8, no. 2, Proposition 2. (2) f(M) = El. 2. FUNCTORIAL PROPERTIES OF THE SYMMETRIC ALGEBRA PROPOSITION 3. Let A be a commutative ring, M and N two A-modules and u: M 4 N an A-linear mapping. There exists one and only one A-algebra homomorphism u': S(M) -+ S(N) such that the diagram MAN 6h I + S(M) 7 S(N) is commutative. Moreover, u' is a graded algebra homomorphism. to the commutative algebra S(N) andf = +fi 0 u: M -+ S(N) ; as The existence and uniqueness of u' follow from no. 1, Proposition 2 applied f(M) c S1(N) = N, the fact that u' is a graded algebra homomorphism follows from no. 1, Remark 1. The homomorphism u' of Proposition 3 will henceforth be denoted by S(U) - If P is an A-module and v: N --f P an A-linear mapping, then (3) S(v 0 u) = S(v) 0 S(u) 3 6.2 for S(v) 0 S(u) is an algebra homomorphism rendering commutative the diagram FUNCTORIAL PROPERTIES OF THE SYMMETRIC ALGEBRA M-------tP uou As S(M) contains M = S1(M), S(u) is sometimes called the canonical ex- tension of u to S(M). The restriction S"(u) : S"(M) -+ S"(N) is such that S"(U)(X&. . .xn) = u(x1)u(x2). . .u(xn) where the xi E M, since S(u) is an algebra homomorphism and S1(u) = u; the restriction So(u) to A is the identity mapping. Note that S"(u) can be obtained from T"(u) : T"(M) -+ T"(N) by passing to the quotients. PROPOSITION 4. rf u: M -+ N is a surjective A-linear mappinx, the homomorphism S(u) : S(M) --f S(N) is surjective and its kernel is the ideal of S(M) generated by the kernel P c M c S(M) of u. We write v = T(u): T(M) + T(N); we know (3 5, no. 2, Proposition 3) that v is surjective and hence it follows from the definitions that v(&) = 3;; if R is the kernel of v, then i1(9b) = R + &. As S(u) : T(M)/af, -+ T(N)/'& is derived from v by passing to the quotients, it is a surjective homomorphism whose kernel is R' = (9 + 3h)/3&. As R is generated by the kernel P of u (4 5, no. 2), so is R'. If u: M --f N is an injective linear mapping, it is not always true that S(u) is an injective mapping (Exercise 1). However it is so when u is an injection such that u(M) is a directfactor in N and then the image of S(u) (isomorphic to S(M)) is a direct factor of S(N) ; the proof is the same as that for the analogous asser- tions for T(u) (8 5, no. 2) replacing T by S. PROPOSITION 5. Let N and P be two submodule; of an A-module M such that their sum N + P is a direct factor in M and their intersection N n P is a direct factor in N and in P. Then the homomorphisms S (N) -+ S (M) , S (P) -+ S (M) and S(N n P) -+ S(M), canonical extensions ?f the canonical injcfions, are injective; ;fS(N), S(P) and S(N A P) are identified with subalgebras of S(M) by means of these homomorphisms, then S(N n P) = S(N) n S(P). The proofreduces to that of 5 5, no. 2, Proposition 4 replacing T by S through - out. The hypotheses of Proposition 5 always hold for arbitrary submodules N, P of M when A is a field. (4) 498 499 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS COROLLARY . Let K be a commutativejeld and M a vector space over K. For every element z E S(M) there exists a smallest vector space N ofM such that z E S(N) and N is finite-dimensional. The proof is derived from that of fj 5, no. 2, Corollary to Proposition 4 re- placing T by S throughout. N is called the vector subspace of M associated with z. 3. tt-th SYMMETRIC POWER OF A MODULE AND SYMMETRIC MULTI - LINEAR MAPPINGS Let X, Y be two sets and n an integer 3 1. A symmetric mapping of X" into Y is any mapping f: X" --f Y such that, for every permutation 0 E 6, and every element (xi) E X", (5) .f(xt~), xa(2), . * ., xu(,)) = f (XI, ~2, * . ., 4. As the transpositions which exchange two consecutive integers generate the group 6, (I, § 5, no. 7), it suffices that condition (5) hold when (I is such a 5 6.3 n-th SYMMETRIC POWER (x, y, ui, v, in M), to say that the function f is of the form g 0 p, means that the corresponding n-linear function f satisfies the relation f(u1,.- .,up,x,y,~i,- * *,~n-p-2) =J(U~,...,U~,Y,~,V~, ...,vn-p-2); in other words, by what has been seen above, this means thatfis symmetric; whence the proposition, taking account of the fact that fin(., 8 22 @ * 1 . 8 xn) = ~1x2. * *xn with the xi E M. The A-module S"(M) is called the n-th symmetric power of M. For every A- module homomorphism u: M + N, the mapping s"(~) : S"(M) + S"(N) which coincides with S(u) on S"(M) is called the n-th symmetricpower of u. Remark. Let CT be a permutation in 6,; as the mapping (XI, ~2, * . *> xn) ++ xu-1(1) 8 xu-1(2) 8 . * * 8 xu-'(,) of Mn into Tn(M) is A-multilinear, it may be written uniquely as (XI, . . ., xn) H uu(x1 8 x2 8 . . . @ xn), where uu is an endomorphism of the A-module T"(M), also denoted by z H 0. z. Clearly, if Q is the identity element of Q,, uu is the identity; on the other hand, writingy, = xU-1(<), weobtain, for every permutation T E 6,,,yT-1(,) =X,,-~(~-I(~)) and hence 7. (0. z) = (TO). 2; in other words, the A-module T"(M) is a left 6,-set under the operation (a, 2) H U. z (I, 5 5, no. 1). The elements of T"(M) such that 0. z = z for all a E 6, are called (contravariant) symmetric tensors Df order n; they form a sub-A-module Sh(M) of T"(M). For all z E Tn(M), we write s. z = 2 0. z and call s . z the symmetrization of the tensor z; clearly s. z is a symmetric tensor and therefore z ++ s. z is an endomorphism of T"(M) whose image S:(M) is contained in S;(M) ; in general, S:(M) # S;(M) (Exercise 5). If z is a symmetric tensor, then s. = n !z; it follows that when n ! is invertible in A, the endomorphism z ++ (n .) ' -1.9.2 is a projector of T"(M) (11, 3 1, no. 8), whose image is S;(M) = Si(M) ; more- Over the kernel on this projector is just 3;. For, obviously 43;) = 3; for all 0 E 6, and 3; is by definition generated by the tensors z - p . z, where p is a transposition exchanging two consecutive numbers in (1, n); also, if 0, T are *opemutationsin6,,thenz - (cT).~ = z - u.z + Q.(Z - ~.z),whence it follows (since every permutation in 6, is a product of transpositions ex- changing two consecutive numbers) that z - 5. z E 3; for all z E Tn(M) and Q E 6,. Therefore (always supposing that n! is invertible in A), it is seen that * - (n!)-ls.z = c (n!)-l(z - 0.z) E 3; for all ZEP(M), which proves our assertion. USE, ace, transposition. When Y is a module over a commutative ring A, clearly the set of symmetric mappings of Xn into Y is a submodule of the A-module Yxn of all mappings of X" into Y. PROPOSITION 6. Let A be a commutative ring and M and N two A-modules. If with every A-linear mapping g: Sn(M) --f N (n 2 1) is associated the n-linear mapping (where on the right-hand side the product is taken in the algebra S(M)), a bijective A-linear mapping is obtained ofthe A-module HomA(Sn(M), N) onto the A-module of symmetric n-linear mappings of M" into N. Recall (11, fj 3, no. 9) that there is a canonical bijection of the A-module HomA(T"(M), N) onto the A-module 9,,(M,. . ., M; N) of all n-linear map- pings of M" into N obtained by associating with every A-linear mapping $ T"(M) -+ N the n-linear mapping (6) (~1, ~2, . . * > xn) ++ g(x1XZ. * .xn) (7) 3 (XI, ~2, . * *, xn) -f (XI 8 ~2 8 * * * 8 xn). On the other hand, the A-linear mappings g: S"(M) -+ N are in one-to-one correspondence with the A-linear mappingsf: Tn(M) --f N such that f is zero on 3;, by associating with g the mapping f = g oPn, where p,: T"(M) -+ S"(M) = T"(M)/3; is the canonical homomorphism (11, linear combination of elements of the form 2, no. 1, Theorem 1). But as 3; is a (~1 ~2 8 .+ * B up) 8 (x 8 y - Y 8 x) 8 (~1 8 . . 8 vn -p - 2) 500 50 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS When n! is invertible in A, the submodules SL(M) and 3; of Tn(M) are therefore supplementary and the restriction to SL(M) of the canonical homo - morphism F(M) -+ S"(M) = Tn(M)/3; is an A-module isomorphism, which allows us in the case envisaged to identify symmetric tensors of order n with the elements of the n-th symmetric power of M. Note however that this identifica - 2 tion is not compatible with multiplication, the product (in T(M)) of two symmetric tensors not being symmetric in general and not therefore having as image in S(M) the product of the images of the symmetric tensors con- sidered. 4. EXTENSION OF THE RING OF SCALARS Let A, A' be two commutative rings, p: A -+ A a ring homomorphism, M an A-module, M' an A-module and$ M -+ M' an A-homomorphism (relative to p) of M into M'. The composite mapping M -+ M' --+ S,.(M') is an A-linear mapping of M into the commutative algebra p*(SA(M')) ; then there exists no. 1, Proposition 2) one and only one A-homomorphism of algebras f': SA(M) -+ S,. (M') rendering commutative the diagram f VMd. f M - M' It follows immediately that if a: A' -+ A" is another ring homomorphism, M" an A"-module, g: M' + M" an A'-homomorphism (relative to a) and g': SAP (M') --f SA-(M') the corresponding A-homomorphism of algebras, then the composite A -homomorphism of algebras Y (9) SA(M) SA*(M') 2 SA.(M") corresponds to the composite A -homomorphism g of: M -+ M" (relative to PROPOSITION 7. Let A, B be two commutative rings, p: A -+ B a ring homomorphism and M an A-module. The canonical extension go PI. $: S,(B @A M) --fB @A SA(M) of the B-linear mapping 1, @ +h: B @A M -+ B @A SA(M) is a graded B -algebra isomorphism. The proof is derived from that of 3 5, no. 3, Proposition 5 replacing T by s and +M by +d. i I I c I I I 1 I k DIRECT LIMIT OF SYMMETRIC ALGEBRAS 3 6.5 5. DIRECT LIMIT OF SYMMETRIC ALGEBRAS Let (A,, +,,) be a directed direct system ofcommutative rings, (Ma, f,,) a direct system of A,-modules, A = lim A, and M = lim Ma. For a < p, we derive canonically from the &-homomorphism fpa: Ma 4 M, an A,-algebra homo - morphism (no. 4, formula (8)) f;,: SAa(Ma) --+ SA,(M,) and it follows from (9) (no. 4) that (SAa(Ma),fLa) is a direct system of A,-algebras. On the other hand, fa: Ma + M be the canonical A-homomorphism; we derive (no. 4, formula (8)) an A,-algebra homomorphism + + fk: S&(') --f SA(M) and it also follows from (9) that the f& constitute a direct system of A,-homo - morphisms. PROPOSITION 8. The A-homomorphism f' = limy:: lim SA.(M,) --f SA(M) is a graded algebra isomorphism. The proof is the same as that of $5, no. 5, Proposition 6 replacing through- out T by S and + by +' and taking account of the fact that a direct limit of commutative algebras is commutative. --++ 6. SYMMETRIC ALGEBRA OF A DIRECT SUM. SYMMETRIC ALGEBRA OF A FREE MODULE. SYMMETRIC ALGEBRA OF A GRADED MODULE Let A be a commutative ring, M = hFLMA the direct sum of a family of A-modules and j,: MA -+ M the canonical injection; we derive an A-homo- morphism of algebras S(jJ : S(MA) -+ S(M). Since S(M) is commutative, Proposition 8 of 3 4, no. 5, can be applied to the homomorphisms S(jA) and there therefore exists a unique algebra homomorphism g: @ S(MA) + S(M) L€L (10) (also denoted by gM) such that S(jJ = g o fA for all A E L, where fA: --t a s(MA) denotes the canonical homomorphism. PROPOSITION 9. The canonical homomorphism g (formula (1 0)) is a graded algebra isomorphism (cf. 3 4, no. 8, Remark 1). TO prove that g is bijective, we define an algebra homomorphism h: S(M) += (8 S(MA) hEL (1 1) 503 502 III TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS such that g 0 h and h 0 g are respectively the identity mappings on S(M) and @ S(MJ. For each A E L, let uA be the composite linear mapping AeL There exists one and only one A-linear mapping u: M -+ (8 S(MJ such that 0 j, = uA for all h E L. As the S(MA) are commutative, so is their tensor product ($4, no. 5) and hence (no. 1, Proposition 2) there exists a unique algebra homomorphism h: S(M) -+ @ S(Mh) such that h 0 & = u; on the other hand, it is immediate that u(M) is contained in the submodule of elements of degree 1 of the graded algebra h3 S(MJ and hence h is a graded algebra homomorphism. For xA E MA, hEL AEL h(g(uA(XA))) = h(g(fh(&A(XA)))) = h(S(jA) (&L(~A))) = h(&(jA(xA))) = uA(xA) ; as the uA(xA) generate the algebra A( s(Mh) ($ 4, no. 5, Proposition 8), h 0 g is certainly the identity mapping. Similarly, idh(%dj&('A)))) = g('&('A)) = g(fA(&h(xA))) = '(jA) (+k. ('A)) = &(jA('A)) and, as the elements +h(jA(xA)) generate the algebra S(M), g o h is certainly the identity mapping. Remark (1) Let N = NA be the direct sum of another family of A-modules with L as indexing set and, for all h E L, let vA: MA-+NA bean A-linear mapping, whence there is an A-linear mapping v = 9 v,: M --f N (11, $ 1, no. 6, Proposition 6). Then the diagram @ S(MJ 5 S(M) AsL is commutative, as follows from the definitions (8 4, no. 5, Corollary to Proposition 8). The sub-A-module of h3 S(M,) with which S"(M) is identified by means of the isomorphism g can be described more precisely. For every finite subset J of L, we write E, = A( S(M,), so that AFL S(MA) = lim EJ relative to the + P SYMMETRIC ALGEBRA OF A DIRECT SUM S 6.6 directed set $(L) of finite subsets of L, by definition (3 4, no. 5). For every family v = (nA) E N(L) (thus having finite support) such that zL nA = n and every finite subset J of L containing the support of the family v, we write (12) SJpV(M) = S"h(MA) so that the submodule EJ,, of elements of degree n in Ej is the direct sum of the SJsV(M) over all the families v of support contained in J and such thatA5L nh = n ($ 4, no. 7, Proposition 10 and $ 4, no. 8). As a convention we write SJ*"(M) = (0) for the families v whose support is not contained in J; then EJ," can also be called the direct sum of all the SJ*V(M), where v runs through the set H, of all families v = (nA)AEL such that zL nA = n. Since SO(MA) is identified with A, clearly also, for two finite subsets J c J' of L and a (restriction of the canonical mapping Ej -+ E, to SJ.'(M)) is bijective. If we write, for all v E H,, Sv(M) = lim SJ*"(M) family v of support contained in J, the canonical mapping SJsV(M) --f SJ'sV (M) (13) --f it is seen that, taking account of 11, $ 6, no. 2, Proposition 5: COROLLARY . The A-module S"(M) is the image under isomorphism (10) of the submodule ofh?, S(MJ the direct sum ofthe submodules Sv(M) for all the families v = (nA) E N(L) such that zL nA = n; if J is the support ofv, Sv(M) is canonically isomorphic to @ S"h(MA). AaJ In general Sv(M), @ Snh(MA) and their image in S"(M) are identified. THEOREM 1. Let A be a commutative ring and M a free A-module with basis (eJAG L. For every mapping a: L --f N ofjnite support, we write u - n u(A) (14) - AsLeh (product in the commutative algebra S(M)). Then, when u runs through the set N(L) of mappings of L into N, ofjinite support, the 6 form a basis of the A-module S(M). As M is the direct sum of the MA = Ae,, it suffices to prove the theorem when L is reduced to a single element and then apply Proposition 9. But when M = Ae (e a free element), then x @t~ - ZJ @ x = 0 for all x, y in M; the ideal 8' is therefore zero, whence T(Ae) = S(Ae) and the theorem then follows from 5 5, no. 5, Theorem 1. AEJ 504 505 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The multiplication table of the basis (14) is given by eaeD = ea+P where u + @ is the mapping A H u(A) + @(A) of L into N. In other words, the basis (eu) of S(M), with the multiplicative law (15), is canonically isomorphic to the free commutative monoid N(L) derived from L; it follows (3 2, no. 9) that the symmetric algebra S(M) of a free module M with bask whose indexing set is L, is canonically isomorphic to the polynomial algebra A[ (XA)AEL], the canonical isomorphism being obtained by mapping e, to X,. In particular (3 2, no. 7, Proposition 7), for every mapping8 L -+ E of L into a commutative A-algebra E, there exists one and only one A -algebra homomorphism f: S(M) -+ E such thatf(e,) = f(h). Remark (2). The above results can equally well be obtained as a consequence of the universal properties of polynomial algebras and symmetric algebras, taking account of 11, 3 1, no. 1 1, Corollary 3 to Proposition 17. COROLLARY . If M is a projective A-module, S (M) is a projective A-module. The proof is the same as that for 3 5, no. 5, Corollary to Theorem 1, re- placing T by S. PROPOSITION 10. Let A be a commutative monoid, M a graded A-module of type A and (MJEE,, its graduation. For every orderedpair (a, n) E A x N, let SU*"(M) be the submodule ofS"(M) the direct sum ofthe submodules 8 S4h(MEh), where (nA)AEL rum through the set offamilies ofintegers 2 0 such that zL n, = n, J is its support and, for each (n,), (aA)AEj rum through the set offamilies of AJ such that zJ u, = u. Then (s"*"(M)),,, xN is the only graduation oftype A x N compatible with the algebra structure on S(M) and which induces on M = S1(M) the given graduation. The fact that S (M) is the direct sum of the Sa. (M) follows from the Corollary to Proposition 9; the rest of the proof is identical with the end of the proof of 3 5, no. 5, Proposition 7. Suppose more particularly that A = Z and let S(M) be given the total gradua - tion (of type Z) (11, fj 11, no. l) corresponding to the graduation of type Z x N (and hence also of type Z x Z) defined above; the homogeneous elements of degree n E Z under this graduation are therefore those of the direct sum of the Sqsrn(M) for q + m = n. Let f be a homogeneous linear mapping of degree 0 of the graded A-module M into a commutative graded A-algebra F of type Z; then the algebra homomorphism g: S(M) -+ F such that f = g 0 +h is a homomorphism of graded algebras of type Z, as follows from the formula g(x,x,. . . x,) = f(x,) f (x,) . . . f (x,) for homogeneous xi in M, from the hypo - thesis on f and from the definition of the graduation of type Z on S(M). 506 (15) AE J 3 7. EXTERIOR ALGEBRAS 1. DEFINITION OF THE EXTERIOR ALGEBRA OF A MODULE DEFINITION 1. Let A be a commutative ring and M an A-module. The exterior algebra ofM, denoted by A(M) or Alt(M) or &(MI, is the algebra over A the quotient ofthe tensor algebra T(M) by the two-sided ideal 3" (also denoted by st) generated by the elements x @ x, where x rum through M. Since the ideal 3" is generated by homogeneous elements of degree 2, it is a graded ideal (11, 3 11, no. 3, Proposition 2) ; we write 3; = 8" n Tn(M) ; the algebra A(M) is therefore graded by the graduation (called canonical) con- sisting of the A"(M) = T"(M)/3;. Then 3; = 3'; = (0) and hence A'(M) is identified with A and A'(M) with T1(M) = M; in what follows we shall always make these identification and the canonical injection M -+ A (M) will be denoted by +" or &. PROPOSITION 1. Let E be an A-algebra and8 M -+ E an A-linear mapping such that (1) There exists one and only one A-algebra homomorphism g: A(M) -+ E such that (f(~))~ = 0 for all x E M. f =go+". In other words, (A (M), v) is a solution of the universal mapping problem (Set Theory, IV, Q 3, no. I), where C is the species of A-algebra structure, the a-mappings being the linear mappings from the A-module M to an A-alge- bra satisfying (1). The uniqueness ofg follows from the fact that +"(M) = M generates A (M). To prove the existence of g, we note that by 3 5, no. 1, Proposition 1 there exists an A-algebra homomorphism g,: T(M) -+ E such that f = g, 0 +; we need to prove that g, is zero on the ideal 3", for then if p: T(M) -+ A(M) = T(M)/~" is the canonical homomorphism, we can write g, = g o p, where g : A (M) -+ E is an algebra homomorphism and the conclusion will follow from the fact that fi 0 + = +". Now, the kernel of g, is a two-sided ideal which, by virtue of (1) and the relation g, 0 =f, contains the elements x @ x for x E M. This completes the proof. 507 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Remarks. (1) Suppose that E is a graded A-algebra of type Z, of graduation (En), and suppose also that the linear mappingf(assumed to satisfy (1)) is such that (2) fW = El. Then the relation g(x1x2. . . x,) = f (XI) f (x2) . . . f (x,) with the xi E M shows that g(Ap(M)) c E, for all p 2 0 and hence g is a graded algebra homo- morphism. (2) To avoid confusion, the product of two elements u, v of the exterior .. algebra A (M) is usually denoted by u A v and is called the exterior product of u by v. The elements of R(M) are therefore the sums of elements of the form x1 A x2 A . a . A n, with xi E M for 1 < i < n and are often called n-vectors. 2. FUNCTORIAL PROPERTIES OF THE EXTERIOR ALGEBRA PROPOSITION 2. Let A be a commutative ring, M and N two A-modules and u : M -+ N an A-linear mapping. There exists one and only one A-algebra homomorphism uc: A(M) +A(N) such that the diagram &M) d- A(N) is commutative. Moreover, u" is a graded algebra homomorphism. The existence and uniqueness of u' follow from no. 1, Proposition 1 applied to the algebra A(N) and f = #& o U: M -+ A (N) ; forf(M) c N and hence fsatisfies condition (1) by definition of 3;: asf(M) c A'(N) = N, the fact that U" is a graded algebra homomorphism follows from Remark 1 of no. 1. The homomorphism U" of Proposition 2 will henceforth be denoted by A(u)- If p is an A-module and v: N -+ P is an A-linear mapping, then (3) A(v 0 u) = A(v> 0 A(u) for A (v) 0 A (u) is an algebra homomorphism which renders commutative the diagram M-P Dll I *; 508 FUNCTORIAL PROPERTIES OF THE EXTERIOR ALGEBRA 4 7.2 Since A(M) contains M = A1(M), A(u) is sometimes called the canonical extension of u to A(M). The restriction An(u): A"(M) -+ A'(N) is such that (4) An(u)(xl A x2 A . . . A x,) = u(x1) A u(x2) A . . A u(xn) with the xi E M, since A(u) is an algebra homomorphism and A'(u) = u; the restriction Ao(u) to A is the identity mapping. Note that An(u) is obtained from P(u) : P(M) -+ P(N) by passing to the quotients. PROPOSITION 3. If u: M -+ N is a surjective A-linear mapping, the homomorphism A(u) : A (M) -+ A(N) is surjective and its kernel is the two-sided ideal of A(M) generated by the kernel P c M = A (M) ofu. The proof is derived from that of 9 6, no. 2, Proposition 4, replacing S by A and 3' by 3". If u: M --f N is an injective linear mapping, it is not always true that A(u) is an injective mapping (4 6, Exercise 3) (see however below no. 9, Corollary to Proposition 12). However this is so when u is an injection such that u(M) is a direct factor of N and then the image ofA(u) (isomorphic to A(M)) is a direct factor of A (N) ; the proof is the same as that for the analogous asser- tions for T(u) (3 5, no. 2) replacing T by A. PROPOSITION 4. Let N and P be two submodules of an A-module M such that their sum N + P is a direct factor in M and their intersection N n P is a direct factor in N and in P. Then the homomorphisms A(N) -+ A(P) -+ A(M) and A (N n P) --f A (M), canonical extensions of the canonical injectionr, are injective; if A(N), A(P) and A(N n P) are identiJied with subalgebras ofA(M) by means of these homomorphism, then (5) A(N n P) = A(N) n A(P). The proof is derived from that of 3 5, no. 2, Proposition 4 replacing T by A throughout. The hypotheses of Proposition 4 are always satisfied by arbitrary submodules N, P of M when A is afield. COROLLARY. Let K be a commutativefield and M a vector space over K. For every element z E A (M), there exists a smallest vector subspace N of M such that z E A (N) and N kJinite-dimensional. The proof is deduced from that of 3 5, no. 2, Corollary to Proposition 4 re- Placing T by A throughout. N is called the vector subspace of M associated with the element z of A (M). 509 Ill TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 3. AN''~ICOMMUTATMTY OF THE EXTERIOR ALGEBRA PROPOSITION 5. (i) Let (xi) <, , be ajnite sequence of elements ofthe module M; for every permutation a in the symmetric group 6,, A X,o, A * * * A XU(,) = €0. Xi A X2 A . * * A X, (6) where E, denotes the signature of the permutation a. (ii) If there exist two distinct indices i, j such that x, = x,, the product Xi A X, A A X, is mo. also, for x, y in M, (i) First of all, since x x = 0 for all x E M by definition of the ideal s", X A y +y A X = (X +y) A (X +y) - X A X -y A y = 0. This establishes (6) in the case n = 2. The general case then follows from 4 4, no. 6, Lemma 3. (ii) Under the hypothesis of (ii), there exists a permutation a E 6, such that ~(1) = i and 42) = j; then the left hand side of (6) is zero for this pemuta- tion and hence so is the right hand side. COROLLARY 1. Let H, K be two complementary subsets of the interval (1, n) of N and let (ih) h k , -p be the sequences of elements of H and K respectively, arranged in increasing order; we write (j,) XH = Xi1 A Xi2 A * * * A Xip, XK = Xjl A Xj2 A . . * A xj,-p; then (7) XH A X, = (-1)"Xl A X2 A ... A X, where v is the number of orderedpairs (i, j) E H x K such that i > j. By Proposition 5 this reduces to proving Lemma 1. If 0 E 6, is the permutation such that a(h) = ih for 1 < h < p, a(h) = jh-,forp + 1 < h < n, then E, = For 1 < h < h' < p orp + 1 < h < h' < n, o(h') > a(h) and the number of ordered pairs (h, h') such that 1 < h < p < h' < n and a(h) > o(h') is equal to v. COROLLARY 2. The graded algebra A (M) is alternating (3 4, no. 9). ting system the set M and using Proposition 5. 510 It suffices to apply Proposition 13 of 5 4, no. 9 to A (M), taking as genera - n-th EXTERIOR POWER OF A MODULE PROPOSITION 6. If M is a finitely generated A-module, A (M) is a finitely generated A-module; also, ifM admits a generating system with n elements, then A'(M) = (0) forp > n. Let be a generating system of M. Every element of Ap(~) is a linear combination of elements of the form 3 7.4 Xi, A Xia A * . * A xi1, where the indices i, belong to (1, n); by Proposition 5, these indices can be assumed to be distinct (otherwise the corresponding element is zero). Ifp > n, there is no such sequence of indices and hence A'(M) = (0). Ifp 6 n, these sequences are finite in number, which completes the proof. 4. n-th EXTERIOR POWER OF A MODULE AND ALTERNATING MULTI - LINEAR MAPPINGS Given two modules M, N over a commutative ring A, an alternating n-linear mapping of M" into N is any n-linear mapping f: M" -+ N such that, for all p<n-2, (8) f(Ul,...,up,x,~,~1,...,vn-p-2) = 0 for all x, the u, (1 < i < p) and the v, (1 6 j 6 n - p - 2) in M. PROPOS~ON 7. Let A be a commutative ring and M and N two A-modules. y with every A-linear mapping g: A"(M) -+ N (n 2) is associated the n-linear mapping (9) (Xi, x2,. . . , X,) - g(x1 A X2 A . . . A X,) a bijective A-linear mapping is obtained of the A-module H~~,(/\"(M), N) onto the A-module of alternating n-linear mappings of M" into N. We consider the canonical bijection of the A-module Hom,(Tn(M), N) onto the A-module $p,(M,. . ., M; N) of all n-linear mappings of Mn into N, obtained by associating with every A-linear mapping f: Tn(M) -+ N the n- linear mapping 3 (xi, * * ., Xn) ++f(xl ~2 @ * * . 8 xn) (11,s 3, no. 9). On the other hand, the A-linear mappings g: /\"(MI --z N are in one-to-one correspondence with the A-linear mappingsf: Tn(M) -+ N such that f zero on 3:, by associating with g the mapping f = g opn, where pn: T~(M) -+ A"(M) = T"(M)/Q: 51 1 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS is the canonical homomorphism (11, $2, no. 1, Theorem 1). But as 3; is a linear combination of elements of the form (~1 8 ~2 @. . . 8 up) @ (X @ X) 8 (01 @ * . 8 vn-p-2) (x, ui, vj in M), to say that f is of the form g o pn means that the corresponding n-linear function] satisfies (8), in other words it is alternating. The A-module A"(M) is called the n-th exterior power of M. For every A- module homomorphism u : M -+ N, the mapping An(U) : A"(M) -+ A\"(N) which coincides with A(u) on A\"(M) is called the n-th exteriorpower ofu. COROLLARY 1. For every alternating n -linear mapping g : M" -+ N, for every permu - tation Q E G,, (10) g(*ya(l), xa(2), . . , xa<n)) = Ea.g(x1, ~2, . . ., xn) for all xi E M; moreover if xi = xj for two distinct indices i,j, then g(x1, ~2, . ., xn) = 0. This is an obvious consequence of Proposition 7 and no. 3, Proposition 5. COROLLARY 2. Let (xi) , be a sequence of it elements of M such that X1 A X2 A A X, = 0; then, for every alternating n -linear mapping g: M" -+ N, g(xl, . . . , x,) = 0. COROLLARY 3. Let f: Mn-l +A be an alternating (n - 1)-linear form. If (xi)lGiQn is a sequence ofn elements of M such that x1 A x2 A . . . A x, =_- 0, then 5 (- l)y-(X1, . . . ,ii, . . . , x,) .xi = 0 (XI,. . *, xn) * i=l 2 (-l)tf(xl, *. .> Gi, ***> xn).xi (11) i=1 (where we write f (xl, . . . , xAi, . . ., x,) =f(xl, . . ., xi-l, xi+l,. . ., x,) for l<i<n. It suffices to prove that the n-linear mapping of M" to M is alternating. Now, if xi = xi + 1, all the terms in the sum on the right hand side have zero coefficients except xi and x,+~, since f is alternating; on the other hand, the coefficient of xi is (-l)ff(xl, .* ., Xi-1, xi+l, xf+2, . . *> xn) 512 3 7.5 EXTENSION OF THE RING OF SCALARS and that of xi+1 is ( -l)i+y(xl,. . ., xi, x,+~, . . ., x,) and they are inverses by hypothesis . Remark. An element z of Tn(M) is called a (contravariant) skew -symmetric tensor of order n if Q. z = E,Z for every permutation 5 E 6, (cf. 3 6, no. 3, Remark); these elements form a sub-A-module AL(M) of Tn(M). For all z E Tn(M), we write a. z = 2 E,(Q. z) and call a. z the skewsymmetrization of z; as E,, = E,E,, it is immediately seen that a. z is a skew-symmetric tensor and therefore z H a. z is an endomorphism of Tn(M) whose image Ai(M) is contained in AL(M); in general AA(M) # Ak(M) (Exercise 8). If z is a hew-symmetric tensor, then a.z = n!z; hence, when n! is invertible in A, the endomorphism z H (n!) -la. z is a projector of Tn(M) whose image is ass, Ak(M) = AA(M). Moreover the kernel of this projector is just 3;; for attention may obviously be confined to the case where n 2 2, hence 2 (a divisor of n!) is invertible in A and x @ x = 2-'(x @I x + x @ x) ; therefore 3; is generated by the ele- ments z + p. z, where p is a transposition exchanging two consecutive num- bers in (1, n); moreover, for two permutations Q, 7 in G,, we can write z - EQT((Q7).Z) = z - EJ7.Z) f EJ7.Z - E05.(7,Z)) whence it follows that z - E~(Q. z) E 3; for all z E Tn(M) and Q E 6,. There - fore (always assuming n! is invertible in A), it is seen that z - (n!)-la.z = C (n!)-l(z - e,(a.z)) E 3; ae6n for all t E T"(M), which establishes our assertion. When n! is invertible in A, the submodules AL(M) and 3; of Tn(M) are therefore supplementary and the restriction to AL(M) of the canonical homo - morphism T"(M) -+ A\"(M) = Tn(M)/3: is an A-module isomorphism, which allows us in the case in question to identify skew-symmetric tensors of order n with the elements of the n-th exterior power of M. Note here also that this identification is not compatible with multiplication, the product in T(M) of two skew -symmetric tensors not being skew-symmetric in general. 5. EXTENSION OF THE RING OF SCALARS Let A, A' be two commutative rings, p :A --f A' a ring homomorphism, M an A-module, M' an A'-module andf: M -+ M' an A-homomorphism (relative to p) of M into M'. The composite mapping M A M' --+ A,.(M') is an *-linear mapping of M into the A-algebra A*(M~) and, as the elements of +L, 513 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS f(M) c M' are of zero square in AA,(~f), there exists (no. 1, Proposition 1) one and only one A-homomorphism ofalgebrasf ": AA(~) -+ AN(~') rendering commutative the diagram f M M' and7 is a graded algebra homomorphism. It is immediately deduced that if 0:A' -+ A" is another ring homomorphism, M" an A"-module, g:M' -+ M" an A-homomorphism (relative to 0) and g":AAa(Mf) -+ A,.(M") the corresponding A"-homomorphism of algebras, then the composite A -homo- morphism of algebras corresponds to the composite A -homomorphism g of: M -+ M" (relative to PROPOSITION 8. Let A, B be two commutative rings, p :A -+ B a ring homomorphism and M an A-module. The canonical extension go PI. +:AB(B @A M) -+ B @A AA(M) of the B -linear mapping 1, @ &:B @A M --f B @A AA(~) is a graded B- algebra isomorphism. The proof is derived from that of 3 5, no. 4, Proposition 5 replacing T by A and +M by &. 6. DIRECT LIMITS OF EXTERIOR ALGEBRAS Let (A,, +,,) be a directed direct system of commutative rings, (Ma, fDa) a direct system of A,-modules, A = lim A, and M = --+ lim Ma. For a < p, we derive canonically from the A,-homomorphism fpa: Ma -+ M, an &-algebra homomorphism (no. 5, formula (12)) fd&: AA=(~,) -+ AAD(~~) and it follows from (1 3) that ( AAe(Ma), f:,) is a direct system of graded A,-modules. On the other hand let fa: Ma -+ M be the canonical A,-homomorphism; we derive (no. 5, formula (12)) a graded A,-algebra homomorphism ~&I:A~,(M,) --f AA(~) and it also follows from (13) that the fc constitute a direct system of A,-homomorphisms. + 514 EXTERIOR ALGEBRA OF A DIRECT SUM 4 7.7 PROPOSITION 9. The A-homomorphism f" = lim fl:lim A&(M,) --f AA(~) is a graded algebra isomorphism. The proof is the same as that of 4 5, no. 4, Proposition 6 replacing throughout T by A and 4 by 4" and taking account of the fact that a direct limit of alter- nating algebras is alternating. 7. EXTERIOR ALGEBRA OF A DIRECT SUM. EXTERIOR ALGEBRA OF A +--+ GRADED MODULE Let A be a commutative ring, M = MA the direct sum of a family of a family of A-modules and &:MA -+ M the canonical injection; an A-homo - morphism of graded algebras A (j,) :A (MA) -+ A (M) is derived. Since A (M) is anticommutative, Proposition 10 of fj 4, no. 7 (or if need be generalized to the case where L is infinite, cf. $4, no. 8, Reamrk 1) can be applied to the homomorphisms A (j,) ; then there exists a unique algebra homomorphism (also denoted by gM) such that A (j,) = g 0 f,, where fh:A(M,) +g$L A(MA) denotes the canonical homomorphism. PROPOSITION 10. The canonical homomorphism g (formula (14)) is a graded algebra isomorphism. To prove that g is bijective, we define a graded algebra homomorphism such that g o h and h 0 g are respectively the identity mappings on A (M) and '(8 A(M,). For each A E L, consider the composite linear mapping A€L There exists one and only one A -linear mapping u: M --f :FL A (MA) such that u 0 j, = u, for all A E L. The skew tensor product '@ A(M,) is an and, for L arbitrary, this follows from the definition of this product given in 8 4, no. 8, Remark 1 and the fact that a direct limit of alternating graded alge- 515 ahemating algebra: for, for L finite, this follows from 4 4, no. heL 9, Proposition 14 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS bras is alternating. As also u(M) is contained in the submodule of elements of '@ A (M,), it follows from no. 1, Proposition 1 and Remark 1 that there exists a unique graded algebra homomorphism. degree 1 in the graded algebra AsL h : (M) --t '@ hsL A (MA) such that h o 4; = u. The verification of the fact that g 0 h and h o g are the identity mappings is then performed as in 3 6, no. 6, Proposition 9 replacing S by A and 4' by 4". Remark. Let N = AqL N, be the direct sum of another family of A-modules with L as indexing set and, for all h E L, let v,:M, + N, be an A-linear mapping, whence there is an A-linear mapping v = @ v,:M -+ N (11, 5 1, no. 6, Proposition 7). Then the diagram A is commutative (cf. 5 4, no. 5, Corollary to Proposition 8). The sub-A-module of '@ A(M,) with which A\"(M) is identified by means of the isomorphism g can be described more precisely. For every finite subset J of L, we write E, = t(8 As J A(M,), so that :qL A(M,) = 3 lim E, relative to the directed set s(L) of finite subsets of L, by definition (5 4, no. 8, Remark 1). For every family v = (n,) EN(^) (therefore with Jinite support) such that zL n, = n and every finite subset J of L containing the support of the family v, we write AsL SO that the submodule EJ,n of elements of degree n in E, is the direct sum of the A"*"(M) for all families v of support contained in J and such that C -n (5 4, no. 7, Proposition 10 and no. 8). By way of convention we write AJvV(~) = (0) for the families v whose support is not contained in J; then hsL - 516 5 7.8 E,, , can also be called the direct sum of all the A"."(M), where v runs through the set H, of all families v = (?zA)AEL such that zL n, = n. Since A'(M,) is identified with A, clearly also for two finite subsets J c J' of L and a family Y of support contained in J, the canonical mapping A"*"(M) +A"'*"(M) (restriction of the canonical mapping E, + E,. to A","(M)) is bijective. If we write, for all v E H,, A"(M) = lim A"."(M) it is then seen that, taking account of 11, 3 6, no. 2, Proposition 5, we have: COROLLARY . The A-module A"(M) is the image under isomorphism (14) of the submodule ofg@ A(MJ the direct sum of the submodules A"(M) for all families v = (n,) E N(L) such that c n, = n; ifJ is the support ofv, A"(M) is canonically isomorphic to (8 AnA(~,). EXTERIOR ALGEBRA OF A FREE MODULE ----f (17) hcL AEL hsJ In general A"(M), 8 A"A(M,) and their image in A"(M) are identified. With this convention : PROPOSITION 11. Let A be a commutative monoid, M a graded A-module of type A and (Mu)uEA its graduation. For every ordered pair (a, n) EA x N, let A"?"(M) be the submodule $A"(M) the direct sum of the submodules @ A,A(M~J, where (nJhpL runs through the set of families of integers 20 such that - n J is its sujport and, for each (n,), (aA)hs J runs through the set of families ofAJ such that 2 ah = a. Then (Aa,n(~))(a,n)EAxN is the onlygraduation Oftype A x N com- Patible with the algebra structure on A(M) and which induces on M = A'(M) the &Den graduation. The fact that A (M) is the direct sum of the AU*"(~) follows from the Corol - lary to Proposition 10; the rest of the proof is identical with the end of the Proof of§ 5, no. 5, Proposition 7. AsJ AEJ *ELn'- , AEJ a EXTERIOR ALGEBRA OF A FREE MODULE THEOREM 1. Let M be an A-module with a basis (eA)hEL. Let L be given the struc - ture Ofa totally ordered set (Set Theory, 111, 3 2, no. 3, Theorem 1) and for every finite subset J OfL we write 1x1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS where (hlc)14kGn is the sequence of elements of J arranged in increasing order (Set Theory, 111, 9 5, no. 3, Proposition 6); we write eB = 1, the unit element of A. Then the e,, where J runs through the set $(L) ofjnite subsets of L, form a basis of the exterior algebra A (M). Since the eh generate the A-module M, every element of A(M) is a linear combination of a finite number of products of the elements eh and hence (taking account of no. 3, Proposition 5) is a linear combination of a finite number of elements e, for J E $(L). It reduces to proving that the e, are linearly independent over A. Otherwise, there would exist between these elements a linear relation with coefficients not all zero; the union of the subsets J which correspond to the e, whose coefficients in this relation are #O is ajnite subset K of L (since there is only a finite number of coefficients # 0). Let N be the submodule of M generated by the eh such that A E K; N is a direct factor in M, hence (no. 2) A(N) is identified with a subalgebra of A(M) and, if we show that the e, with J c K form a basis of A(N), we shall obtain the desired contradiction. It therefore reduces to providing Theorem 1 when the basis of M isjnite; we may therefore suppose that L = (1, m) c N. For each i E L, let M, be the free submodule Ae, of M; M is the direct sum of the M, and A(M,) is the direct sum of A'(M,) = A and A'(M,) = M, (no. 3, Proposition 6). Let A (M) be canonically identified with the A-module the tensor product of the A(M,) (no. 7, Proposition 10); the latter has as basis the tensor product of the bases (1, el) of the A(MJ (11, 9 3, no. 7, Corollary 2 to Proposition 7); thus we obtain all the elements u1 @u2 @ **. @urn where either u, = 1 or u, = e,; if J is the set of indices i such that U, = el, u1 @ u2 @ . . . @ urn is identical with e,, which completes the proof. COROLLARY 1. Suppose that L = (1, m); then the basis (e,) JEB1(L) of A(M) ha 2" elements. For p > m, AP(~) = (0); Arn(~) has a basis consisting of a single element e,; for 0 < p < m the number of elements in the basis (e,) .f R(M) con- sisting of the e, such that Card( J) = p is m! This follows from Set Theory, 111, 3 3, no. 5, Proposition 12 and Set Theory, We return to the case where the set L in Theorem 1 is arbitrary and give 111, 5 5, no. 8, Corollary 1 to Proposition 11. 518 $ 7.9 CRITERIA VOR LINEAR INDEPENDENCE explicitly the multiplication table (9 1, no. 7) of the basis (e,). Given two finite subsets J, K of the totally ordered set L, we write PJ.E = 0 pJ,g = (-1)" ifJnK # 0 ifJnK = 0 (19) where v denotes the number of ordered pairs (A, p) E J x K such that A > F. Then Corollary 1 to Proposition 4 of no. 2 immediately implies the relation e~ A e~ = PJ.K~J~IZ. Note the formula (20) (21) PJ.KPK.J = when J n K = 0, j = Card( J), k = Card(K) (no. 3, Corollary 2 to Proposi- tion 5.) COROLLARY 2. If M is a projective A-module, A (M) is a projective A-module. placing T by A. The proof is the same as that of 9 5, no. 5, Corollary to Theorem 1 re- COROLLARY 3. Let M be a projective A-module and (x~)~~,~,, a Jinite sequence of elements of M. For there to exist on M an alternating n -linear form f such that it is necessary and sujicient that xl A x2 A . . . A x, # 0. f(~1, ~2, * * *, xn) # 0, We know already (with no hypothesis on M) that the condition is necessary (no. 4, Proposition 7). Suppose now that M is projective and that X1 A X2 A ..* A X, # 0. Then A"(M) is projective (Corollary 2) and hence the canonical mapping A\"(M) + (A"(M))** is injective (11, 3 2, no. 7, Corollary 4 to Proposition 13) ; we conclude that there exists a linear formg:A\"(M) --f A such thatg(xl A x2 A - * * A xn) # 0. Iffis the alternating n-linear form corresponding to g (no. 4, Proposition 7), fhenf(x,, . . ., x,) z 0. 9. CRITERIA FOR LINEAR INDEPENDENCE P POSITION 12. Let M be a projective A-module. For elements xl, x2, . . ., x, of M to be linearly dependent, it is necessary and sujicient that there exist A # 0 in A ah that hu1 A X2 A -.. A X, = 0. (22) 519 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The condition is necessary (with no hypothesis on M), for if for example hl (with A # 0) is a linear combination of x2,. . . , x,, relation (22) holds (no. 3, Proposition 5). We show that the condition is sufficient by induction on n; for n = 1, it is a trivial consequence of the definition. Suppose therefore that n > 1 and that condition (22) holds for some A # 0. If h2 A X3 A -1. A X, = 0, then the induction hypothesis implies that x2, . . . , x, are linearly dependent and hence a fortiori xl, x2, . . . , x, are. If h2 A x3 A . . A x, # 0, it follows from no. 8, Corollary 3 to Theorem 1 that there exists an alternating (n - 1)- linear formfsuch thatf(h2 A x3 A . . A x,) = p. # 0. Since Xi A (ha) A A X, = 0, it follows from no. 8, Corollary 3 to Theorem 1 that px, is a linear combina - tion of h2, x3, . . . , x,; hence xl, x2, . . . , x, are linearly dependent. COROLLARY . Let M and N be two projective A-modules and f : M -+ N an A-linear mapping. Iff is injective, then A (f) : A (M) -+ A (N) is injective. We prove this first under the assumption that M is free; let (eJhsL be a basis of M, so that (eJ)JEO(L) (formula (18)) is a basis of A(M). Suppose that the kernel of A(f) contains an element u = uJeJ # 0. Let K be an ele- ment minimal among the finite subsets J such that a,, # 0 and let H be a finite subset of L disjoint from K and such that K U H contains all the J (finite in number) such that u,, # 0; for all J # K such that aJ # 0, we have therefore by definition J n H # 0 and consequently (no. 8, formula (20)). u A eH = +@-KeKuH belongs to the two-sided ideal of A(M), the kernel of A(f). We write eKvH = e,, A e,, A . . A e,,,; then uKf(eh,) A f(e,,) A . . . ~f(e,,) = 0; by virtue of Proposition 12, the elementsf(e,,) (1 < i < n) of N are linearly dependent. But this contradicts the hypothesis thatfis injective (11, 9 1, no. 11, Corollary 3 to Proposition 17). We now consider the general case; let M' be an A-module such that M @ M' = P is free (11,s 2, no. 2, Proposition 4). Consider the linear mapping g : M @ M -+ N @ M @ M' such that g(x, y) = (f(x), 0, y), so that there is a commutative diagram f M-N 520 7 $ 7.9 CRITERIA FOR LINEAR INDEPENDENCE where the vertical arrows are the canonical injections. Since g is injective and P is free, A(g) is an injective homomorphism as has been seen above. Now, A(j):A(M) +A(P) is an injective homomorphism since M is a direct factor in P (no. 2). The composite homomorphism A(M) 3 A(P) ---+ A (g) A(N OP) is therefore injective and, as it is also equal to the composite homomorphism A(M) 3 A(N) r\c,.! A(N OP) we conclude that A(f) is injective. PROPOSITION 13. Let M be an A-module, N a direct factor submodule ofM which is )ee of dimension p and {u} a basis of Ap(~). For an element x E M to belong to N, it is necessary and suficient that u A x = 0. Let P be a submodule of M complementary to N and let y EN, z E P be such that x = y + z. Then u A x = u A z. As I"\'(N) is free of dimension 1, the mapping +:P + P @ Ap(~) such that $(p) = p @ u is bijective (11, 5 3, no, 4, Proposition 4). On the other hand (no. 7, Proposition lo), the com- position of the canonical homomorphisms +:P 8 Ap(~) +A(P) 8 A(N) -+A(M) is injective. The mapping + 0 $ is therefore injective, whence the proposition. THEOREM 2. Let M be an A-module with a Jinite basis (e,),,,,,. For a sequence (XJ~~,~,, Ofn elements Of M to form a basis of M, it is necessary and sujicient that the element A E A such that x1 A x2 A ... A x, = ?,.el A e2 A A en (23) be invertible in A. Recall that el A e2 A + . . A en is the unique element of a basis of A\"(M) (no. 8, Corollary 1 to Theorem 1) so that the element A E A satisfying (23) is determined uniquely. If is a basis of M, x1 A x2 A . - A x, is the unique element in a basis of An(M) (no. 8), then A is invertible. Conversely, suppose A is invertible; then the alternating n-linear form f corresponding to the linear mapping g: /\"(MI -+ A such that g(el A e2 A -. - A en) = A-I is such thatf(x,, x2,. . . , x,) = 1. For all x E M, obviously X A XI A **. A X, = 0 52 1 (no. 3, Proposition 6); applying no. 8, Corollary 3 to Theorem 1, we obtain f (xl, x2, . . . , x,) . x = t=l c ( - l)+y-(x, xl, . . . ,4,, . . . , x,) .xi. As f (xl, . . . , x,) = 1, this shows that every x E M is a linear combination of xl, x2, . . . , x, and, as the latter are linearly independent (since Xi A X2 A ..* A X, # o), they form a basis of M. 3 8. DETERMINANTS 1. DETERMINANTS OF AN ENDOMORPHISM Let M be an A-module with ajinite basis of n elements and u an endomorphism of M. The A-module A\"(M) is a monogenous free module, that is isomorphic to A (no. 8, Corollary 1 to Theorem 1); An(u) is an endomorphism of this module and is therefore a homothety z H At of ratio h E A determined unique - ly (11, § 2, no. 3, Proposition 5). DEFINITION 1. The determinant of an endomorphism u of a free A-module M of jinite dimmsion n (11, 7, no. 2, Corollary to Proposition 3 and Remark l), denoted by det u, is the scalar h such that An(u) is the homothety of ratio A. By formula (4) of 3 7, no. 2, det u is the unique scalar such that (1) u(xI) A u(x2) A A u(x,) = (detu).~, A x2 A A x, for every sequence (xi) , , of n elements of M. If det(u) = 1 , u is said to be unimodular. PROPOSITION 1. (i) If u and v are two endomorphism of a jinite-dimensional free A-module M, then (2) det(u 0 v) = (det u) (det v). (ii) det( 1 M) = 1 ;for every automorphism ti of M, det u is invertible in A and (3) det(u-l) = (det u)-'. If n is the dimension of M, this follows immediately from the relation A"(u 0 v) = (A"(u>) 0 (An(v)> 3 7, no. 2, formula (3)). Let M be a free A-module with a finite basis (el) , ,; given a sequence (xi) < , <,, of n elements of M, the determinant of this sequence with respect to 522 the given basis (ei), denoted by det(x,, x2, . . . , x,) when no confusion can arise over the basis, is the determinant of the endomorphism u of M such that u(e,) = xi for 1 6 i < n. Then by formula (1) (4) and this relation characterizes the mapping (x,) H det(x,, x2,. . . , x,) of Mn into A. It shows that this mapping is an alternating n -linear form. As, by virtue of 3 7, no. 4, Proposition 7, the A-module of alternating n-linear forms is canoni- cally isomorphic to the dual of An(M) and A'(M) is isomorphic to A, it is seen that every alternating n -linear form on M" can be written (x,, . . ., x,) H a det(x,, x2,. . ., x,) x1 A x2 A . . A x, = det(x,, x2,. . ., x,) el A e2 A . . . A en for some u E A. PROPOSITION 2. Let M be a free A-module with a jinite basis (et) endomorphism of M. For every sequence (x,) , of n elements of M, (5) , , and v an det(v(xl), . . ., ~(x,)) = (det v)det(x,, . . ., x,). If u is the endomorphism of M such that u(ei) = x, for all i, then V(Xi) = (v O .)(et) and (5) therefore follows from (2). 2. CHARACTERIZATION OF AUTOMORPHISMS OF A FINITE-DIMEN- SIONAL FREE MODULE THEOREM 1. Let M be a Jinite-dimensional free A-module and u an endomorphism of M. The following conditions are equivalent: (a) u is bijective; (b) u is right invertible (11, 5 1, no. 9, Corollary 1 to Proposition 15); (c) u is left invertible (11, tj 1, no. 9, Corollary 2 to Proposition 15); (d) u is surjective; (e) det u is invertible in A. Let (eJIGtGn be a basis of M. Ifx, = u(e,) for 1 < i 6 n, then x1 A x2 A . a A x, = (det u)el A e2 A By 5 7, no. 9, Theorem 2 a necessary and sufficient condition for the x, to form a basis of M is that det u be an invertible element of A; this proves the equivalence of (a) and (e). Observe that (a) obviously implies each of conditions (b), (c) and (d); it remains to prove that each of conditions (b), (c) and (d) implies (e). Now, if there exists an endomorphism v of M such that vou = 1, or u 0 v = l,, then (det v)(det u) = 1 and hence det u is invert - ible in A. If u is surjective, so is A"(u) (3 7, no. 2, Proposition 3), in other words -. A en. 523 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS the homothety of ratio det u in A is surjective, which immediately implies that det u is invertible. PRO~OSITION 3. Let M be a jnite-dimensional free A-module. For every endomorphism u of M, the following conditions are equivalent: (f) u is injective; (g) det u is not a divisor of zero in A. With the same notation as in the proof of Theorem 1, for u to be injec- tive, it is necessary and sufficient that the xi be linearly independent. By fj 7, no. 9, Proposition 12, it is necessary and sufficient for this that the relation bl A x2 A . . . A x, = 0 (with h E A) imply A = 0. But this is equivalent to A(det u) = 0 since el A . - A e, is a basis of A"(M) ; whence the proposi- tion. Remark. When A is a field, condition (e) of Theorem 2 is equivalent to condi- tion (g) of Proposition 3, since they both mean that det u # 0. In this case therefore all the conditions of Theorem 1 and Proposition 3 are equivalent (cf. 11, fj 7, no. 4, Corollary to Proposition 9). 3. DETERMINANT OF A SQUARE MATRIX DEFINITION 2. Let I be afinite set, A a commutative ring and X a square matrix of type (I, I) over the ring A (11, 5 10, no. 7). The determinant of the endomorphism u of the A-module A', whose matrix with respect to the canonical basis ofA1 is X, is called the determinant of X and denoted by det X. If X = (<LJ)(i,,)EI I and (ei)iEI is the canonical basis of AI, the endomorphism u is then given by When I = (1, n) c N, if we write xi = u(ei) for i E I, the determinant of X is then defined in the relation (6) x, A x2 A . . . A x, = (det X)el A e2 A . . . A e, in other words, det X is equal to the determinant det(x,, x2, . . . , x,) with respect to the canonical basis of A". Consequently: PROPOSITION 4. For n vectors x,, . . ., x, ofAn, let X(x,, . . . , x,) denote the square matrix of order n whose i-th column is xi for 1 < i < n. Then the mapping (xI, . . . , x,) H det(X(x,, . . . , x,) of (A")" into A is alternating and n-linear. In particular, the determinant of a matrix two of whose columns are equal is zero. If a permutation is performed on the columns of a matrix, 5 8.3 DETERMINANT OF A SQUARE MATRIX the determinant of the new matrix is equal to that of the old multiplied by E,. If to one column of a matrix is added a scalar multiple of a column of a different index, the determinant of the new matrix is equal to that of the old. More generally, let M be a free A-module of finite dimension n and Iet (eJiSI be a basis of M; for every automorphism u of M, if X is the matrix of u with respect to the basis (e,) , then ( 7) det(u) = det(X) as follows immediately from the definitions. When I = (1, n), the determinant of X is also denoted by or simply det(&,) if this causes no confusion, or also El1 La .*. Eln 621 Ex2 . . . Ean . . . . . . . En1 Sna *. . 4nn When X = 1, the matrix X is called unimodular. Examples. (1) The determinant of the empty matrix is equal to 1; the determinant of a square matrix of order 1 is equal to the unique element of this matrix. For a matrix of order 2 El1 512 I E21 E22 I = 511E22 - 512E21. We transIate into the language of matrices some of the results of nos. 1 and 2: PROPOSITION 5. If X and Y are two square matrices over a commutative ring A with the same finite indexing set, then det(XY) = (det X)(det Y). (8) 524 525 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS For X to be invertible, it is necessary and suficient that det X be an invertible element of A, and then (9) det(X-l) = (det X)-l. COROLLARY. Two similar square matrices have equal determinants. This follows immediately from no. 1, Proposition 1 and no. 2, Theorem 1. If P is an invertible square matrix, then det(PXP-l) = det X by (8) and PROPOSITION 6. For the columns of a square matrix X of finite order to be linearly independent, it is necessary and suficient that det X be not a divisor of zero in A. (9) * This follows from no. 2, Proposition 3. 4. CALCULATION OF A DETERMINANT Lemma 1. Let A be a commutative ring and M a free A-module with a basis (el),€ J, where the indexing set J is totally ordered. For every integer p < Card(J), every alternating @-linear function f: MP -+ N (where N is an A-module) and every family ofpelementsx, = C E,le, ~JM (1 < i < p), jEJ (10) f(X1, x2, * . -, xp) 8 8.4 CALCULATION OF A DETERMINANT where (j,) , rum through the set of strictly increasing sequences of p elements ofJ. Now f(x1, * - -9 Xp) = (jk) c Ejl.lEjz.Z * * * E,p,Pf(e,l, e,,,. . *> e,,) where (j,),,,,, runs through all the sequences of p elements of J; it then suffices to apply Corollary 1 to Proposition 7 of 5 7, no. 4 to$ Ejtej (1 < i < n) In particular, if J is finite and has n elements and x1 = fEJ are n elements of M, then (11) X1 A X, A -.- A Xn where ( jk)l , 4n is the unique sequence of the n elements of J arranged in increasing order, whence (12) det(x,, ~2, . . . , xn) = c €0 * Eja(l), IEA,(S). z . * . Eja(n). n- With the notation of Lemma 1, comparing formulae (10) and (12) gives Xi A X2 A ' ' * A X - det(XH, 1, XH, 2,. . .) XH,,)~, (13) - HEOp(J) where Sp(J) is the set of subsets of J with p elements and, for every subset HE s,(J), we write xH,t = C Ejiejand eH = ejl A e,, A *. - A ejp, (jk)l<k<p understood that det(XH, ,, . . . , xH,,) is taken with respect to the basis (e,k),,k,p. being the sequence of elements fEH of H arranged in increasing order, it being PROPOSITION 7. Let I be ajnite set and X = (E,jl)(j, l)EIxI a square matrix of type (I, I) over a commutative ring A. Then where (r runs through the group GI of permutations of I and E, is the signature of (I (I, 4 5, no. 7). Attention may be confined to the case where I = (1, n) c N and it then suffices to apply formula (12), where (et)l,lGn is the canonical basis of An and the x1 are the columns of X (cf. no. 3, formula (6)). In particular, for the determinant of a matrix of order 3 E31 f32 533 we have det(X) = El1522533 + 912EZ3531 + 621E32E13 - <13E22<31 - 512<21533 - <11<23532. PROPOSITION 8. For every square matrix X over a commutative ring, the determinant of the transpose matrix tX is equal to the determinant of X. Suppose that X is of type (I, I). For every ordered pair of permutations (r, T of GI, (since multiplication is commutative) JJ ~(t),t = JTT Eo(T(j),,Tt,). In particular take T = cr-,; using the fact that E,-L = E,, it is seen that which proves the proposition. 526 527 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS COROLLARY 1. For n vectors x,, . . . , x, of A”, let Y (x,, . . . , x,) denote the square matrix oforder n whose i-th row is xi, for 1 < i < n. Then the mapping (xl, . . . , x,) H det( Y (xl, . . . , x,) from (A”)” to A is alternating and n -linear. COROLLARY 2. For a square matrix X ofjnite order over a commutative ring A the following conditions are equivalent: (i) the rows of X are linearly independent; (ii) the columns of X are linearly independent; (5) det X is not a divisor of zero in A. This follows immediately from no. 3, Proposition 6 and Proposition 8 MINORS OF A MATRIX 5 8.5 above. COROLLARY 3. Let u be an endomorphism of ajnite-dimensional free A-module M and fu the transpose endomorphism of the dual M* (11, 3 2, no. 5, Definition 5); then (16) det(‘u) = det(u). If X is the matrix of u with respect to a basis of M, ‘X is the matrix of ‘U with respect to the dual basis (II,§ 10, no. 4, Proposition 3) ; as det(u) = det(X) and det(’u) = det(’X), the conclusion follows from Proposition 8. 5. MINORS OF A MATRIX Let X be a rectangular matrix (&,),,,,,,, of type (I, J) whose indexing sets I and J are totally ordered. If H c I and K c J are finite subsets with the same number p of elements, there exists a unique increaring biection +: H --f K (Set Theory, 111, 3 5, no. 3, Proposition 6) ; we shall denote by XH,K the square matrix oftype (H, H) equal to (&b(,))(i,j)EH xH. If the elements of X belong to a commutative ring A, the determinant det(XH,,) is called the minor of the matrix X of indices H, K; these determinants (for all ordered pairs (H, K) of subsets of I and J respectively with p elements) are also called the minors of X of order p. With this notation: PROPOSITION 9. Let M be an A-module with a baris (e,)tEJ (jnite or otherwise) whose indexing set J is totally ordered. For every integer p > 0, let (eH)HE&,(J) be the corresponding basis gAP(~) (3 7, no. 8). Let (x,)~~,,~ be a sequence ofp elements of M; let 528 l and let X denote the matrix (t,,) of type (J, I). Then x1 A x2 A .-. A x - 2 (detXx,,)eH. (17) ’ - HEOp(J) where H runs through the set 5, (J) of subsets of J with p elements. no. 3. This follows immediately from formula (12) of no, 4 and formula (6) of , PROPOSITION 10. Let M and N be two free A-modules of respectiue dimensions m and n, u: M -+ N a linear mapping and X the matrix of u with respect to a basis (ei)i,i,,ofManda basis (fr)l,jsnofN. Then, foreueryintegerp < inf(m, n), the matrix of A”(u) with respect to the basis (eK)KEOp(I) of A’(M) and the basis (fH)HE&,(J) A’(N) (where we have written I = (1, m) and J = (1, n) is the matrix (det(XH,,)) Oftype (S,(J), $,(I)) (and hence with (i) rows and (;) columns). be the sequence of ele- of APliil. hv S 7 no. 2, Hence the element of the matrix of A”(u) which is in the row of index H and the column of index K is the component of index H of the element u(e,,) A . . . A u(e,p); it is therefore equal to det(XH,,) by Proposition 9. The matrix (det(XH,K)) is called the p-th exterior power of the matrix X and is denoted by A”(X). Whenp = m = n, A”(X) is the matrix with the single element det (X). PROPOSITION 11. Let M be a free A-module of jinite dimension n; for every endo- morphism u of M and every orderedpair of elements t, -q of A, I Let (e,),,,,, be a basis of M and let I = (1, n); to calculate the left hand side of (18), we must form the product (Eel + ~(4) A (Eez + ~(ez)) A . . . A (Sen + -qu(e,)) which is equal to the sum of the terms ~“-P~Pz~, where with xi = u(eJ for ~EK, xi = e, for ~EH = I - K, where the integer p runs through the interval (0, n) and, for each p, K runs through the set of 529 ZK = XI A Xz A ..- A X, 1x1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS subsetsof1 withpelements. Ifil < i, < . . < in-, (resp.jl < j, < . . . < j,) are the elements of H (resp. K) arranged in increasing order, we can write (9 7, no. 8, Corollary 1 to Theorem 1 and formula (19)) ZK = pH,Kei, A el, A . . A A u(ej,) A . . A u(ejp). But if X is the matrix of u with respect to the basis (e,) , then by Proposition 10 u(e,i) A * . . A u(efP) = LEOp(I) and hence 2 (det XL,K)eH A eL. zK = pH.K LE apcl, NOW H n L # except for L = K; it therefore follows from 3 7, no. 8, formula (20) that zK = (det XK,K)el A e2 A . . . A en and formula (18) then follows from Proposition 10 and the definition of the trace of a matrix (11, 3 10, no. 11, formulae (49) and (50)). COROLLARY. Under the same hypotheses as in Proposition 11, for the endomorphism A (u) of the A-module A (M) (19) Tr(A(u)) = det(1, + u). It suffices to replace E, and q by 1 in (18) and observe that the matrix of A(u) with respect to the basis of the eH (H E s(1)) is the diagonal matrix of the matrices of the Ak(u) for >/k O (11, 10, no. 7, Example IV). 6. EXPANSIONS OF A DETERMINANT Let I be a totally ordered finite indexing set. For every subset H of I let H' denote the complement I - H. Let X = (Eft) be a square matrix of type (I, I), which can be considered as the matrix of an endomorphism u of M = A' with respect to the canonical basis (eJIEI of M. Let n = Card(1) and let H be a subset of I with q < n elements and K a subset of I with n - q elements; then we can write (no. 5, Proposition 10) (~~(u)) (eH) = 3 det(~R, H)eR (An-q(u))(eK) = det(XS.K)eS where R (resp. S) runs through the set of subsets of I with q (resp. n - q) elements. It follows from 3 7, no. 8, formulae (19) and (20) that eR A es = 0 7 I , I I I 1 530 9 8.6 EXPANSIONS OF A DETERMINANT except when S = R', whence the formula where R runs through the set S,(I) of subsets of I with q elements. If we take K = H', it follows from the definition ofA"(u) (3 7, no. 2, for- mula (4)) and § 7, no. 3, Corollary 1 to Proposition 5 that the right hand side of (20) is pH,A,I\n(u)(eI). Hence (no. 1, formula (1) and 7, no. 2, formula (4) ) (21) WX) = PH,H REOp: 2 PR, R* det(~R, H)det(~R,, w). If on the other hand K # H', then H n K # o ; as the left hand side of (20) is +A"(u)(eH A eK), it is zero, whence 2 PR,R' det(XR,H)det(XR,,K) = 0 for K # H'. (22) R The right hand side of (21) is called the Laplace expansion of the determinant of the matrix X by the q columns whose indices belong to H and the n - q columns whose indices belong to the complement H' of H. The minors det(X,,,) and det (XR,, Hf) are sometimes called complementa y. An important simple case of the Laplace expansion is that where I = (1 , n) and q = 1, hence H = {i}; for every subset R = {j} of I with one element then det XR, = Ell. The minor of det XRt, H' is the determinant of the square matrix derived canonically (no. 5) from the matrix obtained by suppressing in X the row of indexj and the column of index i. We denote this square matrix by X". Obviously pH,w = (-1)l-l and = (-1)'-'; therefore (21) becomes in this case det X = 2 ( -l)l+jE,,l det(X'*) 1=1 (23) and we obtain similarly from (22) 5 (-1)'E,, det(Xfk) = 0 for k # i. Formula (23) is known as the expansion of the determinant of X by the column of index i. The scalar ( - l),+j det(Xfl) is called the cofactor of indices j and i (or, by an abuse of language, the cofactor of Eft) in X. The matrix of cofactors of X is the matrix f=l (24) Y = (( - l)l+' det(Xfl)) (25) 53 1 V(‘51, ‘52, *. ., ‘5n) = S 8.6 2, we subtract from the row of index k the row of the value of the determinant is unaltered EXPANSIONS OF A DETERMINANT For each index k index k - 1 multiplied by 1 1 ... 1 C1 c2 . . . ‘5” <: ‘5; . . . ‘5; <:-I <;-I ... ci-1 . . . . . . . . and hence V(C1, ‘52, . . . > C,) = whence, expanding by the first column and then taking out the factor ck - C1 from the column of index k - 1 from the minor thus obtained (2 < k < n) which establishes (29) by induction. an “upper triangular matrix of matrices” (11, 3 10, no. 7, Example IV) (2) Consider a square matrix of order n which is presented in the form of X= (Y T) 02 We show that det X = (det Y) (det 2). Let n be the order of the matrix X, h that of Y, (e,),,,,, the canonical basis of A“ and xi (1 < i < n) the columns of X; the hypothesis implies that the columns xl,. . ., x, belong to the submodule of An with basis el,. . ., eh and then by definition (no. 3, formula (6)) x1 A x2 A . +. A xh = (det Y)e, A e2 A . . . A e,,. On the other hand, for every index i > h, we can write xi = yi + z,, where yi is a linear combination of e,., e2, . . . , e,, and z, is a linear combination of eh+l,. . ., en. By (SO), ~1 A x2 A A xh A yi = 0 for all i > h, there - fore (30) x1 A X, A But by definition A x, = (det Y)el A e2 A A eh A zh+l A A z,. Zh+l A Zh+2 A **’ A 2, = (det z)eh+l A eh+2 A *.. A e, whence formula (30). 532 533 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS By induction on p, it follows that if X is of the form of an upper triangular matrix of matrices : and hence, if det L is not a divisor of zero in A, systems (34) and (37) are equivalent; if det L is invertible, the unique solution of (34) is given by (39) E,, = (detL)-l(detL,) (1 < i < n). A system (34) such that det L is invertible is also called a Cramer system. In particular let y = 0; it then follows from Proposition 3 of no. 2 that: PROPOSITION 14. For a homogeneous linear system of n equations in n unknowns over a commutative ring to admit a non-zero solution, it is necessary and suficient that 5 8.7 Let L be the square matrix (A,,) of order n; by identifying as usual the matrix with one column consisting of the (resp. the qi) with an element x = (ti) of A" (resp. the element y = (1,) of A"), system (34) may also be written (11, fj 10, no. 3, Proposition 2) APPLICATION TO LINEAR EQUAI'IONS (35) L.x = y. Let u be the endomorphism x ++ L. x of A", with L as matrix with respect to the canonical basis; to say that equation (35) has (at least) one solution for ally E A" means that u is surjective; Theorem 1 of no. 2 then implies the following proposition : (31) det X = (det Xll) (det X22) . . . (det X,,). This can be applied in particular to a triangular matrix (where all the Xi, are of order 1) and more particularly to a diagonal matrix: (32) det(diag(ul, u2,. . ., u,)) = ulu2.. . u,. endomorphism of M and u' an endomorphism of M'. Then (33) det(u @ u') = (det u)"'(det u')". For we can write u @ u' = (u @ 1 M') 0 (1 hl @I u') and are then led to the case where one of the two endomorphisms u, u' is the identity. For example if u' = lw and X is the matrix of u with respect to a basis (e,) of M, then the matrix of u @ lM, with respect to the tensor product of (e,) and a basis of M' can be written as a matrix (with n' rows and n' columns) of matrices with n rows and n columns /x 0 ... o\ (3) Let M, M' be two free A-modules of respective dimensions n, n', u an (0 x ..-. 1) 0 0 ... whence, by virtue of Example 2, det(u @ lM,) = (det X)"' = (det u)"' which immediately gives formula (33). 7. APPLICATION TO LINEAR EQUATIONS Consider a system of n scalar linear equations in n unknowns over a (com- mutative) ring A (11, 3 2, no. 8) : (34) j=l $At- tj j - 71 (1 < ' < n). 534 535 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 8. CASE OF A COMMUTATIVE FIELD All the above applies when the ring A is a commutative field; but there are simplifications and additional results. Thus Proposition 12 of 4 7, no. 9 can be formulated in this case as follows: PRO~OSITION 15. Let E be a vector space over a commutative field; for p vectors xi E E (1 < i 6 p) to be linearly independent, it is necessary and sujicient that x1 A X2 A *.* A xp # 0. COROLLARY . Let X be a matrix of type (m, n) ouer a commutatiue jeld. The rank of x is equal to the greatest integer p such that there exists at least one minor of X of order p which is # 0. The rank of X is the maximum number of linearly independent columns of x (11, $ 10, no. 12, Definition 7). The corollary then follows from Proposition 15 and formula (17) of no. 5. Consider now the case of a system of m linear equations in n unknowns over a commutative jeld K : (41) j=l 2AE- 11 j - ?i (1 < i G 4. PROPOSITION 16. Let L = (Aij) be the matrix (of type (m, n)) of system (41). Let M be the matrix of type (m, n + 1) obtained by adding to L the (n + 1)-th column (ql) (11, $ 10, no. 1). Letp be the rank of L (calculated by applying the Corollary to Proposition 15). Suppose that the minor A of L, the determinant of the matrix by sup- pressing the rows and columns of index >p + 1 in L, is #O (which is alwayspossible by means of a suitable permutation on the rows of L and a suitable permutation on the columns of L). Then, for system (41) to have at least one solution it is necessary and su$fcient that all the minors of order p + 1 of M, the determinants of the submatrices of order p + 1 of M whose columns have indices 1, 2, . . . , p and n + 1, be zero. If this is so, the solutions of system (41) are those of the system consisting ofthejrst p equations; ifthey are written all the solutions of this system are obtained by taking for the Ek of index k > p arbi- tray values and applying Cramer's formulae (no. 7, formulae (37)) to calculate the E, of indexj G p. We know (11, $10, no. 12, Proposition 12) that for the system (41) to have at least one solution, it is necessary and sufficient that the matrices L and M have the same rank. With the rows and columns of L permuted to satisfy the 3 8.9 THE UNIMODULAR GROUP sL(n, A) condition of the statement, let ai (1 < i 6 p) denote the first p columns of L and y = (qi) the (n + 1)-th column of M; since all the columns of L are by hypothesis linear combinations of the a,, to say that M has the same rank p as L means that y is a linear combination of the ai, or also (Proposition 15) that a, A ..- A a, A y = 0. The possibility condition in the statement is the translation of the latter relation, taking account of formula (17) of no. 5. Moreover, since the first p rows of M are linearly independent, the rows of index >p are linear combinations of them and hence every solution of (42) is also a solution of (41). The last assertion is then an immediate consequence of Proposition 13 of no. 7. 9. THE UNIMODULAR GROUP SL(n, A) Let M,,(A) denote the ring of square matrices of order n over A. Consider the mapping det:M,(A) +A. The group GL(n, A) of invertible elements of M,,(A) (isomorphic to the group of automorphisms of the A-module A" (11, $ 10, no. 7)) is just the inverse image under this mapping of the multiplicative group A* of invertible elements of A (no. 3, Proposition 5). Note on the other hand that the mapping det : GL(n, A) -+ A* is a group homomorphism (no. 3, Proposition 5). The mapping det:M,,(A) -+A is moreover surjective (and therefore so is the homomorphism det: GL(n, A) + A*) ; since, for all A E A, det(diag(A, 1,. . ., 1)) = A by virtue of formula (32) of no. 6. The kernel of the surjective homomorphism det:GL(n, A) + A* is a normal subgroup of GL(n, A), which is composed of unimodular matrices; it is denoted by SL,,(A) or SL(n, A) and is often called the unimodular group or special linear group of square matrices of order n over A. In this no. we shall examine the case where A is a jeld. Recall that for 1 < i < n, 1 6 j < n, Eij denotes the square matrix of order n all of whose elements are zero except the one in the row of index i and the column of index j, which is equal to 1 ; with I,, denoting the unit matrix of order n, we write Bij(A) = I,, + hEij for every ordered pair of distinct indices i,j and all A E A (11, $ 10, no. 13). PROPOSITION 17. Let K be a commutativejeld. The unimodular group SL(n, K) is generated by the matrices Bij(A) with i # j and A E K. By 11, 4 10, no. 13, Proposition 14, we know that every matrix in GL(n, K) is a product of matrices of the form B,,(A) and a matrix of the form diag(1, 1,. . ., 1, a) with a E K*. Now it is immediate that det(B,(h)) = 1 and det (diag(1, . . ., 1, a)) = a (no. 6, Example 2); whence the proposition. 536 537 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS COROLLARY . The group SL(n, K) is the group of commutators of GL(n, K), except in the care where n = 2 and where K is afield with 2 elements. As SL(n, K) is the kernel of the homomorphism det of GL(n, K) to a commu- tative group K*, SL(n, K) contains the commutator group I? of GL(n, K) (I, $6, no. 2). To prove that SL(n, K) = I?, it will suffice, by Proposition 17, to show that, for all A E K*, &,(A) belongs to I?. Now, B,,(h) is a conjugate of Bi,(l) in GL(n, K) since B,,(h) = Q.Bi,(l) .Q-l, where Q denotes the matrix with respect to the canonical basis (e,) of the automorphism u of K" such that u(eJ = Aei, v(ek) = e, for k # i. On the other hand, let u,, (for i #j) be the automorphism of K" such that u,,(e,) = -e,, u,,(e,) = e,, uij(tk) = ek for k $ {i,j}, which belongs to SL(n, K); then B,(A) = Ui,B1,( -A)Uil, where u,, is the matrix of ui, with respect to the canonical basis. Similarly, if 1 < i < j, then &,(A) = UliBi,(A)UG1 and finally, for 2 < j, B12(A) = U2,B1,(A)UG1. This proves that all the &,(A) have the same image s in GL(n, K)/r and it re- -ins to show that s is the identity element. Suppose first that K contains an element A distinct from 0 and 1; then 1 = A + (1 - A), the two terms on the right hand side being # 0; the relation &(l) = Bl2(A)Bl2(l - A) shows that s2 = s and hence s is the identity ele- ment. Suppose now that n 2 3. The product B21(l)B31(l) is the matrix of an automorphism u of K" such that .(el) = el + e2 + e3, u(e,) = e, for i # 1. If S is the matrix of the automorphism u' of K" such that u'(e,) = e2 + e3, u'(ei) = e, for i # 2, then S.B21(l)B31(l) .S-l = B21(1); we also deduce that s2 = s, which completes the proof. Remarks. (1) GL(2, F,) = SL(2, F,) ; this is a solvable group of order 6, whose commutator group is of index 2 (11, 9 10, Exercise 14). (2) With the same notation as above, it can be proved as in I, 3 5, no. 7, Proposition 9 that, for i < j, j - i > 1, ui, = u,-~, ,u,, j-luy-ll,,;. hence the group SL(n, K) is generated by the matrices B,,(A) and U,,,,, for 1 < a < n - 1. 10. THE A[X]-MODULE ASSOCIATED WITH AN A-MODULE FNDOMOR- PHlSM Let M be an A-module and u an endomorphism of M. Consider the poly- nomial ring A[X] in one indeterminate X over A. For every polynomial fi E A[X] and all x E M, we write (43) P.x = P(4(4- As (pq) (u) = p(y) o q(u) for two polynomials p, q of A[X], and A[X]-module structure is thus defined on M; the set M, with this structure, will be denoted by Mu; the A-module structure given on M is obtained by restricting the ring of operators of Mu to A. Note that the submodules of Mu are just the submodules of M which are stable under u. 538 7 THE A[X]-MODULE ASSOCIATED WITH AN A-MODULE ENDOMORPHISM 5 8.10 As the mapping (p, x) HP.X of A[X] x M into M is A-bilinear, it defines canonically an A-linear mapping +: A[X] @A M -+ M such that (44) NP 8 4 = P.x = P(u)(+ On the other hand, A[X] @A M has canonically an A[X]-module structure (11, 3 5, no. 1); we shall denote this A[X]-module by M[X]; the mapping +: M[X] --f Mu is A[X]-linear since, for p, q in A[X] and x E M, 4MP 84) = +((?I@) @x) = (qP)*x = du)(P(.)(x)) = 4.+(P @XI* 4P.4 = .(P(u)(x)) = (.P(X))(X) = P.44. zi(P @ x) = p @ 4%). Moreover, u is an A[X]-endomorphism of Mu, for Finally, an A[X]-endomorphism ii of M[X] is defined by writing (11,s 5, no. 1) (45) Moreover it follows from formulae (44) and (45) that that A[X]-I' inear mappings u, ii and + are related by the relation (46) +oii=uo+. Let + denote the A[X]-endomorphism X - 1 of M[X], so that +(p @ x) = (Xp) 8 x - p 8 ~(x). We have the following proposition: PROPOSITION 18. The sequence of A[X]-homomor~hisms (47) M[X] M[X] -----f Mu -+ 0 is exact. + As +(l @ x) = x for all x E M, clearly + is surjective; on the other hand, +(WP @ 4) = X.+(P @ 2) = u(+(P @ XI), in other words, + OX = u 0 + = + o zi by (46) ; this proves that 4 0 4 = 0. It remains to verify that Ker 4 c Im 8. For this note that, since the monomials xk (k > 0) form a basis of the A-module A[X], every element z E M[X] can be written uniquely in the form z = 2 Xk 8 xk, where (xk) is a family of ele- ments of M, of finite support. If z E Ker $I, then +(z) = uk(xk) = 0 and we can write k = 2 (xk 8 xk - 1 8 Uk(xk)) = (xk - ik)(1 @ xk). k But as the A[X]-emdomorphisms X and ii of M[X] are permutable, then 539 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Xk - nk = (X - 6) 0 (2 X’Uk-j-1 1 which proves that there exists a ~EM[X] such that z = +(y). Now let M’ be another A-module and u’ an endomorphism of M‘; let ML,, +‘, ti’, +‘ be the module and mapping obtained from M’ and u’ as Mu, +, ii, + are obtained from M and u. Then: PROPOSITION 19. For a mapping g of M into M’ to be an A[X]-homomorphism ofM, into Mi, it is necessary and suficient that g be an A-homomorphism of M into M’ such that g o u = u’ o g. When this is so, if2 is the A[X]-homomorphism of M[X] into M’[X] equal to lAIx1 8 g (11, 5 5, no. l), the diagram 6 + M[X] + M[X] - Mu--+ 0 I I I ti b, (48) M’[X] + M’[X] --+ Mh. --+ 0 is commutative. The condition g o u = u’ o g is obviously necessary by (43) for g to be an A[X]-homomorphism; it is sufficient, for it implies by induction that g o un = u’n o g for every integer n > 0. On the other hand, for all x E M and all fJ E A[X], 9’(i(P 63 4) = 9”P 63 g(4) = P(U’)(dX)) = g(p(u)(x)) = d+(P 63 4) E’(g(P 63 4) = G’(P €3 g(4) = P 63 u’(g(x)) = P 63 g(u(x)) = E(U(P 63 4) and which proves the commutativity of diagram (48). 11. CHARACTERISTIC POLYNOMIAL OF AN ENDOMORPHISM Let M be a free A-module of dimension n and u an endomorphism of M. Con- sider the polynomial ring in two indeterminates A[X, Y] and the A[X, y]- module M[X,YI = A[X,YI mAM; let u‘ be the endomorphism of the A[X,Y]-module M[X,Y] canonically derived from u (11, $5, no. 1). It follows from no. 5, Proposition 11 that (49) det(X - YE) = A (-I)jTr(A’(u))X”-V for if U is the matrix of u with respect to a basis (eJl of E with respect to the basis (1 63 e,) Tr(A’(Z)) = Tr(A’(u)). of M, U is the matrix of M[X, yl, hence 540 $9.1 NORMS AND TRACES RELATIVE TO A MODULE DEFINITION 3. Let M be aJnite-dimensional free A-module and u an endomorphism of M. The determinant of the endomorphism X - ii of the free A[X]-module M[X] is called the characteristic polynomial of u and is denoted by x,(X). If M is of rank n, it follows from (49) that n x,(X) = C (- l)jTr(A’(u))Xn-j (50) f=O for det(X - YE) = det(X.1, + YU) and det(X - U) = det(X.1, - U). It is therefore seen that xu(X) is a monic polynomial of degree n, in which the coefficient of Xn-l is -Tr(u) and the constant term is ( -l)n det(u). PROPOSITION 20 (“Cayley -Hamilton theorem”). For every endomorphism u of a jinite-dimensional free A-module, xu(.) = 0. In the notation of Proposition 18 (5 3, no. lo), for all x E M, xu(u)(x) is the image under + of xu(X) x. But if v is the endomorphism of M[X], the co- transpose of X - ii (no. 6), then xu(X) 63 x = xu(X)(l 63 4 = (X - U)(v(l 63 4) and the conclusion follows from Proposition 18 of no. 10. $9. NORMS AND TRACES Throughout this paragraph, K will denote a commutative ring and A a unital associative K-algebra. Every A -module will be assumed to be given the K-module structure obtained by restricting the scalars to K. 1. NORMS AND TRACES RELATIVE TO A MODULE DEFINITION 1. Let M be an A-module admitting ajnite basis as a K-module. For all a E A, let aM be the endomorphism x H ax ofthe K-module M. The trace, determinant and characteristic polynomial of aM are called respectively the trace, norm and characteristic polynomial of z relative to M. The trace and norm of a are therefore elements of K, denoted respectively by TrM,(a) and NMIK(a) ; the characteristic polynomial of a is an element of K[X], denoted by PcMIK(a; X). We omit K in the above notation when there is no risk of confusion. From the properties of the trace and determinant of an endomorphism (11, 8 4, no. 3 and 8 8, no. 1) we obtain the relations Tr,(a + u’) = TrM(a) + Tr,(a’) (1) 541 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS TrM(aa’) = TrM(U’U) NM(uu’) = NM(a)NM(a’) for all a, a’ in A. Let (eJIG,<, be a basis of the K-module M and (rntj(a)) the matrix of the endomorphism aM with respect to this basis. The functions mij are linear forms NM(a) = det(m,f(a)) Pc(a; X) = det(GifX - m,,(a)). It follows from the method of calculating a determinant (3 8, no. 11, formula (50)) that *. + c, where (8) cl = -TrM(a), c, = (-l)*NM(u). For h E K, (7) Pc&; X) = xn + clX*-l + (9) TrM(A) = n.h, NM(A) = A”, PcM(A; X) = (X - A),. Let K be a commutative K-algebra. Let M’ = K‘ BK M and A’ = K I& A, so that M’ has an A’-module structure (3 4, Example 2). As a K‘-module M’ admits the basis consisting of the 1 @ e, (1 6 i < n) and the matrix of aM with respect to (e,) is equal to the matrix of (1 @ with re- spect to (e;). Then Trw(l @a) = TrM(a).l, NMr(1 @ a) = NM(a).l PCM,(l @ a; x) = PcM(a; x). 1 (12) for all a E A, where 1 denotes the unit element of K’. If in particular we take K’ = K[X], then (13) 2. PROPERTIES OF NORMS AND TRACES RELATIVE TO A MODULE If M and M’ are two isomorphic A-modules with finite bases over K, then, for all aEA, (14) TrM,(a) = TrM(a), N,.(a) = NM(a), Pc~(u; X) = Pc,(u; x) for iff is an isomorphism of M onto M’, the matrix of aM with respect to a basis B of M over K is the same as the matrix of aw with respect to the basis f (B) of M‘. PcMig(a; X) = NMEwKc~X - a). 542 9 9.3 PROPOSITION 1. Let M = Mo 3 MI 3 . . .I M, = (0) be a decreasing sequence of submodules of an A-module M such that each of the K -modules P, = M,_,/M, (1 6 i < r) admits ajnite basis. Then the K-module M admits ajnite basis and NORM AND TRACE IN AN ALGEBRA Let BI be a basis of P, over K; then a system of representatives B, of B; (mod. M,) is a basis of a supplementary submodule of the K-module M, in the K-module M,.-l (11, 3 1, no. 11, Proposition 21). The union B of the B, (1 < i < r) is a basis of M over K. Let X,, be the matrix of the endomorphism aPc with respect to the basis BI. It is immediate that the matrix of aM with respect to B is of the form and the proposition follows from formulae (4), (5) and (6) of no. 1 and for- mula (31) of § 8, no. 6. PROPOSITION 2. Let A, A’ be two K-algebras, M an A-module and M’ an A’-module. Sufipose that M and M’ are free K-modules of respective dimensions n and n’ and con- sider M mK M’ as an (A BK A‘)-module ($4, no. 3, Example 2). Then, for a E A and a’ E A’, TrMBM(a @ a‘) = TrM(U)TrM,(U’) NM@M,(a @ a’) = (NM(a))n’(NM’(a’))n. (16) (17) Formula (16) follows from 11,s 4, no. 4, formula (26) and formula (1 7) from 8 8, no. 6, formula (33). 3. NORM AND TRACE IN AN ALGEBRA DEFINITI ON 2. Let A be a K-algebra which is ajnite-dimensional free K-module. For every element a E A, the trace (resp. norm,? resp. characteristic polynomial) of a relative to A and K is the trace (resp. determinant, resp. characteristicpolynomial) ofthe endomorphism x I+ ax of the K -module A. t This notion should not be confused with the notion for norm in an algebra over a valued field (General Topology), IX, 3 3, no. 7). 543 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The trace, norm and characteristic polynomial of a E A relative to A and K are denoted by TrAIK(a), NA/K(u) and PcAIK(a; X); we omit K and even A from this notation when there is no risk of confusion. Note that the trace (resp. norm, characteristic polynomial) of a E A is just the trace (resp. norm, charac - teristic polynomial) of a relative to the A-module A,. Suppose that A is the product A, x A, x . . . x A, of a finite number finite -dimensional algebras over K which are free K -modules. Using the above remark and Proposition 1 of no. 2, we have, for every element a = (al,. . . , a,) E A, (18) m pcA/K(a; X) = I-I 1=1 pcA,/K(ai; XI. Similarly, Proposition 2 of no. 2 shows that if A and A' are two algebras, which are free K -modules, of finite dimensions n, n' respectively over K, then, for a E A, a' E A'. (20) NABN(~ @ a') = (NA(a))"'(NA,(a'))". Finally let A be a finite-dimensional algebra over K which is a free K- module, h a homomorphism of K into a commutative ring K' and A' = A,,,, the K'-algebra derived from A by extending the scalars by means of h. It follows from formula (12) of no. 1 that, for all a E A, Tr~,/Kp(l @ a) = h(TrA/K(a)), NA,/r(l @ a) = h(NA/K(a)) PcA,/K,(l @ a; X) = ~(PcA,K(u; x)) (21) where h is the homomorphism K[X] --f K'[X] derived from h. In particular, for K' = K[X], we obtain, writing A[X] = A BK K[X], (22) PcA/K(a; = NAIXI/KIX](X - '1' More generally, if K' is a commutative K-algebra and A' = A @K K', then, for all x E A', Pc,/,(u; X) = NAp/r(~ - a). Examples. (1) Let A be a quadratic algebra over K of type (a, p) and (el, e,) a basis of type (a, p) (5 2, no. 3). For x = Eel + qe,, TrA/K(X) = 25 + pq and NA/K(x) = 5' + f@ - aq,; these functions are therefore identical with the Cayley trace and norm of x (5 2, no. 24). (2) Let A be a quaternion algebra over K. A direct calculation allows US to 544 PROPERTIES OF NORMS AND TRACES IN AN ALGEBRA 8 9.4 verify that TrA/K(X) = 2T(x) and NA/K(x) = (N(x))~, where T and N are the Cayley trace and norm (8 2, no. 4). (3) Let A = M,(K) and let the canonical basis (Eij) ofA (11, 4 10, no. 3) be arranged in lexicographic order. Then it is immediately seen that for every matrix X = 2 EijEi, the matrix (of order n2) of the endomorphism Y H XY is of the form 2. I whence TrA/K(.Y) = n.Tr(X) and NA/K(X) = (det(X))". 4. PROPERTIES OF NORMS AND TRACES IN AN ALGEBRA PROPOSITION 3. Let A be a K-algebra admitting ajnite basis. For an element a E A to be invertible, it is necessary andsuficient that its NA/K(a) be invertible in K. If a admits an inverse a' in A, then NA/K(~)NA/K(~') = N~iK(aa') = NA/K(~) = 1 by formula (3) of no. 1. Conversely, if NA/K(a) is invertible, the endomorphism h: x ++ ax is bijective (4 8, no. 2, Theorem 1). Then there exists a' E A such that aa' = 1 ; then h(a'a - 1) = aa'a - a = (aa' - 1). = 0, whence aa' = 1 since h is injective. Hence a' is the inverse of a. PROPOSITION 4. Let A be a K-algebra admitting a finite basis. For all a E A, PcAIK(a; a) = 0. This follows immediately from the Cayley -Hamilton theorem (5 8, no. 11, Proposition 20). PROPOSITION 5. Let A be a K-algebra and m a two-sided ideal ofA. Suppose that A, = A/m is a free K-module ofjnite dimension n, that there exists an integer r > 0 such that rn' = (0) and that mi- l/mi is a free A,-module ofjnite dimension s, for 1 < i < r. Lets = s1 + . . + s, and for all a E A let a, denote the class ofa mod. m. Then A is a free K-module of dimension ns and, for all a E A, TrA(a) = S*TrAo(aO), NA(a) = (NAo(a0))' (23) PcA(a; x) = (PcA,(a,; x))'. By virtue of 11, 5 1, no. 13, Proposition 25, mi-l/mi is a free K -module of dimension ns,. Hence Proposition 1 of no. 2 can be applied with P, = mf-l/mi; 545 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS this shows in the first place that A is a free K-module of dimension n(sl + . - + s,) = tls. Moreover, the hypothesis implies that the A-module P, is isomorphic to a direct sum of si submodules isomorphic to the A-module A,; by Proposition 1 of no. 2, therefore N,,(a) = NAo(a)Si; finally therefore N*(a) = NAo(@. In this formula NAo(a) is defined by considering A, as a left A-module and it is equal to the determinant of the K-linear mapping x H ax of A, into itself; but, as ax = a,x for x E A,, NAO(a) = NAo(a,), which completes the proof of for- mula (23) for the norm. The two others are shown analogously. PROPOSITION 6. Let A be a commutative K-algebra admitting ajinite basis over K and V an A-module admitting ajinite basis over A. Then V admits ajnite basis over K and for every A-endomorphism u of V, ifuK is the mapping u considered as a K-endo- morphism ofV, Tr(uK) = TrA,x(Tr(~))y det(uK) = NA,K(det(u)) Pc(u,; X) = NAEwKM(Pc(u; XI)- Let (a,) ct < ,,, be a basis of A over K and (e,) <, ,, a basis of V over A ; then (a,e,) is a basis ofV over K (11,s 1, no. 13, Proposition 25). On the other hand the third of formulae (24) can be deduced from the second applied to the endo- morphism X - ii of the Aw]-module A[X] @A V (3 8, no. lo). It will there- fore suffice to show the first two formulae in (24). We shall first establish the following lemma : Lemma 1. Let Xi, (1 < i < n, 1 < j < n) be n2 indeterminutes, X the square matrix (Xi,) of order n and D(Xl,, . . ., X,,,,) €ZIXll,. . ., X,,,,] the determinant det(X). On the other hand let A be a commutative ring, M,, (1 < i < n, 1 < j < n) na matrices of order m over A, which are pairwise permutable, and M the square matrix of order mn over A which can be expressed as a square matrix of matrices (11,s 10, no. 7) (24) Then the determinant ofM is equal to the determinant ofthe square matrix D (MI,, . . ., Mnn) of order m. We proceed by induction on n, the cases n = 0 and n = 1 being trivial. 546 $9.4 PROPERTIES OF NORMS AND TRACES IN AN ALGEBRA Let Z be a new indeterminate and Ni, the matrix Mi, + aijZIm (aij the Kronecker index). If D'j(Xll, . . . , X,,,,) is the cofactor of Xi, in the matrix X (3 8, no. 6), then (25) XjiDk'(X1l, . . . , xnn) = 8jkD(X11, * . , Xnn) (4 8, no. 6, formuIa (28)). We write Nlj = Df,(Nll,. . . , N,,,), which is a square matrix of order m over A[Z] and consider the product N. U, where /Nil 0 ... 0 \ Performing this product in blocks (11, 3 10, no. 5) and using formulae (25), we obtain /P N12 ... Nln\ N.U = (0 '1' .N2:) 0 N,,, . . . N,,,, where we have written P = D(Nll, . . ., N,,,,). Let which is a matrix of order m(n - 1); then (3 8, no. 6, formula (31)) (det N) (det U) = (det P)(det Q) and det U = det Nil. But by the induction hypo - thesis, det Q = det(Dl1(Nll,. . ., If,,,,)) = det Nil and by virtue of the defi- nition of the N,,, clearly det Q is a polynomial in A[Z] of degree m(n - l), whose term in Zrn(,,-l) has coefficient 1 ; it follows immediately that det Q is not a divisor of zero in the graded algebra A[Z]. We therefore conclude that det N = det(D(N,,, . . ., N,,,,)) in A[Z]; if we substitute 0 for Z in these polynomials, then det M = det(D(M,,, . . ., Mnn)). 547 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Having shown this lemma, the K-module V is the direct sum of the K- modules Ae, (1 < j < n); we write u(ej) = k=l i: cjkek. For every element xej E Ae,, with x E A, the component of u(xe,) in Aek is xcjkek; it follows that the matrix ofu, with respect to the basis (aie,) of the K-module V can be expressed in the form of a square matrix of matrices (hfjk), where Mi, is the matrix of the K-linear mapping Xej H xcjkek Of Ae, into Aek with respect to the bases (ate,)lctcm and (atek)lBtQm ofthese two K-modules (11, 8 10, no. 5). If for all t E A, M(t) denotes the matrix, with respect to the basis (ai) ,,, of A over K, of the endomorphism x H xt of A, then Mjk = M(Cjk) ; as t H M(t) is a ring homomorphism the matrices hfjk are permutable with one another. Then det uK = det(D(M,,, . . . , M,,)) by Lemma 1. But as t ++ M(t) is a ring homomorphism, D(Mll,. . . , M,,,,) is the matrix of the K-endomorphism x H x.det(cjk) of A with respect to basis (al) ; by definition its determinant is therelore NAiK(det(u)), which proves the second formula of (24). On the other hand, and the proof of Proposition 6 is complete. COROLLARY . Let A be a commutative K-algebra admitting ajnite basis ouer K and B an A-algebra admitting ajnite basis ouer A. Then B admits ajnite basis ouer K, and, for all 6 E B (“transitivity formulae”) TrB/K(b) = T~A/K(T~B/A(~)), NB/,(b) = NA/K(NB/A(~)) PcB/K(b; X) = NAEwKK~(PcB/A(~; X)). (26) This follows immediately from Proposition 6, setting V = B and u(x) = bx. Remark. Suppose that the homomorphism h H h . 1 of K into A is injective and let K be identified with its image in A; suppose that A admits a finite basis (e,),,,,, as a K-module. Let s be an automorphism of A such that s(K) = K. Let a be an element of A; then, by transporting the structure (27) TrA/K(S(a)) = S(TrA/K(a)) (28) NA,K(s(a)) = S(NA/K(a))* *Consider also a derivation D of A (3 10, no. 2) such that D(K) c K and write D(e,) = zl e,pjt where p,, E K; write ae, = 2 ejhjt with A,i E K. j=l 548 DISCRIMINANT OF AN ALGEBRA Then D(a)et + aD(et) = D(aei) = j=l 2 (D(e,)A,, + ejD(Ajt)). It follows that D(a)e, = i= 2 ejvji 3 9.5 with vjt = D(hjt) + zl (pjkhkt - AjkI*kt). As ,& (ptkhkt - Aikpkt) = O, there- fore TrAiK(D(a)) = $ D(A,,), in other words (29) TrA,K(D(a) = D(T~AIK(~) ). * 5. DISCRIMINANT OF AN ALGEBRA DEFINITION 3. Let A be a K-algebra admitting a finite basis of n elements. The dis- criminant of a sequence (xl, . . . , x,) of n elements of A, with respect to K, denoted by DalK(xl, . . . , x,), is the discriminant of the square matrix t=1 (TrA/K(xixj))l 4 t Bn. 1 <fin* Consider first a basis (e,) , , , of A over K and write (30) Qe, = Ic= i: Ctjkek with Cijk E K. Then TrA,K(et) = s= clss, whence TrA/K(eiej) = C CtjkCkss and therefore k. s 5 (31) DA/K(el,. * *? en) = det(( ~ciikCkss)lQtQn.lB~Qn ) * Now let (xt) <t <,, (XI) , , be two sequences of n elements of A and suppose that there exists a square matrix of order n, M = (mtj), with coefficients in K, such that x, = 2 mijx; for 1 < i < n. We write 3=1 = (TrA/K(XiXj)) 1 B t Bn, 1 CjC n, T‘ = (TrA/K(x~x~)) 16 t Q n, 1 < j< n- Then TrA/K(xtxj) = 2 mtpmjp TrA/K(xLXi), whence T = M. T‘.tM; the rule of multiplication of determinants therefore gives whence finally (32) det T = det M.det T‘.det tM = (det M)2 det T’ DA/K(xl, . . . , x,) = (det M)’DA/K(x(, . . ., x;). 549 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The above formula shows in particular that the discriminants of two bases of A Over K differ by the square of an inuertible element of K and therefore generate the Same (principal) ideal of K. This ideal AA/K is called the discriminant ideal of A Over K; by formula (32) the discriminant of every sequence of n elements of A which differ only in the order of the terms have the same discriminant, for the determinant of a permutation matrix is equal to & 1. Examples. (1) If A is a quadratic algebra of type (a, P) over K, then (in the notation of 3 2, no. 3) Tr(e,) = 2, Tr(e,) = P, Tr(eg) = a Tr(el) + p Tr(e,) = 2a + P2, whence DA/K(el, e,) = pa + 4a. (2) Let A = K[X]/K[X]P, where P(X) = X3 + pX + q, so that if x is the imane of X in A. 1. x, x2 form a basis of A over K and x3 = -px - q. It is seen ___ 0- -~~~ immediately that Tr(1) = 3, Tr(x) = 0, Tr(x2) = -2p, taking account of the relation x3 = -jx - q, Tr(x3) = -3q and Tr(x4) = 2p2, whence easily DA/=(l, X, 2) = -4p3 - 27q2. (3) Let A be a quaternion algebra of type (a, P, y) over K and (1, i,j, k) a basis of A of type (a, P, y); taking account of 3 3, no. 5, formula (30), it is easily found that Tr(1) = 4, Tr(i) = 28, Tr(j) = Tr(k) = 0, then DA/K(l, i,j, k) = - 16y2(P2 + 4a),. (4) LetA = M,(K) and consider the canonical basis (E,j)1,i~,,1,j4nofA over K (11, 3 10, no. 3). It is immediate that TrAlK(El,) = 0 ifj # i and TrAIK(Etf) = n for all i; it follows without difficulty that the matrix (Tr(EifEhk)) of order n2 is of the form n.P, where P is a permutation matrix, whence D*/K((4,)) = knna. 6 10. DERIVATIONS In this paragraph, and unless otherwise mentioned, the algebras considered are not assumed to be associative nor necessarily to possess a unit element; K denotes a commuta - tive ring. 1. COMMUTATION FACTORS When in this paragraph we speak of graduations without specifying them, we shall always mean graduations of type A, where A is a commutatiue group written additively. In this paragraph, a commutation factor over A with values in Z is called a commutation factor over A (3 4, no. 7, Definition 6). A commutation 550 3 10.2 GENERAL DEFINITION OF DERIVATIONS factor over A is therefore identified with a mapping E: (a, p) ++ E,~ = E(a,P) of A x A into the multiplicative group { - 1, l} such that for a, a', P, p' in A, &(a + a', B) = €(a, PI+', P) &(a, P + P') = &(a, P)c(a, P') 4% a) = 4% P). (1) It follows that ~(2a, P) = €(a, 2P) = 1. there are therefore only two such factors, the first defined by When A = Z, every commutation factor E is determined by giving ~(1, 1) ; ~(p, q) = 1 for p, q in Z (2) (3) and the second by ~(p, q) = (-1)pq for p, q in Z. 2. GENERAL DEFINITION OF DERIVATIONS Consider a commutative ring K, six graded K-modules of type A: A, A', A ", B, B', B ", and three K-linear mappings p:A x A'-+A", Al:B x A'-+B", A2:A x B'+B" such that the corresponding K -linear mappings A @KA'+A", B @xA'+B", A @KB'+B" are graded of degree 0. The image p(a, a') for a E A, a' E A is simply denoted by a. a' or even aa', and similarly for the two other bilinear mappings. The degree of a. a' is therefore the sum of the degrees of a and a'. DEFINITION 1. Giuen the above and a commutation factor E on A x A, an &-derivation (or (K, &)-derivation) of degree 6 EA of (A, A', A ") into (B, B', B") is a triple of graded K-linear mappings of degree 6: d: A -+ B, d': A'+ B', d": A" + B" such that, for every homogeneous element a E A and every element a' E A' (4) d"(a.a') = (da) .a' + ~~,~~~(~)a. (d'a'). It obviously suffices by linearity to verify relation (4) when a and a' run through -. respective generating systems of A and A'. It is often convenient to denote the three mappings d, d', d" by the same letter d (which can be justified by denoting equally by d the graded K-linear mapping of degree 6 (a, a', a") ++ (da, d'a', d"d) 551 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS of A @ A' @ A" into B @ B' @ B"). Relation (4) can then be written more simply (5) d (a. a') = (da) . a' + q,, deg(a)a. (da') . K-module of the K-module of graded linear mappings The &-derivations of (A, A', A) into (B, B', B ") of given degree form a sub- Homgr,(A @ A' @ A", B @ B' @ B"). When €(a, p) = 1 for all a, in A, we say simply derivation (or K-derivation) instead of €-derivation. Derivations form a sub-K-module of Hom,(A @A' @A", B @ B' @ B"). When A = Z and ~(p. q) = (- 1)pq, every &-derivation of even degree is a derivation; every E-derivation of odd degree is often called an antiderivation (or K-antiderivation) ; an antiderivation d therefore satisfies (6) d (a. a') = (da) . a' + ( - l)deg(a)a. (da') for a homogeneous element a E A. Remarks. (1) The notion of derivation can be defined for non-graded modules by agreeing to give these modules the trivial graduation. (2) If only &-derivations of given degree 6 are considered, the commutation factor E may be disposed of as follows: the bilinear mapping A,: A x B' + B" is modified by replacing it by the bilinear mapping A;: A x B' + B" such that, for every homogeneous a in A and all b' E B', A;(a, b') = &6.deg(a)A2(a, b')* Then d is a derivation relative to the bilinear mappings p, Al, A;. The general definition of E-derivations given above is especially used in two Case (I) : A = B, A' = B', A" = B" and the three bilinear mappings p, hl, Case (11): A = A' = A", B = B' = B", so that (for p) A is a graded algebra cases : A2 are equal to the same mapping. and the two K -bilinear mappings. (7) A,:B x A-tB, h,:A x B-+B are such that the corresponding K -linear mappings B @, A -+ B, A BK B -+ €3 are graded of degree 0. An €-derivation of degree 6 of A into B is then a graded K-linear mapping d: A -+ B of degree 6, such that for every homogeneous x in A and every y E A, we have the relation (8) (.!/I = (dx)!/ + €6, deg(a>X(d!/) ' 552 EXAMPLES OF DERIVATIONS 9 10.3 Consider in particular in case (11) the case where A is a unital associative K- algebra and Al and A, are the external laws of an (A, A)-bimodule (3 4, no. 3, Definition 3). This holds notably when A and B are two unital associative K- algebras, a unital homomorphism of graded K-algebras p : A --+ B is given and an (A, A)-bimodule structure is considered on B defined by the two external laws A,: (b, a) -bda), A,: (a, b) P(4b for a E A, b E B. Cases (I) and (I I) have the following case in common : consider a graded K- algebra A, take B = A, mappings (7) both being multiplication on A. We then speak of an E-derivation (or (K, &)-derivation) of the graded algebra A: it is a graded K-linear mapping of A into itself, of degree 6, satisfying (8) for every homogeneous x in A and ally E A. In particular if A is a graded ring, considered as an (associative) Z-algebra, we speak of the E-derivation ofthe ring A. Let A be a unital commutative associative K-algebra and B an A-module; when we speak of a derivation ofA into B, it will always be understood that we mean with the A-bimodule structure on B derived from its A-module structure; then the formula (9) d(xy) = x(dy) + y(dx) for x EA, y E A holds for such a derivation d: A -+ B. 3. EXAMPLES OF DERIVATIONS Example 1. *Let A be an R-algebra of differentiable mappings of R into R and let xo be a point of R; R can be considered as an A-module with the external law (f, a) -f(xo)a. Then the mapping f- Df(xo) is a derivation, since (Functions Ofa Real Variable, I, 1, no. 3) Example 2. *Let X be a differentiable manifold of class C" and let A be the graded R-algebra of differential forms on X. The mapping which associates with every differential form w on X its exterior differential dw is an anti- derivation of degree + 1 (Dzyerentiable and Analytic Manifolds, R, § 8) .* Example 3. Let A be an associative K-algebra. For all a E A, the mapping x ++ ax - xu is a derivation of the algebra A (cf. no. 6). Example 4. Let M be a K-module and A the exterior algebra A(M*) with its usual graduation (3 7, no. 1). *It will be seen in 4 11, no. 9 that, for all x E M, the right interior product i(x) is an antiderivation of A of degree - l., 553 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Example 5. Returning to the general situation of Definition 1 of no. 2, let be another commutative ring and p: K -+ K a ring homomorphism; let A, A', A", B, B', g" denote the graded K-modules obtained respectively from A, A', A", B, B', B" by extending the ring of scalars to K (11, 3 11, no. 5) ; we derive from p, A, and A, K-bilinear mappings - ?:A x K'+A", x,:u x A'+B", %,:A x BI-tB" by considering the tensor products by 1~ of the corresponding K -linear mappings to p, A, and A, (IT, 9 5, no. 1). Then, if d is an &-derivation of degree 6 of (A, A', A") into (B, B', B"), the mapping 2 = d @ 1;; ofA @ A' @ A" into B @ B' @ B" is an &-derivation of degree 6 of (A, A', A ") into (B, B', B"). Example 6. Let A be a graded K-algebra of type Z; a graded linear K -mapping of degree 0, d: A -+A, is defined by taking, for x, E A,(n E Z), d(x,) = nx,. This mapping is a derivation of A, since, for xp E A,, xq E A,, dbpx,) = (P + 4)XP", = d(xp)x, + Xpd(X,)* 4. COMPOSITION OF DERIVATIONS We suppose in this no. that case (I) of no. 2 holds, that is that A, A', A are three graded K-modules of type A and that we are given a K-bilinear mapping p: A x A' -+ A" corresponding to a graded K-linear mapping of degree 0, A gK A' -+ A". The graded endomorphisms f of A @ A' @ A such that f (A) c A, f (A') c A' and f (A") c A" form a graded subalgebra of the graded associative algebra Endgr,(A @ A' @ A") (3 3, no. 1, Example 2). In particular two &-derivations of (A, A', A") can be composed, but it should not be thought that the composition of two & -derivations is an E-derivation. On every graded algebra B of type A is defined the &-bracket (or simply bracket when E = 1) of two homogeneous elements u, v, by the formula (10) [u, v], = uv - E~~~~,~~~~~u (denoted simply by [u, v] if E = 1). By extending this mapping by linearity, a K-bilinear mapping (u, v) H [u, v], of B x B into B is obtained. Then, for homogeneous u and u in B [v, ~lc = -%egu,degv[% VIE. Applying this definition to the graded algebra Endgr,(A @ A' @ A"), the &-bracket of two graded endomorphisms is thus defined. PROPOS~ON 1. Let d,, d, be two E-derivations of (A, A', A") of respective degrees S,, 6,. Then the &-bracket is an &-derivation of degree 6, + S,. Moreover, ifd is an E-derivation of (A, A', A") of degree 8 and if c6, 554 cd,, dzlE = '1 dz - &61.6ad2 '1 = - 1, then da = d 0 d is a derivation. COMPOSITION OF DERIVATIONS 8 10.4 Suppose x E A is homogeneous of degree E; for all y E A', (1 1) '1 (d2 ('y)) = ( (d1d2) (') ) Y f &61, 62 + 5 (dZx) (dly) + &6~.€(~1~) (d2Y) + +6z,cx((d1d2) (Y)) taking account of formulae (1) of no. 1. Exchanging the roles of dl and d,, we obtain, after simplifications again using (1) (no. l), (d1d2) ('y) - E61,62(d2d1) ('9) = ((dldZ) ('))Y - E61,6a((d2d1) (.))y + €61 +6a. Ex((dld!2) (Y)) - E61,6,e6,+6a,EX((d2d1) (Y)) that is, writing d = [d,, d,], and S = 6, + S,, ('Y) = (dx) Y + €6, Ex(dy) which proves that d is an E-derivation. %,b = -1, we obtain, since then &6,6+< = -E~,€ by (I), and as E~~,~ = 1 it is seen that d2 is a derivation. COROLLARY . Suppose that A = Z. Then: (i) The square of an antiderivalion is a derivation. (ii) The bracket of two derivations is a derivation. On the other hand, if, in (II), we let d, = d, = d, 6, = 6, = 6 and d2(xy) = (d24Y + %6,,X(dZY) (iii) The bracket-$ an antiderivation and a derivation of even degree is an anti- derivation. (iv) If dl and d, are antiderivations, dld, + d,d, is a derivation. Under the hypotheses of the beginning of this no., consider now a finite sequence D = (d,) , of pairwise permutable derivations of (A, A', A"). For every polynomial P(X,, . . ., X,) in the algebra KIXl, . . ., X,], the element P(d1, . . . , d,) of EndgrK(A 0 A' @ A") is then defined (3 2, no. 9) ; its abbrevia - ted notation is P(D). PROPOSITION 2. With the above hypotheses and notation, con -~ m F(T + T') = F(T1 + Ti,. . ., T, + Tk), Suflpose that P(T + T') = Qi(T)Ri(T') where the Qi and R, belong to KIXl, . . . , X,]. Then, for all x E A and y E A', (12) P(D) (XY) = 7 (QiPIx) (Ri(D)y)- 555 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS We introduce n other indeterminates T;, . . ., T:: and consider the poly- nomial algebra K[T,, . . ., T,, Ti,. . . , T;, TY,. . . , Ti] = B; on the other hand we consider the K-module M of bilinear mappings of A x A' into A"; a B-module structure is defined on M by writing, for every K-bilinear mapping f E M and 1 < i < n, Since the di are permutable with one another, it is seen that, for every poly- nomial F E K[X,, . . ., X,], (F(T)f)(a, a') =f(F(D)a, a'), (F(T')f )(a, a') = f(a, F(D)a') and (F(T")f) = F(D)(f(a, a')). Hence, to prove (12) it suffices to show that (14) or also (P(T") - P(T + T')) . p = 0 in the B-module M. Now, the hypo- thesis that the di are derivations can also be expressed by saying that, for l<i<n, (P(T") - 3 Qi(T)R,(T')) . p = 0 (15) (Tf-Ti -T;).p = 0 in the B-module M. By considering successively the polynomials P(T;, Tg,. . ., Ti) - P(Tl + Ti, TgJ.. ., T:) P(T1 + Ti, Tz, . . ., Ti) - P(T1 + Ti, T2 + T&, . . . , Ti) ......................... P(TI+Ti,...,T,-l + Th-1, T:) - P(T1+ Ti,. . . , T,-1 + Tk-1, T, + T6) it is seen that the difference P(T") - P(T + T') can be written in the form i=1 2 (Tf - Ti - TI)Gi(T, T', T") where the G, are elements of B. Relation (14) is therefore an immediate con- sequence of relations (1 5). COROLLARY (Leibniz's formula). Let di (1 < i < n) be n derivations of (A, A', A ") which are permutable with one another. For all a = (a1, . . . , a,) EN", we write (16) d" = d,"'d,"". . . d,"". 556 DERIVATIONS OF AN ALGEBRA A INTO AN A-MODULE 3 10.5 where we have written (in the notation introduced at the beginning of the chapter) (18) ((P,Y)) = (P + Y)!/(P!Y!)* This follows immediately from the multinomial formula (I, 5 8, no. 2) (T + T')" = 2 ((p,y))T0T" otv=a and Proposition 2. 5. DERIVATIONS OF AN ALGEBRA A INTO AN A-MODULE We suppose in this no. that Case (11) of no. 2 holds. Then there is a graded K- algebra A and a graded K-module E and also two K -linear mappings of degree 0 E @KA+E, A @KE-+E denoted by x8aax.a and a @x++a.x foraEAandxEE. PROPOSITION 3. Let d:A -+ E be an E-derivation of degree 6. Then Ker(d) is a graded subalgebra of A; if A admits a unit element, it belongs to Ker(d). Clearly Ker(d) is a graded sub-K-module of A; further, relation (8) of no. 2 shows that, if x and y are two homogeneous elements belonging to Ker(d), then d(xy) = 0 and hence xy E Ker(d). Finally, if A admits a unit element 1 (of degree 0, cf. 3 3, no. l), relation (8) of no. 2, where x and y are replaced by 1, gives d (1) = d (1) + d (1) and hence d( 1) = 0. COROLLARY . Let d, and d2 be two &-derivations from A to E of the same degree 6. If dl and d2 take the same values at each element of a generating system of the algebra A, then d, = d2. dl - d2 is an &-derivation of degree 6, hence Ker(d, - d2) is a subalgebra of A which contains a generating system of A and hence is equal to A. PROPOSITION 4. Let d :A --+ E be an derivation of degree 6. Suppose that A has a unit element 1 and let x be a homogeneous element of A with an inverse x-l in A. Then (19) d (X- l) = - E6, deg(&- ( (dx)x- ') = - €6, degtx) (X - (dx))x- '. 55 7 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS We have d(xx-l) = d(1) = 0 (Proposition 3), whence (dx)x-l E6,deg(x)X(d(x-1)) = which proves the first formula of (19). On the other hand, x-l is homogeneous of degree -deg(x) and E6, deg(x) = E6, -deg(X) by formulae (1) of no. 1 ; writing d(x-1~) = 0, the second formula of (19) is obtained similarly. PROPOSITION 5. Suppose that A is an integral domain and let L be its field of frac- tions. Every derivation of A into a vector space E over L (considered as an A-module) can be extended uniquely to a dm'vation of L into E. Let d be a derivation of A into E and 2 a derivation of L into E extending d; then, for u E A, v E A, v # 0, of necessity, by virtue of (19), (20) I(u/v) = v-ldu - uv-2dv which proves the uniqueness of 2. Conversely, we show that 2 can be defined by formula (20) ; it must first be verified that if u/v = u'/v' the value of the right hand side of (20) does not change when u is replaced by u' and v by v'. Now, uv' = vu', hence v'(du) + u(dv') = v(du') + u'(dv) and therefore u'(du - uv-ldv) = v(du' - u'v'-ldv'), since uv'v-l = u' and u'u'-~v = u. Thus a mapping 2:L --f E has been defined which extends d; it is immediately verified that it is K-linear and a derivation. PROPOSITION 6. Suppose that A is a unital associative graded K-algebra and E a graded (A, A)-bimodule. If d :A -+ E is an E-derivation of degree 6, then, for every jinite sequence of homogeneous elements of A, of respective degrees ti (1 6 i < n), (21) d(x1x2 . . . x,) = f-1 2 %,cl+...+ct-lX1 - -. xi-i(dxi)xi+i. . . X,. Formula (21) is trivial for n = 0 and is proved by induction on n, taking account of (4) (no. 2). COROLLARY . Suppose that A is a unital commutative associative algebra and E an A-module. If d :A --f E is a derivation, then, for every integer n (22) It suffices to give A the trivial graduation and apply (21) with all the xi equal to x. We return to the general case of an E-derivation d :A -+ E of degree 8. Let ZE be the set of a E A such that for every homogeneous component a, of a of degree u, for every homogeneous x in E, 0, d(x") = tuc"-l(dx) for all x E A. (23) xaa = Ea,deg(x)aa** 558 9 10.6 If A is a unital associative graded algebra and E a graded (A, A)-bimodule it follows immediately from this definition that Z, is a graded subalgebra of A containing the unit element. PROPOSITION 7. Suppose that A is a unital associative graded algebra and E a graded (A, A)-bimodule. Let d :A -+ E be an c-derivation of degree 6 and a a homogeneous element ofZ, of degree u. Then the mapping x I--+ a(dx) is an E-derivation ofdegree 6 + u. We write d'(x) = a(dx) ; for x homogeneous of degree 5 in A and y E A, by virtue of (23) and (1) (no. l), DERIVATIONS OF AN ALGEBRA d'(x!/) = a((dx)y) + E6,€a(x(d!/)) = + E6+a,I:(xa)(dy) = (d'x)y + E6+a,€x(d'y)* Proposition 7 says that the K-module of c-derivations of A into E is a graded Z,-module of type A. 6. DERIVATIONS OF AN ALGEBRA Let A be a graded K-algebra; for every homogeneous element a E A, let ad,(a), or simply ad(a) if no confusion can arise, denote the K-linear mapping of A into A (no. 4, formula (10)) which is graded ofdegree deg a. PROPOSITION 8. Let A be a graded K-algebra. [a, 'IE (i) For every c-derivation d :A -+ A and every homogeneous element a of A, (ii) rfthe algebra A is associative, ad,(a) is an E-derivation of A of degree deg(a). (i) Suppose that d is of degree 6, let u = deg a and write f = [d, ad,(a)],. [d, ad&(a)lE = adE(da)* (24) For every homogeneous element x E A of degree E, we have, by (1) (no. l), f (x> = d(ax - t)xa) - E6,a(a(dx) - Ea,6+S(h)a) = (da)x + E6,aa(dr) - Ea,e(dx)a - E6+a,€x(da) - E6,au(dx) + Ea,t(dx)a = (da)x - ~~+,,~x(da) = [da, XI,. (G) For all x homogeneous of degree E and all y homogeneous of degree q in A, ade(a)(x~) = a(x~) - Ea,t+n(xy)a = (ax - =a. CX~) Y + Ea, tx(a~ - =a, nya) = &(a) Y + Ea, tx.adE(a) (Y) taking account of (1) and the associativity of A. 559 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS When A is associative, adE(a) is called the inner E-derivation of A defined by a. COROLLARY . Let A be an associative graded algebra. For two homogeneous elements, a, b of& (25) Lad,(’), = blE)‘ It suffices to replace d by adE(a) and adE(a) by adE(b) in (24). Ifdega = a, deg b = p, formula (25) is equivalent to the following rela- tion for every homogeneous element c E A of degree y (26) E,,y[a, [b, c]E]E f [c, alEIE + %, Dlc, blEIE = called the Jacobi identity. 7. FUNCTORIAL PROPERTIES In this no., all algebras are assumed to be associative and unital and every algebra homo - morphism is assumed to be unital. PROPOSITION 9. Let A, B be two graded K-algebras, E an (A, A)-bimodule and F a graded (B, B)-bimodule; let p:A -+ B be a graded algebra homomorphism and O:E 3 F a graded A-homomorphism of A-bimodules (relative to p), of degee 0. Then: (i) For every €-derivation d’:B --f F, d’ 0 p: A --f p*(F) is an E-derivation of the same degree. (ii) For every e-derivation d :A -+ E, 8 0 d :A --f p* (F) is an ederivation of the the same degree. The two assertions follow immediately from the relations d’(p(xy)) = d’(p(x)p(y)) = d’(p(x))p(y) + E6’,€p(x)d’(P(Y)) e(d(xy)) = e((dx)y + E6,€X(dy)) = e(dx)P(Y) + 66,CP(x)e(dY) for x E A homogeneous of degree 6 and y E A, 6 and 6‘ denoting the respective degrees of d and d‘. COROLLARY . Let S be agenerating system ofthe algebra A. In order that d’ o p = 8 0 d, it is necessary and suficient that d’(p(x)) = O(d(x)) for all x E S. This is an immediate consequence of Proposition 9 and no. 5, Corollary to Proposition 3. Under the conditions of Proposition 9, we know that B has (by means of p) an (A, A)-bimodule structure (11, 4 1, no. 14, Example 1). PROPOSITION 10. Under the conditions of Proposition 9, for an E-derivation d’ :B --f F to be A-linear for the lgt (resp. right) A-module structures on B and p,(F), it is necessary and suficient that d’ be zero on the subalgebra p(A) of B. 560 4 10.7 FUNCTORIAL PROPERTIES We perform the proof for left A-module structures. For a E A, b E B, d’(p(a)b) = d’(p(a))b + p(a)d’b and hence if d’ 0 p = 0, d’ is linear for the left A-module structures on B and p*(F). Conversely, if this is so, in particular d’(p(a)) = d’(p(a). 1) = p(a)d’(l) = 0 (no. 5, Proposition 3). In particular let DK(B, F) denote the K-module of derivations of B into F (no. 2); those among these derivations which are A-linear, in other words those which are zero on p(A), form a sub-K-module of DK(B, F), denoted by DA, p(B, F) or simply DA(B, F) (obviously D,(B, F) = DK, *(B, F), where C$ : K -+ B is the homomorphism defining the K-algebra structure on B). Now let A, B, C be three graded K-algebras, p:A -+ B, o:B -+ C two graded algebra homomorphisms and G a graded (C, C) -bimodule; if DA(B, G), D,(C, G) and DA(C, G) denote the respective K-modules DA,p(B, o*(G)), D,,.(C, G) and DA,oo,, (C, G), D,(C, G) is clearly a sub-K-module ofDA(C, G) since o(p(A)) c o(B). PROPOSITION 11. Under the above conditions, there is an exact sequence f K-homo- morphimzr where u is the canonical injection and u the homomorphism d ++ d 0 o (Proposition 9). The kernel of v is the set of derivations d: C + G such that d(o(b)) = 0 for all b E B, which is precisely the image of u. 8. RELATIONS BETWEEN DERIVATIONS AND ALGEBRA HOMOMOR - PHISMS We suppose again in this no. that Case (11) of no. 2 holds and the graded K- algebra A is not assumed to be associative. Given an element 8 E A, consider the graded K -module E(6) (11, fj 11, no. 2) such that (E(8))tt = 6 for all p E A. We define on the graded K-module A @ E(8) a graded K-algebra structure by setting, for every homogeneous element a E A and arbitrary elements a’ E A, x, x’ in E(6) the verification of the fact that this multiplication defines a graded ring struc - ture is immediate. 56 1 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The projectionp: (a, x) H a is called the augmentation of the algebra A @ E(6) and is a graded algebra homomorphism. The graded K-linear mappings g:A -+ A @ E(6) of degree 0 such that the composition AAA ~~(61 LA is the identity 1, are the mappings of the form x H (x, f (x)), wheref. A -+ E is a graded K-linear mapping of degree 6. PROPOSITION 12. For a graded K-linear mapping f :A --f E of degree 6 to be an c-derivation, it is necessary and su#cient that the mapping x H (x, f (x)) of A into A @ E(6) be a graded K-algebra homomorphism. Using the fact that for x homogeneous in A and y E A bY,f (XY)) = (x,f (4). (Y,f (Y)), f (XY) = f(x) .Y + E6. deg(xF.f (Y), we obtain, taking account of (28), the equivalent relation whence the proposition. PROPOSITION 13. For the algebra A @ E(6) to be associative and unital, it is neces- sary and sujicient that A be associative and unital, and that the mappings (a, x) H a. x and (a, x) HX.~ dejine on E an (A, A)-bimodule structure; the unit element of A @ E(6) is then (1,O). If an element (u, m) E A @ E(6) is written as unit element of this algebra, it is immediately found that u must be the unit element of A; writing (u,m).(O,x)=(O,x).(u,m)=(O,x), we obtain u.x=x.u=x for XEE and, writing (u, m) .(u, 0) = (u, 0). (u, m) = (u, 0), we obtain m = 0. The fact that A is associative when A @ E(6) is follows from the fact that the aug- mentation is a surjective homomorphism. The condition (x. a’) . a” = x. (a’. a”) is then equivalent to ((0, x) (a’, 0)) (a”, 0) = (0, x) ((a’, 0) (a”, 0)) and similarly the condition a. (a‘. x) = (a. a’) . x is equivalent to (a, 0) ((a’, 0) (O,.)) = ((a, 0) (a’, 0)) (0,x) ; (a, 0) ((0,x) (a’, 0)) = ((a, 0) (0,x)) (a’, 0). finally the condition a. (x. a’) = (a. x) . a’ is equivalent to 9. EXTENSION OF DERIVATIONS PROPOSITION 14. Let A be a commutative ring, M an A-module, B the A-algebra T(M) (resp. S(M), resp. A(M)) and E a (B, B)-bimodule. Let d,:A + E be a 562 EXTENSION OF DERIVATIONS 5 10.9 derivation of the ring A into the A -module E and dl: M + E an additive group homomorphism such that, for all a E A and all x E M, and further, when B = S(M), for all x, y in M, and, when B = A(M), x.d,(x) + d,(x).x = 0 (31) for all x E M. Then there exists one and only one derivation d of B (considered as a Z-algebra) into the (B, B)-bimodule E such that d I A = do and d I M = dl. We take on the Z-module B @ E the associative Z-algebra structure defined by (6, t)(b’, t’) = (bb’, bt’ + tb’) which has (1, 0) as unit element (no. 8, Proposition 13). Under the canonical injection t H (0, t), E is identified with a two-sided ideal of B @ E such that E2 = {O}. On the other hand, the mapping h,: B @ E defined by h,(a) = (a, do(a)) is a unital ring homomorphism (no. 8, Proposition 12); under this mapping, B @ E then becomes an A-algebra. Moreover, if, for all x E M, we write h,(x) = (x, d,(x)), it follows from the definition of h, and (29) that h,(ax) = ho(a)hl(x) ; in other words h, is an A-linear mapping of M into B @ E. Then there exists one and only one A-algebra homomorphism, h: B -+ B @ E such that h I M = h, (and necessarily h I A = h,): for, if B = T(M), this follows from 3 5, no. 1, Proposition 1 ; if B = S(M), condition (30) shows that h(x)h(y) = h(y)h(x) for all x, y in M and the conclusion follows froms6, no. 1, Proposition 2; finally if B = &MI, condition (31) shows that (h(x))2 = 0 for all x E M, since x A x = 0 and the conclusion follows from 3 7, no. 1, Proposition 1. The homomorphism h is such that the composition p 0 h: B -+ B with the augmentation p: B @ E -+ B is the identity I,, for p o h and 1, co- incide by definition for the elements of A and those of M and the set of these elements is a generating system of B. We can therefore write h(b) = (6, d(b)) for all 6 E B and the mapping b d(b) of B into E is a derivation with the required properties, by virtue of Proposition 12 of no. 8. COROLLARY. Let M be a graded K-module oftype A; the K -algebras T(M), S(M) and A(M) are given the corresponding graduations oftype A‘ = A x Z (3 5, no. 5, Proposition 7,§ 6, no. 6, Proposition 10 and 3 7, no. 7, Proposition 11). On the othr knd M is given the graduation of type A’ such that Ma, = M, for all a E A and Ma, = {0} for a E A and n # 1. Let E’ be a commutation factor over A’. 563 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (i) Let E be a graded (lgt and right) T(M)-bimodule of type A'; for all 6 EA and every integer n E Z, every graded K-linear mapping f : M + E of degree 61' = (6, n) can be extended uniquely to an &'-derivation d :T(M) -+ E of degree 6'. (ii) Let E be a graded S(M)-module of type A'; for a graded K-linear mapping f :M -+ E of degree 6' to be extendable to an d-derivation d: S(M) -+ E of degree a', it is necessary and suficient that, for every ordered pair (x, y) of homogeneous ele - ments of M, (33) The d-derivation d is then unique. K-linear mapping f: M -+ E of degree 6' to be extendable to an &'-derivation x.f (y) + E6',(deg(u),l)y.f (x) = y.f + E~',(deiz(x).l)X.f (y)- (iii) Let E be a graded (left and right) A(M)-bimodule of type A'; for a graded d:A(M) -+ E of degree 6', it is necessary and suficient that, for every homogeneous element x of M, (34) ..f (.) + Ek,(deg(x). 1)f (.) sX = O. The d-derivation d is then unique. Remark 2 of no. 2 is applied with one of the external B-module laws on E (with B equal to T(M), S(M) or A(M)) modified; the external law thus modified is still, by (1) (no. l), a B-module law and the B-module law thus obtained on E is still compatible with the other B-module structure. It then suffices to apply Proposition 14 with A = K and do = 0. Example (1). In the application of Proposition 14 note that if do = 0 condition (29) means simply that d, is A-linear. If we take in particular E = B and the (B, B)-bimodule structure derived from the ring structure on B, conditions (30) and (31) are automatically satisfied when dl is taken to be the composi - tion of an endomorphism s of M and the canonical injection M-t B; this is obvious for (30) since S(M) is commutative and for (31) this follows from the fact that x and s(x) are of degree 1 in A(M). It is therefore seen that every endomorphism s of M can be extended uniquely to a derivation D, of T(M) (resp. S(M), resp. A(M)), which is of degree 0. Moreover, for two endomorphisms s, t of M, (35) [D,, Dtl = QS, fl for both sides are derivations of T(M) (resp. S(M), resp. A(M)) which are equal to [s, t] on M. 564 EXTENSION OF DERIVATIONS 5 10.9 The expression for Ds is obtained using formula (21) of no. 5, which gives respectively, for xl, x,,. . ., xn in M, (36) D,(x~x,. . . x,) = t=l 2 . . . xt-1s(xi)x1+1. . . X, D,(x1 A ~2 A ... A x,) n = 2 X1 A .** A Xi-1 A S(X1) A A 0.. A x,. In the case of the algebra A(M), there is the following interpretation of PROPOSITION 15. If M is a free K-module of Jinite rank n, then, for every endo- I 1=1 D, : moqhism s of M, the restriction to A"(M) of the derivation D, is the homothety of ratio Tr(s). .. Let (e,)lsjG, be a basis of M and write e = el A e, A - . A en. If the third formula in (36) gives D,(e) = zlel n A - * . A ei-1 A s(et) A e,,, A . . Example (2). In the Corollary to Proposition 14, part (iii), let A = {0}, the graduation on A(M) then being the usual graduation of type Z; on the other hand take ~(p, q) = (- l)pq. Then, for every linear form x* E M* on M, x IJ (x, x*) is a graded K-linear mapping of degree - 1 of M into A (M) satisfying relation (34); then there exists an antiderivation i(x*) of A(M), of degree -1, such that (by virtue of formula (21) of no. 5) i(X*)(Xl A A x,) = 2 (-l)'-'(Xt, X*)Xl A 0.. A Xi-1 A Xttl A *. . A x, i=1 and which is a special case of the inner product to be defined in 9 11, no. 9, formula (68). PROPOSITION 16. Let A be a commutative K-algebra, M, (1 < i < n) and P A- modules and H the A-module of A-multilinear mappings of M, x M, x . . . x M, 565 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS into P. Suppose that there is given a K-derivation do :A + A of the algebra A, for each i, a K-linear mapping di:Mi --f Mi and a K-linear mapping D:P --f P, so that, for 1 < i < n, (do, di, di) is a K-derivation of (A, Mi, Mi) into itself and (do, D, D) is a K-derivation of (A, P, P) into itself. Then there exists a K-linear mapping D‘ :H -+ H such that (do, D’, D‘) is a K-derivation of (A, H, H) into itself and (37) D(f (xi, - x,)) n = (~~f)(x1,**.,xn) + zlf(xl,** .,X,-l,dixt,xitl,...,X,) for all x1 E M, for 1 < i < n and f E H. defined by (37) is A-multilinear. For a E A, We show that for f E H, the mapping D’fof M, x M, x . . . x M, into P (Dlf)(ax,, xz, -. *? x,) = D(af(x1, * * 4) -f(dl(axl), xz, . ., xn) n --a ,=a x f (xl, . * *, Xl-1, diX,, Xitl, . . ., x,) and by hypothesis D(af(x1, * * * 9 x,)) = (doa)f(x1, . * , Xn) + aD(f (x1, . . . Y Xn)) and dl(axl) = (doa)xl + a.dlxl, which gives (Dlf) (ax,, ~2, * * . > xn) = a * (DLf) (~1, * * . J xn) and linearity in each of the xi is proved similarly. On the other hand, (D’(af))(x,, * f .Y x,) = D(af(x1,. * *, x,)) -5 1=1 af(x1,...,x,-1,dix,,xi+l,...,x,) = (doa)f (~1, * .J 2,)) + aD(f(x1, * . .> xn)) - f: af(xl,. . ., xi-.l, dixi, xitl,. . ., x,) 1= 1 = (doa)f(xl, * * . , xn) + a(DLf) (~1, . * ., xn) in other words D’(af) = (d0a)f + which establishes the proposition. Examples. (3) Applying Proposition 16 to the case n = 1, M1 = M, P = A, then H = M*, the dual of M, and it is seen that for a K-derivation (do, d, d) of (A, M, M) is derived a K-derivation (do, d*, d*) of (A, M*, M*) such that (38) d,<m, m*> = <dm, m*> + <m,d*m*> for mEM and m* EM*. The K-linear mapping of M @M* into itself 566 $ 10.10 PROBLEM FOR DERIVATIONS : NON -COMMUTATVE CASE which is equal to d on M and to d* on M* then satisfies condition (29) and there is therefore a K-derivation D of the A-algebra T(M @ M*), which re - duces to do on A, to d on M and to d* on M*. The restriction d: of D to the sub-A-module C(M) of T(M @ M*) ($ 5, no. 6) is a K-endomorphism of T:(M) such that (do, d:, d:) is a K-derivation of (A, P,(M), T:(M)). Moreover, for i E I, j E J, if we write I’ = I - {i}, J’ = J - {j}, it is immediately verified that for the contraction c: (3 5, no. 6) c:.(d:(z)) = d::(c:(z)) for all z E T:(M). (4) Let M, (1 < i < 3) be three A-modules and, for each i, suppose that (do, di, d,) is a derivation of (A, Mi, Mi) ; applying Proposition 16 again for n = 1, a derivation (do, dij, dij) of (A, Hij, Hij), where Hij = HomA(Mi, M,), is derived for each ordered pair (i,j). With this notation, for u E HomA(Ml, M2) and u E HomA(M2, M3), (39) diS(0 0 u) = (d23~) 0 u + v 0 (dizu) as is immediately verified from the definitions. 10. UNIVERSAL PROBLEM FOR DERIVATIONS; NON -COMMUTATIVE CASE Throughout the rest of 5 10 all the algebras are assumed to be associative and unital and all the algebra homomorphisms are assumed to be unital. Let A be a K-algebra; the tensor product A &A has canonically an (A, A)-bimodule structure under which (40) x.(u €3v).y = (xu) 0 (vy) for all x, y, u, u in A ($4, no. 3, Example 2). The K-linear mapping m:A aK A --f A corresponding to multiplication in A (and hence such that m(x @I y) = xy) is an (A, A)-bimodule homomorphism; its kernel I is therefore a sub-bimodule of A BK A. Lemma 1. The mapping SA:x H x €3 1 - 1 €3 x is a K-derivation of A into I and I is generated, as a lejl A-module, by the image of SA. The first assertion follows from the fact that (xy) €3 1 - 1 €3 (.y) = (x €3 1 - 1 @x).y + x.(y €3 1 - 1 €3Y) by (40). On the other hand, if the element 2 xi €3 yi (for xi, y, in A) belongs to I, by definition 7 xiyi = 0 and hence 7 (Xi €3 Yi) = & 41 @ yi - yi 63 1) by (a), which completes the proof of the lemma. 567 ~ 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS PROPOSITION 17. The derivation 6, has the following universal property : for every (A, A)-bimodule E and every K-derivation d:A -+ E, there exists one and only one (A, A)-bimodule homomorphism f :I -+ E such that d = f 0 SA. Note first that, for every (A, A)-bimodule homomorphism f :I -+ E, f o 6, is a derivation (no. 7, Proposition 9). Conversely, let d: A -+ E be a K- derivation; then we prove first that if there exists an (A, A)-bimodule homo - morphismf: I -+ E such that d = f o S,, f is uniquely determined by this condition for the definition of 6, gives f(x @ 1 - 1 @x) = dx and our assertion follows from the fact that the image of 6, already generates I as a left A -module (Lemma 1) : hence of necessity Conversely, as the mapping (x, y) H -x. dy of A x A into E is K-bilinear, there exists one and only one K -linear mapping g:A @K A -+ E such that g(x By) = -x.dy; it suffices to verify that the restriction f of g to I is A- linear for the left and right A-module structures. The first assertion is obvious since (xx’) . dy = x. (x‘. dy) ; to prove the second, note that, if 7 xi @ y, E I and x E A, then 1 1 xi.d(y,x) = T x,.dy,.x + 7 (xiyt) .& but since 7 x,y, = 0 by definition of I, this completes the proof. We have thus defined a canonical K-module isomorphism f H f 0 6, Horn,,, A)(L E) -+ &(A, E) the left hand side being the K-module of (A, A)-bimodule homomorphisms of A into E. 11. UNIVERSAL PROBLEM FOR DERIVATIONS; COMMUTATlVE CASE Suppose now that A is a commutative K-algebra and E an A-module; E can be considered as an (A, A)-bimodule whose two external laws are identical with the given A -module law. On the other hand the (A, A)-bimodule structure on A @K A is identical with its (A @K A)-module structure arising from the commutative ring structure on A BK A, since in this case, for x, y, u, v in A, x.@ E34.y = (xu) @ (uy) = (xu) €9 (yv) = (. @d(U @v). The kernel 3 of m is therefore in this case an ideal of the ring A @K A and, 568 PROBLEM FOR DERIVATIONS; COMMUTATIVE CASE as m:A @, A -+A is surjective, (A BK A)/3 is isomorphic to A; if also E is considered as an (A @,A) -module by means of m (in other words the (A gK A)-module m,(E)), the (A, A)-bimodule homomorphisms 3 -+ E are identified with the (A BK A)-module homomorphisms 3 -+ E ($ 4, no. 3), in other words there is a canonical K-module isomorphism. $ 10.11 Hom(A,A)(3, E, --f OK A(3, E)* On the other hand, 3E = {0}, for the elements 1 @ x - x @ 1 generate 3 as an (A BK A)-module (no. 10, Lemma 1) and, for all z E E, (1 @x - x @ l)z = 0 by virtue of the definition of the (A BK A)-module structure on E. Since 3 is contained in the annihilator of the (A gK A)-module E and the ((A A)/3)- module structure on E is by definition just the initial A -module structure given on E, there is, taking account of the canonical isomorphism of @K ((A @K A)/3) onto 3/g2 ($4, no. 1, Corollary 1 to Proposition l), a canonical K-module isomorphism HOmA B~A(~, E) --f H0mA(3/s2, E). Taking account of Proposition 17 of no. 10, it is seen that we have proved the following proposition : PROPOSITION 18. Let A be a commutative K-algebra and 3 the ideal the kernel of the surjective canonical homomorphism m :A BK A -+ A, SO that A is isomorphic to (A mK A)/3 and 3/g2 has canonically an A-module structure. Let dA/K:A -+ 1/12 be the K-linear mapping which associates with every x E A the class of x @ 1 - 1 @ x modulo g2. The mapping dAIK is a K-derivation and, for every A-module E and every K-derivation D :A -+ E, there exists one and only one A-linear mapping g: 3/g2 -+ E such that D = g o d,,,. The A-module 8/S2 is called the A-module of K-dzfferentials ofA and is denoted by !&(A) ; for all x E A, dA/K(X) (also denoted by dx) is called the dfferential ofx; it has been seen (no. 10, lemma 1) that the elements dA/K(x), for x E A, form a generating system of the A-mcdule !&(A). Proposition 18 shows that the mapping g H g 0 dA,K is a canonical A-module isomorphism +A:HomA(QK(A), E) -+D,(A, E) (the A-module structure on DK(A, E) being defined by Proposition 7 of no. 5). The ordered pair (RK(A), dA,K) is therefore the solution of the universal mapping problem where C is the species of A-module structure and the a-mappings the K-derivations from A to an A-module (Set Theory, Iv, 3 3, no. 1). 569 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Example. Let M be a K-module; it follows from Proposition 14 of no. 9 that for every S(M)-module E, the mapping D H D I M defines an S(M)-module isomorphism of DK(S(M), E) onto HomK(M, E) ; on the other hand, since E is an S(M)-module, HomK(M, E) is canonically isomorphic to every K-homomorphism of M into E being uniquely expressible in the form x H h(x @ l), where (11, $5, no. 1). Let Do be the K-derivation S(M) +M BK S(M) whose restriction to M is the canonical homomorphism x H x @ 1 ; every K-deriva- tion D:S(M) -+ E can therefore be written uniquely as h o Do with HomS(M)(M @K S(M), HomS(M)(M @K S(M), E, HomSW)(M @K S(M), By the uniqueness of a solution of a universal mapping problem, it is seen that there exists a unique S(M)-module isomorphism @K S(M) --f QK(S(M)) such that Do o o = dS(M)/K; in other words, for all x E M, u(x @ 1) = dx. Q,(S(M)) is a free S(M)-module with basis the set of dfferentials de,. Consider in particular the case where L = (1, n), so that S(M) is identified with the polynomial algebra KIX1,. . ., X,] (5 6, no. 6); for every polynomial P E K[X,, . . ., X,], we can write uniquely dP = t=1 2 D,P.dX, with DIP E KIX1, . . . , X,] and, by virtue of the above, the mappings P 13 D,P are the K-derivations of KIXI, . . . , X,] corresponding to the coordinate forms on QK(S(M)) for the basis (dX,); we also write - instead of DIP and this is called the partial derivative of P with respect to XI. In particular, if M is a free K-module with basis n ap ax* 12. FUNCTORIAL PROPERTIES OF K-DIFFERENTIALS PROPOS~ON 19. Let K & K’ 4 A + A’ be a commutative diagram of commutative ring homomorphism, -q (resp. q’) making U 570 10.12 A (resp. A’) into a K-algebra (resp. K’-algebra). There exists one and only one A-linear mapping v : QK (A) -+ !&(A‘) rendering commutative the diagram A 2 A’ FUNCTORIAL PROPERTIES OF K-DIFFERENTIALS dN/Kr 0 u is a K-derivation of A with values in the A-module Q,.(A’) ; the The mapping v of Proposition 19 will be denoted by Q(u); if there is a existence and uniqueness of v then follow from Proposition 18 of no. 1 1. commutative diagram of commutative ring homomorphisms P’ K & K‘ + K” it follows immediately from the uniqueness’ property of Proposition 19 that Q(u’ 0 u) = Q(U’) 0 Q(u). Since !&(A’) is an A’-module, from Q(u) we derive canonically an A’- linear mapping Q~(u) : QK(A) @A A‘ -+ QKt(A’) (41) such that Q(u) is the composition of Qo(u) and the canonical homomorphism iA: !&(A) -+ QK(A) @A A‘. For every A’-module E’, there is a commutative diagram ’ DK(A,E) DKt(A’, E’) C(4 where C(u) is the mapping D H D o u (no. 7, Proposition 9) and r, is the canonical isomorphism HOm(iA, 1~’) :HomN(QK(A) @A A’, E’) -+ HOmA(!&(A), E’); this follows immediately from Proposition 19 and the definition of the iso- morphisms +A and +A,. 571 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS PROPOSITION 20. Suppose that A‘ = A BK K’, with q‘:K‘ --f A’ and u:A -+ A‘ the canonical homomorphisms. Then the A-linear mapping Q,(u) : Q2,(A) @A A’ 3 Qr(A’) is an isomorphism. By virtue of the fact that in diagram (42) the vertical arrows are bijective, it reduces to proving that, for every A’-module E‘, the homomorphism C(U) :D H D o u in diagram (42) is bijective (11, 9 2, no. 1, Theorem 1). Now Hom(u, lE’) :Hom,,(A gK K’, E’) --f Horn,(& E’) is an isomorphism (11, fj 5, no. 1, Proposition 1) and C(u) is its restriction to D,.(A’, E’) and hence is injective; moreover, iff :A’ -+ E’ is a K’-linear mapping such that f 0 u: A + E’ is a K-derivation, it follows immediately from the fact thatf is K’-linear and the fact that f ((x @ l)(y C3 1)) = (y C3 1)f (x @ 1) + (x @ 1)f (y @ 1) for x, y in A, that f is a K’-derivation, the elements x @ 1 for x E A forming a generat - ing system of the K’-module A‘; this completes the proof that C(u) is bijec- tive. From now on we confine our attention to the case where p : K -+ K’ is the identity ma@ing of K; every K -algebra homomorphism u:A -+ B is therefore mapped to a B-linear mapping (43) %~(u) :%(A) @K B --f %(B). On the other hand, we can consider the B-module of A-dfferentials QA(B) since B is an A-algebra by means of u; the canonical derivation dBBIA: B -+ Q,(B) being a fortiori a K-derivation, it factorizes uniquely into QU B Q,(B) ---+ QA(B) where Ru is a B-linear mapping (no. 1 1, Proposition 18). For every B-module E, there is a commutative diagram (44) ’ DK(B,E) DA(B, iu where j, is the canonical injection (no. 7); this follows immediately from Proposition 18 of no. 1 1. PROPOSITION 2 1. The sequence of B-linear mappings (45) !&(A) BAB R,(u? QK(B) ---+ Q2,(B) +O is exact. 572 FUNCTORIAL PROPERTIES OF K-DIFFERENTIALS 5 10.12 It reduces to verifying that, for every B-module E, the sequence Horn(&, IE) 0 -+ HomB(OA(B), E) A HomB(GK(B), E) Hom(%(u). 1~) @A B, is exact (11, 32, no. 1, ‘Theorem 1); but by virtue of the fact that in the commutative diagrams (42) and (44) the vertical arrows are isomorphisms, it suffices to show that the sequence 0 + D,(B, E) L DK(B, E) C(U) DK(A, E) is exact, which is just Proposition 11 of no. 7. We consider now the case where the K-algebra homomorphism u:A -+ B is surjective; if 3 is its kernel, B is then isomorphic to A/3. We consider the restriction dl 3: 3 + !&(A) of the canonical derivation d = dA,K and the composite A-linear mapping d‘:3 + d13 !&(A) + iA !&(A) @AB. Then d‘(32) = 0, since, for x, y in 3, d’(xy) = d(q) @ 1 = (x.dy + y.dx) @ 1 = dy @ U(X) + dx @ ~(y) = 0 since u(x) = u(y) = 0. Hence we derive from d‘, by passing to the quotient, an A-linear mapping 2: 3/V + !&(A) @A B and as 3 annihilates the A-module 3/S2, 2 is a B-linear mapping. PROPOSITION 22. Let 3 be an ideal of the commutative K-algebra A, B = A/3 and u: A -+ B the canonical homomorphism. The sequence of B-linear mappings - 8/S2 - d @A - now QK(B) 0 (6) is then exact. Note that !&(A) @AB is identified with QK(A)/3QK(A) and that the image of 2 is the image of d(3) in this quotient module; the quotient of QK(A) @AB by Im(2) is therefore identified with the quotient Q,(A)/N, where N is the sub-A -module generated by 3QK(A) and d(3). Moreover, the composite mapping A C&(A) - !&(A)/N is a K-derivation (no. 7, Proposition 9) and, since it is zero on 3 by definition ofN, it defines, when passing to the quotient, a K-derivation Do :B -+ Q,(A)/N. 573 I1 I TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Taking account of the uniqueness of the solution of a universal mapping problem, it reduces to proving that, for every B-module E and every K- derivation D: B --+ E, there exists a unique B-linear mapping g: QK(A)/N --f E such that D = g o Do. But, the composite mapping D 0 u: A -+ E is a K- derivation (no. 7, Proposition 9) and hence there exists one and only one A- linear mappingf: !&(A) -+ E such that f 0 dAIK = D 0 u. This relation shows already thatfis zero on d (3) ; as also 3E = (0) since E is a B-module, f is zero on gQK(A) ; hence f is zero on N and defines, when passing to the quotient, a &linear mapping g: !&(A)/N -+ E such that g 0 Do = D; the uniqueness of g follows from the uniqueness off. It must not be thought that, even if u: A -+ B is an injective homomorphism, Q,(u) : Q,(A) @A B -+ QK(B) is injective (Exercise 5). However we have the following proposition : PROPOSITION 23. Let A be an integral K-algebra, B itsJeld offractions and u:A + B the canonical injection. Then Qo(u) : QK(A) @A B -+ QK(B) is an isomorphism. Using the fact that in diagram (42) the vertical arrows are bijective, it reduces to proving that, for every vector space E over B, the mapping C(u) :DK(B, E) -+ DK(A, E) is bijective. But this follows from the fact that every K -derivation of A into E can be extended uniquely to a K-derivation of B into E (no. 5, Proposition 5). 3 11. COGEBRAS, PRODUCTS OF MULTILINEAR FORMS, INNER PRODUCTS AND DUALITY In this paragraph, A is a commutative ring with the trivial graduation. For a graded A-module M of type N, M*gr will denote the graded A-module of type N, whose homogeneous elements of degrees n are the A-linear forms which are zero on M, for all k # n. 1. COGEBRAS DEFINITION 1. A cogebra over A (or A-cogebra, or simply cogebra if no confusion can arise) is a set E with a structure dejined by giving the following: (1) an A-module structure on E; (2) an A-linear mapping c:E --f E @A E called the coproduct of E. DEFINITION 2. Given two cogebras E, E‘, whose coproducts are denoted respectively by c and c’, a morphism of E into E’ is an A-linear mapping u: E -+ E’ such that (1) (u @ u) 0 c = c’ 0 u, 574 COGEBRAS $ 11.1 in other words, it renders commutatiue the diagram of A-linear mappings Ed E’ & c It is immediately verified that the identity mapping is a morphism, that the composition of two morphisms is a morphism and that every bijective morphism is an isomorphism. Examples. (1) The canonical isomorphism A+A BAA (11, 3 3, no. 5) defines an A-cogebra structure on A. (2) Let E be a cogebra, c its coproduct and Q the canonical automorphism of the A-module E @A E such that Q(X @ y) = y @ x for x E E, y E E; the A-linear mapping Q 0 c defines a new cogebra structure on E. With this struc - ture E is called the opposite cogebra to the given cogebra E. (3) Let B be an A-algebra and let m:B @A B -+ B be the A-linear mapping defining multiplication on B (3 1, no. 3). The transpose tm is then an A- linear mapping of the dual B* of the A-module B into the dual (B @A B)* of the A-module B @A B. If also B is a Jinitely generated projective A-module, the canonical mapping p : B* @A B* -+ (B @A B) * is an A-module isomorphism (11, 3 4, no. 4); the mapping c = p.-l 0 tm is then a coproduct defining a cogebra structure on the dual B* of the A-module B. (4) Let X be a set, A(x) the A-module of formal linear combinations of elements of X with coefficients in A (11,s 1, no. 11) and (ex)xsx the canonical basis of A(x). An A -linear mapping c : A(x’ --f Acx’ @A A‘x) is defined by the condition c(e,) = ex @ ex and a canonical cogebra structure is thus obtained on A(x). (5) Let M be an A-module and T(M) the tensor algebra of M (3 5, no. 1) ; by 11, 3 3, no. 9 there exists one and only one A-linear mapping c of the A-module T(M) into the A-module T(M) aAT(M) such that, for all n 2 0, (3) c(x1xz. . .x,) = z (x1xz.. .xp) @ (xp+l.. .x,) OGpCn for all xi E M (x1x2. . . x, denotes the product in the algebra T(M)). Thus T(M) is given a cogebra structure. (6) Let M be an A-module and S(M) the symmetric algebra of M ($6, no. 1) ; the diagonal mapping A: x ++ (x, x) of M into M x M is an A-linear mapping to which there therefore corresponds a homomorphism S(A) of the A-algebra S(M) into the A-algebra S(M x M) (3 6, no. 2, Proposition 3). On the other hand, in 5 6, no. 6 we defined a canonical graded algebra isomor- 5 75 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS phism h: S(M x M) -+ S(M) @A S(M); by composition we therefore obtain an A-algebra homomorphism thus defining on S(M) a cogebra structure. For all XEM, by definition s( A) (x) = (x, x) and the definition of h given in 5 6, no. 6 shows that It follows that c is the unique algebra homomorphism such that, for all x E M, (4) As c is an algebra homomorphism, it follows that, for every sequence (xi) of n elements of M, c = h 0 S(h) :S(M) 3 S(M) S(M), h((x, x)) = x @ 1 + 1 @ X. C(X) = x @ 1 + 1 @ x. , n (5) c(x,x,. I. x,) = n 1=1 (Xi @ 1 + 1 @ x,) = c (Xil . . . Xi,) @ 1 * * Xj"J the summation on the right hand side of (5) being taken over all ordered pairs of strictly increasing sequences (in some cases empty) of elements of (1, n), whose sets of elements are complementary. The element c(xlx2. . . x,) is an element of total degree n in S(M) @A S(M) and its com- ponent of bidegree (p, n - p) is i, < i, < < i,, j, < j, < -.. <jn-, (6) . * * xo(p)) 8 (xo(p + 1) . . . where the summation is taken over all permutations (J E S, which are increas - ing in each of the intervals (1, p) and (p + 1, n). (7) Let M be an A-module and proceed with the exterior algebra A(M) as with S(M) in Example 6; the diagonal mapping A:M 3 M x M this time defines a homomorphism A(A) of the A-algebra A(M) into the A-algebra A(M x M) (3 7, no. 2, Proposition 2); on the other hand there is a canonical graded algebra isomorphism (3 7, no. 7, Proposition lo), whence by composition there is an algebra homomorphism c = h o A (A) :A (M) 3 A (M) B@A A (M), which can be considered as an A-module homomorphism A(M) -+ A (M) @A A(M) and which therefore defines on A(M) a cogebra structure. It can be proved as in Example 6 that c is the unique algebra homomorphism such that, for all XEM, ~:A(M x M)+A(M)~@I.,A(M) (7) C(X) = x @ 1 + 1 @ x, 576 3 11.1 COGEBRAS whence, for every sequence (xi) , ,, of elements of M, C(X1 A X2 A . . * A X,) = (Xi @ 1 4- 1 @ Xi) A . . . A (X, @ 1 f 1 @ X,) where the product on the right hand side is taken in the algebra A\(M) '@A A(M); to calculate this product, consider, for every ordered pair of strictly increasing sequences i, < i, < <j,-= of elements of (l,n), whose sets of elements are complementary, the product yly, . . . y,, where y,,, = xi,, @ 1 (1 < h 6 p) and y,, = 1 @ x,, (1 < k < n -p) and the sum is taken over all these products. As the graded algebra A(M) B@A A(M) is anticommutative and the elements x, @ 1 and 1 @I xi are of total degree 1, by fj 4, no. 6, Lemma 3 and Lemma 1, < i,, j, <j, < (8) C(X1 A X2 A *** A X,) = 2 (-1)'(xi1 A 0.. A xi,) @ (xjl A -.* A xjn-p) v being the number of ordered pairs (h, k) such thatj, < i, and the summation being taken over the same set as in (5). The element c(xl A . . . A x,) is of total degree n in A(M) "@,A(M) and its homogeneous component of bi- degree (p, n - $) is equal to 3 Eo(Xuc1, A . . . A Xacp)) 8 (xo(p+1) A * . . A xo(n)) (9) the summation being taken over permutations (J E 6, which are increasing in each of the intervals (1, p) and (p + 1, n). When in future we speak of A(X), T(M), S(M) or A(M) as cogebras, we shall mean, unless otherwise mentioned, with the cogebra structures defined in Examples 4, 5, 6 and 7 respectively. (8) Let E, F be two A-cogebras and c, c' their respective coproducts. Let T: (E @A E) @A (F @A F) -+ (E @A F) @A (E @A F) denote the associativity isomorphism such that T((X @ x') @ (y @ y')) = (x @ y) @ (x' @ y') for x, x' in E and y, y' in F. Then the composite linear mapping E BAF - (E BAE) @A (F @AF) --L (E BAF) @A (E @AF) defines a cogebra structure on the A-module E @A F, called the tensor product of the cogebras E and F. on the A-module E is said to be compatible with the coproduct c of E if c is a graded homomorphism of degree 0 of the graded A-module E into the graded A-module (of type A) E @A E, in other words (11, c c3 c' Let E be a cogebra and A a commutative monoid. A graduation 11, no. 5) if 577 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS In what follows, we shall most often limit our attention to graduations of type N compatible with the coproduct; a cogebra with such a graduation will also be called a graded cogebra. If F is another graded cogebra, a graded cogebra morphism + : E -+ F is by definition a cogebra morphism (Definition 2) which is also a graded homomorphism of degree 0 of graded A-modules. Examples. (9) It is immediate that the cogebras T(M), S(M) and A(M) defined above are graded cogebras. 2. COASSOCIATIVITY, COCOMMUTATIVITY, COUNIT Let E be a cogebra, c its coproduct, N, N', N" three A-modules and m a bi- linear mapping of N x N' into N". Let fi: N @, N' -+ N" denote the A-linear mapping corresponding to m. If u : E -+ N, u : E 4 N' are two A-linear map- pings, we derive an A-linear mapping u @ u : E @, E -+ N @, N' and a com- posite A-linear mapping of E into N": (11) m(u, v):E E @AE 2 N BAN' $_ N". Clearly we have thus defined an A-bilinear mapping (u, u) H m(u, u) of Hom,(E, N) x Hom,(E, N') into Hom,(E, N"). When E is a graded cogebra, N, N', N" graded A-modules of the same type and h a graded homomorphism of degree k of N @, N' into N", then, if u (resp. u) is a graded homomorphism of degree p (resp. q), m(u, u) is a graded homomorphism of degree p + q + k. Examples. (1) Take E to be the graded cogebra T(M) (no. 1) and suppose that N, N', N" have the trivial graduation. A graded homomorphism of degree -p of T(M) into N (resp. N', N") then corresponds to a multilinear mapping of MP into N (resp. N', N"). Given a multilinear mapping u: MP 4 N and a multilinear mapping u: Mq + N', the above method allows us to deduce a multilinear mapping m(u, u) : MP+q -+ N" called the product (relative to m) of u and u. Formulae (3) (no. 1) and (1 1) show that, for xl,. . . , in M, (m(u, u))(xl,' * '> 'Ptq) = m(u(xl, * * *J 'P), u(xp+lY * "J 'Ptq)). (2) Take E to be the graded cogebra S(M) (no. I), preserving the same hypotheses on N, N', N". A graded homomorphism of degree -p of S(M) into N then corresponds to a symmetric multilinear mapping of Mp into N (5 6, no. 3). Then we derive from a symmetric multilinear mapping u: MP --f N and a symmetric multilinear mapping u: MQ 4 N' a symmetric multilinear mapping m(u, v): Mp+q --f N", also denoted (to avoid confusion) by u.,u (or even U.V) and called the symmetricproduct (relative to m) of ii and u. Formulae (6) (no. 1) and (1 1) show that, for xl,. . . , xPcq in M, 578 the summation being taken over permutations cr E Q, which are increasing on each of the intervals (1 , p) and (p f 1, p + q). (3) Take E to be the graded cogebra A(M) (no. 1). Then we deduce similarly from an alternating multilinear mapping u: MP -+ N and an alter- nating multilinear mapping u: MQ -+ N' an alternating multilinear mapping m(u, u) : MP +q 4 N", also denoted by u A u or u A u and called the alternating product (relative to m) of u and u. Formulae (9) (no. 1) and (1 1) show in this case that, for xl,. . . , xp + in M, = 3 Eom(U(-vo(i), . . .? X~(PJ, u(xa(p+i), . . ., x,(ptq))) (u Am u)(xi,. . the summation again being taken over permutations cr E 6, + increasing on each of the intervals (1, p) and (p + 1 , p + q). We return to the case where E is an arbitrary graded cogebra (of type N) and assume that the three modules N, N', N" are all equal to the underlying A-module of a graded A-algebra B of type Z, the mapping m being multiplica - tion in B, so that fi : B @, B + B is a graded A-linear mapping of degree 0. Thus a graded A-algebra structure is obtained on the graded A-module Homgr,(E,B) = c. In particular, suppose that B = A (with the trivial graduation), so that Homgr,(E, A) is the graded dual E*Br, which thus has a graded A-algebra structure. Let F be another graded cogebra c' its coproduct and +:E -+ F a graded cogebra morphism (no. 1) ; then the canonical graded morphism which are 6 = Horn(+, lB) : Homgr,(F, B) + Homgr,(E, B) is a graded algebra homomorphism. For u, u in Homgr,(FJ B) and x E E, and therefore ~(uu) = $(u)&(u), which proves our assertion. homomorphism. In particular, the graded transpose $C$:F*~' -+ E*g' is a graded algebra 579 1x1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Remark. Suppose that the E, are Jinitely generated projective A-modules, so that the graded A-modules (E @A E)*gr and E*gr E*gr can be canonic - ally identified (11, 5 4, no. 4, Corollary 1 to Proposition 4). If also the A-modules A A and A are then canonically identified (11, 4 3, no. 4), the linear mapping E*Pr E*gr --f E*gr which defines multiplication in E*gr can be called the graded transpose of the cobroduct c. PROPOSITION 1. Let E be a cogebra over A. In order that, for every associative A- algebra B, the A -algebra HomA(E, B) be associative, it is necessary and suficient that the coproduct C: E -+ E @A E be such that the diagram ELE@.,E be commutative. Let B be an associative A-algebra and u, v, w three elements of C = Hom,(E, B). Let m3 denote the A-linear mapping B @A B @A B -+ B which maps b @I 6' @I 6" to bb'b". By definition of the product on the algebra C, (uu)w is the composite mapping 68 IE u@u@w m3 ELE@IE-E@IE@E--+B@B@IB---+B EA EBE - E~E~E -+B@B@B -B. whilst u(vw) is the composite mapping IE@C U@V@W m3 It follows that if diagram (12) is commutative, the algebra Hom,(E, B) is associative for every associative A-algebra B. To establish the converse, it suffices to show that there exists an associative A-algebra B and three A-linear mappings u, u, w of E into B such that the mapping m3 o (u @ v @ w) of E @ E @ E into B is injective. Take B to be the A-algebra T(E) and u, v, w the canonical mapping of E into T(E). The mapping m3 0 (u @ v @ w) is then the canonical mapping E @ E @ E = T3(E) -+ T(E), which is injective. When the cogebra E satisfies the condition of Proposition 1, it is said to be coassociative. Examples. (4) It is immediately verified that the cogebra A (no. 1, Example (1) the cogebra A(x) (no. 1, Example 4) and the cogebra T(M) (no. 1, Example 5) are coassociative. If B is an associative A-algebra which is a finitely generated projective A-module, the cogebra B* (no. 1, Example 3) is coassociative: for the commutativity of diagram (12) then follows by transposition from that of the diagram which expresses the associativity of B (3 1, no. 3). Conversely, the 580 COASSOCIATIVITY, COCOMMUTATIVITY, COUNIT 3 11.2 same argument and the canonical identification of the A-module B with its bidual (11,s 2, no. 7, Corollary 4 to Proposition 13) show that if the cogebra B* is coassociative, the algebra B is associative. Finally, the cogebras S(M) and A(M) (no. 1, Examples 6 and 7) are coassociative; this follows from the com- mutativity of the diagram M-MxM A 11~~. AxlM MxM-MxMxM the functorial properties of S(M) (3 6, no. 2) and A(M) (4 7, no. 2), which give the corresponding commutative diagrams and the existence and functoriality of canonical isomorphisms for symmetric and exterior algebras of a direct sum (3 6, no. 6 and 5 7, no. 7). PROPOS~ON 2. Let E be a cogebra over A. In order that, for every commutative A-algebra B, the A-algebra HomA(E, B) be commutative, it is necessary and su&cient that the coproduct c: E -+ E @A E be such that the diagram E (where Q is the symmetry homomorphism such that (~(x @ y) = y @ x) is commutative (in other words, it suffices that the cogebra E be identical with its opposite (no. 1, Example 2). Let B be a commutative A-algebra and u, u two elements of C = HomA(E, B) . 58 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS By definition of the product in C, uv and uu are respectively equal to the com- posite mappings E&E@E%B@B-~~,B E E @ E 2 B @ B 5 B. and It follows that if diagram (15) is commutative the algebra Hom,(E, B) is commutative for every commutative A-algebra B. To establish the converse, it suffices to show that there exist a commutative A -algebra B and two A- linear mappings u, v of E into B such that m o (u @ v) :E @ E + B is injec- tive. Take B to be the algebra S(E @ E) and u (resp. u) the composition of the canonical mapping E @ E --f S(E @ E) and the mapping x H (x, 0) (resp. x H (0, x)) of E into E @ E. If h:S(E) @ S(E) -+ S(E @ E) is the canonical isomorphism (3 6, no. 6, Proposition 9) and A: E -+ S(E) is the canonical mapping, then h-l o m 0 (u @ v) = A @ A. Now A @3 A is injec- tive, for A(E) is a direct factor of S(E) (11, 3 3, no. 7, Corollary 5 to Proposi- tion 7). When the cogebra E satisfies the condition of Proposition 2, it is said to be cocommutative. Examples. (5) It is immediate that the cogebra A (no. 1, Example 1) and the cogebra A(x) (11, 9 11, no. 1, Example 4) are cocommutative. It follows from formula (5) of no. 1 that the cogebra S(M) is cocommutative. Finally, for an A-algebra B such that the A-module B is projective and finitely generated to have the property that the cogebra B* (no. 1, Example 3) is cocommutative, it is necessary and sufficient that B be commutative; for (using the canonical identification of the A-module B with its bidual (11, 3 2, no. 7)), this follows from the fact that the commutativity of diagram (14) is equivalent by transposi - tion to that of the diagram which expressed the commutativity of B (3 1, no. 3). PROPOSITION 3. Let E be a cogebra over A. In order that, for every unital A -algebra B, the A -algebra Hom,(E, B) be unital, it is necessary and su@cient that there exist a linear form y on E rendering commutative the diagrams E-LE@,E E_~,E@,E (16) A 6AE E €3.4-4 where c:E + E 8, E is the coproduct and h' and h" the canonical isomorphism 582 4 11.2 COASSOCIATIVITY, COCOMMUTATIVITY, COUNIT (11, 3 3, no. 4, Proposition 4). The unit of Hom,(E, B) is then the linear mapping x H y(x) 1 (where 1 denotes the unit element of B). Let y be a linear form on E rendering diagram (16) commutative. Let B be a unital A-algebra with unit element 1, q :A + B the canonical mapping and v = q o y the element of the A-algebra C = Horn,@, B). For every element u E C, uv is the composite mapping (17) E~E@E---tE@A-B@B+B. 1EmY uC971 m Then uv = nz o (u @ r)) o h" = u. It is similarly proved that uu = u and hence u is unit element of C. Conversely, let the A-module A @ E be given a unital algebra structure such that (a, x)(a', x') = (ad, ax' + a'x) for a, a' in A and X, x' in E. Let B denote the A-algebra thus defined and let C be the A-algebra Hom,(E, B). Suppose that C is unital and let e:x H (y(x), A(x)) be its unit element (where y(x) E A and A(x) E E). On the other hand letfbe the element x H (0, x) of C. An immediate calculation shows thatfe is the element (OJ (h") -'(('I3 @3 y)(c(x)))) of C. The condition fe = f implies the commutativity of the second diagram of (16) and it is similarly seen that the condition ef = f implies the commuta- tivity of the first diagram of (16). A linear form y on E rendering diagrams (16) commutative is called a counit of the cogebra E. A cogebra admits at most one counit: for it is the unit element of the algebra Hom,(E, A). A cogebra with a counit is called counital. Examples. (6) The identity mapping is the counit of the cogebra A; on the cogebra A(x) (no. 1, Example 4) the linear form y such that y(e,) = 1 for all x E Xis the counit. On the cogebra T(M) (resp. S(M), A(M)) the linear form y such that y(1) = 1 and y(z) = 0 for z in the T"(M) (resp. S"(M), A"(M)) for n 3 1 is the counit. Finally, let B be an A-algebra which is a finitely generated projective A-module and has a unit element e; then on the cogebra B* (no. 1, Example 3) the linear form y: x* H (e, x*) is the counit, for this form is just the transpose of the A-linear mapping qe : 6 H Se of A into B and by transposition the commutativity of diagrams (16) follows from that of the diagrams which express (using ye) the fact that e is unit element of B (3 1, no. 3); the same argument moreover shows that conversely, if the cogebra B* admits a counit y, the transpose of y defines a unit element e = %( 1) of B. PROPOSITION 4. Let E be a cogebra admitting a counit y and suppose that there exists in E an element e such that y(e) = 1 ; then E is the direct sum of the sub -A-modules Ae and E, = Ker(y) and c(e) = e @ e (mod. E, @ E,) (18) {c(x] = x @ e + e @ x (mod. E, @ E,) for all x E E,. 583 I 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The first assertion is immediate, for y(x - y(x)e) = 0 and the relation y(.e) = 0 implies tc = 0. Let c(e) = 7 si @ ti, so that e = 7 y(Si)ti = Y(ti)si by (16) and 1 = y(e) = $. y(si)y(t,). Therefore C i (si - y(si)e) @ (ti - y(ti)e) = 7 st 8 ti - 7 e @ Y(St)ti - 7 Y(th 8 e + 7 Y(si)e @ Y(ti)e which, by the above relation, is just c(e) - e @ e; this therefore proves the first relation of (18). On the other hand the decomposition of E 8 E as a direct sum A(e @ e) 0 ((A4 @ E,) 0 (E, @ (Ae)) 0 (E, @ E,) allows us to write, for x E E,, C(X) = A(e @ e) + (e @ y) + z @ e) + u where u = 2 v, @ w,, y, z and the v, and w, belong to E,. The definition of the counit y then gives x = Ae + y = Ae + z and, as y(x) = 0, necessarily A = 0, x = y = z, whence the second relation of (18). Remark. Let C be a counital coassociative A-coalgebra, B a unital associative A-algebra and M a left B-module. The A-bilinear mapping (b, m) H bm of B x M into M defines an A-bilinear mapping I HOmA(C, B) X HOmA(C, M) --f HOmA(C, M) by the general procedure described at the beginning of this no. It is immedi- ately verified that this mapping defines on HomA(C, M) a left module struc - ture over the ring HomA(C, B). 3. PROPERTIES OF GRADED COGEBRAS OF TYPE N PROPOSITION 5. (i) Let E be a graded cogebra admitting a counit y; then y is a homogeneous linear form of degree 0. (ii) Suppose further that there exists an element e E E such that Eo = Ae and y(e) = 1. Then the kernel E, ofy is egual to E, = zl En, c(e) = e @ e and (19) for all x E E,. ~(x) = x @ e + e @ x (mod. E, 63 E,) 584 BIGEBRAS AND SKEW -BIGEBRAS 11.4 (i) It suffices to verify that y(x) = 0 for x E En, for all n 2 1. Since c is a graded homomorphism of degree 0, with, for allj such that 0 6 j < n, ytj and zij in E,; applying (16) (no. 2) we obtain whence, equating the components of degree 0 and degree n on the two sides x = 7 Y(Yio)zin = 7 Y(zio)yin 0 = 7 Y(Yin)Zio = 7 Y(Zin)yio and therefore y(x) = 7 y(yi,)y(zio) = y(0) = 0. (ii) Since Ker(y) and E, are both supplementary sub-A-modules of Ae = Eo and E, c Ker(y) by (i), E, = Ker(y) (11, 3 1, no. 8, Remark 1); the other assertions follow from Proposition 4 of no. 2. PROPOSITION 6. Let E be a graded cogebra over A. In order that, for every commutative A-algebra B, with the trivial graduation, thegraded A -algebra of type Z Homgr,(E, B) (no. 2) be anticommutative (3 4, no. 9, Definition 7), it is necessary and sujficient that, ;fog denotes the automorphism of the A-module E @A E such that for x, E E,, xq E E,, where p and q are arbitrary elements of N, thc diagram .&p @ x,) = (- l)pqx, @ x, E E~AE+E@AE be commutative. The proof is analogous to that of Proposition 2 of no. 2. When the graded cogebra E satisfies the condition of Proposition 6, it is said to be anticocommutatiue. Example. It follows immediately from formula (8) of no. 1 that for every A- module M, the graded cogebra A (M) is anticocommutative. DEFINITION 3. A graded bigebra (resp. skew graded bigebra) over a ring A is a set E With a graded A-algebra structure of type N and a graded A-cogebra structure of Q@e N, with the same underlying graded A-module structure and such that : 4. BIGEBRAS AND SKEW -BIGEBRAS (1) The A-algebra E is arsociatiue and unital. 585 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (2) The A-cogebra E is coassociative and counital. (3) The coproduct c:E -+ E @A E is a homomorphism of the graded algebra E into the graded algebra E @A E (resp. graded algebra E (4) The counit y of E is a homomorphism of the graded algebra E into the algebra A (with the trivial graduation) such that, $ e denotes the unit element of the A-algebra E (cf. $ 4, no. 7)). E, y(e) = 1. If E is a graded bigebra whose graduation is trivial, E is called simply a bigebra. A graded bigebra is called commutative (resp. cocommutative) if the underlying algebra is commutative (resp. if the underlying cogebra is cocommutative) ; a skew graded bigebra is called anticommutative (resp. anticocommutative) if the underlying graded algebra is anticommutative (resp. if the underlying graded cogebra is anticocommutative) . It follows from Definition 3 and no. 2, Proposition 5 that, for a graded bigebra or a skew graded bigebra E, If E and F are two graded bigebras (resp. two skew graded bigebras), a mapping +: E -+ F is called a graded bigebra morphism (resp. skew graded bigebra morphism) if: (1) + is a graded algebra morphism (and hence maps the unit element of E to the unit element of F) ; (2) + is a graded cogebra morphism such that, if y and y' are the respective counits of E and F, then y = y' 0 +. Examples. (1) Let S be a monoid with identity element u, so that the algebra E = A(s) of the monoid S over A admits the unit element e, (3 2, no. 6) ; it has been seen on the other hand that E has canonically a coassociative cocommuta- tive A-cogebra structure with a counit y such that y(e,) = 1 for all s E S (no. 1, Example 4 and no. 2, Examples 4, 5 and 6). The formula c(es) = e, @J e, giving the coproduct shows also immediately that c is an algebra homomorphism. Thus a cocommutatiw bigebra structure has been defined on E and E, with this structure, is called the bigebra of the monoid S over A. If T is another monoid with unit element v, f :S +T a homomorphism such that f (u) = v andf,A,:A(S) -+ A(T) the A-algebra homomorphism derived from f (3 2, no. 6), it is immediately verified thatf,,, is a bigebra homomorphism. (2) Let M be an A-module. The graded A-algebra ($ 6, no. 1) and graded A-cogebra (no. 1, Example 6) structures defined on S(M) define on this set a commutative cocommutative graded bigebra structure; for we have seen (no. 1, Example 6) that the coproduct on S(M) is an algebra homomorphism and it 586 THE GRADED DUALS T(M)*gr, S(M)*g' AND A (M)*gr $ 11.5 follows from the definition of the counit y (no. 2, Example 6) that y(1) = 1 and that y is an algebra homomorphism of E into A. (3) Let M be an A-module. We see as in Example 2 that the graded A-algebra ($ 7, no. 1) and graded A-cogebra (no. 1, Example 7) structures on A(M) define on this set an anticommutative anticocommutative skew graded bigebra structure. Remark. If M is an A-module such that M BAM # (0)) the graded A- algebra ($5, no. 1) and graded A-cogebra (no. 1, Example 5) structures on T(M) do not dejine a bigebra structure, for in general C(XlXZY~Y2) # C(XlXZ)~(YIY,) for four elements xl, x,, y,, y, of M, as formula (3) of no. 1, shows. 5. THE GRADED DUALS T(M)*=', S(M)*gr AND A (M)*Kr From now on we return to the general conventions of the chapter on algebras, which will therefore be assumed (unless otherwise mentioned) to be associative and unital. Let M be an A-module; the graded A-cogebra structures defined on T(M) (no. 1, Example 5), S(M) (no. 1, Example 6) and A(M) (no. 1, Example 7) allow us to define canonically on the graded duals T(M)*gr, S(M)*gr and A(M)*@~ graded algebra structures of type N, by virtue of no. 2, Propositions 1 and 3 and the convention made on the graduation of the graded dual of a graded module (no. 1). Moreover, the graded algebra S(M)*gr is commutative (no. 2, Proposition 2 and Example 5) and the graded algebra A (M) *gr is anticommuta- tive (no. 3, Proposition 6 and Example). In A (M) *gr every element of degree 1 is of zero square; such an element is identified with a linear form f on M and its square is the alternating bilinear form f A f on M2 such that (fAf)(XYY) =f(x)f(Y) -f(Y)f(4 (no. 2, Example 3). know that u defines canonically graded algebra homomorphisms Let N be another A-module and u an A-linear mapping of M into N. We (u) :T(M) -+ T(N) S(u) :S(M) -+ S(N) A(u) :&MI -+ A(N) f (5 5, no. 2, 5 6, no. 2 and 3 7, no. 2). It is immediately verified in formula (3) of no. 1 that T(u) is also a cogebra morphism. On the other hand, if AM (resp. A,) 587 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS denotes the diagonal mapping M --f M x M (resp. N -+ N x N), there is the relation (u x u) 0 AM = AN 0 u; it follows that S(u x u) 0 S(&) = S(A,) 0 S(u) (resp. A(u x u) 0 A(A~) = &A,) o A(u)). Using the definition of coproduct in S(M) and A(M) (no. 1, Examples 6 and 7) and the functorial character of the canonical isomorphisms S(M X M) --f S(M) @A S(M) and A(M x M) -+ A(M) B@, &MI, it is seen that S(u) and A(u) are also cogebra morphismt (and hence in this case bigebra morphisms). It follows immediately that the graded transposes (11, 3 11, no. 6) of the homomorphisms 'T(u) :T(N)*Br -+ T(M)*g' tS(u):S(N)*gr -+ S(M)*,' (23) 'A (u) :A (N) *gr --f A (M) *gr are graded algebra homomorphisms. We now note that the dual M* of M is identified with the submodule of elements of degree 1 in T(M)*Kr (resp. S(M)*8', A(~)*gr). It therefore follows from the universal property of the tensor algebra (9 5, no. 1) and the universal property of the symmetric algebra (3 6, no. 1) that there exists one and only one graded algebra homomorphism 8r:T(M*) -+ T(M)*gr which extend the canonical injection M* 3 T(M)*g', and one and only one graded algebra homomorphism which extends the canonical injection M* -+ S(M)*gr. On the other hand, the canonical injection of M* in the opposite algebra to A(M)*gr is such that the square of every element of M* is zero; hence (9 7, no. 1, Proposition 1) there exists one and only one graded algebra homomorphism Bs:S(M*) -+ S(M)*Br o,, :A(M*) --f (A(M)*=)o which extends the canonical injection M* -+ A(M)*gr.S These homomorphisms t This also follows from formulae (5) and (9) of no. 1. $ This injection is extended to a homomorphism into the opposite algebra to A(M)*gr instead of a homomorphism into A(M)*~? for reasons of convenience in the calculations. 588 THE GRADED DUALS T(M)*'?, S(M)*g' AND A (M)*gr 9 11.5 are functorial : for example, for every A-module homomorphism u : M --f N, the diagram T(N*) T(M*) T(N) *gr -> T(M) *gr is commutative, as follows immediately from the universal property of the tensor algebra (3 5, no. 1); there are analogous commutative diagrams for 8, and O,,. We shall find the homomorphisms &, 05 and 0 A explicitly. For this we con - sider more generally a coassociative A-cogebra E with coproduct c and define by induction on n, for n > 2, the linear mapping c, of E into Emn by c, = c and tT(n) c, = (cn-l 8 1E) 0 c. (24) On the other hand we denote by m,:A@" -+ A the canonical linear mapping such that m,(E1 8 Ez C3 . . . C3 En) = EIE,. . . E,, and note that, for n 2 2, mn = mo (mn-l@ 1A) (25) writing m = m,. With this notation: Lemma 1. (i) In the associative algebra E* = Horn,(& A), the product of n elements ul, u2, . . . , u, of degree 1 is given by uluz. . . u, = m, 0 (ul @ uz @ . . . @ u,) 0 c,. (ii) Suppose also that the cogebra E is graded. Then, in the graded associative algebra E*gr = Homgr,(E, A), the product of n elements ul, u,, . . . , u, Of degree 1 is given by (26) UlU2. . . u, = mn 0 (~1 @ ~2 8 * . * @ u,) 0 6, (27) where 8,:E -+ EBn is the linear mapping which maps each x E E to the component of c,(x) ofmultidegree (1, 1, . . . , 1). Formula (26) is just the definition of the product in E* for n = 2; to prove it by induction on n, observe that u1u2. . . u, = m 0 ((uluz. . . u,-~) @ u,) 0 c = m 0 ((mn-l 0 (~1 8 uz 8 * . * @ un-1) 0 cn-1) 8 un) 0 = 0 (mn-1 @ 1,) 0 (u1 @ ~2 8 . . . C3 un - 1 @ un) 0 (Cn- 1 @ 1E) 0 c = m, 0 (ul @ u2 8 -. - 8 u,) 0 c, by virtue of (24), (25), 11, 3 3, no. 3, formula (5) and the relation u, = 1,o u, 0 1,. 589 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS When E is graded and the elements ui E E*g' homogeneous of degree 1, then by definition for homogeneous elements xi E E, (u1 €3242 €3 .-. @ufl)(x1 €3x2 €3 ... €3xfl) = 0 unless all the xi are of degree 1, whence formula (27). It followsfrom formulae (3), (5) and (7) ofno. 1 and formula (24) that when E is taken to be one of the three graded cogebras T(M), S(M) and A(M), we obtain respectively by induction on n (using the fact that the coproduct is a graded homomorphism of degree 0), for xl, x2,. . . , x, in M: when E = T(M), when E = S(M), when E = &MI, S,(x,xz. . . x,) = OE c 6n E,.X,(l) €3 %(Z) €3 . . . @ Xu(,). It suffices to note, for example when E = A(M), that in the expression c,(x1x2 . . . x,) = (cfl-l arising from formula (8) of no. 1, the only terms which can gwe a term of multidegree (1 , 1,. . . , 1) are those for which n - p = 1 and hence S,(xlxz.. . x,) = x1 @ x2 €3 . . * €3 X, 8,(x1x2.. . x,) = OE6n c XO(1) @ %(Z) €3 . . . @ Xdfl) 1,)CC ( - i)v(xil . . . xtp) €3 p,, . . . x,"-,)> ~n(XlX2. . . xfl) is the term of multidegree (1, 1, . . . , 1) in the sum n 1=1 c (-l)n-*C ,~l(X1...Xi~lXi+l...X,) @Xi i: (-l),-%-l(xl.. .Xi_lX,+l.. .x,) 63 xi, (XTX;. . . xx, XlXZ. . . x,) = n i=1 (x?, Xi) and this term is necessarily equal to i=l whence the result by the induction hypothesis. of M* is given by * Using Lemma 1, the product in T(M)*gr of n linear forms x:, xg, . . . , x, n (28) for xi E M (1 < i < n) ; the product of these n forms in S(M)*g' is given by 590 <x:x: . . . x;, XlXZ. . . x,) = UEBn i: (I3 i=l <X8,i,, Xi)) ; THE GRADED DUALS T(M)*'', S(M)*gr AND A (M)*gr 5 11.5 finally, the product of these forms in A (M) *g' is given by (30) (xfx?. . . x:, xlxz.. . x,) = det((x:, xj)). In each of the three cases, we have respectively e,(x: g x; €3 . . . g g) = .T.;. . . x,* eS(x;x:. . . x:) = .fX;. . . .,* 8A (X: A Xz A . . . A Xz) = .,*.,*-I. . . = n(n-1)12 XlXZ * * . . . x, * (-1) and hence we deduce from (28), (29) and (30) the relations n (28 bis) (eT(xT €3 X; @ *. * @ X;), X1 @ xz @ . . . €3 x,) = i=1 (in other words OT restricted to T2(M*) is just the canonical homomorphism of 11, 3 4, no. 4) (29 bis) (30 bis) (~A(x: A X; A . . . A X:), X1 A X2 A . . . A x,) = ( -l)n(n-l)izdet((x.i*, x,)). PROPOSITION 7. Let M be a jnitely generated projective A-module. Then the canonical homomorphisms eT:T(M*) +T(M)*g' and OA:A(M*) + (A(M)*g')o are bi- jective. Also the graded dual A(M)*g' is then equal to the dual A(M)* of the A- module A (M) . Suppose first that M has a,finite basis (e,),,,,, and let (e:),,,,, be the dual basis of M* (11,s 10, no. 4). Formula (28 bis) shows that, for every finite sequence s = (jJ1 Ck,n of n elements of the interval (1, m) of N, @ '. . @ ej*n) is the element of index s in the basis of (T"(M))*, dual to the basis of Tfl(M) consisting of the e, = erl 63 . - . @ e,,, (5 5, no. 5, Theorem 1). Hence eT is bijective. Similarly, formula (30 bis) shows that, for every finite subset H of (1, m) with n elements, ( - l)fl(fl-1)/20A (e:) (notation of 5 7, no. 8, Theorem 1) is the element of index H in the basis of (A"(M)) *, dual to the basis of Afl(M) consis - ting of the e,. Hence O,, is bijective. Suppose now only that M is finitely generated and projective; then M is a direct factor of a finitely generated free A-module L, so that there exist two 59 1 1x1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS A-linear mappings M 4 L -% M whose composition is the identity 1,. We deduce a commutative diagram T(M*) To\ T(L*) T(M*) eTl 'T(fi) OTl 'W) OT/ T(M)*gr + T(L)*gr + T(M)*gr and an analogous commutative diagram where T is replaced by A. The proposition then follows from the following lemma : Lemma 2. Let XIIYIl-X 4 4 X' __f Y' + x Id' U' be a commutative diagram of sets and mappings such that u o u and v' o u' are the identity maipings of X and X' respectively. Then, if g is injective (resp. surjective, resp. bijective), so is$ u is injective since u o u is, hence, if g is injective, u' of = g o u is injective and therefore f is injective. Similarly v' is surjective since v' 0 u' is; hence, if g is surjective, f 0 v = u' o g is surjective and therefore f is surjective. The last assertion of Proposition 7 follows from the fact that A(M) is then a finitely generated A-module (5 7, no. 3, Proposition 6 and 11, 3 11, no. 6, Remark). We now examine what can be said concerning the homomorphism 8s when M is projective andjnitely generated. Suppose first that M admits a finite basis (el),,,,,. In the notation at the beginning of the chapter the A-module Sn(M) admits as basis the family of elements ea such that la1 = n. Let u, (for la1 = n) denote the element of index a in the basis of (S"(M))* dual to (ea). The elements u,, for a E N", therefore form a basis of the algebra S(M) *gr and we shall obtain the multiplication table of this basis explicitly. We write u,uD = yENm c aaDYuY with aapY E A. Then by definition aapy = (UaUp, ey> = m((ua @ uB)(c(eYY))), where m: A 8 A -+ A defines the multiplication on A and c is the coproduct of S(M). In other words, aapV is just the coefficient of eu @ eD when c(ey) is 592 THE GRADED DUALS T(M)*gr, s(M)*gr AND A (M)*" 3 11.5 written in terms of the basis of S(M) @ S(M) consisting of the et @ en, where t and q run through N". But since c is an algebra homomorphism, rn m c(eY) = I3 (c(e,))Yi = FI (ei 8 1 + I g ei)yl i=l i=l by formula (4) of no. 1 ; this gives c(e y- ) - c+tl=u c (5, Ylb' G3 en (31) where we write n ! (cf. 5 10, no. 4, formula (18)). Hence we obtain the multiplication table uauB = ((a, P))ua+o- (33) On the other hand, if (e:),,,,, is the basis of M*, dual to (ei), it follows from formula (29 bis) that, for all a E Nrn, OS(e*'*) = a!u, in the notation of 5 6, no. 6. Hence the homomorphism Os is bijective if and only if the a!u, form a basis of S(M) *gr, or also if the elements u ! 1 are invertible. PROPOSITION 8. Suppose that the ring A is an algebra over the jeld Q of rational numbers; then, for every jinitely generated projective A-module M, the homomorphism Os:S(M*) -+ S(M)*gr is bijective. It amounts to proving this when M is finitely generated and free; we pass from this to the general case using Lemma 2 as in the proof of Proposition 7. Remark. Let M be an A-module and p :A + B a commutative ring homomorph- ism. Then there is a commutative diagram of graded B -algebra homomorph- isms (34) T((M*)(B)) - (T(M)*gr)(B, VUM) UT(M) T((M(B,)*) 7 T(M(B))*gr where the first row is a homomorphism composed of the homomorphism @r @ 1B:T(M*) @A B + T(M)*gr and the canonical isomorphism T((M*)(B,) --f T(M*) @A 593 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (3 5, no. 3, Proposition 5). It is immediately verified, using formula (28) and the definition of the homomorphism uE (11, 5 5, no. 4), that this diagram is commutative. When M is a finitely generated projective A-module, M,,, is a finitely generated projective B-module (11, 9 5, no. 1, Corollary to Proposition 4) and all the homomorphisms of the above diagram are bijective (Proposition 7 and 11, 9 5, no. 4, Proposition 8). There are analogous commutative diagrams with T replaced by S or A; the diagram for A also consists of bijective homomorphisms when M is projective and finitely generated (Proposition 7) ; if further A is an algebra over Q, the diagram for S also consists of bijective homomorphisms (Proposition 8). 6. INNER PRODUCTS: CASE OF ALGEBRAS Let E = @ Ep be a graded A-algebra of type N and P a graded A-module of type Z; for every homogeneous element x E E,, lgt multiplication by x is an A-linear mapping e(x) of E into itself which is graded of degree p. For every ele- ment u E Homgr,(E, P), the right inner product of u by x, denoted by u L x, is the element u o e(x) of Homgr,(E, P). We also write (i(x))(u) = u L x and we see that i (x) is a graded endomorphism of degree p of the graded A-module Homgr,(E, P). If now x = 2 xp is an arbitrary element of E (with x, E E, for all p 2 0, x, = 0 except for a finite number of values of p), we write i(x) = 5 i(x,), which is therefore an endomorphism of the A-module Pa0 pa0 ,=o HOmgr,(E, P). To remember which element, in the expression u L x, “operates” on the other, observe that the element x which “operates” on u is placed at the free end of the horizontal line in L. The associativity of the algebra E goes over to the relation e(xy) = e(x) 0 e( y) for x, y homogeneous; whence, by definition of i(x), (35) i(xy) = i(y) o i(x) first for x, y homogeneous and then, by linearity, for arbitrary x, y in E; this may also be written (36) (u L x) L y = u L (xy) for x, y in E and u E Homgr,(E, P) ; as on the other hand clearly i (1) is the identity mapping (since this follows from e( 1) = lE) and x H i (x) is A-linear, it is seen that the external law of composition (x, u) HU L x (~EE, u E Homgr,(E, P)) defines, with addition, a right E-module structure on Homgr, (E, P) . 594 4 11.6 INNER PRODUCTS: CASE OF ALGEBR.4S In particular we consider the case P = A, Homgr,(E, P) being in this case the graded dual E*g’ of E; i (x) is then the graded transpose of the A-linear mapping e(x) (11, 9 11, no. G), in other words, for all si, y in E, u E E*gr, (u X,Y> = (u, XY). (37) With the convention at the beginning of the paragraph, note that, if x E E,, i(x) is an endomorphism of E*gr of degree -p. For every homogeneous element x E E,, right multiplication by x is simi- larly denoted by e’(x) and the element u 0 e’(x) of Homgr,(E, P), called the 1eJt innerproduct of u by x, by x -I u; we write i‘(x) = x J u and i’(x) is therefore a graded endomorphism of Homgr,(E, P) of degree p; as above this definition can be extended to the case where x is an arbitrary element of E. As in this case e‘(xy) = e’(y)e‘(x), i’(xy) = i‘(x) 0 i’(y) (38) (39) which may also be written x -1 (y -1 u) = (xy) -I u and shows that the external law of composition (x, u) ++ x -I u defines, with addition, a lgt E-module structure on Homgr,(E, P). The associativity of E implies on the other hand that e(x) 0 e’(y) = e’(y) 0 e(x) for x, y homogeneous in E, whence the relation (y -1 U) L X = y -1 (U L x) so that the two external laws of composition on Homgr,(E, P) define on this set an (E, E)-bimoduZe structure (11, 6 1. no. 141. (40) ~ ,y I When we take P = A, i‘(x) is the graded tramp& of e’(x) ; in other words, for all x, y in E, u E E*gr, <Y, x -I u> = (YX, u>. (41) When the graded algebra E is commutative, obviously u L x = x -I u. When E is anticommutative and P = A, then for x E E,, y E E, and u E E,*, yx = (- l),‘xy, whence, by (37) and (41), (u L x, y) = (- l)pr(y, x -I u). But as the two sides of this relation are zero except for r = q - p, X -1 U = ( -l)p(q-p)U L X. Let F be another graded A-algebra and +: E -+ F an A-homomorphism of I graded algebras; then 6 = Horn(+, lP) : Homgr,(F, P) --f Homgr,(E, P) is a graded A-homomorphism of degree 0; by definition, for x, y in E and 21 E Homgr,(F, P) 595 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS or also (42) (43) $@ L +(4) = b(u, L x &(+(x) A u) = x -I &(u). and similarly In other words, when Homgr,(F, P) is considered as an (E, E)-bimodule by means of the ring homomorphism +: E + F, it is seen that 6 is an (E, E)- bimodule homomorphism (or also an E-homomorphism of the (F, F)-bimodule Homgr,(F, P) into the (E, E)-bimodule Homgr,(E, P)). Examples. In particular the above may be applied when E is one of the graded algebras T(M), S(M) or A(M) for an A-module M and P an A-module (with the trivial graduation). To find explicitly the bimodule structures thus ob- tained, note that the elements of degree --n of Homgr,(T(M), P) (resp. Homgr,(S(M), P), resp. Homgr,(/\(M), P)) are identified with the n-linear mappings (resp. symmetric n-linear mappings, resp. alternating n -linear mappings) of M" into P. It suffices to express the products f L (x1 (3 x, @.. .@ x,) (resp.fL (x1x2. . .xp), resp.fL (xl A x2 A . . . A x,)) for every finite sequence (x~)~~~~~ of elements of M and the analogues for the left inner product. It follows immediately from the definitions that (44) fL (x1 @x2 @...@x,) = (xl @x2 @*..@x,) As= 0 ;f@ > n and that, for p 6 n,f~ (xl @ x2 @. . . @ x,) (resp. (x1 63 x2 €9.. * @ .,) Jf) is the (n - p)-linear mapping defined by (f L (x1 @ x2 63. . .@ 4) (Yl, . . . , Yn -,I ((x1 c3.2 @....Xx,) .Jf)(Y1,...,Yn-,) =f(x1,. . *, x,,y1,. . .,Yn-,) =~(Y~,...,Y~-P,x~,...,xP). For p > n, there are also in Homgr,(S(M), P) (rcsp. HoxngrA(A(M), P)) formulae (44) with x1 @ x2 . . @ x, replaced by xlx2. . . x, (resp. A x2 A . . . A xp). For p < n, the same substitutions in (45) define the symmetric (n - @)-linear mappings f L (x1x2. . .xp) and (x1x2. . .x,) J f (resp. the alternating (n - @)-linear mappings and (xl A xa A. . . A xP) ~f). f~ (xl A x2 A . . . A xP) 596 4 11.7 When n = p, the above products are equal to the constant function on M equal If u: M --f N is an A-module homomorphism, T(u): T(M) --f T(N) is a graded A-algebra homomorphism, then it follows from what we have seen above that (T(u)) is a T(M)-homomorphism of the (T(N), T(N))-bimodule Homgr,(T(N), P) into the (T(M), T(M))-bimodule Homgr,(T(M), P), relative to the ring homomorphism T(u). There are analogous results for (S(u)) - and INNER PRODUCTS : CASE OF COGEBRAS tof(x1, . * . > 2,) * (A(u>)-. 7. INNER PRODUCTS: CASE OF COGEBRAS Let E = @ E, be a coassociative counital graded cogebra. Then we know (no. 2, Propositions 1 and 3) that the graded dual E*gp has (with the convention on graduations made at the beginning of the paragraph) a graded algebra structure of type N over A, the product of two elements u, u of this algebra being defined by uu = m 0 (u @ u) 0 c, where c: E -+ E 8, E is the coproduct and m:A @,A+A defines the multiplication. In other words, if, for ~EE, C(X) = ,LO yt (3 zi, we can write (canonically identifying A €9, E and E) = u(((u @ 'E) ')(')) = (((' @ 'E) c)(x), v>* This can be interpreted by saying that, for all u homogeneous of degree p in E*g', the left multiplication e(u) : u - uu in E*gp is the graded transpose of the graded endomorphism of degree -p (46) i(u) = (u 8 lE) oc of E; hence, in the above notation, Formula (46) also defines an element i(u) E Endgr,(E) for every element u E E*"; for all x E E and all u E E*g', we write (47) SO that, for u and v in E*gr, x L u = (i(u))(x) (x, uv> = (x L u, v>. The element x L u of E is called the right inner product oj'x by u. 59 7 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMF.TRIC ALGEBRAS Here again the element u which "operates " on x is placed at the fiee end of the horizontal line in L . For any two elements u, v of E*gr, (48) x L (uu) = (x L u) L u, in other words i(uu) = i(u) 0 i(u). As above let ~(x) = 3 yi @ zi, so that x L (uv) = (uv)(yl)zi. If then (49) x L (uv) = c i. 1 u(y;l)v(y;)z,. On the other hand, if c(zi) = zik @ zyk, then (50) (x L u) L v = 2 1. k u(y{)u(zik)z:k. Now, the coassociativity of E shows that (no. 2, Proposition 1) and the equality of expressions (49) and (50) follows from the fact that they are respectively the image of the left and right hand sides of (51) under the linear mappingffrom E @ E @ E to E such thatf(x 63 y @ z) = u(x)v(~)z. We recall on the other hand (no. 2, Proposition 3) that the unit element of the algebra E*gr is the linear form e: x H y(x) . 1 ; hence x L e = C y(yi)zt = x t by virtue of the definition of counit. As the mapping u I+ i(u) is linear, it is seen that the external law of composition (u, x) H x L u defines a right E*gr- module structure on E. Similarly we define, for all u E E*gr, the endomorphism of E (52) (53) i'(u) = (1, @ u) 0 c (i'(u))(x) = u -1 x and, for all x E E, we write and this element of E is called the lejl inner product ofx by u. As above it is seen 598 INNER PRODUCTS: CASE 01' COGEURAS 5 11.7 that the external law (u, v) ++ u -I x defines a lejit E*gr-module structure on E. Moreover, these two structures are compatible, in other words, (u -I x) L u = u -1 (x L u) (54) for u, u in E*gr (11,s 1, no. 14). With the same notation as above, the left hand side of (54) is 2 u(zi)v(ylj)yyj and the right hand side is 2 u(yi)u(zh)&; their right hand sides of (51) under the linear mapping g of E @ E @ E into E such that g(x @ y 8 z) = u(x)u(z)y. It is therefore seen that the two external laws of composition on E defines on this set an (E*gr, E*g')-bimodule structure. When the cogebra E is cocommutahe, then u -1 x = x L u for all x E E and u E E*gr; when it is anticocommutative (5 4, no. 9) and u E E,* and x E E,, we can write ~(x) = 2 (& yi, @ zf, -,) with yir and zij in E, for all j and then by hypothesis equality follows f,l from the fact that they are the respective i, images k of the left and 06164 As u(yi,) = 0 (resp. u(zi,q-,) = 0 unlessj = p (resp. q -j = p), it is seen by the above that u A x = ( - l)p(-)x L u. Finally, let +: E -+ F be a graded cogebra morphism; then it has been seen (no. 2) that the graded transpose t~: F*gr -+ E*gr is a graded algebra homomor - phism; therefore, for x E E, u, v in F*gr, <+ (x L % (4) 9 v> = (x L t+ (u) , t+ (4 ) = (x, t+ w+ (4 ) = (x, t+ (4 > = <+(XI, uv> = (+(XI L u, u> +(XI L 24 = +b L "(4); -I +(XI = +("((.I -1 4. whence (55) (56) and similarly In other words, 4 is an F*g'-homomorphism of the (E*=', E*g')-birnodule E into the (F*gr, F*g')-birnodule F, relative to the ring homomorphism 599 @ 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Examples. In particular the above can be applied when E is one of the graded cogebrasT(M), S(M) orA(M) for an A-moduleM (no. I,Examples5,6and 7). To find explicitly the bimodule structures thus obtained, we again identify a homogeneous element f of degree n in T(M)*g' (resp. S(M)*gP, resp. A (M)*gr) with an n-linear form (resp. symmetric n -linear form, resp. alternating n -linear form, also called an ngorrn) on Mn. It suffices to express the products (x163x263-..@x,) Lf(resp. (x1x2...xp) LA resp. (xl A x2 A . . A x,) L f) for every finite sequence (xt)lGtGp of elements of M and the analogues for the left inner product. Now, definitions (46) and (52) and formulae (3), (6) and (9) of no. 1 give respectively: (X,@~x,@--.€wp)Lf =f-I(x163x263-*.c3X,) =o (x1x2.. .xp) L f = f -I (x1x2.. .xp) = 0 for p < n. (Xi A XZ A*..A XP) L f = f-l (Xi A X2 A...A X,) = 0 (57) { Fort 2 n, we have respectively (58) (59) (XI 63x2 @*..@xp) Lf=f(Xl,...,Xn)Xn+1 @...@xp (x1x2. . ..,) Lf = f (XU(l), * . . J xO(n))~dn+l). . .XU(P) (60) (x1 A x2 A . . . A xp) Lj = z Euf(x,(l), . . . , xo(n))xu(n+ 1) A . . * A XO(,) (where, in (59) and (60), the summations are taken over the permutations Q E 6, which are increasing on each .f the intervals ( 1, n) and (n + 1, p) of N) ; and similarly (61) f~ (~1 63 ~2 @*. . @ xp) =f(Xp-n+ly *. .J xp)x1 8 ~2 @.. * 8 xp-n (62) f ('1'2' ' "P) = z f ('U(P-fl+l)J * * 'J xO(P))xO(l)' ' .XO(p-n) (63) f -1 (XI A X2 A . A Xp) = &f (xO(p-n+l), . . ., Xu(,))%1) A . . . A IC,(p-n) (where, in (62) and (63) , the summations are taken over the permutations Q E 6, which are increasing on each ofthe intervals (1 , p - n) and (p - n + 1 , p) of N) - 8. INNER PRODUCTS: CASE OF BIGEBRAS Let E be a graded bigebra (resp. skew graded bigebra) (no. 4, Definition 3) ; then the results of nos. 6 and 7 can be applied to define the right (resp. left) inner products x L u E E and u L x E E*gp (resp. u -I x E E and x -I u E E*=I') 600 9 11.8 for all x E E and all u E E*gr. Thus an (E, E)-bimodule structure and an (E*"', E*g')-bimodule structure are obtained on E. Further: PROPOSITION 9. Let E be a graded bigebra (resp. skew graded bigebra). For every element x of degree 1 in E, the le$t and right inner products by x are derivations (resp. antiderivations) (5 10, no. 2) of the algebra E*gr. In the notation of no. 6, for every homogeneous element x of degree 1 in a graded bigebra (resp. a skew graded bigebra) E, by Proposition 5 of no. 3 and the fact that c is a homomorphism of degree 0. Suppose first that E is a graded bigebra. For ally E E, by definition INNER PRODUCTS: CASE OF BIGEBRAS c(x) = x @ 1 + 1 @ x, ((4 X,Y> = (uv, XY) = m((u €9 V)(C(XY))) and since c is an algebra homomorphism, c(xy) = c(x)c(y). Let c(y) = 2 si @ti with si and t, in E; therefore 60 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Then also ,<c< .i.r (; u(xs,,)v(ti,,-j)) = ((u L x)~, y). On the other hand, ~(s,,) = 0 unlessj = -p and hence we may also write OGIGT c ( - l)j(T u(sij)v(xti,r-J) = (- l)”(u(v L x), y). We therefore conclude that (65) (i(4)(UV) = ((i(x))(4b + (-1)”4(i(4)(4), in other words i (x) is an antiderivation in E*g’. The assertions relating to the left inner product by an element x of degree 1 in E are proved similarly. Remarks. (1) Let E be a graded bigebra over A and N, N’, N” three graded A-modules. Let m be an A-bilinear mapping of N x N‘ into N”; for UE Homgr,(E, N) and v E Homgr,(E, N’), let u.v denote the graded homo - morphism m o (u @ v) o c of E into N”. On the other hand, let i(x) denote the (right or left) inner product by ~EE in the A-modules Homgr,(E,N), Homgr,(E, N’) and Homgr,(E, N”). Then, if x is of degree 1, for all u E Homgr,(E, N) and v E Homgr,(E, N’). geneous of degree p, then (i(x))(u.v) = ((i(x))(4) .v + fl. ((iM)(V)) Under the same conditions, if E is a skew graded bigebra and u is homo- (i@))(U.V) = ((i(x))(.)).v + (-1)pu.((i(4)(v))* The proofs are the same as in Proposition 9. for all x E E, if ~(x) = formula” (2) The same argument as in the above proof proves, more generally, that xi €4 x; then, for all u, v in E*gr, the “Leibniz f (i (4) (4 = c 1 (i (4) (4 . (i (4)) (4 holds. In particular, for every primitive element of a graded bigebra E, that is such that ~(x) = x @ 1 + 1 @ x, i(x) is a derivation of E*gr. PROPOSITION 10. Let E be a graded bigebra (resp. skew graded bigebra). For every element f of degree 1 in E*gr, the le$t and right inner products are derivations (resp. antiderivations) of the algebra E. Let x E E,, y E E, (p 2 1, q 2 1). By Proposition 5 of no. 3, we can write 602 1 9 11.9 INNER PRODUCTS where xi, and x;, belong to E,, y$ and & to Ek. If E is a graded bigebra, the component of c(xy) = c(x)c(y) belonging to El @ E, is equal to 7 4.1 €4 4,,-1Y + c Yl, c3 xy;,-1 (XY) Lf = p(4.1)4.,-lY + c , f(Y;*l)xY;q-l &;,1 @x;l.p-lY + (-1),&;*l €4xy;,q-1 and hence by definition = (2 Lf)Y + X(Y Lf) and the right inner product by f is a derivation. If on the other hand E is a skew graded bigebra, the component of c(xy) belonging to El @ E is equal to and this time we obtain (Y) Lf = (x Lf)Y + (-l)px(Y Lf) which shows that i (f) is then an antiderivation. The argument is similar for the left inner product by$ Examples. Propositions 9 and 10 apply in particular to the graded bigebra S(M) and the skew graded bigebra A(M). The inner products by elements of degree 1 in S(M) (resp. S(M)*g’) are derivations which commute with one another, since S(M) (resp. S(M)*g’) is commutative. Similarly, the inner products by elements of degree 1 in A(M) (resp. A(M)*~’) are antiderivations, which are of zero square, for the square of an element of degree 1 in the algebra A (M) (resp. A(M) *gr) is * zero. 9. INNER PRODUCTS BETWEEN T(M) AND T(M*), S(M) AND s(M*), A(M) AND A(M*) The right inner product defines on T(M) (resp. S(M), resp. A(M)) a right module structure over the algebra T(M) *gr (resp. S(M) *gr, resp. A(M) *gr) (no. 7, Examples). Using the canoriial homomorphisms OT (resp. 85, resp. 0,) of no. 5, we derive a right T(M*)-module structure on T(M) a right S(M*)-module structure on S(M) a left /\(M*)-module structure on A(M). The external law of any of these structures is also denoted by (z*, t) ++ i(z*) .t 603 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (by an abuse of language) ; we also write t L z* instead of i (t*) . t in the case of T(M) or S(M) ; on the other hand, we write z* J t in the case of A(M) and say that this is a left inner product oft by z*, since then we have a left A(M*)- module law. For Z* homogeneous of degree n and t homogeneous of degree p, i(z*).t = 0 if@ < n and, for X,EM (1 6 i <p), $EM* (1 <j < n) and p 2 n, by virtue of formulae (58), (59) and (60) of no. 7, (68) i(xT A xz A . . . A xz). (xl A x2 A . . . A .,) where, the formulae (67) and (68), (r runs through the set of permutations c E 6, which are increasing on the intervals (1, n) and (n + 1, p). We can also write, in the inner product notation, (t L u*, v*) = (t, OT(U*U*)) for t E T(M), u*, v* in T(M*) (t L u*, v*) = (t, Os(u*v*)> for t E S(M), u*, v* in S(M*) (u*,u* J t) = (OA(u* A u*), t> for t~/l(~),u*, v* in A(M*). We leave to the reader the task of finding explicitly the analogous formulae for left inner products, this time using formulae (61), (62) and (63). The above can be applied with M replaced by its dual M*; M* must then be replaced by the bidual M** and T(M*), for example, thus has a right module structure over the algebra T(M**). But the canonical mapping cM:M -+ M** defines an algebra homomorphism T(c,) :T(M) -+ T(M**), by means of which T(M*) has a right T(M)-module structure. Similarly S(M*) (resp. A(M*)) has a right S(M)-module (resp. left A(M)-module) structure. The explicit formulae giving the external laws of these modules are derived im - mediately from the above by exchanging the roles of M and M*. Note that, for all x E M, i(x) is always a derivation (resp. antiderivation of zero square) of the graded algebra S(M*) (resp. A(M*)). PROPOSITION 11. The canonical homomorphism eT:T(M*) -+ T(M)*gr (resp. Os:S(M) -+ S(M)*gr, resp. OA :A(M*) -+ A(M)*=) is a right T(M)-module (resp. right S(M)-module, resp. left A(M)-module) homomorphism. 604 INNER PRODUCTS IN FINITELY GENERATED FREE MODULE 5 1 1.10 We show first that, for z* E T(M*) and t E T(M), (69) oT(Z* L t) = &(z*) L t. Since M is a generating system of the algebra T(M), we need only prove (69) when t = XEM; moreover we can restrict our attention to the case where z* = xf 8 x: €3. . . @ x,*, where the x; E M*, and then, by (66) with the roles of M and M* interchanged, z* L x = (x, x:>x,* 8. . @ x,*. Therefore, for all y2, . . . , y, in M, P = <oT(z*), = (oT(z*) Lx,yz @“‘@!lp) @ yz €3. * . @ Yp> whence (69). We prove secondly that for z* E S(M*) and t E S(M), L t) = e+*) t. (70) As above, we can limit ourselves to the case t = x E M. But further, here i (x) is a derivation of S(M*) and a derivation of S(M) *gr. Therefore (3 10, no. 7, Corollary to Proposition 9) it suffices to verify (70) for t* = x* E M*, since M* is a generating system of S(M*) ; but this is trivial, the two sides then being equal to (x*, x). A similar argument proves the relation o,4,(t z*) = t J oi\(z*) (71) for z* EA(M*) and ~EA(M): observe then that, for XEM, i(x) is an antiderivation in A (M*) as well as in A(~)*gr and use 4 10, no. 7, Corollary to Proposition 9. There is an analogous result for left inner products. 10. EXPLICIT FORM OF INNER PRODUCTS IN THE CASE OF A FINITELY GENERATED FREE MODULE Let M be a finitely generated free A-module, (e,)lGi,n a basis of M and (e:),,,,, the dual basis of M*. For every finite sequence s = (il, . . . , i,) of elements of (1, n), let e, = e,, @ e,, @. . . @ eip (resp. e: = ec @. . . @ e;). We know (5 5, no. 5, Theorem 1) that the e, form a basis of the A-module T(M) and the ez a basis of the A-module T(M*). If s, t are two finite sequences of elements of (1, n), let s.t denote the sequence obtained as follows: if s 7 (ii, . . . , i,) and t = (jl, . . . ,jJ, s. t is the sequence (il, . . . , z,, j1, . . . ,jq) with@ + q terms. Then e,,, = e, @ e,. It then follows from (66) that .. e, L e: = 0 et,, L eF = e,. if s is not of the form t.u (72) 605 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS Similarly, the symmetric algebra S(M) has as basis the set of monomials em with cc E N" (5 6, no. 6, Theorem 1) and S(M*) the set of monomials e*, with E N"; recall (no. 5) that u,, for la1 = k, denotes the element of the basis of (Sk(M))*, dual to the basis (ea),,,=, of Sk(M); the u,, for a EN", therefore form a basis of S(M)*gr. The definition of right inner product by eP in S(M)*gr as the transpose of multiplication by e0 in S(M) then shows that (73) U,L eo = 0 if cc + p u, L e0 = u,-~ if cc 2 p. Similarly, since S(M) is here canonically identified with the graded dual of S(M)*gr, i(up) is the graded transpose of multiplication by u0 in S(M)*gr and hence from the multiplication table (33) (no. 5) of the basis (u,) we deduce that eaLuD=O if a+p ea L ug = (p, cc - p)ea-o if a 2 p. (74) As for the right inner product of an element of S(M) by an element of S(M*), the definition of this product (no. 9) and formula (34) of no. 5 allow us to deduce from (74) the formulae rea L e*P = 0 (75) There are analogous formulae for the inner product of an element of S(M*) by an element of S(M) interchanging the roles of M and M* (since M** is here identified with M). Remark. Being given the basis (eJl 41Gn allows us to identify the algebra S(M) with the polynomial algebra AIXl, . . ., X,] (5 6, no. 6); formula (75) shows that the inner product by e*, is just the differential operator D' = DY'DF. . .D>, where D, = a/aX, for 1 6 i 6 n (5 10, no. 11, Example). Consider finally the exterior algebra A (M) , which has as basis the set of elements e,, where J runs through the set of subsets of the interval (1, n) of N (9 7, no. 8, Theorem 1) ; similarly A(M*) has as basis the elements eJ*. It fol- lows from formula (68) of no. 9 that eh -I e, = O if KQ.J & -I e, = (- 1)~(~-1)12 pK,J-KeJ-K if K c J and P = Card(K), (76) { where pK,J-K is the number defined by formula (19) of 5 7, no. 8. There are analogous formulae with the roles of M and M* interchanged. 606 5 11.11 ISOMORPHISMS 11. ISOMORPHISMS BETWEEN Ap(M) AND A"-'(M*) FOR AN n-DIMEN- PROPOSITION 12. Let M be a free A-module of dimension n; let e E A"(M) be an element forming a basis of A"(M) and let e* be the element of A"(M*) such that { ( - l)"("- l)lZO,, (e*)} is the dual basis of{e} in (A"(M)) *. Let +:A (M*) +A (M) be the mapping t H z -I e* and +':A (M*) + A(M) the mapping z* H z* -I e. Let +p (resp. &) be the restriction of+ (resp. +') to A'(M) (resp. A'(M*)). Then: (i) The mapping + is a left A (M)-module isomorphism and the mapping +' is a left A(M*)-module isomorphism; moreover the mappings + and +' are inverses of one another. (ii) The mapping +p is an isomorphism ofthe A-module A"(M) onto the A-module A"-P(~*) and the mapping +; is an isomorphism ofthe A-module Ap(~*) onto the A-module A"- '(M). (iii) If we write B(u, u*) = (u, OA(v*)) for u EA(M) and u* EA(M*) then, for u* E P(M*) and v* E Pp(~*), SIONAL FREE MODULE M (77) B(+;(u*), u*) = (- l)P(n-P)B(~* , +' n-p(~*))* The fact that + is A(M)-linear and +' is A(M*)-linear follows from the formulae (u A v) -I e* = u -I (v -I e*) and (u* A v*) -I e = u* -I (v* -I e) (no. 6, formula (37), using the fact that OA is an isomorphism of A(M*) onto the opposite algebra to A (M) *) . On the other hand there exists a basis (e,) , ,, of M such that e*= (-l)n(n-1)/2* el A e,* A +. . A en*, e=e,Ae2A.-.Ae, and where (e:) is the basis dual to (e,). We write I = (1, n); it follows from (76) that, for every subset J of I with p elements, +(eJ) = (-1)n(n-1)/2+~(~-1)/2 * PJ. I-JeI J { +'(eJ*) = (- 1)~(~-1)/2 PJ. I-J~I-J. (78) This proves that + and +' are bijective; moreover pJ,I-JpI-J, , = (-l)p(n-p) (9 7, no. 8, formula (21)); as the number is even, it follows that + and 9' are inverses of one another. Finally, to prove (77), it suffices to take u* = e: and v* = eF-,; the verification also follows from the definition of Or\, formulae (78) and the relation pJ,I-JpI-J, , = ( - l)P("-p) (3 7, no. 8, formula (21)). Note that, for u* E A'(M*) and v* E A"-=(M*), 607 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS B(+;(u*), v*) is, to within a sign, the coefficient of u* A v* with respect to the basis {e*} of K(M*). PROPOSITION 13. With the hypotheses and notation of Proposition 11, for every endo- morbhism g of the A-module M. (79) (detg)+ = A(") O + O &g)- Clearly A ("9) = 8; 1 o ('A (9)) 0 8 A ; since A (9) is an endomorphism of the algebra A(M) and by definition, for all z E Ap), Qp,(A(g)(Z) -I e*) = OA(~*) L oA(A(g)(Z)), we deduce from formula (42) of no. 6 that ((exi 0 ("(g)) 0 0,) 0 + 0 A(g))(z) = eil(tA(g)(OA(e*)) L 2) = z J (A(tg)(e*)) = (detg)(z J e*) = (detg)+(z) taking account of 3 8, no. 4, Proposition 8. COROLLARY . For every automorphism g of E, (80) A&-1) = (detg)-l+ 0 (A(g)) 0 9-l. 12. APPLICATION TO THE SUBSPACE ASSOCIATED WITH A p-VECTOR Let K be a field and E a vector space over K. Recall that with every p-vector z E E(E) is associated a finite -dimensional subspace M, of E, namely the smallest vector subspace M of E such that z E K(M) (9 7, no. 2, Corollary to Proposition 4). PROPOSITION 14. (i) The orthogonal of M, in E* is the set of x* E E* such that x* J z = 0. (ii) The subspace M, associated with z is the image of A\"-~(E*) under the mapping A,:u* H u* J z ofAP-'(~*) into E. Let N denote the image of A,. For x* E E* and u* E W'(E*), Therefore, for x* to be orthogonal to N, it is necessary and sufficient that x* -I z be orthogonal to ~,,(A(E*)). Now, the latter condition is equivalent to saying that x* J z = 0; for let (eA)AoL be a basis of E; giving L a total order - ing, it has been seen (§ 7, no. 8, Theorem 1) that the eJ, for J running through the set $(I,) of finite subsets of L, form a basis of A(E); it then follows from 608 5 11.13 formula (30) of no. 5 that the elements On (e;) are, to within a sign, the coordi - nate forms on A (E) relative to the basis (e,) ; whence our assertion. The orthogonal of N therefore consists of the x* E E* such that x* J z = 0 and the conclusion of (i) will therefore follow from (ii). We show first that N c M,. Let M be a vector subspace of E such that z E A (M) and let j: M -+ E be the canonical injection; let p, denote the mapping v* ++ v* J z of A"-'(M*) into M; it follows from formula (60) of no. 7 that there is a canonical factorization PURE p-VECTORS. GRASSMANNIANS which proves that N c M and hence N c M, by definition of M,. It remains to verify that N = M,. Suppose the converse: there would then exist a basis (ei)l<isn of M, and an element x* E E* such that (x*, el) = 1, (x*, ej) = 0 for 2 < j < n and such that x* is orthogonal to N and hence x* J z = 0. We write z = & aHeH, where the sum is taken over the subsets of (1, n) with p elements. By (68) (no. 9), x* J eH = 0 if 1$H x* J e{l)uH = eH if H c (2, n) which shows that the relation x* -I z = 0 implies uH = 0 for 1 E H. But this is impossible, for z would then belong to A'(M~), where M' is the subspace of M generated by e2, . . . , en. 13. PURE p-VECTORS. GRASSMANNIANS Let K be a field and E a vector space over K. A p-vector t E P(E) is called pure (or sometimes decomposable) if it is non-zero and there exist vectors xl, . . . , x, in E such that z = x1 A x2 A . . . A xp. For this, it is necessary and sufficient that the subspace M, associated with z (which is always of dimension >p for z # 0) be exactly of dimension p (since A\"(M,) is then of dimension 1). In particular, every non-zero scalar, every non-zero element of E = A'(E), every non-zero element of A\"(E), when E is of dimension n, is pure. PROPOSITION 15. Let E be a vector space of dimension n and let e be an element #O of &(E) (hence forming a basis of this vector space). Let +:A(E) --f A(E*) be the vector space isomorphism associated with e (no. 1 1, Proposition 12). If z is a pure ele- ment ~A"(E), then +(z) is apure element gAn-P(~*) and the subspaces associated with z and +(z) are orthogonal. 609 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The casesp = 0 andp = n are trivial. Suppose therefore that 1 < p < n - 1 and let z = x1 A . . . A xp # 0. Then there exists a basis (e,),,, ,,, of E such that e, = xi for 1 < i < p and e = el A e2 A . . . A en. It then follows from formula (78) of no. 11 that +(z) = & e$+, A . .. A e,*, whence the proposition. COROLLARY . IfE is of dimension n, every non-zero (n - 1)-vector E is pure. PROPOSITION 16. For an element z # 0 &”(E) to bepure, it is necessary and suficient that,for all u* E A’-,(E*), (81) 1. If z = x1 A . + - A x,, formula (68) (no. 9) with n = p - 1 shows that u* J z is a linear combination of the xi (1 < i < p), whence (81). If on the other hand the subspace M, asso- ciated with z is of dimension >p, consider a basis (e,)lG,gn of this subspace with n > p. It follows from no. 11, Proposition 13 that each of the e, is of the form u* -I z for some u* EA’-~(E*) and relation (81) therefore implies e, A z = 0 for 1 < j < n. It follows that in the expression z = aHeH (where H runs through the set of subsets of (1, n) with p elements) all the coefficients aH are zero, whence z = 0, contrary to the hypothesis. The criterion of Proposition 16 is equivalent to writing conditions (81) when u* runs through a basis of AP-l(~*). In particular, suppose that E is of finite dimension n and let (eJl be a basis of E. Conditions (81) are then equiva - lent to the conditions (82-(Jy HI) {e:, (eg J z) A z) = O for all subsets J, H of (1, n) such that Card( J) = p + 1 and Card(H) = p - 1. Now, if I and I‘ are two subsets of (1, n) with p elements, formulae (76) of no. 10 and multiplication table (20) of 3 7, no. 8 show that {e:, {eg J e,) A er) = 0 unless there exists an i E (1, n) such that I - H = {i} and J - I’ = {i}, in which case (U* -I 2) A z = 0. The case p = 0 is trivial and we assume p where E,, J,H = ~(,),~p(,),~,; it can then be said that for i E J n C H, q, J,H is equal to + 1 if the number of element of J which are < i and the number of elements of H which are < i have the same parity, and - 1 otherwise. a,eI, where I runs through the It follows immediately that if we write z = 610 APPENDIX set of subsets of (1, n) withp elements, relation (82-(J, H)) is equivalent to the relation (84-(J, H) 1 2 ‘i, J,HaJ-{i)aHu(i) = 0. iEJ u CH Relations (84) are called Grassman’s relations; these are therefore necessary and sufficient conditions (when J describes the set of subsets with p + 1 ele- ments and H the set of subsets with p - 1 elements of (1, n)) for an element z # 0 of A,(E) to be pure. Note that relations (84) are not independent. For example, for n = 4 and p = 2, Grassmann’s relations reduce to the single relation a12a34 - a13a24 + al4aZ3 = 0. (85) Let D,(E) be the subset of A’(E) consisting of the pure p-vectors; clearly D,(E) is saturated with respect to the equivalence relation between u and v: “there exists A E K* such that u = Au” and two elements u, v of D,(E) are equivalent under this relation if and only if the subspaces Mu and M, of E which are associated with them are the same. Therefore we thus obtain a canonical bijection of the set of p-dimensional vector subspaces of E onto the image G,(E) ofD,(E) in theprojective space P(A”(E)) associated with A,(E). The sub- set G,(E) ofP(Ap(E)) is called the Grassmannian of indexp of the vector space E. When E is finite-dimensional and (e,) , , , is a basis of E, the Grassmanian of index p is the set of points of P(Ap(E)) for which a system of homogeneous coordinates (a,) (relative to the basis (e,) of Ap(E)) satisfies Grassmann’s relations (84). When E = K”, we sometimes write G,,”(K) instead of G,(Kn), so that G,,,(K) = P,,-,(K). The mapping M I-+ Mo, which associates with every p- dimensional subspace of K” the orthogonal subspace in E* (identified with K“ under the choice of basis dual to the canonical basis of K“) therefore defines a canonical bijection of G,,, ,(K) onto G,,, ,,-,(K) ; Proposition 15 shows that this bijection is the restriction to G,,, ,(K) of a canonical isomorphism of the projective space P(A,(K”)) onto the projective space P(A”-~(K~)). APPENDIX ALTERNATIVE ALGEBRAS. OCTONIONS 1. ALTERNATIVE ALGEBRAS Let A be a commutative ring and F a (not necessarily associative) A-algebra. For any three elements x, y, z of F, we write a(x,y, 2) = X(YZ) - bY)Z (1) 61 1 III TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (associator of x, y, z) ; a is obviously an A-trilinear mapping of F x F x F into F. Lemma 1. For all p, q, r; s in the algebra F, (2) a&, r, s) - a@, qr, s) + 4, q, ~$1 = q, r)s + Mq, r, $1. The verification follows immediately from definition (1). (a) For every ordered pair of elements x, y of F, the subalgebra generated by x and y (b) The trilinear mapping (x, y, z) t+ a(x, y, z) is alternating (3 7, no. 3). (c) For every orderedpair ofelements x, y of F, x2y = x(xy) andyx2 = (yx)x. Clearly (a) implies (c). We show that (c) implies (b): by definition (3 7, no. 3) to prove (b), it suffices to verify that a(x, x, y) = 0 and a(x, y, y) = 0, which is precisely (c). Lemma 2. Let E be an A-algebra such that the trilinear mapping (x, y, z) +> a(x, y, z) is alternating, S a generating system of E and U a sub-A-module of E containing S and such that sU c U and Us c U for all s E S. Then U = E. The set U' of x E E such that XU c U and Ux c U is obviously a sub-A - module of E, which contains S by hypothesis. On the other hand, for x, y in U' and u E U, by hypothesis (xy>u = x(yu) + a(x,y, u) = x(yu) - a(x, u,y) = x(yu) - (xu)y + 4.9) E U; on passing to the opposite algebra, we have similarly u(xy) E U. Hence U' is a subalgebra of E and, since it contains S, U' = E. Hence EU c U and a fortiori UU c U, which proves that U is a subalgebra of E; as it contains S, U = E, which proves the lemma. A subset H of F is called strongly associative if a(u, v, w) = 0 when at least two of the elements u, u, w belong to H. Lemma 3. Suppose that the mapping a is alternating. IfH is a strongly associative subset of F, the subalgebra of F generated by H is strongly associative. As the set of strongly associative subsets of F is inductive, it suffices to prove that if H is a maximal strongly associative subset of F, H is then a subalgebra of F. As H is obviously a sub-A-module of F, it suffices to verify that for any two elements u, u of H, H u {uv} is also strongly associative, for by virtue of the definition of H, this will imply uv E H. Now, for all z E H and all t E F, by (2) a(uu, t, z) - a(u, vt, z) + a(u, v, tz) = 0 PROPO~ITION 1. For an A-algebra F, the following conditions are equivalent: is associative. To prove that (b) implies (a), we use the following 4 lemmas: 612 APPENDIX since H is strongly associative; as u, v, z are in H, also a(u, vt, z) = a(u, v, tz) = 0, whence a(uv, t, z) = 0. Using the fact that a is alternating, this shows that a(p, q, r) = 0 whenever at least two of the elements p, q, r belong to H u {UV} whence the lemma. Lemma 4. Suppose that the mapping a is alternating. Then, for all x E F, the subalgebra of F generated by x is strongly associative. a(u, v, w) = 0 whenever two of the three elements u, v, zp) are equal to x and it suffices to apply Lemma 3. Lemma 5. Suppose that the mapping a is alternating and let X, Y be two strongly associative subalgebras of F. Then the subalgebra of E generated by X u Y is associative. Let Z be the set of z E E such that a(u, v, z) = 0 for all u E X and v E Y, this is obviously a sub-A -module containing X and Y since X and Y are strongly associative; by Lemma 2, it will suffice to verify that, for u E X and u E Y, uZcZ,vZ CZ,Zuc ZandZvc Z.Now,foru,u'inX,v~Yandz~Z,by a(u'u, z, u) - a(u', uz, v) + a(u', u, zv) = a(u', u, z)v + u'a(u, z, v) = O by virtue of the fact that X is strongly associative and the definition of Z. But as X is strongly associative, a(u', u, zv) = 0 and since u'u E X, a(u'u, z, v) = 0 by definition of Z. Hence a(u', uz, v) = 0, which shows that uZ c Z. Applying (2) now with (p, q, r, s) = (u, z, u, u'), we obtain similarly Zu c Z. Inter- changing the roles of X and Y and using the fact that a is alternating, we obtain uZ c Z and Zu c Z; whence the lemma. It now suffices, to prove that (b) implies (a) in Proposition 1, to take X = {XI and Y = {y}, using Lemma 4. (2) DEFINITI ON 1. An algebra F is called alternative ifit satisjies the equivalent conditions of Proposition 1. An associative algebra is obviously alternative. In no. 3 we shall give an example of an alternative algebra which is not associative. If F is an alternative A-algebra, every A -algebra F A' obtained from F by extending the scalars (§ 1, no. 5) is an alternative A'-algebra, as follows from condition (b) of Proposition 1. 2. ALTERNATIVE CAYLEY ALGEBRAS PROPOSITION 2. Let A be a ring, F a Cayley A-algebra, e its unit element, s:x H x its conjugation and N: F -+ A its Cayley norm (5 3, no. 4). (i) For F to be alternative, it is necessary and suficient that, for euey orderedpair of elements x, y of F, x2y = x (q) . 613 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (ii) IfF is alternative, then N(xy) = N(x)N(y) for all x, y in F. (iii) Suppose that F is alternative. For an element x E F to be invertbile, it is necessary and su$cient that N(x) be invertible in Ae; the inverse of x is then unique and equal to N(x)-'Z; denoting this by x-l, x-'(xy) = x(x-'y) = y or ally E F. The condition x2y = x(xy) is obviously necessary for F to be alternative (no. 1, Proposition 1). Conversely, if it holds for every ordered pair of elements ofF, applying it to and 3, it gives X2j = %(Xy); applying the conjugation s to this relation, we obtain yx2 = (yx)x, so that conditions (c) of Proposition 1 of no. 1 are satisfied. Obviously a(e, x, y) = 0 for all x, y in F. If F is alternative, we therefore deduce from Proposition 1 (no. 1) that the subalgebra G of F generated by e, x and y is associative. As ?i = -x + T(x) E -X + Ae, X E G and similarly ij E G. Then N(xy) = (xy)(Xy) = xy.y.j? = N(y)xj? = N(y)N(x), using the fact that N(y) E Ae. This proves (ii). Finally we prove (iii). If N(x) is invertible in Ae and we write x' = N(x) -I%, then xx' = x'x = e, for N(x) = XX = Xx. Conversely if x admits a left inverse x", then N(x")N(x) = N(e) = e by (ii) and N(x) is invertible in Ae; further, as x' = N(x) -1. is in the subalgebra generated by x and e, the elements x, x', x" belong to the associative subalgebra generated by x, x" and e and hence X" = ~"(xx') = (x"x)x' = x', whence the uniqueness assertion. The formulae x-'(xy) = x(x-'y) = y follow from the fact that x-', x and y are elements of the subalgebra generated by x, y and e, which is associative. PROPOSITION 3. Let E be a Cayley A -algebra, y an element of A and F the Cayley extension of E deJined by y and the conjugation of E (8 2, no. 5, Proposition 5). For F to be alternative, it is necessary and suficient that E be associative. Let u = (x, y), v = (x', y') be two elements of F (where x, y, x', y' are in E). Then (5 2, no. 5, formula (27)) u2v = ((x" + yyy).' + yU'(yX + yx), (y. + yx).' + y'(x2 + yyy)) u(uv) = (x(xx' + yj'y) + y(x'j + X.ij')y,y(f'. + yjy') + (y.' + y'x).). (3) { Using the fact that ijy and 2 + x are in Ae, examining these formulae shows that the associativity of E implies u2v = u(uv) and hence the fact that F is alternative (Proposition 2). Conversely, if F is alternative, the equation u2v = u(uv) applied when y' = 0 gives (y?i + yx).' = y(X'X) + (yF')x. Now the left hand side is equal to (yT(x))jz' = y(.'T(x)) = ~(X'X + 2%); 614 APPENDIX comparing with the right hand side, we obtain (y?i')x = y(%'x), which proves the associativity of E, since x, y and 2' are arbitrary elements of E. 3. OCTONIONS Let E be a quaternion algebra of type (a, p, y) over A (5 2, no. 5, Example 2) and let S E A. The Cayley extension F of E by 8 and the conjugation of E is called an octonion algebra over A and is said to be of type (a, P, y, S) . By Propo - sition 3 of no. 2, F is an alternative algebra. It has a basis (ei)o,<r,<7 of 8 elements, defined by eo = (e, 01, ei = (i, 01, ea = (j, 01, e3 = (k 0) e4 = (O, e), e5 = (O, i), e6 = (o,j), e7 = (O, k) where (e, i,j, k) is the basis of E defined loc. cit. ; clearly e, (also denoted by e) is the unit element of F. If u = $ &ei is an element of F (with the Ei E A), formulae (23), (24) and (31) of 3 2, no. 5, give for the conjugate, trace and norm of the octonion u i - + PLE5 - d52) + yS(G + PE6E7 - 4;). t=0 = (Eo + PEJeo - *=1 t Eiti (4) TFb) = 2Eo + BE, NF(u) = E: + PEoEi - at? - Y(G + PLE3 - aE$) Now let u = (x, y), u' = (x', y') and U" = (x", y") be three octonions (where the elements x, x', x", y, y', y" belong to E). Formulae (24) and (27) of 9 2, no. 5 give TF((uu')u") = T(xx'x") + ST(y'yx") + ST(j"yZ) + ST(y"y'x) TF(u(u'u")) = T(xx'x") + ST(x"ij'y) + ST(5'y"y) + ST(xj7'y') (where T denotes trace in E and use is made of the fact that E is associative). As T(xy) = T(yx) for all quaternions x, y (3 2, no. 4, formula (17)), it follows that (5) T~((uu')u") = TF(u(u'u")). We study in particular octonions of type ( - 1, 0, - 1, - 1) ; formulae (4) then simplify to 615 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS *If we take A to be the field R of real numbers, the octonions of type ( - 1,0, - 1, - 1) over R are called Cayley octonions (or octaves). It follows from Proposition 2 (ii) of no. 2 that every Cayley octonion #O is invertible.* PROPOSITION 4. Let F be an octonion algebra of type (-1, 0, - 1, -1) over A. There exists a vector space V of dimension 3 ouer the field with two elements 2/22 and a bijection A ++ ei of V onto the basis (e,),,,,, such that (7) eh = e,, eie; = fei+u for all A, p. in V. In order that ei(eLe{) = (ele,)e{, it sufices that, in V, A, p, v be linearly dependent over 2/22; this condition is necessary if 2 # 0 in A. We preserve the notation at the beginning of this no, It follows from formulae (33) of 4 2, no. 5 that the set S consisting of the elements f e,, f el, fe,, -+e, is stable under multiplication. Moreover, for x, y, y' in E, by formula (22) of 5 2, no. 5, (8) (x, 0) (0, Y') = (0, Y'X), (0, Y') (x, 0) = (0, Y'?), (0, Y) (0, Y') = ( -Y?, 0) so that the set T consisting of the elements +el (0 < i < 7) is stable under multiplication ; moreover, its multiplication table is independent of the ring A. In particular, let A" be the field 2/22 with two elements and let E" be the quaternion algebra of type (1, 0, 1) over A" and F" the algebra of octonions of type (1, 0, 1, 1) over A"; let (e;)o$i,7 be the basis of F" formed as described above. Since -el = er, the set T" of el has 8 elements and is stable under multiplication; moreover, it follows immediately from the above that the mapping O:T+T" such that O(ei) = @(-el) = elfor 0 < i < 7 is a homo- morphism for multiplication. Moreover the quaternion algebra E" is in this case commutative and hence F" is associative (3 2, no. 5, Proposition 5) ; further, conjugation in F" is in this case the identity. Hence T" is a group and formulae (8) show that it is commutative; these formulae and formulae (33) of 5 2, no. 5 show that the square of every element of T" is the unit. If V is used to denote the group T" written additively, V can be given a unique vector space structure over 2/22, necessarily of dimension 3 since Card(V) = 8 = Zdirn("). For all A E V, let ei then denote the element of (eJOsiQ7 such that O(e;) = A; then eh = e,; moreover, as 0 is a homomorphism and the relation O(x) = O(y) is equivalent to x = fy, eie; = _+e;+u. If A, p, v are linearly independent over 2/22, they form a basis of V and hence all the elements el (0 < i < 7) would belong to the subalgebra generated by el, e; and e:; when 2 # 0 in A, el,(e;e:) = (el,e;)e: is therefore impossible, for F would be associative and hence E would be commutative ($2, no. 5, Proposition 5), which contra - dicts relations (33) of § 2, no. 5. On the other hand, if A, p, v are linearly 616 APPENDIX dependent in V, the three elements el, e:, ei belong to a subalgebra with 2 generators of F, which is therefore associative (no. 1, Proposition 1); whence the conclusion. Remark. As ii = -el for A # 0, eLz = -e , for A # 0, eLcl = -eleL for A # 0, p # 0 and p # A. 617 EXERCISES ----- el el 0 ez 0 ez ---- ---- Fez 31 1. Let E be an algebra over a (commutative) field K and L a commutative extension field of K. Show that, if the algebra E,,, admits a unit element, so does E (cf. 11, 3 3, no. 5, Proposition 6). e3 0 e4 "!!I §2 1. Let E be an A-algebra with a basis of two elements el, ez and a unit element e = he, + Fez. (u) Show that the ideal a of A generated by h and p is the whole ring A (in the contrary case, note that E/aE = E @A (A/a) = (0) would hold con- trary to the fact that A/a # (0) and E is a free A -module). (b) If a, p are two elements of A such that aA. + pp = 1, show that the elements e and u = pel - aez also form a basis of E and deduce that E is a quadratic A-algebra. 2. Show that every quadratic algebra over Z is isomorphic to an algebra of type (0, n) with n E N, or an algebra of type (m, 0) with m a non-square, or an algebra of type (m, l), where m is an integer which is not of the form k(k - 1) ; moreover, no two of these algebras are isomorphic. 3. (u) Let K be a commutative field. Show that, for K to be the centre of a quaternion algebra of type (a, p, y) over K, it is necessary and sufficient that one ofthe following cases hold: (1) py # 0; (2) y = 0, p # 0 and pz + 4a # 0; (3) p = 0, a # 0 or y # 0 and 2 # 0 in K. (b) Let K be a commutative field such that 2 # 0 in K; show that the quaternion algebra over K of type (1, Y) is isomorphic to the matrix algebra Mz(K) (consider the basis of this algebra consisting of the elements +(I + i), (c) Let K be a commutative field such that 2 # 0 in K and let E be the quaternion algebra over K of type (a, y). A quaternion z E E is called pure if Z = -z (or also T(z) = 0). Show that if z is pure, so is every quaternion tzt -I, where t E E is invertible. - i), (3. + k/2Y), t(j - 4). 618 EXERCISES (d) If u, u are any two quaternions, uu - uu is pure. Deduce that if u, u, w are any three quaternions, then (uv - UU)ZW = w(uv - my. (e) Show that if there exists in E a quaternion z # 0 such that N(z) = 0, there also exists a pure quaternion z' # 0 such that N(z') = 0. (Note, in the notation of formula (33) (no. 5), that, if a is not a square in K, the exis- tence of t # 0 in E such that N(z) = 0 is equivalent to the existence of an element y E K + Ki such that y = N(y).) 5. Over a commutative field K in which 2 = 0, every quaternion algebra E of type (a, 0, y) is commutative and the square of every x E E belongs to K; the subalgebra K(x) of E generated by an element x E K is therefore a quad - ratic algebra of type (A, 0) over K and E is a quadratic algebra over K(x). Show that the set C of x E E whose square is equal to the square of an ele- ment of K is a vector subspace of E whose dimension is equal to 1, 2 or 4. If C is of dimension 1 (in which case C = K), E is a field. If C is of dimen - sion 2, there exists a quadratic algebra K(x) contained in E which is a field and E has a basis over K(x) consisting of two elements 1 and u with u2 = 0; the set of y E E such that yz = 0 is an ideal a of dimension 2 over K and E/a is isomorphic to K(x). Finally, if C is of dimension 4 (and hence equal to E), the set a of y E E such that y2 = 0 is an ideal of dimension 3 and E/a is iso- morphic to K; there exists in E a basis (1, el, ez, e,) such that e: = ef = eg = 0, e1e2 = e,, e,e3 = e2e3 = 0; Ke, = b is the only ideal of dimension 1 in E, b is the annihilator of a and a/b is the direct sum of two ideals of E/b, of dimen - sion 1, which annihilate one another. 6. Let K be a commutative field in which 2 # 0 and E a quaternion algebra over K of type (0, 0, y). (u) If y is not a square in K, there exists no left (resp. right) ideal in E of dimension 1 over K; the set a of x E E such that x2 = 0 is a two-sided ideal of dimension 2 over K and E/a is a field, a quadratic algebra over K. (b) If y is a square #O in K, there exists in ,E a basis (e,),,,,4 with the following multiplication table : 619 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The set a of x E E such that x2 = 0 is a two-sided ideal of dimension 2, the direct sum of the two two-sided ideals Ke, and Ke, which annihilate one another; the latter are the only (left or right) ideals of dimension 1 in E. The quotient algebra E/a is isomorphic to the product of two fields isomorphic to K. (c) If y = 0, the set a of x E E such that x2 = 0 is a two-sided ideal of dimen- sion 3; Kk = b is the only (left or right) ideal of dimension 1 in E; it is a two- sided ideal, the left and right annihilator of a. In the quotient algebra E/b, a/b is the direct sum of two two-sided ideals of dimension 1 which annihilate one another; finally E/a is a field isomorphic to K. 7. Let K be a commutative field in which 2 # 0 and E the algebra over K with a basis of four elements 1, i, j, k (1 the unit element) with multiplication table .. .. i2 - -J '2 - - k 2 = 1, ZJ =JZ = k, jk = kj = i, ki = ik =j. Show that E is isomorphic to the product offour fields isomorphic to K (consider the basis of E consisting of the elements (1 + Ei)(l + E?), where E and E' are equal to 1 or -1). The algebra E is the algebra (over K) of the product of two cyclic groups of order 2. Generalize this to the algebra of the product group of n cyclic groups of order 2 (cf. 5 4, no. 1, Example 2). 8. The quaternionic group Q (I, 5 6, Exercise 4) is isomorphic to the multi- plicative group of the eight quaternions f 1, i, f j, 5 k in the quaternion algebra of type ( - 1, - 1) over a field in which 2 # 0. Show that the algebra of the group Q over a field K in which 2 # 0 is isomorphic to the product of four fields isomorphic to K and the quaternion algebra of type ( + 1, - 1) over K (if c is the element of Q corresponding to the quaternion - 1, the elements of Q can be written as e, i, j, k, c, ci, cj, ck; consider the basis of E consisting of the elements +(e + c), +(c - c), +(e + c)i, +(e - c)i, +(e + c) j, $(c - c) j, 9. (a) Show that the algebra E of the dihedral group D4 of order 8 (I, 5 6, Exercise 4) over a field K in which 2 # 0 is isomorphic to the product of four fields isomorphic to K and the algebra of matrices of order 2 over K. (If a, b are the two generators of D, introduced in I, 3 6, Exercise 4, the elements of D4 can be written as a'bj with 0 < i < 3, 0 < j < 1; consider the basis of E consisting of the four elements +(e + a2), +(e - u2), +(a + u"), +(a - a") and these four elements multiplied on the right by b; use Exercise 4.) Deduce that, if - 1 is a square in K, the algebras over K of the non-isomorphic groups Q and D, are isomorphic. (b) Show similarly that the algebra F of the dihedral group D3 of order 6 over a field K in which 2 # 0 and 3 # 0 is isomorphic to the product of two fields isomorphic to K and the matrix algebra of order 2 over K (consider here 620 He + c)k, He - c)4. EXERCISES the basis of F consisting of the elements e + u + 2, a + a2 - 2e, a - a2 and these three elements multiplied on the right by 6). 10. Let G be a group and H a normal subgroup of G. Show that, if a is the two-sided ideal of the group algebra A(G), generated by the elements ts - s, where t runs through H and s runs through G, the group algebra 11. Let A be a commutative ring and S a commutative monoid with iden - tity element e. Suppose that an external law (s, x) H xs is given on A with S as domain of operators, such that, for all s E S, the mapping x H xs is an automorphism of the ring A and (xS)' = xSt for all x E A and s and t in S (which implies that e is the identity operator for the external law considered). Then a multiplicative internal law of composition is defined on the A-module A@), whose canonical basis is denoted by (bJ, by the relation is isomorphic to the quotient algebra A(G)/a. where the as, belong to A. provided the as, satisfy the conditions Show that this law and addition on A(') define a ring structure on this set, as, tast, u = a;uas* tu for all s, t, u in S. The element be is unit element of this ring if also 1 ae,s = as,e = for all s E S; in that case A can be identified with a subring of A") and if C denotes the subring of A consisting of the elements invariant under all the automorphisms x H xs, A@) is a C-algebra, called the crossed product of the ring A and the monoid S, relative to thefactor system (as, t). If the factor system (as, t) is replaced by , where, for all s E S, c, is an invertible element of A and c, = 1, the new crossed product obtained is isomorphic to the crossed product defined by the factor system (asvt). If in particular A is taken to be a quadratic algebra over a ring C and S a cyclic group of order 2, say {e, s>, such that 9 = X for x E A, show that every crossed product of A and S is a quaternion algebra over C. 12. Let I be a non-empty set. Show that the elements of the free magma M(1) can be identified with the sequences (n an arbitrary integer 2 1) where the a, are either elements of I or a special symbol P (representing the "open brackets"), where these sequences satisfy the following conditions : (i) The length n of the sequence is equal to twice the number of symbols P which appear in it, plus 1. 62 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS liil For all k < n - 1, the number of symbols P which appear in the sub- \--/ sequence (ai) 1 (Use Set Theory, I, Appendix to associate with each sequence (ai) an element of M(1) ; for example, to the sequence PPPxyzPxy corresponds the word k is 3 k/2. (((&j(xy)).) 7 13. Let A be a commutative ring and E an A-module. An A-module En is defined inductively by writing E, = E, E, = 0 (E, gA~q) for n > 2. p+q=n W We write ME = @ En and an A-algebra structure is defined on ME by setting, for xp E E, and xq E E,, xpxq = xp @ xq E E, + 4. (u) Show that for every A-algebra B and every A-linear mappingf: E -+ B, there exists one and only one A-homomorphism of algebras ME -+ B extend - ing$ ME is called the free algebra Ofthe A-module E. n=1 (b) An integer a(.) is defined by induction on n by the formulae C a(p)a(q) ifn > 2. P+q=n 4), 4.1 = Show that En is isomorphic to the direct sum of a(.) modules isomorphic to Em". (c) Letf(T) be the formal power series a(n)Tn. Show that f(T) = T + (f(T))2. *Deduce the formulae 1.3.5.. . (2n - 1) '* - " -4T and = 2n-1 f(T) = 2 n! (d) If I is any set and E = A(I), show that LibA(I) is identified with ME. There are canonical homomorphisms M(1) -% "I) -+ N whose composition is length in M(1) ; w(m) is called the multidegree of m E M(1) ; the mapping w defines on LibA(I) a graduation of type N"). Show that the set (LibA(I))a of homogeneous elements of multidegree a E "I) under this graduation is a free A-module of rank equal to 1 n! 1.3.5.. . (2n - 1) .(a) = a(.) -i = 2"-' a. a! where a! = n (a(i)) ! and n = 1EI 2 a(i) = 1 (a) is the length of a. 1EI 622 EXERCISES 14. (a) Let M be a monoid whose law of composition is denoted by T; suppose further that M is totally ordered by an order relation x < y such that the relations x < y and x' < y', or x < y and x' < y', imply x T x' < y T y'. Show that if A is an integral domain, the algebra over A of the monoid M has no divisors of 0. (b) Deduce that if A is an integral domain, every polynomial algebra A[(Xi),EI] is an integral domain. (c) If A is an integral domain and f and g two polynomials in A[(Xi)i,I] such that fg is homogeneous and #O, show that f and g are homogeneous. In particular, the invertible elements of the ring A[(Xi)lE1] are the invertible elements of the ring A. 15. Let A be a commutative ring and n an integer 32 such that for all a E A the equation nx = a has a solution x E A. Let m be an integer 2 1 and u a polynomial in A[X] of the form Xmn + f (X) wherefis of degree < mn - 1 ; show that there exists a polynomial u E A[X] of the form Xm + w, where w is of degree <m - 1, such that u - un is zero or of degree < m(n - 1). 16. Let A be a commutative ring and u = 2 akXk a divisor of 0 in the ring A[X]. Show that if v = 2 bkXk is an element #O of A[X] of degree n k=O k=O such that uu = 0, there exists a polynomial w # 0 of degree <n - 1 such that uw = 0. (Reduce it to the case where b, # 0; if aku = 0 for 0 < k 6 m - 1, show that we can take w = b,; if aku = 0 for O<k <p<m- 1 and apu # 0, show that apbo = 0 and therefore we can take w = 2 apbk + lXk.) Deduce that there exists an element c # 0 of A such that cu = 0. n-1 k=O 17. Let A be a commutative ring. Show that, in the algebra A[X] of poly- nomials in one indeterminate over A, the mapping (u, u) H U(V) is a law of composition which is associative and left distributive with respect to addi- tion and multiplication in A[X]. If A is an integral domain, show that the relation u(u) = 0 implies that u = 0 or that u is constant and that, if u # 0 and the degree of u is >O, the degree of u(v) is equal to the product of the degrees of u and u. 18. Let K be a commutative field and K[X] the ring of polynomials in one indeterminate over K. Show that every automorphism s of the ring K[X] leaves K invariant (Exercise 14(c)) and induces on K an automorphism of this field; show further that necessarily s(X) = aX + b, with a # 0 in K 623 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYhfMETRIC ALGEBRAS and b E K (cf. Exercise 17). Conversely, show that being given an auto- morphism 0 of K and two elements a, b of K such that a # 0 defines one and only one automorphism s of K[X] such that s I K = 0 and s(X) = aX + b. If G is the group of all automorphisms of the ring K[X] and N the subgroup of G consisting of the K-algebra structure of K[X], show that N is a normal subgroup of G and that G/N is isomorphic to the automorphism group of the field K; the group N is isomorphic to the group defined on the set K* x K by the law of composition (A, p) (A', p') = (Ah', A'p + p'). 19. Prove the polynomial identity in 8 indeterminates (X? + xi + xi + XZ)(Y? + Y; + Yi + Yq) = (X,Y, - X2Y2 - X3Y3 - X4Y4)2 + (X,Y, + x,y, + X3Y4 - X4Y3)' + (Xly3 + x3y1 f x4y2 - X7,y4)' + (xIy4 + x&yl f x2y3 - X3y2)'. (Apply formula (32) of no. 5 to a suitable quaternion algebra.) 20. Show that there exists no polynomial identity in 6 indeterminates (X: + Xi +Xg)(Yl + Y: + Yz) = P2 + Q2 + R2 where P, Q, R are three polynomials with coefficients in Z with respect to the indeterminates Xi and Y,. (Observe that 15 = 3.5 cannot be written in the formp2 + q2 + r2, wherep, q, r are integers.) 21. Let S be a multiplicative monoid with identity element e, satisfying condition (D) of no. 10 and further satisfying the two following conditions: (1) the relation st = e implies s = t = e: (2) for all s E S, the number 12 of terms in a finite sequence (t,), GiGn of elements distinct from e and such that t,t,. . . tn = s is less than a finite number v(s) depending only on s. Show that, for an element x = zs a2 of the total algebra of S over a ring A to be invertible, it is necessary and sufficient that a, be invertible in A (reduce it to the case where a, = 1 and use the identity - p+l = (e - z)(e + t + . . + 2")). 22. If K is a commutative field, show that in the ring of formal power series K[[X,, X,, . . . , X,]] there is only a single maximal ideal, equal to the set of non-invertible elements. On the other hand, show that in the poly- nomial ring K[X,, . . . , X,] for p 2 1 there always exist several distinct maximal ideals. 624 EXERCISES 23. Let K be a commutative field; show that there exists no formal power series u(X, Y) E K[[X, Y]] such that, for an integer m > 0, (X + Y)u(X,Y) = XmY". 24. Let K be a commutative field and let k be an integer such that k # 0 in K. Show that for every formal power series u of K[[X]] with constant term equal to 1, there exists a formal power series v E K[[X]] such that vk = u. 25. Let A be a ring, Commutative or otherwise, and let cs be an endomorphism of A. On the additive product group E = AN an internal law of composition is defined by writing (an)@,,) = (yJ, where yn = 2 apcp(Pq) (with the convention a0(Q = E,). (a) Show that this law defines, with addition on E, a ring structure on E, admitting as unit element e the sequence (an) where a. = 1 and a, = 0 for n > 1. The mapping which associates with every E E A the element (an) E E, such that cto = E,, a, = 0 for n > 1, is an isomorphism of A onto a subring of E, with which it is identified. Then we write 2 anXn instead of (a,,) and Xpp = crp(p)Xp for all p E A; if u = ctnXn # 0, the smallest (b) If A is a ring with no divisor of 0 and 0 is an isomorphism of A onto p+q=n n=O integer n such that a, # 0 is called the order of u and n=o denoted by w(u). a subring of A, show that E is a ring with no divisor of 0 and that w(uv) = a(.) + w(v) for u # 0 and u # 0. (c) For u = 2 a,Xn to be invertible, it is necessary and sufficient that a. be invertible in A. (d) Suppose that A is a field and cs an automorphism of A. Show that E admits a field of left fractions (I, 3 9, Exercise 15) and that in this field F every element #O can be written uniquely in the form uX-~, where u is an element of E of order 0. 7 *26. (a) Let A and B be two well-ordered subsets of R (which are neces - sarily countable; cf. General Topolopy, IV, 3 2, Exercise 1). Show that A + B is well-ordered and that, for all c E A + B, there exists only a finite number of ordered pairs (a, b) such that a E A, b E B and a + b = c (to show that a non-empty subset of A + B has a least element, consider its greatest lower bound in R). (b) Let K be a commutative field. In the vector space 1cR, consider the vector subspace E consisting of the elements (a,) such that the set (depending 625 n=O 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMEI'RIC ALGEBRAS on (a,.)) of x E R such that a,. # 0 is well-ordered. For any two elements (a,.), ayp, (a sum which is (p,.) of E, we write (a,)(@,.) = (y,), where y,. = meaningful by virtue of (a)). Show that this law of composition defines, with addition, ajeld structure on E; the elements of E are also denoted by teR 2 a,Xt and called formal power series with well-ordercd real exponents, with coefficients in K.* c y+z=,. §3 1. Let A be a commutative monoid written additively with the identity element denoted by 0, all of whose elements are cancellable. Let E be a graded A-algebra of type A and suppose that E admits a unit element e = zAea where e, E E,. Show that necessarily e = e, E E,. §4 7 1. Let A be a commutative ring, E, F two A-algebras, M a left E-module, N a left F-module and G the algebra G = E @A F. For every ordered pair of endomorphisms u E End,(M), u E End,(N), there exists one and only one G-endomorphism w of M gAN such that w(x @y) = ~(x) @ u(y) for x E M, y E N and we can write w = +(u @ u), where 4 is an A-linear mapping (called canonical) of End,(M) @A End,(N) into End,(M @A N). Suppose in what follows that A is ajeld. (a) Show that the canonical mapping + is injective (consider an element z = 2 ui @ vi, where the v, E End,(N) are linearly independent over A and write (+(z))(x, @y) = 0 for every element x, of a basis of M over A and all (b) Suppose that M is ajinitely generatedfree E-module. Show that the mapping is bijective in each of the following cases : (1) E is of finite rank over the field A; (2) N is a finitely generated F-module. (c) Suppose that M is a free E-module and that there exist a y E N and an infinite sequence (v,) of endomorphisms of the F-module N such that the vector subspace (over A) of N generated by the v,(y) is of infinite rank over A. Show that, if the mapping + is bijective, every basis of M over E is necessarily jinite. (d) Suppose still that M is a free E-module. Show that, for the G- module M BAN to be faithful, it is necessary and sufficient that the F- module N be faithful (consider a basis of E over A). 'I[ 2. Let K be a commutative field, E, F two fields whose centre contains t YEN). 626 EXERCISES K, M a left vector space over E and N a left vector space over F; let (a) Let x # 0 be an element of M and y # 0 an element of N. Show that x @ y is free in the G-module M @lK N (use Exercise 1 (d) and the transi - tivity of End,(M) (resp. End,(N)) in the set of elements of M (resp. N) distinct from 0). N is a free G -module. (If (m,) is a basis of M over E and (nD) a basis of N over F, show that the sum of the G-modules G(m, @ no) is direct, using a method analogous to that of (u).) G = E gKF. (b) Show that M §5 1. Let N be the Z-module 2/42, M the submodule 22/42 of N and j:M -+ N the canonical injection. Show that T(j) :T(M) -+ T(N) is not injective. 2. Let I and J be two finite sets, I, = I U {a}, J, = J u {p}, where a and are two elements not belonging to I and J respectively, and suppose that I, and J, are disjoint. Let E be a vector space of finite rank over a commuta - tive field K and let z be an element of T:(E). For all i E I, let Wi be the sub- space of E* consisting of the x* E E* such that cb(z @ x*) = 0 (tD being the contraction of index i E I and index (3 E Jo in T:,(E)) and let Vi be the sub- space of E orthogonal to Wl. For all j E J, let W, be the subspace of E con- sisting of the x E E such that C;(Z @ x) = 0 (6; being the contraction of index a E I, and index j E J in Ty(E)) and let V; be the subspace of E* orthogonal to W,. Show that z belongs to the tensor product (@ Vi) @ V;) canonic- subspaces of E, (U;),€ a family of subspaces of E* such that z belongs to the tensor product (g U,) @ (9 U;), identified with a subspace of T:(E), then V, c U, and V; c U; for all i E I andj E J (cf. 11, 3 7, no. 8, Proposition 17). 3. Let M be a finitely generated projective A-module. For every endo- morphism u of M, let u" denote the tensor O,'(u) E M* @A M corresponding to it. (a) For every element x E M, show that ~(x) = C;(X @ u"), where x @ u" belongs to M @ M* @ M = T::i3'(M). (6) Let u, v be two endomorphisms of M and w = u o u; show that ZZ = cg(v" @ u"), where v" @ u" belongs to T::;$;(M). (c) For every automorphism s of M and every tensor z E T:(M), a tensor S. z E T:(M) is defined by the condition that if z = @ zk where zk E M for k E I and zk E M* for k E J, then s. z = @ zf, with zf, = s(zk) if k E I, ally identified with a subspace of T:(E). Moreover, iEI if (Ui)IEI is a family of ksIuJ keIuJ 627 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS zl, = ~S-~(Z~) if kEJ. Show that the group GL(M) operates by the law (s, z) ++ s. z on the A-module T:(M), the mappings z H s. z being automorph- isms of this A-module. Moreover, for every ordered pair of indices ~EI, j€J, ci(s.2) = s.c:(z). If M is projective and finitely generated, show, in the notation of Exercise 2, that s.ii = (s 0 u o S-')". 4. Let M be a finitely generated projective A-module; for every bilinear mapping u of M x M into M, let E denote the tensor OM-'(u) in Ti:!2,(M). Show that for the algebra structure on M defined by u to be associative, it is necessary and sufficient that the tensors c:(U" @ 6) and ci(E @ ii) correspond under the canonical associativity isomorphism of Ti;! 2, 3)(M) onto Ti;! 3, 4)(M). 7 5. Let E be a vector space over a commutative field K and let T(E) be the tensor algebra of E. (a) Show that T(E) contains no divisors of 0 and that the only invertible elements in T(E) are the scalars # 0. If dim(E) > 1, T(E) is a non-commuta- tive K -algebra. (b) Let u, v be two elements of T(E). Show that, if there exist two elements a. b of T(E) such that ua = vb # 0, one of the elements u, u is a right multiple of \I the other. (Consider first the case where u and v are homogeneous; reduce it to the case where E is finite-dimensional and write u = elul + . . . + e,u,, v = elvl + . . . + enun, where (e,) is a basis of E. Conclude by arguing by induction on the smallest degree of the homogeneous components of ua.) Deduce that, if dim(E) > 1, the ring T(E) admits no field of left fractions (I, 5 9, Exercise 15). (c) Show that for two non-zero elements u, u of T(E) to be permutable, it is necessary and sufficient that there exist a vector x # 0 in E such that u and v belong to the subalgebra of T(E) generated by x (use (6)). Deduce that, if dim(E) > 1, the centre of T(E) is equal to K. 7 6. Let K be a commutative field and E, F two K-algebras each with a unit element which is identified with the unit element 1 of K. Consider the K-algebra T(E @ F), where E @ F is considered as a vector space over K; E (resp. F) is identified with a vector subspace of T(E @ F) under the canonical injection. Let 3 be the two-sided ideal of T(E @ F) generated by the ele- ments xI @ x2 - (x1x2) and yl @I y2 - (y1y2), where xi E E, yi E F, i = 1,2; let E * F denote the quotient algebra T(E @ F)/3; it is called the free product of the K-algebras E and F; let + denote the canonical homomorphism of T(E @ F) onto E * F and a and p the restrictions of + to E and F; a and p are K-algebra homomorphisms. (a) Show that for every ordered pair of K-homomorphisms u: E -+ G, 628 EXERCISES u:F + G into a K-algebra G, there exists one and only one K-homomorphism w:E * F -+ G such that u = ~e) o a and v = w 0 p (which justifies the termino- (6) Let R, be the vector subspace of T(E @ F) the sum of the Tk(E @ F) for 0 < k < n and write 3, = 3 n R,. Show that, ifwe consider a basis of E consisting of 1 and a family (eJheL and a basis of F consisting of 1 and a family ( f,)GeM, then a supplementary subspace of 3, + R,-' is obtained in R, by considering the subspace of R, with basis the elements of the form z, @ z2 @ . . . 8 z, where, either z2, + is equal to one of the ea for 2j + 1 6 n and z2, is equal to one of the f, for 2j < n, or z2,+, is equal to one of the f, €or 2j + 1 < n and z2, is equal to one of the e, for 2j 6 n (argue by induc - tion on n). (c) Let P, = c$(R,) C E * F; then P, C p,+,, Po = K, PhPk C Phik and E * F is the union of the P,. Show that there is a canonical vector space iso- morphism, for n 2 1, lorn) * where we write G, = E/K for k odd, Gk = F/K for k even. In particular the homomorphisms a and p arc injective, which allows us to identify E and F with sub -K-algebras of E * F. Pk (resp. PL) is used to denote the inverse images in P of G, @ G2 @ .. . @ G, (resp. G2 @ G3 @ .-. @ G,+,); the height of an element z E E * F, denoted by h(z), is the smallest integer such that t E P,; if h(z) = n and z E Pk (resp. z E P:), z is called pure odd (rcsp. pure even). Then P, = Pk + P: and Pk n P: = Pn-l for n > 1. (d) Show that, if u, v are two elements of E * F such that h(u) = r, h(u) = s, then h(uv) < h(u) + h(v) and that h(uv) = h(u) + h(v) unless u and u are pure and, either r is even and u, u are not of the sameparity, or r is odd and u, v are of the same parity. (If for example r is even, u EP:, UEP~, show that uu E P,+s-l is possible only if u E Pr-, or v E P,-,.) (e) Suppose that E has no divisors of zero and, in the notation of (d), sup- pose that r is even, u pure even and u pure odd. Show that h(uv) = r + s - 1, unless there exist two invertible elements x, y of E such that uy E Pr-, and xb E P,-,. (Let (ai) (resp. (b,)) be the basis of the supplementary subspace of 3, + Rr-l in R, (resp. of 3, + R,-, in R,) considered in (b); write u = 2 +(ai)xi and v = c yj$(b,), where xi and y, are in E, and show that, where h(uv) < h(u) + h(v) when E and F are assumed to have no divisors of zero. if uv E f P,+,-,, then necessarily I xiyj E K.) Treat similarly the other cases (f) Deduce from (d) and (e) that, when E and F have no divisors of zero, nor does E * F. (g) Generalize the above definitions and results to any finite number of unital K-algebras. 629 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 7 7. Let E be a vector space of finite dimension n over a field K. For every integer r 3 1, the symmetric group 6, operates linearly on the tensor product E@, by the action (6, z) H 0. z such that 0.(~1 @ x2 @ ' . @ xr) = xo-1(1) @ x0-1(2) @ . . . @ xo-I(r) for every sequence of r vectors (x,)~<~<, of E. Let uo(z) = 0. z. Show that for every sequence (vi) < , of r endomorphisms of E, (*) Tr(u, 0 (vl @ v2 @ . . . @ v,)) = Tr(w,)Tr(w,) . . . Tr(wk) where the endomorphisms wj of E are defined as follows: 0-l is decomposed into cycles, 0-l = All,,. . . Ak and, if Aj = (il i, . . . ih), we write wj = vil 0 viz 0 . . 0 Vth. (Reduce it to calculating the left hand side of (*) when each ui is of the form x H (x, &ak, where (aj) < 78. Let A be an integral domain and K its field of fractions; for every A-module M, let T(M) denote its torsion module (11, 3 7, no. 10). (a) Show that for the algebra T(M) to have no divisors of zero, it is neces- sary and sufficient that the A-module T(M) be torsion-free (observe that T(M)(K, is isomorphic to T(Mo,) and use Exercise 5). (b) Let u:M + M' be an injective A-module homomorphism. Show that Ker(T(u)) c T(T(M)); if M' is torsion-free, then Ker(T(u)) = T(T(M)) (cf. , is a basis of E and (a;) the dual basis.) 11, 3' 7,. no. 10, Proposition 27). (c) Give an example of a torsion-free A-module M such that T(T(M)) \, is non-zero (11, 3 7, Exercise 31). 7 9. Let A be a commutative ring, F the free associative algebra over A of the set of integers I = (1, n), where n 2 2, and Xi (1 < i < n) the corre- sponding indeterminates, so that a canonical basis of F over A consists of the products X,,X,, . . . Xi,, where (il, . . ., ir) is an arbitrary finite sequence of elements of I. (a) Let C, denote the set of commutators [X,,Xj] = X,Xj - XjXi for 1 < i < j < n. Assuming that C,-l is defined, C, denotes the set of elements [Xi, PI = X,P - PX,, where 1 6 i < n and P runs through C,-l. Let U be the graded subalgebra of F generated by 1 and the union of the C, for m 2 2. Show that, for 1 6 i 6 n, [X,, PI E U for all P E U (confine it to the case where P is a product of elements of C, and argue by induction on the number of factors). (b) Suppose from now on that A is afield. For every integer m 2 0, let u, be the vector A-space of homogeneous elements of degree m in U (SO that Uo = A, U1 = {0}) ; choose a basis R,, 1, . . . , R,, in each U,, consisting 630 EXERCISES of products of elements of u ck. For every ordered pair of integers (rn, k) such that rn 2,1 Q k < d,, let Em, k be the vector sub-A-space of U generated by the RD,, such thatp > m or p = m andj 2 k; show that Em, is a two-sided ideal of u and that, if L,, k is the left ideal of F generated by Em, k, L,, k is a two-sided ideal of F. k4m (c) For every multiindex M = (q, . . . , a,) E N", we write X" = xyx7.. . x;. Show that the elements X"R,, k (m > 0, 1 < k 6 d,) form a basis of F over K. (To prove that these elements form a generating system of the vector space F, prove that every element X,, . . . Xi, is a linear combination of them by induction on the degree r. To see that the X"R,,, are linearly independent, argue by reductio ad absurdurn by considering the maximum degree with respect to one of the Xi of the monomials Xu which would appear in a linear relation between the XUR,,, and proceed by induction on this maximum degree; for this, reduce it to the case where K is infinite (considering if necessary F K' for an extension field K' of K) and observe that when X, + A is substituted in an R,, for Xi, for some A E K', the element R,, is unaltered). (d) Show that the two-sided ideal 3(F) of F generated by the commutators [u, v] = uv - uu of elements of F is the sum of the U, for m 2 2; F/3(F) is isomorphic to AIXl, . . . , X,] and hence is an integral domain. (e) Show that the algebra U is not a free associative algebra by proving that the quotient U/3(U), where 3(U) is generated by the commutators of elements of U, admits nilpotent elements #O. (Consider the image z' in U/3(U) of the element z = [X,, X,] of U; zf2 # 0 but ti4 = 0.) 10. Let A be a commutative ring, M'-& M A M" -+ 0 an exact sequence of A-modules and n an integer > 0. Let L = Coker(Tn(u)) SO that there is an exact sequence T"(M') T"(M) + L -to. For every subset J of the set {I, 2, . . ., n>, we write PJ =N, @N2 @ @N,, whereN, = M'ifiEJ,Nt = Mifi$J (P, is therefore equal to Tn(M)) and QJ= Nl @N2 @ ... @Nn, whereN, = M'ifiEJ,N, = M"ifi$J (Q, is therefore equal to T"(M").) For every integer i such that 0 < i < n, let Per) denote the submodule of T"(M) generated by the union of the images 63 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS ofthe PJ for all the subsets J such that Card(J) = i and let Lei) be the canonical image of PCi) in L. (a) Show that there exists a canonical surjective homomorphism (cf. 11, 3 5, no. 6, Proposition 6). (6) If the sequence 0 -+ M' the homomorphism defined in (u) is bijective. M 4 M" -+ 0 is exact and splits, show that §6 1. If M is a monogenous A-module, then S(M) = T(M). Deduce an exam - ple of an injective homomorphism j :N -+ M such that S(j) : S(N) + S(M) is not injective (5 5, Exercise 1). 2. Establish for symmetric algebras the properties analogous to those of Exercise 8(a) and (b) of 5 5. 3. Let K be a field, B the polynomial ring K[X,, X,] and p the ideal of B generated by the polynomial X: - X;. (u) Show that p is prime and therefore A = B/p is an integral domain. Let u and b denote the canonical images of XI and X, in A; a and b are not invertible in A and u3 = b2. (b) Let m be the (maximal) ideal of A generated by a and b. Show that the symmetric algebra S(m) is not an integral domain. (Observe that m is identified with the quotient module A2/N, where N is the submodule of A' generated by (b, -a); conclude that S(m) is isomorphic to the quotient ring A[U, V]/(bU - aV) and show that in the polynomial ring A[U, V] the principal ideal (bV - aU) is not prime, by considering the product T 4. Let A be a commutative ring and a an ideal of A. The Rees algebra of the ideal a is the subalgebra R(a) of the polynomial algebra A[T] in one in - determinate, consisting of the polynomials co + c,T + . - - + c,Tn, where cQ E A and c, E ak for every integer k 2 1. (u) Show that there exists one and only one surjective A -algebra homo - morphism r:S(a) -+ R(a) such that the restriction of r to a is the A-linear mapping x -+ xT. (b) Suppose that the ring A is an integral domain. Let a,, . . . , a, be ele- ments of A with a, # 0; consider the A-algebra homomorphism b2(bV3 - U3).) u:A[X,, . . . , X,] -+ A[T] such that u(X,) = atT. The kernel r = Ker(u) is a graded ideal whose homo- 632 EXERCISES geneous component r, of degree m consists of the homogeneous polynomials f of degree m such that f (al, . . . , a,) = 0. If r' c r is the ideal of A[X,, . . . , X,] generated by the homogeneous component r, of r, show that the A-module r/r' is a torsion A-module. (To verify that, for all f E P,, there exists c # 0 in A such that 4~ r', argue by induction on m, by writing where the fr are homogeneous of degree rn - 1 ; consider the polynomial observe that g E rl c r' and form aF-'f - X;-'g.) (c) Deduce from (b) that when A is an integral domain, the kernel of the homomorphism r of (u) is the torsion module of S(a). Deduce that, for S(a) to be an integral domain, it is necessary and sufficient that S(a) be a torsion- free A-module. In particular, if the ideal a is a projective A -module, S(a) is an integral domain and isomorphic to R(a). gW,, 3 . ., Xn) = XlfI(a1,. . -3 an) + Xzfi(az, . . .> an) + . . . + Xnfn(an), 7 5. Let E be a vector space of finite dimension n over a field K. (a) Show that for every integer m the symmetric power Sm(E) and the vector space Sk(E) of symmetric contravariant tensors of order rn have the same .. n+m-1 rn dimension + n- ') = ( *(b) If I< is of characteristic p > 0, show that the vector space S;(E) of symmetrized tensors of order p has dimension (observe that if, in a tensor product x1 @ x, @I . . . @I xp ofp vectors of E, xi = xj for an ordered pair of distinct indices i,j, then the symmetrization of this product is zero; use the fact that if (Hi)lG1<r is a partition of {1,2,. . .,p} into non-empty sets, com- prising at least two sets, the subgroup of 8, leaving each of the H, stable has order not divisible by p). On the other hand, the canonical image in SP(E) of the space SL(E) of symmetric tensors of order p is of dimension n and is generated by the images of the tensors x @ x @ . . . 8 x (p times), where x E E (use the same remark) ; the canonical mapping of SL(E) into SP(E) is therefore not bijective although the two vector spaces have the same dimen- sion.* (2 -~ §7 1. Let A be an integral domain and K its field of fractions. (a) Show that the exterior power AK of the A-module K reduces to 0. 2 (b) For every A -module E c K, show that every exterior power AE P is a torsion A-module for p 2. 633 I11 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 2 (c) If E is the A-module defined in 11, 5 7, Exercise 31, show that AE is non-zero. Give an example of an integral domain A and an A-module E contained in the field of fractions K of A such that no exterior power AE reduces to 0. 2. For every ideal a of a ring A and every A-module E, the exterior algebra A(E/aE) is isomorphic to (AE)/a(AE). 3. Give an example of a ring A with the following property: in the A- module E = A2, there exists a submodule F such that, if j : F + E is the canonical injection, Aj is identically zero even when neither AE nor AF reduces to 0 (11, $4, Exercise 2). 4. Let E be a free A-module ofdimension n 2 3. Show that if 1 < p < n - 1 the A-modules AE and A E are isomorphic; but if 2p # n, there exists no isomorphism + of AE onto A E such that for every automorphism u of E, (p o (hu) = rx u) o + (if (e,) is a basis of E, consider the automorphism u such that u(e,) = e, + e,, y(ek) = e, for k # i, and give i and j all possible values). 5. Let K be a commutative field, E, F two vector spaces over K and u a linear mapping of E into F of rank r. Show that if p < r the rank of Au is equal to (i) and that ifp > r, iu is identically zero (take in E a basis r vectors of which form a basis of a subspace supplementary to u-l(O) and the rest a basis of i'(0)). P 2 2 2 P n--P P n-p P 6. Let K be a commutative field and E a vector space over K. m (a) For an element z = zp (with zp E AE) of the exterior algebra AE of E to be invertible, it is necessary and sufficient that zo # 0 (prove it first when E is finite-dimensional and then pass to the general case, noting that for all z E AE there exists a finite -dimensional subspace F of E such that (6) Suppose that K is such that 2 # 0 in K. Show that, if E is infinite- dimensional or of even finite dimension, the centre of the algebra AE consists of the elements such that zp = 0 for all odd indices p. If E is of odd dimension n, the centre of AE is the sum of the above subspace and AE. In all cases, p=o EAF). n 634 EXERCISES the centre of AE is identical with the set of elements of AE permutable with all the invertible elements of AE. 7. For every A-module E, show that, if we write [u, v] = u A v - u A u for any two elements of AE, then for any three elements u, v, w of AE, "u, VI, WI = 0. 7 8. Let E be a free A-module. (u) Show that in Tn(E), 3; (no. 1) is a free sub-A-module equal to the kernel of the endomorphism z H Q. z and admits a supplement which is a free A- module; AE is therefore canonically isomorphic to A;(E). If also in A the equation 25 = 0 implies < = 0, then AE(E) = Ak(E). * (b) Suppose that A is a field of characteristic p > 0 and that p # 2. Then the canonical image of Ai(E) = Ai(E) in AE is * zero. (c) Suppose that A is a field of characteristic 2. Then Ak(E) = Sk(E), AE(E) = S;(E), but in general Ak(E) # AL(E)., n P 9. Let E be an A-module. The skewsymmetrization of a linear mapping g of Tn(E) into an A-module F is by definition the linear mapping Qg of Tn(E) into F such that ug(z) = g(az) for all z E Tn(E) ; if 9: En --f T"(E) is the canon - ical mapping, the n-linear mapping ug 0 + is also called the skewsymmetriza - tion of the n-linear mapping g 0 +. Deduce from Exercise 8(a) that if E is free every alternating n-linear mapping of En into F is the skewsymmetriza - tion of an n-linear mapping of En into F. 7 10. Let E be an A-module with a generating system of n elements. Show that every alternating n-linear mapping of En into an A-module F is the skewsymmetrization of an n-linear mapping of En into F. (Let u, (1 < j < n) be the generators of E and C#I :An -+ E the homomorphism such that +(e,) = a, for allj, where (e,) is the canonical basis of An. Iff:En -+ F is an alternating n-linear mapping, so isf,: (xi, . . . , x,) -f(+(xl), . . . , +(xn)) ; show that if go is an n-linear mapping of (An)" into F such that fo = ago, go can also be written as (xl, . . ., x,) -g(+(xl), . . ., +(x,)), where g is an n-linear mapping of En into F.) 7 11. Let K be a commutative field in which 2 = 0, A the polynomial ring K[X, Y, Z] and E the A-module A3/M, where M is the submodule of A3 generated by the element (X, Y, Z) of A3; finally, let F be the quotient A-module of A by the ideal AX2 + AY2 + AZ2. Show that there exists an alternating bilinear mapping of E2 into F, which is the skewsymmetrization of no bilinear mapping of E2 into F, but that in T2(E) the module 3; is equal to the kernel of the endomorphism z * uz (consider T2(E) as a quotient 635 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS module of T2(A3) and show that the kernel of t H at is the canonical image of the module of symmetric tensors SL(A3)). 7 12. In the notation of Exercise 11, let B be the quotient ring A/(AX2 + AY2 + AZ2) and let G be the B-module E @A B. Show that in T2(G), the module 8; is distinct from the kernel of the endomorphism z H QZ (method analogous to that of Exercise 1 1). 13. (a) Let M be a graded A-module of type N. Show that there exists on the algebra S(M) (resp. A(M)) one and only one graduation of type N under which S(M) (resp. A(M)) is a graded A-algebra and which induces on M the given graduation. Show that if the homogeneous elements of even degree in M are all zero (in which case the graduation on M is called odd), the graded algebra A (M) thus obtained is alternating (5 4, no. 9, Definition 7). (b) Given a graded A -module of type N, consider the following universal mapping problem: C is the species of anticommutative graded algebra struc- ture ($4, no. 9, Definition 7), the morphisms are A-homomorphisms of graded algebras ($ 3, no. 1) and the a-mappings are the graded homo - morphisms of degree 0 of M into an anticommutative graded A-algebra. Let M- (resp. M+) denote the graded submodule of hi generated by the hom- geneous elements of odd (resp. even) degree. Show that the graded algebra G(M) = A(M-) s(M+) is anticommutative; a canonical injection j of M into this algebra is defined by writing,j(x) = x @ 1 ifx E M-,j(x) = 1 @ x if x E M+. Show that the graded algebra G(M) and the injection j constitute a solution of the above universal mapping problem. G(M) = A(M-) @AS(M+) is called the universal anticommutative algebra over the graded A-module M. (c) Prove the analogues of Propositions 2, 3 and 4 for the universal anti- commutative algebra. Consider the case where M admits a basis consisting of homogeneous elements: find explicitly in this case a basis of the universal anticommutative algebra. 14. Let M be a projective A-module and let xl, . . . , x, be linearly indepen - dent elements of M; for every subset H = {i,, . . ., i,,J of m < n elements of the interval (1, n), where i, < i, < . . . < i,, we write XH = Xil A Xi2 A . . . A Xi,. Show that when H runs through the set of subsets of (1, n) with m elements, the xH are linearly independent in A (M). 1 1 1 ... 1 a1 6, a, a, . . . 6, b, a, . . . a, b, b, b, . , . . . . . . . . an 636 = (a1 - bl)(a2 - b,) EXERCISES 15. Let M be an n-dimensional free A-module. Show that every generating system of M with n elements is a basis of M. §8 1. In a matrix U of order n, if, for each index i, the column of index i is replaced by the sum of the columns of index # i, the determinant of the matrix obtained is equal to ( - l)n-l(n - 1) det U. If in U, for each i, the sum of the columns of index #i is subtracted from the column of index i, the determinant obtained is equal to -(n - 2)P-l det U. 2. Let A = det(aij) be the determinant of a matrix of order n; for 1GiGn-l and 1 <j < n - 1, we write Show that the determinant det(pu) of order n - 1 is equal to a12a13 ..*a~,n-iA- 3. Show the identity Deduce the following identities: . . . (a, - b,) 637 V*T TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 111 albl alb2 a163 . . . ~16, a162 a262 a263 . . . a26, a163 a263 a363 . , . a36, albn ~26, ~36, . . . an6, .......... u2a3. . . a, a3a4 . . . an61 b1b2b3 . . . 6,-1 6263 . . .6, a3a4 . . . anal a16263 . . . b,-l a26,. . .6, b36,. . .6,61 a4. . . anala2 . . , ~1~263, . . 6n-l ..................... a2a3.. . b, a3a4.. . 6,b, a4.. .b,61b2 . . . a1a2a3.. . anTl a4 . . . ~~6162 a4 . . . ~~~162 . . . . . . x a: la1 x x a, a2 ... a,-l 1 a, x u2 ... a,-l 1 ala2x ... an-1 1 . . . . . . . . . ala2a ,... x 1 a, a2 a3 . . . an 1 a2 ... = (a1u2.. . a, - 6162.. . 6,Jn-l. = (x - al)(x - a2). . . (X - an) ala2x ... an . . . . . . . a, a2 a3 . . . x = (x + a1 + a2 + . . . + a,)(x - q)(x - a,) (reduce the last determinant to the preceding one). 4. Calculate the determinant I a, + 61 61 61 . . . 61 I bn 6, 6, . . . a, + b, (express A, in terms of A, - l). 638 .. . (x - an) EXERCISES 5. If the a, and 6, are elements of a commutative field such that a, + 6, # 0 for every ordered pair of indices (i,j), prove that c (a, - 4(6, - 6,) - i<I 1 deti;lr+a,) - 2 (aI + 6,) t.i (“Cauchy’s determinant”). 6. Show that, if X is a matrix with n rows and m columns, Y a matrix with p rows and n columns and Z = YX the product matrix with p rows and m columns, the minors of 2 of order q are zero if n < q; if q < n, they are given det(ZL, 1x1 = F det(Y~, ~)det(X~. H) where K runs through the set of subsets of (1, n) with q elements (use formula (3) of 9 7, no. 2). 7. Let A = det(a,,) be the determinant of a matrix U of order n; for each index i (1 < i 6 n) let Al denote the determinant obtained by multiplying the element a,, in U by P, for 1 < j 6 n. Show that the sum of the n deter - minants At (1 < i < n) is equal to (gl + P2 + ... + P,)A (expand A, by the row of index i). 8. Let A = det(a,,) be the determinant of a matrix U of order n and D a permutation belonging to 6,; for each index i (1 6 i < n) let A, be the determinant obtained by replacing the element a,, in U by at, ,,(,) for 1 < j < n; if p is the number of indices invariant under the permutation 0, show that the sum of the n determinants A, (1 < i 6 n) is equal to PA (same method as in Exercise 7). 9. Let A be a square matrix of order n, B a submatrix of A with p rows and q columns and C the matrix obtained by multiplying in A, each of the elements belonging to B by the same scalar K. Show that each term in the total expan - sion of det C is equal to the corresponding term in the total expansion of det A multiplied by a scalar of the form a”, where r 2 p + q - n and r depends on the term considered (perform a suitable Laplace expansion of det C). 10. Let I?, A be the determinants of two matrices U, V of order n; let H and K be any two subsets of (1, n) with $ elements and (ik) (resp. (j,)) the sequence obtained by arranging the elements of H (resp. K) in increasing order; let rH, be the determinant obtained by replacing in U the column of index ik by the column of v of index jk, for 1 < k < p; similarly let AKSH be the determinant obtained by replacing in V the column of indexj, by the 639 by 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS column of U of index i,, for 1 < k < p. Show that, for every subset H of (1, n) with p elements, where K runs through the set of subsets of (1, n) withp elements (use formulae (21) and (22) of no. 6). 11, Let A be the determinant of a square matrix X of order n and A, the determinant of the square matrix AX, of order (i). Show that ApAn-p = A(;) (use formulae (21) and (22) of no. 6). 12. A matrix (at,) of order n is called centrosymmetric (resp. skew-centrosym- metric) if an-l+l,n-,+l = ctij (resp. an-i+l.n-j+i = -atj) for 1 < i < n, 1 <j<n. (a) Show that the determinant of a centrosymmetric matrix of even order 2p can be expressed in the form of a product of two determinants of order p and the determinant of a centrosymmetric matrix of order 2p + 1 in the form of a product of a determinant of order p and a determinant of order p + 1. (b) Show that the determinant of a skew-centrosymmetric matrix of even order 2p can be expressed in the form of a product of two determinants of order p. The determinant of a skew -centrosymmetric matrix of odd order 2p + 1 is zero if in A the relation 2E = 0 implies 5 = 0; otherwise it can be expressed in the form of the product of a, + + by two determinants of order P* n - 1, the complement of aij. Show that 13. Let A = det(ccij) be a determinant of order n and Atj the minor of order all a12 ... aln XI a21 M22 ..* aZn 22 = Az - 2 ( -l)itfAip,yj. . . . . . . . . t. j an1 @,z ... an, Xn Y1 y2 ... Yn If A = 0 and the aij belong to ajeld, show that the above determinant is the product of a linear form in xl, x2, . . . , x, and a linear form in yl, yz, . . . , y, (use Exercise 5 of 4 5 and Exercise 8 of 11, 8 6). Give an example where this result fails to hold when the ring of scalars A is not a field (take A to be the ring Z/(S) and n = 2). 640 EXERCISES 14. Show the identity 01 1 1 ... 1 1 0 a,+a, a,+a, ... a,+a, 1a2+a, 0 a,+a, ... 1 a,+a, a,+a, 0 ... a3+an 1 an+a, an+a, a,+a, ... 0 a2 + an ................ n = ( - 1y2n-1 c a,a,. . . a, _lUi+l. . . a, i=l (use Exercise 13). 15. Show that if the columns of a square matrix of order n over A are linearly independent, the rows of the matrix are also linearly independent (cf. 11, tj 10, Exercise 3). 16. Let E, F be two free A-modules of respective dimensions m and n. For a linear mapping u: E -+ F to be injective, it is necessary and sufficient that m < n and that, if X denotes the matrix of u with respect to any two bases of E and F, there exist no scalar p # 0 such that the products by p of all the minors of X of order m are zero. 17. Let (*> be a system of m linear equations in ri unknowns over a commutative ring A. Let xj (1 6 j < n) be the columns of the matrix X = (atj) of this system and y = (pi); suppose that in X all the minors of order >p are zero but that x1 A x2 A . . . A x, # 0. For the system (*) to have a solution, it is necessary that x1 A x2 A . . . A xp A y = 0. Conversely, if this condition holds, there exist n + 1 elements t, (1 < j < n + 1) of A such that En+, # 0 and j=laijEj = PiEnti (1 Q i < m). 18. Let E and F be two free A -modules ofrespective dimensions m and n, u:E -+ F a linear mapping and X the matrix of u with respect to any two bases of E and F. (a) If m 2 n and there exists a minor of X of order n which is invertible, u is surjective. (b) Conversely, if u is surjective, show that m 2 n and that there exists 64 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS a minor of X of order n which is non-zero. If further, in the ring A, the ideal generated by the non-invertible elements is distinct from A, there exists a minor of X of order n which is invertible (consider the exterior power Au). (c) Let B be a commutative ring and A the product ring B x B. Give an example of a surjective linear mapping of the A-module A' onto A, all the elements of whose matrix (with respect to the canonical bases of A2 and A) are divisors of zero. 19. Let X be a matrix over a commutative field. For X to be of rank p, it suffices that there exist a minor of X of order p which is #O and such that all the minors of order p + 1 containing this minor of order p are zero (show that every column of X is then a linear combination of the p columns to which the elements of the minor of order p in question belong). 20. Let A be a commutative ring, M an arbitrary A-module and U = (a,,) a matrix of type (m, n) with elements in A. (a) Suppose that m = n and let A = det(U); then, if xl,. . ., x, are ele- ments of M such that 5 CL,~X, = 0 for 1 < i < n, then Ax, = 0 for every index i. (b) Suppose that m and n are arbitrary, that all the minors of U of order > r are zero and that a minor of order r is invertible, for example that of the matrix (uU) where i < r andj < r; then if a, denotes the row of index i in U, the a, of index < r form a basis of the submodule of A" generated by all the rows of U; for n j= 1 i > I, therefore a, = Zl hkuk. Then let yl, . . . , ym be elements of M; for there to n .r- exist elements x,, . . . , x, of M satisfying the system of equations j=1 2 ui,xl = y, for 1 6 i < m, it is necessary and sufficient that yi = 7 2 1. Let K be an infinite commutative field, m, n, r three integers 2 0 such that r < n < m, M,,,"(K) the vector K -space (of dimension mn) con- sisting of the matrices of type (m, n) over K and V a uector subspace of M,, .(K) such that r is the greatest value of the rank of matrices X E V. We propose to prove the inequality (*I dim(V) < mr. (a) Show that we can assume that rn = n and that the matrix 642 EXERCISES belongs to V. Deduce that every matrix X E V is then of the form (where X,, is of order r) with the condition X2,X1, = 0 (use the fact that for all E K, the matrix &Yo + X is of rank < r). (b) Deduce from (u) that, for two matrices X, Y in V, X2lY12 + y21x12 = 0. (c) For every matrix XE V, we write F(X) = (X11X12) E Mr,"(K); f is a linear mapping and dim(V) = dim(Im(f)) + dim(Ker(f)). On the other hand, let u, denote the linear form (Y11Y12) H Tr(X2,Y1,) on Mr,,(K). Show that the linear mapping XH u, of Ker(f) into the dual (M,,,(K))* is injective; complete the proof of (*) by noting that the image under XH ux of Ker(f) is contained in the orthogonal of Im(f). 7 22. Let F be a mapping of M,(K) into a set E, where K is a commutative field, such that, for every triple of matrices X, Y, 2 in Mn(K), F(XY2) = F(X2Y). We except the case M2(F2) ; show then that there exists a mapping CD of K into E such that F(X) = CD(dct(X)). (Using the Corollary to Proposition 17 (no. 9), show first that, for U E SL,(K), F(X) = F(XU); deduce that, in GLn(K), F(X) depends only on det(X). On the other hand, if, for r < n - 1, we write show that there exists a matrix Y of rank r such that X, - = YX, and Y = X,Y and deduce that F(X) has the same value for all matrices of rank < n.) 23. In every A-algebra E, if xl, . . . , x, are elements of E, we write c * * .xnl = a 6n ~a*a(l)xa(Z) * - . xu(,)* Show that if E admits a finite generating system over A, there exists an integer N such that [x1x2. . .xN] = 0 for all elements x, (1 < j < N) of E. 24. Let Xi (1 < i < rn) be square matrices of the same order n over the commutative ring A. Show that if rn is even then (Exercise 23) Tr[X1X2. . . X,] = 0 643 1x1 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS and if m is odd Tr[X,X,. . .X,] = m .Tr(X,[X,X,. . .X,-,]). (If A is the cyclic permutation (1,2,. . ., m), C the cyclic subgroup of 8, generated by hand H the subgroup of 6, consisting of the permutations leaving m invariant, note that every permutation in 8, can be written uniquely as 67 with Q E H and T E C; then use the fact that the trace of a product of ma- trices is invariant under cyclic permutation of the factors.) 7 25. Let A be a commutative ring containing the field Q of rational num - bers and let X be a square matrix of even order 2n 2 2 over A. Let L be the subset {1,2,. . ., n + l} of {I, 2,. . ., 272) and L' its complement. Show the following formula ("Redei's identity ") : (*) det(&,L) det(X~,~) = & (-1) n(" 11 det(XH,K) det(X€I',K') where H runs through the set of subsets of n elements of (1, 2, . . . ,272) contain- ing 1 and K the set of subsets of n elements of L and r = Card(H n L'). (Evaluate the coefficient of a term xa(,), ,x0(,), ,. . . x,(,,,). ,,, in the right-hand side of (*) for an arbitrary permutation o E QZn, noting that this coefficient can only be #O if H = o(K) ; show that it can only be #O if o(L) = L.) 26. (a) In the notation of Proposition 19 (no. lo), suppose that there exist two A[X]-linear mappingsg,, g, ofM[X] into M'[X] such that +' o g, = g, o +; show that there then exists an A[X]-linear mapping g of M, into ML, such that 4' 0 g, = 4' o 9. Further, if g, and g, are A[X]-isomorphisms of M[X] onto M'[X], g is an A[X]-isomorphism of Mu onto ML,. (6) In the notation of no. 10, the endomorphisms u, u' are called equivalent if there exist two isomorphisms f,, f, of M onto M such that u' 0 f, = f, o u; they are called similar if there exists an isomorphism f of M onto M' such that u' 0 f = f 0 u. Deduce from (a) that, for u and u' to be similar, it is necessary and sufficient that the endomorphisms X - ii and X - ii' (of the A[X]-modules M[X] and M'[X] respectively) be equivalent. §9 1, Let K be a commutative field with at least 3 elements and A the algebra over K with a basis consisting of the unit element 1 and two elements e,, e, such that e; = el, e1e2 = e,, e,e, = ei = 0; let A' be the opposite algebra to A. Show that there exists in A elements x such that TrA/K(x) # TrAoIK(x) and NA&) # NAo/K(x)* 644 EXERCISES s 10 1. If A and B are K-algebras and a and P two K-homomorphisms of A into B, an (a, P)-derivation of A into B is a K-linear mapping d of A into B satisfying the relation d(xy) = (dx)a(y) + P(x)(dy) for x,y in A, in other words, a derivation in the sense of Definition 1 (no. 2) where A = A' = A", B = B' = B", d = d' = d" , p: A x A -+ A is multiplication, Al : B x A -+ B is the K-bilinear mapping (z, x) ++ za(x) and A,: A x B -+ B the K-bilinear mapping (x, z) ++ P(x)z. Suppose henceforth that A and B are commutativejelds and that a # p. Show that there then exists an element b E B such that d is the (a, P)-derivation x ++ ba(x) - P(x)b. (It can be assumed that d # 0. Show first that the kernel N of d is the set of x E A such that a(.) = P(x) and then show that for x 4 N the element b = d(x)/(a(x) - P(x)) is independent of the choice ofx.) 2. Let K be a commutative field of characteristic #2, A and B two K- algebras and a a K-linear mapping of A into B; an a-derivation of A into B is a K-linear mapping d of A into B satisfying the relation d(q) = (dx)a(y) + a(x)(dy) for x,y in A, in other words, a derivation on the sense of Definition 1 with A = A' = A", B = B' = B", d = d' = d", p multiplication in A, h, the mapping (2, x) - za(x) of B x A into B and A, the mapping (x, z) H a(x)z of A x B into B. Show that if B is a commutativefield, an extension of K, and d # 0, there exists b # 0 in B such that a(xy) = a(x)a(y) + b(dx)(dy) for x,y in A. If b = c2 with c E B, c # 0, there exist two homomorphismsf, g of A into B such that a(%) = (f (x) + g(x))/2, dx = (f (x) - g(x))/2c. Otherwise, there exists a quadratic algebra (tj 2, no. 3) B' over B, a II. E B' such that u2 = c. and a homomorphism f of A into B' such that -. a(.) = (f(x) +f(x))/2, ,- -7 dx = (f (4 - f(x))/2. 7 3. Let K be a field, commutative or otherwise, E the left vector space K:" and let be the canonical basis of this space. On the other hand let Q be an endomorphism of K and d a (0, l,)-derivation (Exercise 1) of K into itself, in other words an additive mapping of K into itself such that d(Ed = (dE)O(?) + E(dyi). 645 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS The elements of E are linear combinations of products a,e, and a multiplication is defined on E (Z-bilinear mapping of E x E into E) by the conditions (aen)(Pem) = (aen-d(c(P)Crn+l + (dB)em) fern > 1 (~eo) (P4 = (aP)em* Thus an (in general non-commutative) ring structure is defined on E with eo as unit element (identified with the element 1 of K). Ifwe write X = el, then en = Xn for n 2 1, so that the elements of E can be written as a. + alX + . . . + a,X”, with the multiplication rule Xcc = o(a)X + (da) for aEK. This ring is denoted by K[X; LT, d] and is called the ring of non-commutative polynomials in X with coegicients in K, relative to IS and d. For a non-zero element z = 2 ajX’ of E, the degree deg(z) is the greatest integerj such that a, # 0. j (a) Show that if u, u are two elements #O of E, then: uv # 0 and deg(uv) = deg(u) + deg(v); deg(u + u) 6 sup(deg(u), deg(v)) if u + u # 0. Moreover, if deg(u) 2 deg(v), there exist two elements w, r of E such that u = wu + r and, either r = 0, or deg(r) < deg(v) (“Euclidean division” in (b) Conversely, let R be a ring such that there exists a mapping u H deg(u) of the set R of elements #O of R into N, satisfying the conditions of (a). Show that the set K consisting of 0 and the non-zero elements u E R such that deg(u) = 0 is a (not necessarily commutative) field. If K # R and x is an element of R for which the integer deg(x) > 0 is the smallest possible, show that every element of R can be written uniquely in the form 2 i ajx’ with a, E K. (To see that every element of R is of this form, argue by reductio ad absurdurn by considering an element u E R which is not of this form and for which deg(u) is the smallest possible.) Prove that there exist an endomorphism 0 of K and a (IS, 1,)-derivation d of K such that xa = c(a)x + da for all GC E K. Deduce that R is isomorphic to a field or a ring K[X; IS, d]. 4. Let M be a graded K-module of type N; show that every graded endo- morphism of degree r can be extended uniquely to a derivation of degree r of the universal anticommutative algebra G(M) (3 7, Exercise 13). * 5. Let K be a field of characteristic p > 0, B the field K(X) of rational functions in one indeterminate and A the subfield K(X*) of B. Show that the canonical homomorphism !&(A) @A B -+ Q,(B) is not injective., El. 646 EXERCISES § 11 1. If V is a finitely generated projective A-module, so is End(V), canonically identified with V @A V* (11, 5 4, no. 2, Corollary to Proposition 2). If every element u €End(V) is associated with the A-linear form v~Tr(uv) on End(V), an A-linear bijection is defined of End(V) onto its dual (End(V))*. Identifying End(V) and its dual A-module under this mapping, the A- algebra structure on End(V) defines by duality (no. 1, Example 3) an A- cogebra structure on End(V), which is coassociative and has as counit the form Tr: V + A. In particular, for V = A”, a cogebra structure is defined on M,(A) for which the coproduct is given by c(Eij) = Ekj @ E,k. This cogebra structure and the algebra structure on Mn(A) do not define a bigebra structure. 2. Show that in Definition 3 (no. 4), condition (3) (relative to graded bige- bras) is equivalent to the following: the product m: E @ E + E is a morphism of the graded cogebra E @ E into the graded cogebra E. 3. Show that if M is an A-module there exists on T(M) one and only one cocommutative bigebra structure whose algebra structure is the usual structure and whose coproduct c is such that, for all x E M, c(x) = 1 @ x + x @ 1. 4. Let E be a commutative bigebra over A, m the product E @ E -+ E, c the coproduct E -+ E @ E, e the unit element and y the counit of E. (a) Show that for every Commutative A-algebra B, the law of composition which associates with every ordered pair (u, ’) HomA-slg.(E, B, HomA-slg.(E, B, the A-algebra homomorphism E E@E %B@B 3 B (where mB is the product in B) defines a monoid structure on HomA-a,g,(E, B). (b) In order that, for every commutative A -algebra B, HomA-alg,(E, B) be a group, it is necessary and sufficient that there exist an A-algebra homomor - phism i: E -+ E such that i (e) = e and the composite mappings m 0 (1 @ i) o c and m 0 (i @ lE) 0 c are both equal to the mapping x ++ y(x)e. i is then called an inversion in E; it is an isomorphism of E onto the opposite bigebra EO. 5. Let El, E, be two bigebras over A. Show that on the tensor product El @A E, the algebra and cogebra structures the tensor product of those on El and E, define on El @A Ez a bigebra structure. 6. Let M be a graded A -module of type N. Define a skew graded bigebra structure on the universal anticommutative algebra G(M) (3 7, Exercise 647 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 13); deduce a canonical graded algebra homomorphism G(M*) --f G(M)*gr and notions of inner product between elements of G(M) and elements of G(M*). 7. With the hypotheses and notation of Proposition 11 (no. 9), prove the formula (detg)+-l = A(g) 0 +-I oA(tg). Form this formula and (79) (no. 11) derive a new proof of the expression for the inverse of a square matrix using the matrix of its cofactors ($8, no. 6, formula (26)). 8. Let E be a free A-module of dimension n. For every A-linear mapping u of A(M) into an A-module G, we can write u = p=o 2 up, where up is the restriction of u to R(M) ; we then set qu = p=o 2 (- 1)%,, so that 1% = u. Let u be an isomorphism of M onto its dual M* and let A be the determinant of the matrix of u with respect to a basis (e,) of M and the dual basis of M* (a deter - minant which does not depend on the basis of M chosen). If + is the isomor- phism of A(M) onto A(M*) defined starting with e = el A e2 A . . . A en, show the formula qn+lA+ = A(tu) o +-I oA(u). 9. The adjoint of a square matrix X of order n over A is the matrix 2 = (det(Xji)) of minors of X of order n - 1. Show that if X is invertible then det(2) = (det X)n-l and that every minor det (.fH,K) of 2 of order p is given by the formula (1) det (iH,K) = (det X)P-l(XH,,K,) where H’ and K’ are the complements of H and K respectively in (1, n) (“Jacobi’s identities”; use relation (80) of no. 11). 10. From every identity Q, = 0 between minors of an arbitrary invertible square matrix X of order n over A another identity 6 = 0 can be derived called the complement of 0 = 0, by applying the identity Q, = 0 to the minors of the adjoint 2 of X and then replacing the minors of x” by functions of the minors of X using identity (1) of Exercise 9. In this way show the following identity : where Xtj. hk denotes the minor of X of order n - 2 obtained by suppressing in X the rows of index i, j and the columns of index h, k. det(Xih) det(Xjk) - det(Xik) det(Xjh) = (det X) det(XtfShk) for i < j 648 EXERCISES 7 11. Let @ = 0 be an identity between minors of an invertible square identity (Exercise 10) and k an integer >O. Let Y be an invertible square matrix of order n + k and let F0 be the submatrix of the adjoint matrix P 6 k. If ko is assumed to be invertible and the identity 6 = 0 is applied to the minors of yo and then each minor which appears in this identity (considered as a minor $ f) is replaced by its expression as a function of the minors of Y, using identity (1) of Exercise 9, an identity @k = 0 is obtained between minors of the matrix Y (valid if Y and F0 are invertible) which is called the extension of order k of the identity Q, = 0. In particular, Let A = (atj) be an invertible square matrix of order n + k, B the submatrix of A of order k obtained by suppressing in A the rows and columns of index > k and A,, the determinant of the matrix of order k + 1 obtained by suppressing in A the rows of index > k except that of index k + i and the columns of index >k except that of index k +j; if C denotes the matrix (Al,) of order n, show the identity matrix X of order n over a commutative ring A, @ w = 0 the complementary formed by suppressing in Y w the rows of index < k and the columns of index det C = (det A)(det B)n-l valid when A and B are invertible (show that it is the extension of order k of the total expansion of a determinant of order n). State the identities obtained by extending the Laplace expansion (5 8, no. 6) and the identity of Exercise 2 of 5 8. 7 12. Let X = ( Eu) be a square matrix of order n over a commutative field K; let H be a subset of (1, n) with p elements and H’ the complement of H in (1, n) ; suppose that, for every ordered pair of indices h E H, k E H’, Show that, for all subsets L, M of (1, n) with fi elements, PM, M‘ det(-%, H) det(X~,, H’) - PL,L, det(X~, K) det(XL, H,) = 0. (Consider the columns x, of index h E H in X as vectors in E = Kn and the columns xz of index k E H’ as vectors in E* and show that the (n - p)-form xz- is proportional to &,(xH).) 13. Let I’ and A be two determinants of invertible matrices of order n over a commutative ring and r,, the determinant obtained by replacing in r the column of index i by the column of A of indexj. Show that det(I’,,) = P-lA. (Expand Pi, by the i-th column and use Exercise 9.) 649 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS 14. Let M be a free A-module of dimension n > 1. Show that every iso- morphism J, of A(M) onto A (M*) such that A (6) 0 + = + 0 A(u) for every automorphism u of M, of determinant equal to 1, is one of the isomorphisms I$,, defined in Proposition 12 of no. 11 (proceed as in 3 7, Exercise 4). P n-p n-p P 15. Let E be a free A -module of dimension n. Let z be a p-vector over E and z* a (p + q)-form on E; z can be canonically identified (4 7, Exercise 8) with the skewsymmetrization uzl of a contravariant tensor z1 of orderp and z* with the skewsymmetrization of a covariant tensor of order p + q. If, in the mixed tensor zl. z*, the k-th contravariant index and the (p + k)-th covariant index are contracted for 1 < k < p, show that the covariant tensor of order q thus obtained is skewsymmetrized and is canonically identified with the q- form z* L z, up to sign. 16. Let E be a free A-module of dimension n. The regressive product of x E A (E) and y E A (E) with respect to a basis {el of A (E), denoted by x v y, is the element +(+(x) A +(y)), the isomorphism c$ being relative to e. This product is only defined to within an invertible factor, depending on the basis {e} chosen. Show that if x is ap-vector and y a q-vector, x v y = 0 ifp + q < n and that, ifp + q > n, x v y is a (p + q - n)-vector such that n -1 y v x = (-l)(n-P)(n-% v y. The regressive product is associative and distibutive with respect to addition and defines on A(E) a unital algebra structure isomorphic to the exterior algebra structure. Express the components of x v y as a function of those of x and y for a given basis on E. 17. Let K be a commutative field and E a vector space over K. (a) Let z be a p-vector #O over E and let V, be the vector subspace of E consisting of the vectors x such that z A x = 0. Then dim(V,) < p; if (xJl idQ is a free system of vectors of V,, there exists a (p - q)-vector -1 v such that z = v A x1 A . . A x,. If E is finite -dimensional, then V, = +(MZ), where MZ is the orthogonal in E* of the subspace M, associated with z. For z to be a pure p-vector, it is necessary and sufficient that dim(V,) = p and then V, = M,. (b) Let v be a purep -vector, w a pure q-vector and V and W the subspaces of E associated respectively with v and w. In order that V c W, it is necessary and sufficient that q p and that there exist a (q - p)-vector u such that w = u A v. In order that V n W = (01, it is necessary and sufficient that 650 EXERCISES v A w # 0; the subspace V + W is then associated with the pure (p + q)- vector v A w. (c) Let u and u be two pure p-vectors and U and V the subspaces associated respectively with u and v. For u + u to be pure, it is necessary and sufficient that dim(U n V) 2 p - 1. 18. Let z = c aHeH be a non-zero purep-vector of an n-dimensional vector (eJlSidn of E. Let G be a subset of (1, n) withp elements such that aG # 0; let (ih)lGhdp be the sequence of indices in G arranged in increasing order and (jk)l<k<n-p the sequence of indices in the complement G’ of G, arranged in increasing order. For every ordered pair (h, k) of indices such that 1 < h < p, 1 < k < n - p, let Phk be the component aH of z corresponding to the subset H of (1, n) consisting of the p - 1 indices in G distinct from in and of index jk E G’; let x be the matrix (Phk) with p rows and n - p columns. Given an arbitrary subset L of (I, n) with p elements such that L n C G has q 2 1 elements, show that (aG)q-laL is equal, up to sign, to the minor ofXoforder q, consisting of the rows of index h such that ih E G n C L and the columns of index k such thatj, E L n c G. (Write z as being of the form aGyl A yz A . . . ~y,, where the vectors y, are such that, in the matrix Y with n rows and p columns whose columns are the y,, the submatrix consisting of the rows whose indices belong to G is the unit matrix.) 19. Over a finite-dimensional vector space E, let z = x1 A x2 A . . . A x, be a purep-vector #O; let u* be a q-form (q < p) and u* an arbitrary element of A(E*). Show that space E, expressed H using its components with respect to an arbitrary basis where H runs through the set of subsets of (1, p) with q elements, K is the com- plement of H in (1, p) and x, (resp. xK) denotes the exterior product of the xi such that i E H (resp. i E K). 20. Let E be an n-dimensional vector space, z a pure p-vector over E, V the vector subspace of E associated with z, u* a pure q-vector over E*, W‘ the subspace of E* associated with u* and W the subspace of E, of dimension n - q, orthogonal to W’. If q < p, u* -I z is a pure (p - q)-vector over E, which is zero if dim(V n W) > p - q; if dim(V n W) = p - q, V n W is associated with u* -I z. 21. (u) Show directly (without using the isomorphisms +p) that every (n - 1)-vector over an n-dimensional vector space is pure. (b) Let A be a commutative algebra over a field K with a basis consisting of the unit element 1 and three elements a,, a2, u3 whose products with one 65 1 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS another are zero. Let E be the A-module A3 and (e,) show that the bivector its canonical basis; 6(e~ A e3) + a2(e3 A ‘1) + %(‘I A ez) is not pure. 22. Let E be an ordered set in which every interval (a, b) is finite. Let S be the subset of E x E consisting of the ordered pairs (x, y) such that x < y. Show that a coassociative cogebra structure is defined on the A-module A(‘) of formal linear combinations of the elements of S by taking as coproduct For this cogebra the linear form y such that is a counit. 23. Let C be a cogebra over a ring A. A sub-A -module S of C is called a (a) If C is coassociative (resp. cocommutative), so is every subcogebra. If y (b) Which are the subcogebras of the cogebra A(x) (no. 1, Example 4).? (c) If (S,) is a family of subcogebras of C, 2 S, is a subcogebra of C (the subcogebra of C if c(S) c Im(S @ S). is a counit of C, the restriction of y to any subcogebra of C is a counit. smallest containing all the S,) . 24. Let C be a cogebra over a ring A. A sub-A-module S of C is called a (a’) A subcogebra (Exercise 23) is a right and left coideal. The converse is right (resp. left) coideal if c(S) c Im(S @I C) (resp. c(S) c Im(C @ S)). true‘if A is a field. . (6) Every sum of right (resp. left) coideals is a right (resp. left) coideal. (c’) If S is a right (resp. left) coideal of C, the orthogonal So is a right (resp. 1eft)’ideal of thedual algebra C*. (d) If A is a field and 3‘ a right (resp. left) ideal of the dual algebra C*, the orthogonal 3’0 in C is a right (resp. left) coideal of C. (Argue by reductio ad absurdum by assuming that there exists an x E 8“ such that c(x) $ 8’O @ C; write c(x) = 2 yr @ ti, where the zi are linearly independent, and show that there would bey’ E 8’ and z‘ E C* such that ( y’z’, x) # 0.) Deduce that for a vector subspace S of C to be a right (resp. left) coideal, it is necessary and sufficient that So be a right (resp. left) ideal of C*. f 652 EXERCISES (e) Deduce from (d) that if A is a field every intersection of right coideals (resp. left coideals, resp. subcogebras) of C is a right coideal (resp. left coideal, resp. subcogebra). For S to be a subcogebra of C, it is necessary and sufficient that its orthogonal So be a two-sided ideal of the algebra C*. 25. Let C be a cogebra over a ring A. A sub-A-module S of C is called a coideal if c(S) c Im(S @ C) + Im(C @ S). If C is counital, with counit y, a coideal S is called conull if y(x) = 0 for x E S. (a) A right or left coideal is a coideal. Every sum of coideals is a coideal. (b) If S is a coideal of C, the orthogonal So is a subalgebra of the dual algebra C*. If C is counital and S is conull, So is unital. (c) Suppose that A is a field and C a counital cogebra. Show that if E’ is a unital subalgebra of C* with the same unit element as C*, the orthogonal E’O in C is a conull coideal. (d) If S is a coideal of C, show that there exists on CjS one and only one cogebra structure such that the canonical mapping p: C --f CjS is a cogebra morphism. If C is counital and S is conull, CjS is counital. (e) For every cogebra morphismf: C + C’, S = Ker( f) is a coideal of C, S‘ = Im(f) a subcogebra of C’ and, in the canonical factorization c -L cjs 4 S’ + I C‘ ofJ g is a cogebra isomorphism. (26) Let C be a coassociative counital cogebra over a commutative field K. (a) For every vector subspace E of C which is a left C*-module under the law (u, x) ++ u A x, the left annihilator of this C* -module is a two-sided ideal 3’ of the algebra C*. Show that if E is finite-dimensional over K, C*/8’ is finite -dimensional. (6) Deduce from (a) that for every element x E C, the subcogebra of C generated by x is finite -dimensional. (Note that the C*-module E generated by x is finite -dimensional and that, if 3‘ is its left annihilator, x E S’O, the ortho - gonal of 8’ in C.) (27) Let B be an associative unital algebra over a commutative field K and let m: B BK B -+ B be the K-linear mapping defining the multiplication on B. The tensor product B* gK B* (where B* is the dual vector space of the vector space B) is canonically identified with a vector subspace of (B gK B)* (11, 3 7, no. 7, Proposition 16). (a) Show that for an element w E B* the following conditions are equiva - lent: (1) ‘m(w) E B* @ B*; (2) the set of x -I to, where x runs through B, is a finite-dimensional subspace of B* ; (3) the set of w L x, where x runs through B, is a finite -dimensional subspace ofB*; (4) the set of x -I w L y, where x and y run through B, is contained in a finite-dimensional subspace of B*. 653 111 TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS (6) Let B‘ c B* be the set of w E B* satisfying the equivalent conditions in (a). Show that tm(B’) c B‘ & B’. (For w E B’, write “(w) = u, @vi, where for example the u, are linearly independent in B*; show that then 11, E B‘ for all i using the associativity of B and arguing by reductio ad absurdum.) B‘ is therefore the largest cogebra contained in B*, for which tm is the coproduct. B’ is called the dual cogebra of the algebra B. When B is finite-dimensional, (c) Show that if B is a commutative field of infinite dimension over K, B’ = {O}. (Note that the orthogonal of the set of x -I w, for same w E B*, x running through B, is an ideal of B.) (d) Let B,, B, be two K-algebras and f: B, -+ B, a K-homomorphism of algebras. Show that Y(B&) c B; and that restricted to B&, is a cogebra homomorphism of Bk into B;. (e) If C is a coassociative counital cogebra over K, the canonical injection C -+ C** of the vector space C into its bidual maps C onto a subspace of (C*)’, the dual cogebra of the algebra C* dual to C, and is a cogebra homo- morphism of C into (C*)’. B‘ = B*. APPENDIX 1. Show that in alternative Cayley A-algebra (notation of fj 2, no. 4) T((xy)z) = T(x(yz)). (Reduce it to proving that a(x,y, z) = a(Z,y, 2)’) 2. Let F be an alternative A-algebra, in which the relation 2x = 0 implies x = 0. Show that, for all sc, y, z in F, x(yz)x = (xy) (zx) (Moufang’s identity). (In the identity (xy). +y(z.) = 4y.I + (Y4X successively replace y by xy and z by zx.) implies x = 0. Show that if x E F is invertible then, for ally, z, t in F, 3. Let F be an alternative Cayley A-algebra, in which the relation 2x = 0 (N(x) + N(y))(N(z) + N(t)) = N(xZ + ~1) + N(xt - (~z)(~-’y)) (use Exercises 1 and 2). 654 HISTORICAL NOTE (Chapters I1 and 111) (N.B. Numbers in brackets refer to the bibliography at the end of this Note.) Linear algebra is both one of the oldest and one of the newest branches of mathematics. On the other hand, at the origins of mathematics are the problems which are solved by a single multiplication or division, that is by calculating a value of a functionf(x) = ax, or by solving an equation ax = 6: these are typical problems of linear algebra and it is impossible to deal with them, indeed even to pose them correctly, without “thinking linearly”. On the other hand, not only these questions but almost everything concern- ing equations of the first degree had long been relegated to elementary teach - ing, when the modern development of the notions of field, ring, topological vector space, etc. came to isolate and emphasize the essential notions of linear algebra (for example duality) ; then the essentially linear character of almost the whole of modern mathematics was perceived, of which “linearization” is itself one of the distinguishing traits, and linear algebra was given the place it merits. To give its history, from our present point of view, would therefore be a task as difficult as it is important; and we must therefore be content to give a brief summary. From the above it is seen that linear algebra was no doubt born in response to the needs of practical calculators; thus we see the rule of three? and the rule of false position, more or less clearly stated, playing an important role in all the manuals of practical arithmetic, from the Rhind papyrus of the Egyptians to those used in our primary schools, by way of Aryabhata, the Arabs, Leonard of t Cf. J. TROPFKE, Geschichte der Elementar -Mathematik, 1. Band, 2te Ausgabe, Berlin -Leipzig (W. de Gruyter), 1921, pp. 150-155. 655 HISTORICAL NOTE ON CHAPTERS I1 AND 111 Pisa and the countless “calculation books” of the Middle Ages and the Renais- sance; but they never constituted more than a small part, for the use of practical men, of the most advanced scientific theories. As for mathematicians proper, the nature of their research on linear algebra depends on the general structure of their science. Ancient Greek mathematics, as expounded in the Elements of Euclid, developed two abstract theories of a linear character, on the one hand that of magnitudes ([2], Book V; cf. Histo- rical Note to General Topology, IV) and on the other hand that of integers ([2], Book VII). With the Babylonians we find methods much more akin to our elementary algebra; they know how to solve, and most elegantly ([l], pp. 181-183), systems of equations of the first degree. For a very long time, nevertheless, the progress of linear algebra is mainly confined to that of alge- braic calculations and they should be considered from this point of view, foreign to this Note; to reduce a linear system to an equation of the type ax = 6, it suffices, in the case of a single unknown, to know the rules (already, in substance, stated by Diophantus) for taking terms from one side to the other and combining similar terms ; and, in the case of several unknowns, it suffices to know also how to eliminate them successively until only one is left. Also the Treatises on algebra, until the XVIIIth century, think that all is accomplished as far as the first degree is concerned, when they have expounded these rules; as for a system of as many equations as unknowns (they do not consider others) where the left hand sides are not linearly independent forms, they are content to observe in passing that this indicates a badly posed problem. In the treatises of the XIXth century and even certain more recent works, this point of view is only modified by the progress of notation, which allows writing systems of n equations in n unknowns, and by the introduction of determinants which allow formulae of an explicit solution to be given in the “general case”; this progress, the credit for which would have belonged to Leibniz ([7], p. 239) had he developed and published his ideas on this subject, is mainly due to the mathe- maticians of the XVIIIth and early XIXth centuries. But we must first study various currents of ideas which, much more than the study of linear equations, contributed to the development of linear algebra in the sense in which we understand it. Inspired by the study of Appollonius, Fermat [4(a)], having conceived, even before Descartes [5], the principle of analytic geometry, has the idea of classifying plane curves according to their degree (which, having become little by little familiar to all mathematicians, can be considered to have been definitely grasped towards the end of the XVIIth century) and formulates the fundamental principle that an equation of the first degree, in the plane, represents a line and an equation of the second degree a conic: a principle from which he deduces immediately some “very beautiful” consequences relating to geometric loci. At the same time, he enun- ciates [4(b)] the classification of problems into problems with a single solution, problems which reduce to an equation in two unknowns, an equation in three 656 HISTORICAL NOTE ON CHAPTERS I1 AND I11 unknowns, etc. ; and he adds : the first consist of determining a point, the second a line or plane locus, the others a surface, etc. (“. . . such a problem does not seek only a point or a line, but the whole of a surface appropriate to the question; here surfaces as loci have their genesis and similarly for the rest ”, loc. cit., p. 186; here already is the germ of n-dimensional geometry). This paper, formulating the principle of dimension in algebra and algebraic geometry, indicates a fusion of algebra and geometry in absolute conformity with modern ideas, but which, as has already been seen, took more than two centuries to penetrate into men’s minds. At least these ideas soon result in the expansion of analytic geometry which reaches its fulness in the XVIIIth century with Clairaut, Euler, Cramer, Lagrange and many others. The linear character of the formulae for trans - formation of coordinates in the plane and in space, which Fermat cannot have failed already to have perceived, is put in relief for example by Euler ([8(a)], Chapters 11-111 and Appendix to Chapter IV), who here lays the foundation of the classification of plane curves and that of surfaces according to their degree (invariant precisely because of the linearity of these formulae) ; he it is also (loc. cit., Chapter XVIII) who introduces the word “affinity” to describe the relation between curves which can be derived one from the other by a transformation x’ = ax, y‘ = by (but without perceiving anything geometri - cally invariant in this definition which remains bound to a particular choice of axes). A little later we see Lagrange [9(a)] devoting a whole memoir, which long remained justly famous, to typically linear and multilinear problems of analytic geometry in three dimensions. Around about this time, in relation to the linear problem constituted by the search for a plane curve passing through given points, the notion of determinant takes shape, first in a somewhat empirical way, with Cramer [lo] and Bezout 11 11 ; this notion is then developed by several authors and its essential properties are definitively established by Cauchy [13] and Jacobi [16(a)]. On the other hand, whilst .mathematicians had a slight tendency to despise equations of the first degree, the solution of differential equations was con- sidered a capital problem; it was natural that, among these equations, linear equations, with constant coefficients or otherwise, should early be distinguished and their study contributed to emphasize linearity and related properties. This is certainly seen in the work of Lagrange [9(b)] and Euler [8(b)], at least as far as homogeneous equations are concerned; for these authors see no point in saying that the general solution of the non-homogeneous equation is the sum of a particular solution and the general solution of the corresponding homo - geneous equation and they make no use of this principle (known however to d’Alembert) ; we note here also that, when they state that the general solution of the homogeneous linear equation of order n is a linear combination of n particular solutions, they do not add that these must be linearly independent and make no effort to make the latter notion explicit; it seems that only the teaching of Cauchy at the Ikole Polytechnique throws some light ([14], 65 7 HISTORICAL NOTE ON CHAPTERS I1 AND I11 PP. 573-574) on these points as on many others. But already Lagrange (lac. cit.) introduces also (purely by calculation, it is true, and without giving it a name) the adjoint equation L*(y) = 0 of a linear differential equation L( y) = 0, an example typical of duality by virtue of the relation szL(y) dx = s L*(z) y dx, valid for y and z zero at the extremities of the interval of integration; more precisely, and 30 years before Gauss defined explicitly the transpose of a linear substitution in 3 variables, we see here the first example without doubt of a “functional operator” L* the transpose or “adjoint” of an operator L given by means of a bilinear function (here the integral At the same time and again with Lagrange [9(c)J, linear substitutions, in 2 and 3 variables at first, were in the process of conquering arithmetic. Clearly the set of values of a function F(x,y), when x and y are given all integral values, does not change when a linear substitution with integral coefficients, of determinant 1, is performed on x and y; on this fundamental observation Lagrange founds the theory of representations of numbers by forms and that of the reduction of forms; and Gauss, by a step whose boldness it has become difficult for us to appreciate, isolates the notion of equivalence and that of class of forms (cf. Historical Note to I) ; on this subject, he recognizes the necessity of certain elementary principles relating to linear substitutions and introduces in particular the notion of transpose or adjoint ([12(a)], p. 304). From this moment onwards, the arithmetic study and the algebraic study of quadratic forms, in 2, 3 and later n variables, that of bilinear forms which are closely related to them and more recently the generalization of these notions to an infinity of variables were, right up to the present, to constitute one of the most fertile sources of progress for linear algebra (cf. Historical Note to IX) . But a perhaps still more decisive progress was the creation by Gauss, in the same Disquisitiones (cf. Historical Note to I), of the theory of finite commutative groups, which occur there in four different ways, in the additive group of integers modulo m (for rn an integer), in the multiplicative group of integers prime to rn modulo m, in the group of classes of binary quadratic forms and finally in the multiplicative group of m-th roots of unity; and, as we have al- ready noted, it is clearly as commutative groups, or rather as modules over 2, that Gauss treats all these groups and studies their structure, their relations of isomorphism, etc. In the module of “complex integers” a + 6i, he is later seen studying an infinite module over Z, whose isomorphism he no doubt perceived with the module of periods (discovered by him in the complex domain) of elliptic functions; in any case this idea already appears neatly in Jacobi’s work, for example in his famous proof of the impossibility of a function with 3 periods and his views on the problem of inversion of Abelian integrals 658 yt dx). HISTORICAL NOTE ON CHAPTERS I1 AND 111 [16(6)], to result soon in the theorems of Kronecker (cf. Historical Note to General Tofiology, VII) . Here, another current joins those whose course and occasional meanders we have sought to trace, which had long remained underground. As will later be expounded in more detail (Historical Note to IX), “pure” geometry in the sense understood in the last century, that is essentially projective geometry of the plane and space without using coordinates, had been created in the XVIIth century by Desargues [6], whose ideas, appreciated in their true value by a Fermat and put into practice by a Pascal, had then been buried in oblivion, eclipsed by the brilliant progress of analytic geometry; it was revived towards the end of the XVIIIth century, with Monge, then Poncelet and his rivals Brianchon and Chasles, sometimes completely and voluntarily separated from analytic methods, sometimes (especially in Germany) closely intermixed with them. Now projective transformations, from whatever point of view they are considered (synthetic or analytic), are of course just linear substitutions on the projective or “barycentric” coordinates; the theory of conics (in the XVIIth century) and later that of quadrics, with whose projective properties this school is principally concerned for a long time, are just that of quadratic forms, whose close connection with linear algebra we have already pointed out earlier. To these notions is added that of polarity: also created by Desargues, the theory of poles and polars becomes, in the hands of Monge and his suc - cessors and soon under the name of the principle of duality, a powerful tool for transforming geometric theorems; if it cannot be affirmed that its relation with adjoint differential equations was perceived during that period (they are indicated by Pincherle at the end of the century), then at least Chasles did not fail [17] to perceive its relation with the notion of reciprocal spherical tri- angles, introduced into spherical trigonometry by Vihte ([3], p. 428) and Snellius as early as the XVIth century. But duality in projective geometry is only an aspect of duality of vector spaces, taking account of the modifications imposed when passing from the affine space to the projective space (which is a quotient space of it, under the relation “scalar multiplication”). The XIXth century, more than any period in our history, was rich in mathe - maticians of the first order; and it is difficult in a few pages, even restricting ourselves to the most salient features, to describe all that is produced in their hands by the coming together of these movements of ideas. Between the purely synthetic methods on the one hand, a species of Procrustean bed where their orthodox protagonists put themselves to torture, and the analytic methods related to a system of coordinates arbitrarily imposed on the space, the need is soon felt for a geometric calculus, dreamed of but not created by Leibniz and imperfectly sketched by Carnot: first appears addition of vectors, implicit in Gauss’s work in his geometric representation of imaginary numbers and the applications he makes of this to elementary geometry (cf. Historical Note to General Topology, VIII), developed by Bellavitis under the name of “method of 659 HISTORICAL NOTE ON CHAPTERS I1 AND 111 equipollences” and taking its definitive form with Grassmann, Mobius and Hamilton; at the same time, under the name of “barycentric calculus”, Mobius gives a version of it suitable for the needs of projective geometry [ 181. At the same period, and by the same men, the step, so natural (once engaged on this path), already announced by Fermat, is taken from the plane and “ordinary” space to n-dimensional space; indeed an inevitable step, since the algebraic phenomena which can in two or three variables be interpreted geo- metrically are still valid for an arbitrary number of variables; thus to impose, in using geometric language, the limitation to 2 or 3 dimensions, would be for the modern mathematician just as tiresome a yoke as that which always pre- vented the Greeks from extending the notion of number to ratios of incom- mensurable magnitudes. Hence the language and ideas relating to n-dimen- sional space appear almost simultaneously on all sides, obscurely in the work of Gauss, clearly in the work of the mathematicians of the following generation; and their greater or less assurance in using them was perhaps less due to their mathematical inclinations than to their philosophical or even purely practical outlook. In any case, Cayley and Grassmann, around 1846, handle these concepts with the greatest of ease (and this, says Cayley quite contrary to Grassmann ([22 (a)], p. 32 I), “without recourse to any metaphysical notion”) ; Cayley is never far away from the analytic interpretation and coordinates, whereas in Grassmann’s work from the start, addition of vectors in n-dimen- sional space and the geometric aspect take the upper hand, to result in the developments of which we shall speak in a moment. Meantime the impulse given by Gauss was pushing mathematicians, in two different ways, towards this study of algebras or “hypercomplex systems”. On the one hand, it was inevitable to try to extend the domain of real num- bers otherwise than by introducing the “imaginary unit” i = 4 - 1 and per- haps thus open up vaster domains just as fertile as that of the complex numbers. Gauss himself was convinced ([12(b)], p. 178) of the impossibility of such an extension, as long as one wants to preserve the principal properties of complex numbers, that is, in modern language, those which make it into a commutative field ; and, either under his influence or independently, his contemporaries seem to have shared this conviction, which was only justified much later by Weierstrass [23] in a precise theorem. But, once multiplication of complex numbers is interpreted by rotations in the plane, then, if it is proposed to extend this idea to three dimensional space, (since the rotations in space form a non- Abelian group) non-commutative multiplications have to be envisaged ; this is one of Hamilton’sf guiding ideas in his discovery of quaternions [20], the first example of a non-commutative field. The singular nature of this example (the only one, as Frobenius was later to show, which can be constructed over the t Cf. the interesting preface of his Lectures on quaternions [ZO] where he retraces the whole history of his discovery. 660 HISTORICAL NOTE ON CHAPTERS I1 AND I11 field of real numbers) somewhat restricts its import, in spite of or perhaps even because of the formation of a school of fanatical “quaternionists” : a strange phenomenon, which was later reproduced around the work of Grassmann, and then by the vulgarizers who draw from Hamilton and Grassmann what is called “vector calculus”. The abandoning a little later of associativity, by Graves and Cayley who construct the “Cayley numbers”, opens up no very interesting path. But after Sylvester had introduced matrices and (without giving it a name) had clearly defined their rank [21], again it was Cayley [22(b)] who created the calculus of matrices, not without observing (an essen- tial fact often lost sight of later) that a matrix is only an abridged notation for a linear substitution, just as Gauss denoted the form ax2 + 2bXY + cY2 by (a, b, c). This is just one aspect, the most interesting for us of course, of the abundant production by Sylvester and Cayley on determinants and every - thing connected with them, a production full of ingenious identities and im- pressive calculations. Also (amongst other things) Grassmann discovers an algebra over the reals, the exterior algebra which still bears his name. His work, earlier even than that of Hamilton [19(a)], created in an almost complete moral solitude, remained for a long time little known, no doubt because of its originality, because also of the philosophical mists, in which it begins by enveloping itself and which for example at first deterred Mobius. Moved by preoccupations analogous to those of Hamilton but of greater import (and which, as he soon sees, are the same as those of Leibniz) , Grassmann constructs a vast algebraico-geometric edifice, resting on a geometric or “intrinsic” conception (already more or less axioma- tized) of n-dimensional vector space; among the more elementary results at which he arrives, we quote for example the definition of linear independence of vectors, that of dimension and the fundamental relation dim V + dim W = dim(V + W) + dim(V W) (loc. cit., p. 209; cf. [19(b)], p. 21). But it is especially exterior multiplication, then inner multiplication, of multivectors which provide him with the tools with which he easily treats first the problems of linear algebra proper and then those relating to the Euclidean structure, that is orthogonality of vectors (where he finds the equivalent of duality, which he does not possess). The other path opened up by Gauss in the study of hypercomplex systems is that starting from the complex integers a + bi; after these follow quite naturally algebras or more general hypercomplex systems, over the ring Z of integers and over the field Q of rationals, and first of all those already envisaged by Gauss which are generated by roots of unity, then algebraic number fields and modules of algebraic integers : the former are the principal topic in the work of Kummer, the study of the latter was undertaken by Dirichlet, Hermite, Kronecker and Dedekind. Here, in contrast to what happens with algebras over the reals, it is not necessary to abandon any of the characteristic properties 66 1 HISTORICAL NOTE ON CHAPTERS I1 AND 111 of commutative fields and attention was confined to the latter throughout the XIXth century. But linear properties and for example the search for the basis for the integers of the field (indispensible for a general definition of the dis- criminant) play an essential role at many points; and with Dedekind at any rate the methods are destined to become typically “hypercomplex” ; Dedekind himself moreover, without setting himself the problem of algebras in general, is conscious of this character of his works and of what relates them for example to the results of Weierstrass on hypercomplex systems over the reals ([24], in particular vol. 2, p. 1). At the same time the determination of the structure of the multiplicative group of units in an algebraic number field, effected by Dirichlet in some famous notes [15] and almost at the same time by Hermite, was vitally important in clarifying ideas on modules over Z, their generating systems and, their bases (when such exist). Then the notion of ideal, defined by Dedekind in algebraic number fields (as a module over the ring of integers of the field), whilst Kronecker introduces in polynomial rings (under the name of “systems of modules”) an equivalent notion, gives the first examples of modules over more general rings than Z; and in the work of the same authors, and then Hilbert, in particular cases the notion of group with operators is slowly isolated, and the possibility of constructing always from such a group a module Over a suitably defined ring. At the same time, the arithmetico-algebraic study of quadratic bilinear forms and their “reduction” (or, what amounts to the same, of matrices and their ‘‘invariants”) leads to the discovery of the general principles on the solution of systems of linear equations, principles which due to the lack of the notion of rank, had escaped Jac0bi.t The problem of the solution in integers of systems of linear equations with integral coefficients is attacked and solved, first in a special case by Hermite and then in all its generality by H. J. Smith [25] ; the results of the latter are found again, only in 1878, by Frobenius, in the frame - work of a vast programme of research instituted by Kronecker and in which Weierstrass also participates ; incidentally during the course of these works, Kronecker gives definitive form to the theorems on linear systems with real (or complex) coefficients, which are also elucidated, in an obscure manual, with the minute care characteristic of him, by the famous author of Alice in Wonder - land; as for Kronecker, he disdains to publish these results and leaves them to his colleagues and disciples ; the word “rank” itself is only introduced by Frobenius. Also in the course of their teaching at the University of Berlin Kronecker [26] and Weierstrass introduce the “axiomatic” definition of determinants (as an alternating multilinear function of n vectors in n-dimensional space, normed so that it takes the value 1 at the unit matrix), a definition equivalent to that t Concerning the classification of systems of n equations in n unknowns when the determinant is zero, he says ([16(u)], p. 370) : ‘‘fiuullo prolixum uidetur nqotium” (it could not be elucidated briefly). 662 e HISTORICAL NOTE ON CHAPTERS I1 AND I11 derived from Grassmann’s calculus and to that adopted in this Treatise; again during his courses Kronecker, without feeling the need to give it a name and in a still non -intrinsic form, introduces the tensor product of spaces and the “Kronecker” product of matrices (the linear substitution induced on a tensor product by given linear substitutions applied to the factors). This research cannot be separated from the theory of invariants created by Cayley, Hermite and Sylvester (the “invariant trinity” of which Hermite later speaks in his letters) and which, from a modern point of view, is above all a theory of representations of the linear group. Here there comes to light, as the algebraic equivalent of duality in projective geometry, the distinction between series of cogredient and contragredient variables, that is vectors in a space and vectors in the dual space; and, after attention has been turned first to forms of low degree and then of arbitrary degree, in 2 and 3 variables, almost without delay bilinear, then multilinear forms are examined in several series of “co- gredient” or “contragredient” variables, which is equivalent to the intro- duction of tensors; the latter becomes explicit and is popularized when, under the inspiration of the theory of invariants, Ricci and Levi-CivitA, in 1900, in- troduce into differential geometry “tensor calculus” [28], which later came into great vogue following its use by the “relativist” physicists. Again the progressive intermingling of the theory of invariants, differential geometry and the theory of partial differential equations (especially the so-called problem of Pfaff and its generalizations) slowly leads geometers to consider alternating bilinear forms of differentials, in particular the “bilinear covariant” of a form of degree 1 (introduced in 1870 by Lipschitz and then studied by Frobenius), to result in the creation by E. Cartan [29] and PoincarC [30] of the calculus of exterior differential forms. PoincarC introduces the latter, in ordcr to form his integral invariants, as the expressions which appear in multiple integrals, whilst Cartan, guided no doubt by his research on algebras, introduces them in a more formal way, but without failing to observe that the algebraic part of their calculus is identical with Grassmann’s exterior multiplication (whence the name which he adopts), thus definitively restoring the work of the latter to its rightful place. The translation, into the notation of tensor calculus, of exterior differential forms, moreover shows immediately their connection with skew- symmetric tensors, which, once a purely algebraic point of view is adopted, shows that they are for alternating multilinear forms what covariant tensors are for arbitrary multilinear forms; this aspect is further clarified with the modern theory of representations of the linear group; and thus, for example, the substantial identity between the definition of determinants given by Weierstrass and Kronecker and that resulting from Grassmann’s calculus is recognized. We thus arrive at the modern period, where the axiomatic method and the notion of structure (at first vaguely perceived, and defined only recently) allow us to separate concepts which until then had been inextricably mixed, to formulate what was vague or left to intuition and to prove with proper 663 HISTORICAL NOTE ON CHAPTERS I1 AND I11 generality theorems which were known only in special cases. Peano, one of the creators of the axiomatic method and also one of the first mathematicians fully to appreciate the work of Grassmann, gives as early as 1888 ([27], Chapter IX) the axiomatic definition of vector spaces (finite -dimensional or otherwise) over the field of real numbers and, in a completely modern notation, of linear map- pings of one such space into another; a little later, Pincherle seeks to develop applications of linear algebra, thus conceived, to the theory of functions, in a direction it is true which has not been very fruitful; at least his point of view allows him to recognize “Lagrange’s adjoint” as a special case of the trans - position of linear mappings: which appears soon, still more clearly, and for partial differential equations as well as for differential equations, in the course of the memorable works of Hilbert and his school on Hilbert space, and its applications to analysis. It is on that occasion that Toeplitz [31], also intro - ducing (but by means of coordinates) the most general vector space over the reals, makes the fundamental observation that the theory of determinants is not needed to prove the principal theorems of linear algebra, which allows these to be extended without difficulty to infinite -dimensional spaces ; and he also indicates that linear algebra thus understood can of course be applied to an arbitrary commutative base field. On the other hand, with the introduction by Banach, in 1922, of the spaces bearing his name,t spaces not isomorphic to their dual are encountered, albeit in a problem which is topological as much as it is algebraic. Already between a finite -dimensional vector space and its dual there is no “canonical” isomorphism, that is determined by its structure, which had long been reflected in the distinction between cogredient and contragredient. Nevertheless it seems beyond doubt that the distinction between a space and its dual was definitively established only after the work of Banach and its school; also in these works the importance of the notion of codimension comes to light. As for duality or “orthogonality” between the vector subspaces of a space and those of its dual, the way in which it is formulated today presents not just a superficial analogy with the modern formulation of the principal theorem of Galois theory (cf. Algebra, V) and with so-called Pontrjagin duality in locally compact Abelian groups; the latter goes back to Weber, who, in the course of arith - metical researches, lays in 1886 its foundations for finite groups; in Galois theory the “duality” between subgroups and subfields takes form in the work of Dedekind and Hilbert ; and orthogonality between vector subspaces derives visibly, first from duality between linear varieties in projective geometry and also from the notion and properties of completely orthogonal varieties in a Euclidean space or a Hilbert space (whence its name). All these strands are reassembled in the contemporary period, in the hands of algebraists such as t These are complete normed vector spaces over the field of real numbers or that of complex numbers. 664 BIBLIOGRAPHY E. Noether, Artin and Hasse and topologists such as Pontrjagin and Whitney (not without the ones influencing the others) to arrive, in each of these fields, at the stage of knowledge whose results are expounded in this Treatise. At the same time a critical examination is made, which is destined to eliminate, on each point, the hypotheses which are not completely indispensable, and especially those which would close the way to certain applications. Thus the possibility is perceived of substituting rings for fields in the notion of vector spaces and, creating the general notion of module, of treating at the same time these spaces, Abelian groups, the particular modules already studied by Kronecker, Weierstrass, Dedekind and Steinitz and even groups with operators and for example of applying the Jordan-Holder theorem to them; at the same time, with the distinction between right and left modules, we pass to the non- commutative, which arises from the modern development of the theory of algebras by the American school (Wedderburn, Dickson) and especially the German school (E. Noether, E. Artin). BIBLIOGRAPHY 1. 0. NEUGEBAUER , Vorlesungen iiber Geschichte der antiken Mathematik, Vol. I : 2. Euclidis Elementa, 5 vols., ed. J. L. Heiberg, Lipsiae (Teubner), 1883-88. 2 bis. T. L. HEATH , The thirteen books of Euclid’s Elements. . ., 3 vols., Cambridge, 1908. 3. FRANCISCI VIETAE , Opera mathematica . . ., Lugduni Batavorum (Elzevir), 1646. 4. P. FERMAT , Oeuvres, vol. I, Paris (Gauthier-Villars), 1891: (u) Ad locos planos et solidos Isagoge (pp. 91-1 10; French transl. ibid., vol. 111, p. 85); (b) Appendix ad methodum. . . (pp. 184-188; French transl., ibid., vol. 111, p. 159). 5. R. DESCARTES , La gkomi‘trie, Leyde (Jan Maire), 1637 (=Oeuvres, ed. Ch. Adam and P. Tannery, vol. VI, Paris (L. Cerf), 1902). 6. G. DESARGUES , Oeuvres. . ., vol. I, Paris (Leiber), 1864: Brouillon proiect d’une atteinte aux tutnemens des rencontres d’un c8ne auec un plan 7. G. W. LEIBNIZ , Mathematische Schriften, ed. C. I. Gerhardt, vol. I, Berlin (Asher), 1849. 8. L. EULER : (a) Introductio in Anulysin Infinitorum, vol. 2, Lausannae, 1748 ( = Opera Omnia (I), vol. IX, Zurich-Leipzig -Berlin (0. Fussli and B. G. Teubner), 1945) ; (b) Institutionum Calculi Integralis, vol. 2, Petropoli, 1769 (=Opera Omnia (l), vol. XII, Leipzig-Berlin (B. G. Teubner), 1914). 9. J.-L. LAGRANGE , Oeuvres, Paris (Gauthier -Villars), 1867-1892 : (a) Solu- tions analytiques de quelques probl&mes sup les pyramides triangulaires, Vorgriechische Mathematik, Berlin (Springer), 1934. (pp. 103-230). 665 BIBLIOGRAPHY vol. 111, pp. 661-692; (6) Solution de diffkrents probltmes de calcul intkgral, vol. I, p. 471; (c) Recherches d'arithmktique, vol. 111, pp. 10. G. CRAMER , Introduction d 1 'analyse des lignes courbes, Geneva (Cramer and 11. E. BEZOUT , Thkorie ginirale des iquations algkbriques, Paris, 1779. 12. C. F. GAUSS, Werke, Gottingen, 1870-1927 : (a) Disquisitiones arithmeticae, vol. I ; (b) Selbstanzeige zur Theoria residuorum biquadraticorum, Commentatio secunda, vol. 11, pp. 169-178. 13. A.-L. CAUCHY , Mtmoire sur les fonctions qui ne peuvent obtenir que deux valeurs tgales et de signes contraires par suite des transpositions optrtes entre les variables qu'elles renferment, J. Ec. Polytech., cahier 17 (volume X), 1815, pp. 29-112 (=Oeuvres compl2tes (2), vol. I, Paris (Gauthier -Villars), 1905, pp. 91-169). 14. A.-L. CAUCHY , in Legons de calcul dzj'iirentiel et de calcul inte'gral, rkdigies principalement d'apr2s les mithodes de M. A.-L. Cauchy by AbbC Moigno, vol. 11, Paris, 1844. 15. P. G. LEJEUNE -DIRICHLET , Werke, vol. I, Berlin (G. Reimer), 1889, pp. 619 -644. 16. C. G. J. JACOBI , Gesammelte Werke, Berlin (G. Reimer), 1881-1891: (a) De formatione et proprietatibus determinantium, vol. 111, pp. 355-392; (6) De functionibus duarum variabilium . . ., vol. 11, pp. 17. M. CHASLES, Apergu historique sur l'origine et le diveloppement des mithodes en gom'trie . . ., Bruxelles, 1837. 18. A. F. MO BIUS, Der baryzentrische Calcul . . ., Leipzig, 1827 (=Gesammelte Werke, vol. I, Leipzig (Hirzel), 1885). 19. H. GRASSMANN: (a) Die lineale Ausdehnungslehre, ein neuer Zweig der Mathe - matik, dargestellt und durch Anwendungen auf die iibrigen Zweige der Mathe - matik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Kristallonomie erlautert, Leipzig (Wigand), 1844 ( = Gesammelte Werke, vol. I, 1st Part, Leipzig (Teubner), 1894) ; (b) Die Ausdehnungslehre, vollstandig und in strenger Form bearbeitet, Berlin, 1862 ( = Gesammelte Werke, vol. I, 2nd Part, Leipzig (Teubner), 1896). 695-795. Philibert), 1750. 25-50. 20. W. R. HAMILTON , Lectures on quaternions, Dublin, 1853. 21. J. J. SYLVESTER , Collected Mathematical Papers, vol. I, Cambridge, 1904: No. 25, Addition to the articles . . ., pp. 145-151 (=Phil. Mag., 1850). 22. A. ~AYLEY, Collected Mathematical Papers, Cambridge, 1889-1898 : (a) Sur quelques thkor&mes de la gComttrie de position, vol. I, pp. 317-328 (=Crelle's J., vol. XXXI (1846), pp. 213-227); (6) A memoir on the theory of matrices, vol. 11, pp. 4751196 (=Phil. Trans., 1858). 23. K. WEIERSTRASS, Mathematkche Werke, vol. 11, Berlin, (Mayer und Muller), 666 BIBLIOGRAPHY 1895: Zur Theorie des aus n Haupteinheiten gebildeten complexen Grossen, pp. 31 1-332. 24. R. DEDEKIND , Gesammelte mathematkche Werke, 3 vols., Braunschweig (Vieweg), 1930-32. 25. H. J. SMITH , Collected Mathematical Papers, vol. I, Oxford, 1894: On systems of linear indeterminate equations and congruences, p. 367 ( =Phil. Trans., 186 1 ) . 26. L. KRONECKER, Vorlesungen iiber die Theorie der Determinanten . . ., Leipzig (Teubner'l. 1903. ,I 2 7. G. PEANO, Calcolo geometric0 second0 1 'Ausdehnungstheorie di Grassmann preceduto dalle operazioni della logica deduttiva, Torino, 1888. 28. G. RICCI and T. LEVI-CIVITA, Mtthodes de calcul diffkrentiel absolu et leurs applications, Math. Ann., vol. LIV (1901), p. 125. 29. E. CARTAN, Sur certaines expressions diffkrentielles et le probl&me de Pfaff, Ann. E.N.S. (3), vol. XVI (1899), pp. 239-332 (=Oeuvrescompl2tes, vol. II,, Paris (Gauthier -Villars), 1953, pp. 303-396). 30. H. PO IN CAR^, Les me'thodes nouvelles de la micanique ce'leste, vol. 111, Paris (Gauthier-Villars), 1899, Chapter XXII. 3 1. 0. TOEPLITZ , Ueber die Auflosung unendlichvieler linearer Gleichungen mit unendlichvielen Unbekannten, Rend. Circ. Mat. Pal., vol. XXVIII (1909), pp. 88-96. 667 INDEX OF NOTATION T f xi: I, 5 1, no. 2. P4i<Q i=P nn T x, I xn, nx (n EN) : I, fj I, no. 3. T T x,,: I, 3 1, no. 5. O<i<j<nXil, i<j 5 d xi,, 2 2 xil: I, 9 1, no. 5. i=p j=r 1=r i=p 0 <i1<i2< ... < ip<n x~ltz***lp, il c iz< ... < tp Ya, &a, Y(4, S(a) : I, 9 2, no. 2. T T xiltz..ip: I, 9 1, no. 5. 0, 1: I, fj 2, no. 1. E, (S a subset of a commutative monoid E) : I, 9 2, no. 4. Z, + (addition in Z) : I, 3 2, no. 5. < (order relation on Z) : I, fj 2, no. 5. N*: I, 3 2, no. 5. i (for n E Z) : I, 9 2, no. 7. -x,x -y,x + y - z,x -y - Z,X -y + z - t: I, $2,110. 8. nx (n E Z) : I, 3 2, no. 8. xn (n E Z) : I, 5 2, no. 8. lx -9 -9 xly: I, 3 2, no. 8. XY 669 INDEX OF NOTATION a.~, %.a, xu (a an operator) : I, 3 3, no. 1. a I X, M. I X, 6 I X (a an operator, 6 a set of operators) : I, 3 3, no. 1. 6,: I, $4, no. 1. (G: H), G/H (H a subgroup of G) : I, 3 4, no. 4. x Kerf, Imf (fa group homomorphism) : I, 3 4, no. 5. n G, (G, groups) : I, 3 4, no. 8. G1 x G,: I, 3 4, no. 8. n G, (G, groups) : I, 3 4, no. 9. x G y (mod. a), x = y (a) (a, x, y rational integers): I, 4 4, no. 10. ~,(a) (p a prime number, a a rational integer) : I, 3 4, no. 10. Aut(G), Int(G), Int(x) (G a group, x E G) : I, 3 5, no. 3. N,(A), N(A) (G a group, A c G): I, 3 5, no. 3. C,(A), C(A) (G a group, A c G): I, 3 5, no. 3. E/G, G\E (G a group operating on E) : I, 3 5, no. 4. G\E/H (G, H groups operating on E by commuting actions) : I, 3 5, no. 4. 6,: I, $ 5, no. 7. 7s. Y (transposition of support {x, y}) : I, 3 5, no. 7. &(a), E, (a a permutation) : I, 3 5, no. 7. a,,, a,: I, $ 5, no. 7. F&E%G (E, F, Ggroups): I, 36, no. 1. F x G, 6, (T a homomorphism of G into Aut(F)) : I, 3 6, no. 1. "f(fEF,gEG): I, 36, no. 1. (x, y), (A, B) (x, y elements, A, B subsets of a group G) : I, 4 6, no. 2. D(G): I, 3 6, no. 2. Cn(G) : I, 3 6, no. 3. Dn(G) : I, 4 6, no. 4. EG (G a group operating on E) : I, 3 6, no. 5. M,(X), M(X) (X a set): I, fj 7, no. 1. 1 (w) (w an element of M(X)) : I, 3 7, no. 1. ww', w. w' (w, w' elements of M(X)): I, 3 7, no. 1. 1 (w) (w a word over X) : I, 3 7, no. 2. ww', w. w' (w, w' words over X) : I, 3 7, no. 2. Mo(X) (X a set) : I, $ 7, no. 2. I(.) (a a decomposition) : I, fj 7, no. 3. * G,, GI * G, (Gl, G,, Gi groups) : I, 3 7, no. 3. (71, . . ., 7,; rl, . . . , r,) (T, generators, r, relators) : I, 3 7, no. 6. (-rl,. . ., T,; u1 = ul,. . ., u, = u,) (T, generators, ui, u, elements of a free ZCx), N(x) (X a set): I, 3 7, no. 7. 670 y (mod. H), x = y (H) (H a normal subgroup) : I, 3 4, no. 4. ,€I t €1 (5 5) .T(f', 9') (Lf' in F, 57, g' in G): I, 3 6, no. 1. group) : I, $ 7, no. 6. 1 INDEX OF NOTATION 0, 1 (elements of a ring) : I, 3 8, no. 1. Ao (A a ring) : I, 3 8, no. 3. (a) (a an element of A) : I, 3 8, no. 6. 2 ah. (ah ideals) : I, 3 8, no. 6. x e y (mod a), x = y (a) (a an ideal) : I, 3 8, no. 7. A/a (a a two-sided ideal) : I, 3 8, no. 7. ab (a, b two-sided ideals) : I, 3 8, no. 9. A[S-I] (S a subset of a ring A) : I, 3 8, no. 12. F, (p a prime number) : I, 3 9, no. 1. Q: I, 3 9, no. 4. Q+: I, 3 9, no. 4. 1x1, sgn x (x a rational number) : I, 3 9, no. 4. in(G) : I, 3 4, Exercise 13. D,: I, $ 6, Exercise 4. Q: I, 3 6, Exercise 4. A z B: I, 3 6, Exercise 39. e(G) : I, $ 7, Exercise 39. d,,(X) : I, 3 7, Exercise 39. A,, A, (A a ring): 11, CX,((X~),~~ a family of elements of a module of finite support: 11, 5 1, Hom,(E, F), Hom(E, F), (E, F, A-modules): 11, 4 1, no. 2. End,@), End(E), Aut(E), GL(E) (E an A-module) : 11, 3 1, no. 2. Hom,(u, u), Hom(u, u) (u, u linear mappings): 11, 3 1, no. 2. 1, (E a module): 11, 3 1, no. 2. 0 (zero module) : 11, 3 1, no. 3. Ker u, Im u, Coim u, Coker u (u a linear mapping) : 11, 3 1, no. 3. 1, no. 1. LEI no. 1. (A: E, -+ F, linear mappings) : 11,s 1, no. 5. @EL,E,OE,+l @.*.@Eq((EJ,EIafamilyofA-modules): 11,s 1, no. 6. 2 2 fLf, Of,+l 0. . . Of, (A: EL -+ FL linear mappings) : 11, 3 1, no. 6. E(I) (E a module) : 11, 3 1, no. 6. 0 M, ((MJL a family of submodules) : 11, 3 1, no1 7. long,(M), long(M) (M an A-module of finite length) : 11, 3 1, no. lo. gst (Kronecker symbol) : 11, 3 1, no. 11. C&.t(Taset, &elementsofaring):II,§l,no. 11. LEI (A: EL -+ F, linear mappings) : 11, 4 1, no. 6. LEI L€I teT 67 I INDEX OF NOTATION Ann@), Ann(x) (S a subset of a module, x an element of a module) : 11, $ 1, p*(E), EIB, (E an A-module, p: B --f A a ring homomorphism): 11, $ 1, no. 13. p*(u) (p: B -+ A a ring homomorphism, u an A-linear mapping) : 11, $ 1, E* (E a module): 11, $ 2, no. 3. (x, x*> (x an element of a left module E, x* an element of its dual E*) : 11, no. 12. no. 13. fj 2, no. 3. (x*, x} (x an element of a right module E, x* an element of its dual E*) : 11, $ 2, no. 3. tu (u a linear or semi-linear mapping) : 11, $ 2, no. 5. 6 (u an isomorphism) : 11, $ 2, no. 5. E @ F, E @A F (E a right A-module, F a left A-module) : 11, $ 3, no. 1. x @ y (x E E (a right module), y E F (a left module)) : 11, $ 3, no. 1. u @ u (u, u linear mappings) : 11, $ 3, no. 2. u 8 u (u, u semi-linear mappings) : 11, $ 3, no. 3. A,, (A ring) : 11, $ 3, no. 4. Y2(E, F; G) (E, F, G modules over a commutative ring) : 11, $ 3, no. 5. 8 G,, (8 x, ((G,) a family of Z-modules, x, E G, for all A) : 11, $3, no. 9. 8 uh (u,: G, -+ GL Z-linear mappings) : 11,s 3, no. 9. A h€L A€L h€L @v, (u, Z-linear mappings) : 11, 3, no. 9. x1 @ x2 8.. . @ x,: 11, $ 3, no. 9. u1 @ u2 @. . . @ u, (ut linear mappings) : 11, fj 3, no. 9. Pn(El, . . ., En; G) (El, . . ., En, G modules over a commutative ring): 11, Tr(u) (u an endomorphism of a module over a commutative ring) : 11, $ 4, no. p*(E), E,, (E an A-module, p: A --f B a ring homomorphism) : 11, $ 5, no. 1. p*(u), qB) (p : A -+ B a ring homomorphism, u an A-module homomorphism) : dim, E, dim E, [E: K] (E a vector K-space) : 11, $ 7, no. 2. dim, E, dim E (E an A-module any two bases of which are equipotent) : 11, codim,F, codim F (F a vector subspace of a vector space E) : 11,s 7, no. 3. rg(u) (u a linear mapping of vector spaces) : 11, $ 7, no. 4. rg(u) (u an element of a tensor product of vector spaces) : 11, § 7, no. 8. dim,E, dim E (E an affine space over a field K) : 11, $ 9, no. 1. a + t, t + a (a a point, t a translation of an afKne space) : 11, Q 9, no. 1. EI @A~ Ez @A2 E, 8. . . @A,,-a En-1 @A,,-l En: 11, $ 3, no. 9. $ 3, no. 9. 3. 11, $5, no. 1. $ 7, no. 2. 672 INDEX OF NOTATION 6 - a (a, 6 points of an affine space) : 11, $ 9, no. 1. 2 A,x, ((xl)ler a family ofpoints of an affine space, (AJ, of finite support, such that 2 A, = 1 or 2 A, = 0) : 11, $ 9, no. 1. P(V), A(V) (V a vector space) : 11, $ 9, no. 5. P,(K), A,(K) (K a field) : 11, $ 9, no. 5. dim, P(V), dim P(V) (V a vector K -space) : 11, $ 9, no. 5. R, 00 (K a field): 11, $ 9, no. 9. PGL(V), PGL,(K), PGL(n, K) (K a field, V a vector space) : 11,s 9, no. 10. tM (M a matrix) : 11, $ 10, no. 1. M' + M" (M', M" matrices over a commutative group): 11, $ 10, no. 2. f(M', M"), M'M " (M', M" matrices) : 11, $ 10, no. 2. E,, (matrix units): 11, $ 10, no. 3. a(M), Ma (M a matrix, a a ring homomorphism) : 11, $ 10, no. 3. M (x), x (x an element of a finitely generated free module) : 11, $ 10, no. 4. M(u) (u a homomorphism of a free module into a free module) : 11, $10, no. 4. M(x), M(u) (matrices relative to decompositions as direct sums): 11, $ 10, M (u) (u a semi-linear mapping) : 11, $ 10, no. 6. M,,(A), I,, 1, (A a ring) : 11, $ 10, no. 7. GL,(A), GL(n, A) (A a ring): 11, $ 10, no. 7. tX-l (X an invertible square matrix) : 11, $ 10, no. 7. diag(aJle,, diag(al, a,, . . ., a,) : 11, $ 10, no. 7. Xl @ X, (Xl, X, matrices over a commutative ring) : 11, $ 10, no. 10. Tr(X) (X a square matrix over a commutative ring) : 11, $ 10, no. 11. rg(X) (X a matrix over a field) : 11, $ 10, no. 12. deg(x) (x an element of a graded group) : 11, $ 11, no. 1. M(A,) (M a graded module, A, an element of the monoid of degrees): 11, HomgrA(M, N) (M, N graded modules over a graded ring A) : 11,s 11, no. 6. EngrA(M), M*gr (M a graded module) : 11, fj 11, no. 6. M: N (M, N modules) : 11, $ 1, Exercise 24. [: t] (a, 6, c, d points on a projective line) : 11, $ 9, Exercise 11. SL(E) (E a vector space) : 11, $ 10, Exercise 12. PSL(E) (E a vector space) : 11, $ 10, Exercise 14. x.y, xy (multiplication in an algebra) : 111, $ 1, no. 1. Eo (E an algebra): 111, $ 1, no. 1. HomA-als.(E, F) (E, F A-algebras) : 111, $ 1, no. 1. E/b (b a two-sided ideal of an algebra E) : 111, $ 1, no. 2. e (E an algebra): 111, fj 1, no. 2. qc, qE, q (E an algebra, c a unit) : 111, fj 1, no. 3. a family of scalars, c E I, no. 5. $ 11, no. 2. 673 INDEX OF NOTATION T(u), N(u) : 111, $ 2, no. 3. u: 111, $ 2, no. 4. H: 111, $ 2, no. 5. LibA(I), Libas,(I), Libasc,(I): 111, 3 2, no. 7. U((xJie1): 111, $2, no. 8. A[(xt)t€I], Nxilier: 111, $2, no. 9- A[(&){eI] : 111, $ 2, no. 9- AIXl, . . ., X,,]: 111, $ 2, no. 9. A[X], A[X, Y], A[X, Y, Z]: 111, $ 2, no. 9. Xv (v a multiindex) : 111, $ 2, no. 9. 2 ?&, 2 ts.s (s elements of a monoid) : 111, $ 2, no. 10. A[[X,]lt I: 111, $ 2, no. 11. C ~xV: 111, 2, no. 11. w(u), oK(u) (u a formal power series) : 111, $ 2, no. 11. 8 E,, El @A Ez 8. . . @A En, El 8 Ez 8. . . 8 En (E, A-algebras) : 111, E@, (E an algebra) : 111, $4, no. 1. @ El (I an infinite set, Et algebras) : 111, $ 4, no. 5. (8 '8 E,, '85, 'GBn (Et, G graded algebras, phisms, E a system of commutation factors) : 111, $ 4, no. 7. 8 IEI $4, no. 1. (€1 (8 xi (ut algebra homomorphisms, xi elements, I infinite) : 111, $ 4, no. 5. graded algebra homomor- ,€I ui, t€I IEI IS1 '8 E,, EK F, ~GB,, '(8 ,€I ft,fi g@fz, yan: 111, 4, no. 7. IEI @' M, Tfl(M), Tensn(M), C(M), T(M), Tens(M),T, (M) (M an A-module): T(u), Tn(u) (u a linear mapping) : 111, $5, no. 2. T:(M), C(M) (M a module) : 111, fj 5, no. 6. aL(z), 4: 111, $ 5, no. 6. a?;;!' 111, $ 5, no. 6. c:: 111, $5, no. 6. S(M), SA(M), Sym(M) : 111, $ 6, no 1. a', &, 3:: 111,s 6, no. 1. Sn(M), Sn(u), S(u) (M a module, u a linear mapping) : 111, 5 6, nos. 1 and 2. S. z: 111, $ 6, no. 3. k (a a multiindex) : 111, $ 6, no. 6. A(M), A,(M), Alt(M) : 111, 3 7, no. 1. a", 3f$, 3:: 111, $ 7, no. 1. 111, $ 5, no. 1. 674 INDEX OF NOTATION A,(M): 111, 7, no. 1. A(u>, An(u) (u a linear mapping) : 111, xE (H a subset of (1, n)) : 111, $ 7, no. 3. u (xl, . . ., i,, . . ., x,): 111, $ 7, no. 4. u.2: 111, $ 7, no. 4. Ak(M), AA(M): 111, $ 7, no. 4. U A V, X1 A X2 A * * * A X,: 111, $ 7, no. 1. 7, no. 2. eJ: 111, $ 7, no. 8. pJ, K: 111, 3 7, no. 8. det(u) (u an endomorphism) : 111, (i 8, no. 1. det(xl, x,, . . . , x,) (x, vectors in an n-dimensional free A-module) : 111, 3 8, det(X) (X a matrix) : 111, $ 8, no. 3. no. 1. det(St,)l<t<n, l<j<n, det(Etj) : 111, 3 8, no* 3. 611 Eln .... : 111, $ 8, no. 3. .... En1 . . . Em XE, K, X" (X a matrix) : 111, $ 8, nos 5 and 6. SL,(A), SL(n, A) : 111, $ 8, no. 9. p.x (p E A[X], x an element of an A-module) : 111, $ 8, no. 10. Mu, M[X] (M an A-module, u an endomorphism) : 111, $ 8, no. 10. xu(X) (u an A-module endomorphism) : 111, $ 8, no. 11. TrM/K(a), NM,K(a), Pc,/,(a; X) (A a K-algebra, M an A-module, a E A) : 111, TrAIK(a), NAiK(a), Pc,/,(u; X) (A a K-algebra, a E A) : 111, $ 9, no. 3. DA/K(xl, . . . , x,) (xi elements of a K-algebra A) : 111, (i 9, no. 5. [u, u],, [u, u] (u, u elements of a graded algebra) : 111, (i 10, no. 4. P(D), P(dl, . . ., d,) (P a polynomial, d, derivations): 111, 5 10, no. 4. ad,(a), ad(a) (a a homogeneous element of a graded algebra) : 111, tj 10, no. 6. D,(B, F) (B a K-algebra, F a (B, B)-bimodule) : 111, (i 10, no. 7. DA, p(B, F), D,(B, F): 111, $ 10, no. 7. D, (s an endomorphism) : 111, $ 10, no. 9. QK(A), dAIK(x), dx (x an element of a K-algebra A) : 111, (i 10, no. 11. D,P, BP/BX, (P a polynomial) : 111, $ 10, no. 11. Q(u), Q,(u) (u a ring homomorphism) : 111, 3 10, no. 12. Q, (u a K-algebra homomorphism) : 111, $10, no. 12. M*gr (M a graded module) : 111, 3 11. u. u, u. ,,,v (u, v symmetric multilinear mappings) : 111, 5 11, no. 2. u A v (u, v alternating mutilinear mappings) : 111, 5 1 I, no. 2. $9, no. 1. OT, €Is, O,, : 111, $ 11, no. 5. 675 INDEX OF NOTATION u L x, i (x) : 111, 3 11, no. 6. x J u, i’(x): 111, $ 11, no. 6. x u, i (u) : 111, $ 11, no. 7. u J x, i’(u) : 111, $ 11, no. 7. G,(E), G,,, ,(K) : 111, 3 11, no. 13. a(%, y, z) : 111, Appendix, no. 1. ME: 111, $ 2, Exercise 13. E * F: 111, $ 5, Exercise 6. R[a] : 1111, $ 6, Exercise 4. K[X; Q, d]: 111, 3 10, Exercise 3. 2: 111, 9 11, Exercise 9. 676 INDEX OF TERMINOLOGY Abelian group: I, $ 4, no. 2. Absolute value of a rational number: I, 3 9, no. 4. Action of one set on another: I, 3 3, no. 1. Action, canonical : I, 3 3, no. 1. Action, distributive: I, $ 3, no. 4. Action, induced: I, $ 3, no. 2. Action, quotient: I, $ 3, no. 3. Action, right, left, derived from a law of composition: I, 3 3, no. 1. Actions which commute: I, $5, no. 4. Addition: I, $ 1, no. 1. Addition of rational integers: I, $ 2, no. 5. Additively (law written): I, $ 1, no. 1. Adjoint of a matrix: 111, $ 11, Exercise 9. Adjunction of a unit element (algebra derived by) : 111, 3 1, no. 2. AfFine function: 11, $ 9, no. 4. Affine group: 11,$9, no.4. Affine line, plane, hyperplane: 11, 3 9, no. 3. AfFine mapping: 11, $ 9, no. 4. AfFine space: 11, 9, no. 1. AfFine subset, linear variety: 11, $ 9, no. 3. AfFinely free, affinely related family: XI, $ 9, no. 3. Affinely independent points: 11, $ 9, no. 3. Algebra, A-Algebra: 111, $ 1, no. 1. Algebra, alternative: 111, Appendix, no. 1. Algebra, alternating graded: 111, $ 4, no. 9. Algebra, anticommutative graded: 111, $4, no. 9. Algebra, associative: 111, 4 1, no. 1. Algebra, Cayley: 111, $ 2, no. 4. Algebra, commutative: 111, 4 1, no. 1. 677 INDEX OF TERMINOLOGY Algebra derived from an algebra by the adjunction of a unit element: 111, 4 1, Algebra, exterior, of a module: 111, 3 7, no. 1. Algebra, free: 111, 3 2, no. 7. Algebra, free associative: 111, 5 2, no. 7. Algebra, free associative and commutative: 111, Algebra, free, of a module: 111, 5 2, Exercise 13. Algebra generated by generators subject to relations: 111, 4 2, no. 8. Algebra, graded, over a graded ring: 111, 5 3, no. 1. Algebra, total, of a monoid: 111, fj 2, no. 10. Algebra obtained by extension of scalars: 111, 3 1, no. 5. Algebra obtained by restriction of scalars: 111, 5 1, no. 5. Algebra of a group, of a magma, of a monoid: 111, 3 2, no. 6. Algebra of dual numbers: 111, 5 2, no. 3. Algebra of formal power series: 111, 3 2, no. 11. Algebra of Hamiltonian quaternions: 111, 4 2, no. 5. Algebra of octonians of type (a, p, y, 6) : 111, Appendix, no. 3. Algebra of quaternions: 111, 3 2, no. 5. Algebra of quaternions of type (a, 6, y), of type (a, y) : III,§ 2, no. 5. Algebra, opposite: 111, 4 1, no. 1. Algebra, product: 111, 3 1, no. 4. Algebra, quadratic: 111, 3 2, no. 3. Algebra, quadratic, of type (a, p) : 111, 3 2, no. 3. Algebra, quotient: 111, 3 1, no. 4. Algebra, Rees: 111, fj 6, Exercise 4. Algebra, restricted, of a monoid: 111, 3 2, no. 10. Algebra, symmetric, of a module: 111, 3 6, no. 1. Algebra, tensor, of a module: 111, 3 5, no. 1. Algebra, unital: 111, 3 1, no. 1. Algebra, universal, defined by a generating system related by a family of Algebra, universal, generated by a set subjected to identities: 111, 3 2, no. 8. Algebra, universal unital associative, defined by a generating system related by Algebras, linearly disjoint: 111, 3 4, no. 4. Alternating graded algebra: 111, 3 4, no. 9. Alternating group: 111, 3 5, no. 7. Alternating multilinear mapping: 111, 3 7, no. 4. Alternative algebra: 111, Appendix, no. 1. Amalgamated sum: I, 3 7, no. 3. Annihilated by a scalar (element) : 11, 5 1, no. 12. Annihilator, left, right: I, 3 8, no. 6. Annihilator, of a subset, of an element of a module: 11, 3 1, no. 12. no. 2. 2, no. 7. relators: 111, 5 2, no. 8. a family of relators: 111, 3 2, no. 8. 678 INDEX OF TERMINOLOGY Antiautomorphism: 111, 3 1, no. 1. Anticocommutative graded cogebra: 111, 3 11, no. 3. Anticocommutative skew graded bigebra: 111, 4 11, no. 4. Anticommutative graded algebra: 111, fj 4, no. 9. Anticommutative skew graded bigebra: 111, 8 11, no. 4. Antiderivation, K-antiderivation: 111, 3 10, no. 2. Antiendomorphism of a ring: 11, 4 10, no. 6. Associated (B-module) with an A-module and a ring homomorphism B + A: Associated (faithful module) with a module: 11, 3 1, no. 12. Associated (law of action) with an action: I, 5 3, no. 1. Associated (linear mapping) with an affine linear mapping: 11, 5 9, no. 4. Associated (vectorspace) with a module over an integral domain: 11, 3 7, Associated (vector subspace) with a homogeneous element of an exterior Associated (vector subspace) with a homogeneous element of a symmetric Associated (vector subspace) with a homogeneous element of a tensor algebra : Associative algebra: 111, 3 1, no. 1. Associative algebra, free: 111, 5 2, no. 7. Associative and commutative algebra, free: 111, 3 2, no. 7. Associative law: I, 5 1, no. 3. Associativity relations in a multiplication table: 111, 3 1, no. 7. Associativity theorem: I, 3 1, no. 3. Associator: 111, Appendix, no. 1. Attached (affine space) to a vector space: 11, 3 9, no. 3. Augmentation: 111, 3 10, no. 8. Automorphism, inner of a group: I, 3 5, no. 3. Automorphism, inner, of a ring: I, 3 8, no. 4. Automorphism with no fixed point: I, 3 6, Exercise 23. Barycentre of m points, barycentre of a family of weighted points: 11, 3 9, Barycentric coordinate: 11, 3 9, no. 3. Bases dual to one another: 11, 3 2, no. 7. Basic family in a group: I, 3 7, no. 6. Basis dual of a basis of a module: 11, 3 2, no. 6. Basis, Hamel: 11, fj 7, no. 1. Basis of a module: 11, 3 1, no. 11. Basis of an algebra: 111, 3 1, no. 7. Basis of T:(M) associated with a basis of M: 111, 5 5, no. 6. 11, 3 1, no. 13. no. 10. algebra: 111, 5 7, no. 2. algebra: 111, 5 6, no. 2. 111, 3 5, no. 2. no. 2. 679 L INDEX OF TERMINOLOGY Basis of type (a, p) of a quadratic algebra: 111, $ 2, no. 3. Basis of type (a, p, y), of type (M, y), of a quaternion algebra: 111,s 2, no. 5. Basis, projective: 11, $ 9, Exercise 10. Biadditive, Z-bilinear mapping: 11, $ 3, no. 1. Bicentralizer: I, $ 1, no. 5. Bicentralizer of a subalgebra: 111, $ 1, no. 2. Bicyclic group: I, $ 6, Exercise 26. Bidual of a module: 11, 3 2, no. 7. Bigebra, anticommutative, anticocommutative, skew graded: 111, $ 11, no. 4. Bigebra, cocommutative, commutative, graded: 111, 3 11, no. 4. Bigebra, graded bigebra, skew graded bigebra, 111, $ 11, no. 4. Bigebra of a monoid: 111, $ 11, no. 4. Bigraded group, ring, module: 11, $ 11, no. 2. Bigraduation: 11, 3 11, no. 1. Bilinear mapping: 11, $ 3, no. 5. Bimodule, (A, B)-bimodule: 11, $ 1, no. 14. Bimodule over algebras: 111, $ 4, no. 3. Binomial formula: I, 3 8, no. 2. Block product of matrices: 11, $ 10, no. 5. Boolean ring: I, $ 9, Exercise 8. Bordered matrix: 11, 3 10, no. 1. Bracket, €-bracket of two derivations: 111, $ 10, no. 4. Cancellable, left, right, cancellable, element: I, 3 2, no 2. Cartan-Brauer-Hua Theorem: I, 3 9, Exercise 18. Cayley algebra: 111, $ 2, no. 4. Cayley extension of an algebra: 111, $ 2, no. 5. Cayley -Hamilton theorem: 111, 3 8, no. 11. Cayley norm, trace: 111, $ 2, no. 4. Cayley octonians: 111, Appendix, no. 3. Central element: I, 3 1, no. 5. Central extension: I, 3 6, no. 1. Central homothety: 11, 3 1, no. 2. Central ring homomorphism: 11, $5, no. 3. Central series, lower: I, $ 6, no. 3. Centralizer: I, 8 5, no. 31 Centralizer of a subalgebra of an associative algebra: 111,s 1, no. 2. Centralizer of a subset: I, $ 1, no. 5. Centralizer of a subset of a field: 11, $ 7, no. 7. Centralizer subalgebra: 111, $ 1, no. 2. Centralizing element: I, $ 5, no. 3. Centralizing subset: I, $ 5, no. 3. Centre: I, $ 1, no. 5. 680 INDEX OF TERMINOLOGY Centre of an algebra: 111, $ 1, no. 2. Centre of a projective linear mapping: 11, $ 9, no. 10. Change of coordinates, formulae of 11, $ 10, no. 8. Characteristic of a field : I, $ 9, Exercise 4. Characteristic polynomial of a matrix: 111, $ 8, no. 11. Characteristic subgroup: I, 4 5, no. 3. Class, conjugacy: I, $ 5, no. 4. Class, nilpotency, of a group: I, $ 6, no. 3. Class, solvability, of a group: I, $ 6, no. 4. Coassociative cogebra: 111, $ 11, no. 2. Cocommutative bigebra: 111, $ 11, no. 4. Cocommutative cogebra: 111, $ 11, no. 2. Codiagonal mapping: 11, $ 1, no. 6. Codimension of an affine linear variety: 11, $ 9, no. 3. Codimension of a vector subspace: 11, 4 7, no. 3. Coefficients of a formal power series: 111, $ 2, no. 11. Coefficients of a linear combination: 11, $ 1, no. 1. Coefficients of a polynomial: 111, 3 2, no. 9. Coefficients of a system of linear equations: 11, $ 2, no. 8. Cofactor of an element of a square matrix: 111, $ 8, no. 6. Cogebra, A-cogebra: 111, $ 11, no. 1. Cogebra, anticocommutative graded: 111, $ 11, no. 3. Cogebra, coassociative: 111, $ 11, no. 2. Cogebra, cocommutative: 111, 3 11, no. 2. Cogebra, counital: 111, $ 11, no. 2. Cogebra, graded: 111, $11, no. 1. Cogebra, opposite: 111, 3 11, no. 1. Coimage of a linear mapping: 11, $ 1, no. 3. Coincidence group: I, $ 4, no. 8. Cokernel of a linear mapping: 11, $ 1, no. 3. Column of a matrix: 11, $ 10, no. 1. Combination, linear: 11, tj 2, no. 5. Combinations, formal linear (module of) : 11, $ 1, no. 11. Commutation factor: 111, $ 10, no. 1. Commutative algebra: 111, $ 1, no. 1. Commutative field: I, $ 9, no. 1. Commutative graded bigebra: 111, $ 11, no. 4. Commutative group, free, over X: I, 3 7, no. 5. Commutative group with operators: I, $ 4, no. 2. Commutative law: I, $ 1, no. 5. Commutative magma: I, 9 1, no. 5. Commutative monoid, free, over X: I, 3 7, no. 7. Commutative ring: I, 9 8, no. 1. 68 1 INDEX OF TERMINOLOGY Commutativity relations in a multiplication table: 111, 5 1, no. 7. Commutativity theorem: I, 3 1, no. 5. commutator group: I, 3 6, no. 2. Commutator of two elements: I, 3 6, no. 2. Commute, actions which: I 3 5, no. 4. Commute, elements which: I, 3 1, no. 5. Compatible (equivalence relation) with a law of composition: I, 5 1, no. 6. Compatible (equivalence relation) with an action: I, 3 3, no. 3. Compatible (graduation) with a coproduct: 111, fj 11, no. 1. Compatible (graduation) with an algebra structure: 111, 3 3, no. 1. Compatible (graduation) with a ring, module, structure: 11, 3 11, no. 2. Compatible law of composition and equivalence relation: I, 3 1, no. 6. Compatible, left, right (equivalence relation), with a law of composition: I, Compatible (mapping) with an action: I, 3 3, no. 1. Compatible (mapping) with the operation of a monoid: I, 3 5, no. 1. Compatible module or multimodule structure: 11, 3 1, no. 14. Complementary minors: 111, 3 8, no. 6. Component, homogeneous, of an element in a graded group: 11, 3 11, no. 1. Component of an element in a direct sum: 11, 5 1, no. 8. Component of an element with respect to a basis: 11, $ 1, no. 11. Component, S-connected: I, fj 7, Exercise 18. Component submodule of a direct sum: 11, 3 1, no. 6. Composition in M(X) : I, 3 7, no. 1. Composition, law of: I, fj 1, no. 1. Composition of a family with finite support: I, 3 2, no. 1. Composition of an ordered sequence: I, 3 1, no. 2. Composition of the empty family: I, fj 2, no. 1. Composition of words: I, 3 7, no. 2. Composition series: I, fj 4, no. 7. Condition, maximal (resp. minimal) (set of subgroups satisfying the) : I, 4 4, Congruence modulo a rational integer: I, 3 4, no. 10. Congruent (elements) modulo an ideal: I, 3 8, no. 7. Conjugacy class in a group: I, 3 5, no. 4. Conjugate elements in a group: I, 3 5, no. 4. Conjugate elements under the operation of a group: I, 3 5, no. 4. Conjugate subsets in a group: I, 3 5, no. 4. Conjugation, conjugate in a Cayley algebra: 111, 3 2, no. 4. Conjugation, conjugate in a quadratic algebra: 111, 3 2, no. 3. Constants of structure of an algebra with respect to a basis: 111, 3 1, no. 7. Contraction of two indices in a mixed tensor: 111, 3 5, no. 6. Contragredient of an invertible square matrix: 11, $ 10, no. 7. 3 3, no. 3. Exercise 15. 682 INDEX OF TERMINOLOGY Contragradient of an isomorphism: 11, 3 2, no. 5. Contravariant tensor: 111, fj 5, no. 6. Coordinate, barycentric: 11, fj 9, no. 3. Coordinate form: 11, fj 2, no. 6. Coordinate of an element with respect to a basis: 11, 4 1, no. 11. Coordinates, homogeneous (system of), of a point in a projective space: 11, $9, no. 6. Coordinates of a tensor over M with respect to a basis of M: 111,s 5, no. 6. Coproduct: 111, 8 11. no. 1. Coset (right, lefty modulo a subgroup : I, $ 4, no. 4. Cotranspose of an endomorphism: 111, $ 8, no. 6. Counital cogebra: 111, 3 11, no. 2. Counit: 111, 3 11, no. 2. Covariant tensor: 111, 3 5, no. 6. Cramer formulae, system: 111, 3 8, no 7. Cross -ratio: 11, 3 9, Exercise 11. Crossed homomorphism: I, $ 6, Exercise 7. Crossed product: 111, 3 2, Exercise 11. Cyclic group: I, fj 4, no. 10. Cycle of a permutation: I, 3 5, no. 7. Decomposablep -vector: 111, 3 11, no. 13. Decomposition, direct, of a ring: I, 4 8, no. 11. Decomposition, reduced decomposition of an element in an amalgamated sum of monoids: I. S 7. no. 3. lo I Defined (group) by generators and relations: I, 4 7, no. 6. Defined (monoid) by generators and relations: I, 3 7, no. 2. Degree of a homogeneous element in a graded group: 11, 4 11, no. 1. Degree of a polynomial with respect to the indeterminates Xj such that j E J : Degree, total degree, of a monomial: 111, 3 2, no. 9. Degree, total degree, of an element of a free algebra, of a free associative algebra: 111, 3 2, no. 7. Degree, total degree, of a polynomial: 111, $2, no. 9. Denominator: I, 8 2, no. 4. &-derivation, inner: 111, 3 10, no. 6. Derivation, K-derivation: 111, 3 10, no. 2. (K, &)-derivation of degree 8, &-derivation of degree 6: 111, lo, 110. 2. &-derivation of a graded algebra, &-derivation of a ring: 111,s 10, no. 2. Derivation of a ring A into a ring B: 111, 5 10, no. 2. Derivative, partial: 111, 6 10, no. 11. 111, 3 2, no. 9. Derived (element) from an element of the free algebra by substituting elements for the indeterminates: 111, 5 2, no. 8. 683 INDEX OF TERMINOLOGY Derived (element) from an element of the free associative algebra by sub - Derived group of a group: I, 3 6, no. 2. Derived (left action, right action) from a law of composition: I, 3 3, no. 1. Derived series of a group: I, 4 6, no. 4. Determinant, Cauchy: 111, 4 8, Exercise 5. Determinant of a matrix: 111, 3 8, no. 3. Determinant of an endomorphism: 111, 3 8, no. 1. Determinant of a sequence of vectors: 111, fj 8, no. 1. Determinant, Vandermonde: 111, 3 8, no. 6. Diagonal elements of a square matrix: 11, 3 10, no. 7. Diagonal matrix of matrices: 11, 3 10, no. 7. Diagonal of a square matrix, diagonal matrix: 11, Diagram, exact: 11, 3 1, no. 4. Differences, group of I, 3 2, no. 4. Differences, monoid of I, K-differential: 111, 3 10, no. 11. Differential of an element: 111, 8 10, no. 11. Dihedral group: I, 3 6, Exercise 4. Dilatation: 11, 3 10, Exercise 11. Dimension of a free module: 11, Dimension of an affine linear variety: 11, Dimension of an affine space: 11, 3 9, no. 1. Dimension of a projective space: 11, 3 9, no. 5. Dimension of a vector space: 11, 7, no. 2. Dimorphism: 11, 1, no. 13. Direct decomposition of a ring: I, 5 8, no. 11. Direct factor: I, 3 4, no. 9. Direct limit: see Limit, direct. Direct product: I, 3 4, no. 9. Direct product, internal: I, 3 4, no. 9. Direct sum: I, 3 4, no. 9. Direct system: see System direct. Direction of an affine linear variety: 11, 3 9, no. 3. Direction parameters of an affine line: 11, 3 9, no. 3. Direction vector of an affine line: 11, 3 9, no. 3. Discriminant ideal of an algebra: 111, 3 9, no. 5. Discriminant of a finite sequence in an algebra: 111, 3 9, no. 5. Distributive action: I, 3 3, no. 4. Distributive, left distributive, right distributive, law: I, 3 3, no. 4. Distributive (mapping) with respect to an index: I, 3 3, no. 4. Distributivity of one law of composition with respect to another: I, 3 3, no. 5. Divisor, left, right: I, 9 8, no. 1. stituting elements for the indeterminates: 111, § 2, no. 8. 10, no. 7. 2, no. 4. 7, no. 2. 9, no. 3. 684 INDEX OF TERMINOLOGY Divisor of zero, left, right: I, 3 8, no. 1. Domain, integral, domain of integrity: I, 3 9, no. 2. Double coset with respect to two subgroups: I, Dual bases: 11, 3 2, nos. 6 and 7. Dual, graded, of a graded module: 11, 3 11, no. 6. Dual numbers, algebra of: 111, Dual of a module: 11, tj 2, no. 3. 5, no. 4. 2, no. 3. Element, central: I, Element centralizing a subset: I, 3 5, no. 3. Element derived from an element of the free algebra by substituting elements Element derived from an element of the free associative algebra by substituting Element, free, of a module: 11, 1, no. 11. Element, homogeneous (homogeneous of degree n), of a graded group: 11, Element, identity: I, 4 2, no. 1. Element invariant under an operator: I, Element, isobaric, of a graded group: 11, Element, left cancellable, right cancellable, cancellable: I, 3 2, no. 2. Element, left invertible, right invertible, invertible: I, Element, left regular, right regular, regular: I, Element normalizing a subset: i, Element, p-regular: I, 3 6, Exercise 28. Element, p-unipotent : I, Element, primitive, in a free group: I, Element, primitive, of a graded bigebra: 111, tj 11, no. 8. Element resulting from substituting elements for indeterminates in a free Element, s-neighbouring: I, 5 7, Exercise 18. Element, torsion, of a module: 11, 3 7, no. 10. Element, unit: I, 3 2, no. 1. Element, unit, of an algebra: 111, Element, zero: I, 5 2, no. 1. Elements congruent modulo an ideal: I, 8 8, no. 7. Elements, conjugate, in a group: I, 3 5, no. 4. Elements, conjugate, under the operation of a group: I, Elements, diagonal, of a square matrix: 11, fj 10, no. 7. Elements, linearly dependent (linearly independent) in a module: 11, 8 1, Elements, orthogonal: 11, 3 2, no. 4. Elements, permutable, elements which commute: I, 5 1, no. 5. 1, no. 5. for indeterminates: 111, elements for indeterminates: 111, 2, no. 8. 2, no. 8. 3 11, no. 1. 3, no. 2. 11, no. 1. 2, no. 3. 2, no. 2. 5, no. 3. 6, Exercise 28. 7, Exercise 26. group: I, 3 7, no. 5. 1, no. 1. 5, no. 4. no. 11. 685 INDEX OF TERMINOLOGY Empty matrix: 11, 5 10, no. 1. Endomorphism: I, 5 1, no. 1. Endomorphism of a module: 11, 0 1, no. 2. Endomorphism of a ring: I, 3 8, no. 4. Endomorphism, unimodular : 111, 4 8, no. 1. Endomorphisms, equivalent, similar: 111, 3 8, Exercise 26. Ends, number of ends: I, 5 7, Exercise 37. Envelope, injective, of a module: 11, 5 2, no. 1. Equation, linear, homogeneous linear equation, homogeneous linear equation Equation of a hyperplane: 11, 5 7, no. 5. Equation, scalar linear: 11, 5 2, no. 8. Equations, linear (system of) : 11, 5 2, no. 8. Equations (system of) of a vector subspace: 11, 5 7, no. 5. Equivalent composition series: I, 9 4, no. 7. Equivalent endomorphisms: 111, 3 8, Exercise 26. Equivalent matrices: 11, 5 10, no. 9. Even permutation: I, 3 5, no. 7. Exact diagram: 11, 5 1, no. 4. Exact sequence: 11, fj 1, no. 4. Expansion by a column: 111, 3 8, no. 6. Expansion by a row: 111, 5 8, no. 6. Expansion, Laplace: 111, 4 8, no. 6. Extension, Cayley, of an algebra: 111, 5 2, no. 5. Extension, central: I, $6, no. 1. Extension, essential, of a module: 11, 3 2, Exercise 15. Extension of laws of operation: I, 5 5, no. 1. Extension of one group by another: I, 5 6, no. 1. Extension of one module by another: 11, 5 1, no. 4. Extension of scalars (module obtained by) : 11, 5 5, no. 1. Extension of scalars (algebra obtained by) : 111, 5 1, no. 5. Extension, trivial: I, 5 6, no. 1. Extension, trivial, of a module: 11, 5 1, no. 9. Exterior algebra of a module: 111, 5 7, no. 1. Exterior power, p-th, of an endomorphism: 111, Exterior power, p -th, of a matrix: 111, 3 8, no. 5. Exterior power, p-th, of a module: 111, 5 7, no. 4. Exterior product of a@-vector and a q-vector: 111, 3 7, no. 1. External law of composition: I, 5 3, no. 1. associated with a linear equation: 11, 4 2, no. 8. 7, no. 4. Factor, direct, of a group: I, 5 4, no. 9. Factor, direct, of a module: 11, 5 1, no. 9. Factor of a product: I, 9 1, no. 2. 686 INDEX OF TERMINOLOGY Factors, system of: 111, 5 2, Exercise 11. Faithfully (monoid operating) : I, 3 5, no. 1. Faithful module: 111, 9 1, no. 12. Family, affinely free, affinely related, of points in an affine space: 11, 5 9, Family, basic, free, generating, in a group: I, 5 7, no. 6. Family, free, related, of elements of a module: 11, 5 1, no. 11. Family, generating, of an algebra: 111, 5 1, no. 2. Family, orthogonal, of projectors: 11, 5 1, no. 8. Family, projectively free, projectively related, of points in a projective space: 11, 5 9, no. 7. Fibre product: I, 5 4, no. 8. Field: I, 5 9, no. 1. Field, commutative, skew field: I, 5 9, no. 1. FieId of fractions of an integral domain: I, 5 9, no. 2. Field of left fractions: I, 5 9, Exercise 15. Field of rational numbers: I, 5 9, no. 4. Field, projective: 11, 5 9, no. 9. Finer composition series: I, 5 4, no. 7. Finite group: I, 5 4, no. 1. Finitely generated group: I, 5 7, Exercise 16. Finitely generated module: 11, 5 1, no. 7. Fixer of a subset of a set: I, 5 5, no. 2. Fixing a subset of a set (operator, set of operators) : I, $5, no. 2. Form, canonical bilinear: 11, 5 2, no. 3. Form, coordinate: 11, 5 2, no. 6. Form, linear: 11, 5 2, no. 3. Form, n-linear: 11, 5 3, no. 9. n-form: 111, 5 11, no. 7. Formula, binomial: I, 5 8, no. 2. Formula, Leibniz: 111, 4 10, no. 4. Formulae, Cramer’s: 111, 5 8, no. 7. Formulae of change of coordinates: 11, 5 10, no. 8. Formulae, transitivity, of norms and traces: 111, fj 9, no. 4. Fractions (field of) of an integral domain: I, 5 9, no. 2. Fractions, group of, of a monoid: I, 4 2, no. 4. Fractions, monoid of, with denominators in S: I, 4 2, no. 4. Fractions, ring of, with denominators in S: I, 3 8, no. 12. Fractions, total ring of I, 5 8, no. 12. Free algebra, associative algebra, associative and commutative algebra : 111, Free algebra of a module: 111, 3 2, Exercise 13. Free commutative group: I, 0 7, no. 5. no. 3. 5 2, no. 7. 687 a INDEX OF TERMINOLOGY Free commutative monoid: I, 3 7, no. 7. Free element, family, module, subset, system: 11,s 1, no. 11, and (by an abuse Free family in a group: I, 5 7, no. 6. Free group: I, 3 7, no. 5. Free magma: I, 3 7, no. 1. Free monoid: I, 3 7, no. 2. Free product of algebras: 111, 3 5, Exercise 6. Free product of groups: I, 5 7, no. 3. Free vector in an affine space: 11, fj 9, no. 1. Freely, group operating: I, 3 5, no. 4. Function, linearly affine, affine function: 11, 3 9, no. 4. Generated by a family of ordered pairs (equivalent relation): I, 3 1, of language) 11, 4 9, no. 7. no. 6. Generated by a subset (ideal) : I, 3 8, no. 6, and 111, 3 1, no. 2. Generated by a subset (stable subgroup) : I, 3 4, no. 3. Generated by a subset (stable subset) : I, 3 1, no. 4. Generated by a subset (subalgebra) : 111, 3 1, no. 2. Generated by a subset (subfield) : I, 3 9, no. 1. Generated by a subset (submagma) : I, 4 1, no. 4. Generated by a subset (subring) : I, 3 8, no. 5. Generated by a subset (unital submagma, submonoid) : I, 3 2, no. 1. Generating family of a group: I, CJ 7, no. 6. Generating family of an algebra: 111, 3 1, no. 2. Generating set, system, of a field: I, 3 9, no. 1. Generating set, system, of a magma: I, 3 1, no. 4. Generating set, system, of a module: 11, 4 1, no. 7. Generating set, system, of an ideal: I, fj 8, no. 6. Generating set, system, of a ring: I, Generating set, system, of a stable subgroup: I, 3 4, no. 3. Generating set, system, of a stable subset: I, 3 1, no. 4. Generating set, system, ofa unital submagma, of a submonoid: I, 3 2, no. 1 Generators of a presentation: I, Graded algebra over a graded ring: 111, 3 3, no. 1. Graded bigebra: 111, 11, no. 4. Graded bigebra, skew: 111, Graded cogebra: 111, fj 11, no. 1. Graded group, module, ring: 11, fj 11, nos. 1 and 2. Graded homomorphism: 11,s 11, no. 2. Graded subalgebra: 111, fj 3, no. 2. Graded subring, submodule, ideal: 11, 9 11, no. 3. Graded tensor product of type A,: 111, 3 4, no. 8. 8, no. 5. 7, no. 6. 11, no. 4. 688 INDEX OF TERMINOLOGY Graduation compatible with a coproduct: 111, 3 11, no. 1. Graduation compatible with an algebra structure: 111, 3 3, no. 1. Graduation induced, quotient graduation: 11, 9 11, no. 3. Graduation of type A: 11, 4 11, no. 1. Graduation, partial, total graduation: 11, 3 11, 110s. 1 and 2. Graduation, trivial: 11, 3 11, no. 1. Grassmannian: 111, 3 11, no. 13. Grassmann relations: 111, 3 11, no. 13. Greatest common divisor (g.c.d.) of two integers: I, Group: I, 3 2, no. 3. Group, additive, of a ring: I, 3 8, no. 1. Group, affine: I, 9 9, no. 4. Group, alternating: I, 3 5, no. 7. Group, bicyclic: I, 4 6, Exercise 26. Group, bigraded: 11, tj 11, no. 2. Group, concidence, of two homomorphisms: I, 3 4, no. 8. Group commutator: I, tj 6, no. 2. Group, cyclic: I, 3 4, no. 10. Group defined by generators and relations: I, 3 7, no. 6. Group, derived: I, 4 6, no. 2. Group, dihedral: I, 3 6, Exercise 4. Group, finite, infinite group: I, 4 4, no. 1. Group, finitely generated: I, 5 7, Exercise 16. Group, finitely presented: I, 7, Exercise 16. Group, free commutative, over a set: I, 3 7, no. 7 and 11, 3 1, no. 11. Group, free, over a set: I, 3 7, no. 5. Group, graded: 11, 3 11, no. 1. Group, linear: 11, 9 2, no. 6. Group, minimal simple: I, 4 6, Exercise 27. Group, monogenous: I, 3 4, no. 10. Group, multiplicative, of a ring: I, 4 8, no. 1. Group, nilpotent, nilpotent group of class n: I, 3 6, no. 3. Group of differences, group of fractions: I, 3 2, no. 4. Group of exponential type: I, 3 7, Exercise 39. Group operating faithfully: I, 9 5, no. 1. Group operating freely: I, 3 5, no. 4. Group operating simply transitively: I, 3 5, no. 6. Group operating transitively: I, 5 5, no. 5. p-group: I, 3 6, no. 5. Group, projective: 11, Group, residually finite: I, 3 5, Exercise 5. Group, solvable, solvable group of class n: I, 3 6, no. 4. Group, special linear: 111, 8, no. 11. 9, no. 10. 8, no. 9. 689 INDEX OF TERhlINOLOGY Group, supersolvable: I, 3 6, Exercise 26. Group, symmetric: I, $ 4, no. 1. Group, unimodular: 111, 5 8, no. 9. Group with operators, Abelian, commutative: I, $4, no. 2. Group with operators: I, 5 4, no. 2. Group with operators, product: I, 4 4, no. 8. Group with operators, product (or internal product), of a family of quotient Group with operators, quotient: I, 5 4, no. 4. Group with operators, simple: I, $ 4, no. 4. Groupoid: I, 5 4, Exercise 23. Hall's Theorem: I, $ 6, Exercise 10. Hamel basis: 11, $ 7, no. 1. Hamiltonian quaternions: 111, 5 2, no. 5. Homogeneous element in a graded group: 11, 3 11, no. 1. Homogeneous G-set: I, 5 5, no. 5. Homogeneous linear equation, linear systcm: 11, 9 2, no. 8. Homogeneous subset of degree p in a formal power series: 111, 5 2, no. 11. Homogeneous subset of degree p with respect to certain indeterminates in a formal power series: 111, 4 2, no. 11. Homomorphism, algebra: 111, 5 1, no. 1. Homomorphism, A-moduIe, A-homomorphism: 11, 5 1, no. 2. Homomorphism, central ring: 11, 4 5, no. 3. Homomorphism, crossed: I, 5 6, Exercise 7. Homomorphism, essential: 11, 3 2, Exercise 15. Homomorphism for two laws of composition: I, 0 1, no. 1. Homomorphism, graded: 11, 3 11, no. 2. Homomorphism, graded algebra: 111, 5 3, no. 1. Homomorphism, group: I, 5 4, no. 1. Homomorphism, monoid: I, 3 2, no. 1. Homomorphism, M -set: I, 5 5, no. 1. Homomorphism, multimodule: 11, 5 1, no. 14. Homomorphism of groups with operators : I, $ 4, no. 2. +-homomorphism: I, 5 3, no. 1. Homomorphism, projection: I, 3 4, no. 8. Homomorphism, ring: I, 5 8, no. 4. Homomorphism, trivial: I, 5 2, no. 1. Homomorphism, unital: I, 5 2, no. 1. Homomorphism, unital algebra: 111, 5 1, no. 1. Homothety: I, $ 4, no. 2 and 11, 3 1, no. 1. Homothety, central: 11, 4 1, no. 2 and 111, 4 9, Exercise 6. Hyperplane, affine: 11, $ 9, no. 3. groups: I, 5 4, no. 8. 690 INDEX OF TERbIIKOLOGY Hyperplane at infinity: 11, $ 9, no. 8. Hyperplane passing through 0 in a vector space: 11, 5 7, no. 3. Hyperplane, projective: 11, $ 9, no. 7. Hyperplane, projective, taken as hyperplane at infinity: 11,s 9, no. 10. Ideal, discriminant: 111, 5 9, no. 5. Ideal, graded: 11, 5 11, no. 3 and 111, 3 3, no. 2. Ideal, irreducible two-sided : I, 4 8, Exercise 1 1. Ideal, left, right ideal, two-sided ideal in an algebra: 111, 5 1, no. 2. Ideal, left, right ideal, two-sided ideal in a ring: I, 5 8, no. 6. Ideal, maximal: I, fj 8, no. 6. Ideal, prime: I, $ 9, no. 3. Ideal, principal: I, $ 8, no. 6. Ideal of relators: 111, $ 2, no. 8. Ideal, zero: I, 3 8, no. 6. Idempotent: I, 5 1, no. 4. Identities, Jacobi, between minors of a determinant: 111, $ 11, Exercise 9. Identities, polynomials: 111, $ 2, no. 9. Identity element: I, $ 2, no. 1. Identity, Jacobi: 111, $ 10, no. 6. Identity, Redei: 111, $ 8, Exercise 25. Image (inverse image) of a projective linear variety under a projective map- Image of a homomorphism: 11, $ 1, no. 3. Indeterminate, indeterminate of index i: I, Index of a subgroup: I, $ 4, no. 4. Induced action: I, 3 3, no. 2. Induced graduation: 11, 3 11, no. 3. Induced K-structure: 11, 5 8, no. 1. Induced law: I, 5 1, no. 4. Infinity, hyperplane at: 11, 5 9, no. 8. Infinity, points at: 11, 5 9, no. 8. Inner automorphism of a group: I, $ 5, no. 3. Inner automorphism of a ring: I, 5 8, no. 4. Inner product, left, right: 111, 5 11, nos. 6 and 7. Integer, negative, positive, strictly negative, strictly positive: I, 4 2, no. 5. Integer, rational: I, 3 2, no. 5. Integers, rational, modulo a (ring of) : I, 5 8, no. 11. Integers, relatively prime: I, $ 8, no. 11. Integral domain: I, $ 9, no. 2. Integrity, domain of: I, 3 9, no. 2. Internal direct product, restricted sum: I, 3 4, no. 9. Invariant element: I, $ 5, no. 2. ping: 11, § 9, no. 10. 7, no. 5 and 111, 3 2, no. 7. 69 1 INDEX OF TERMINOLOGY Invariant subgroup, stable subgroup: I, $ 4, no. 4. Inverse, left inverse, right inverse, element: I, $ 2, no. 3. Inverse, left, right inverse, of a linear mapping: 11, $ 1, no. 9. Inverse limit: see Limit, inverse. Inverse system: see System, inverse. Inversion in a bigebra: 111, $ 11, Exercise 4. Inversion of a permutation: I, $ 5, no. 7. Invertible, left invertible, right invertible, element: I, $ 2, no. 3. Invertible, left, right invertible, linear mapping: 11, $ 1, no. 9. Invertible square matrix: 11, $ 10, no. 7. Irreducible ideal : I, 3 8, Exercise 1 1. Isobaric element: 11, $ 1 1, no. 1. Isomorphic magmas: I, 3 1, no. 1. Isomorphism, magma: I, 3 1, no. 1. Jacobi identity: 111, $ 10, no. 6. Jordan -Holder series: I, $ 4, no. 7. Jordan-Holder Theorem: I, $ 4, no. 7. Juxtaposition: I, $ 7, no. 2. Kernel of a group homomorphism: I, $ 4, no. 5. Kernel of a linear mapping: 11, 4 1, no. 3. Kronecker symbol: 11, 3 1, no. 11. Krull’s Theorem: I, $ 8, no. 6. Laplace expansion: 111, $ 8, no. 6. Law and equivalence relation, compatible: I, $ 1, no. 6. Law, associative: I, $ I, no. 3. Law, commutative: I, $ 1, no. 5. Law, group: I, $ 4, no. 1. Law, induced: I, $ 1, no. 4. Law, left distributive, right distributive, distributive with respect to another: Law not everywhere defined : I, $ 1, no. 1. Law of composition: I, $ 1, no. 1. Law of left, right, operation of a monoid on a set: I, 3 5, no. 1. Law of right, left action associated with an action: I, $ 3, no. 1. Law, opposite: I, $ 1, no. 1. Law, quotient: I, $ 1, no. 6. Law written additively, multiplicatively: I, 3 1, no. 1. Least common multiple (1.c.m.) of two integers: I, $ 8, no. 11. Leibniz formula: III, 9 10, no. 4. Lemma, Zassenhaus’s: I, $4, no. 7. I, 3 3, nos. 4 and 5. 692 INDEX OF TERMINOLOGY Length of a decomposition in an amalgamated sum: I, 3 7, no. 3. Length of a group: I, $ 4, no. 7. Length ofa module: 11, $ 1, no. 10. Length of an element in a free group: I, $ 7, Exercise 19. Length of an element in a. free magma : I, $ 7, no. 1. Length of a word: I, 3 7, no. 2. Limit, direct, of A,-algebras: 111, $ 1, no. 6. Limit, direct, of A,-modules: 11, $ 6, no. 2. Limit, direct, of actions: I, $ 10, no. 4. Limit, direct, of graded algebras: 111, $ 3, no. 3. Limit, direct, of groups, of groups with operators, of monoids: I, 3 10, no. 3. Limit, direct, of magmas: I, $ 10, no. 3. Limit, direct, of rings: I, $ 10, no. 3. Limit, inverse, of A,-algebras: 111, fj 1, no. 6. Limit, inverse, of A,-modules: 11, $ 6, no. 1. Limit, inverse, of groups, of groups with operators, of monoids: I, $ 10, no. 1. Limit, inverse, of magmas: I, $ 10, no. 1. Limit, inverse, of rings: I, 3 10, no. 1. Line, affine: 11, $ 9, nos. 1 and 3. Line (passing through 0) in a vector space: 11, $ 7, no. 3. Line, projective: 11, $ 9, no. 5. Linear equation, system: 11, $ 2, no. 8. Linear equation, system, homogeneous: 11, $2, no. 8. Linear form: 11, $2, no. 3. Linear group: 11, $ 2, no. 6. Linear group, special: 111, $ 8, no. 9. Linear mapping: 11, 3 1, no. 2. Linear mapping, function, afine: 11, $ 9, no. 4. Linear mapping, projective: 11, 3 9, no. 10. Linear problem: 11, $ 2, no. 8. Linear variety: 11, 3 9, nos. 3, 7 and 11. Linear variety, affine: 11, $9, no. 3. Linear variety, projective: 11, $ 9, nos. 7 and 11. Linearly dependent, linear independent, elements : 11, 3 1, no. 1 1. Linearly disjoint subalgebras: 111, 3 4, no. 4. Linearity, principle of extension by: 11, 3 1, no. 7. Magma: I, 3 1, no. 1. Magma, associative: I, $ 1, no. 3. Magma, commutative: I, 5 1, no. 5. Magma, defined by generators and relations: I, 3 7, no. 1. Magma, free, over a set: I, 5 7, no. 1. Magma of mappings into a magma : I, 3 1, no. 1. 693 INDEX OF TERMINOLOGY Magma, opposite: I, 3 1, no. 1. Magma, product: I, 1, no. 1. Magma, quotient: I, 3 1, no. 6. Magma, unital: I, 3 2, no. 1. Magmas, isomorphic: I, 3 1, no. 1. Mapping, a%e linear, affine: 11, 3 9, no. 4. Mapping, alternating multilinear: 111, fj 7, no. 4. Mapping, biadditive, Z-bilinear: IT, 3 3, no. 1. Mapping, C -bilinear: 11, 3 3, no. 5. Mapping compatible with an action: I, 3 3, no. 1. Mapping compatible with the operation of a monoid: I, Q 5, no. 1. Mapping distributive with respect to a variable: I, 3 3, no. 4. Mapping, linear, A -linear: 11, 3 1, no. 2. Mapping, linear, A,(T) --f E determined by a mappingT -+ E: 11, 4 I Mapping, linear, associated with an afine mapping: 11, 8 9, no. 4. Mapping, multiadditive, Z-multilinear: 11, 3 3, no. 9. Mapping, C -multilinear: 11, 3 3, no. 9. Mapping, orbital: I, 0 5, no. 4. Mapping, projective linear, projective: 11, 4 9, no. 10. Mapping, right invertible linear, left invertible linear: 11, 3 1, no. 9. Mapping, semi-linear: 11, 3 1, no. 13. Mapping, symmetric multilinear: 111, 3 6, no. 3. Matrices differing only by the order of the columns, rows: 11, 3 10, no. 9. Matrices, equivalent: 11, 3 10, no. 9. Matrices, similar square: 11, 8 10, no. 9. Matrix, contragredient, of an invertible matrix: 11, 3 10, no. 7. Matrix, diagonal: 11, 3 10, no. 7. Matrix, empty: 11, $10, no. 1. Matrix, invertible: 11, 3 10, no. 7. Matrix, lower triangular, upper triangular: 11, 3 10, no. 7. Matrix, matrix of type (p, q) : 11, 3 10, no. 1. Matrix, monomial: 11, 3 10, no. 7. Matrix obtained by bordering a matrix: 11, 3 10, no. 1. Matrix obtained by suppressing columns, rows: 11, 3 10, no. 1. Matrix of a linear mapping with respect to two bases: 11, 3 10, no. 4. Matrix of a linear system: 11, 6 10, no. 4. Matrix of an element with respect to a basis: IT, 3 10, no. 4. Matrix of an endomorphism with respect to a basis: 11, 3 10, no. 7. Matrix of a permutation: 11, fj 10, no. 7. Matrix of a semi-linear mapping with respect to two bases: 11, 3 10, no. 6, Matrix of passing from one basis to another: 11, 3 10, no. 8. Matrix, scalar: 11, 3 10, no. 7. no. 11. 694 INDEX OF TERMINOLOGY Matrix, square, square matrix of order n: 11, 3 10, no. 7. Matrix, unimodular: 111, 3 8, no. 4. Matrix units: 11, 3 10, no. 3. Matrix with only zeros below (above) the diagonal: 11, 3 10, no. 7. Matrix, zero: 11, 3 10, no. 2. Maximal ideal: I, 3 8, no. 6. G-mean: I, $6, Exercise 8. Minimal normal stable subgroups: I, 3 4, Exercise 15. Minimal simple group: I, 3 6, Exercise 27. Minor, minor of orderp of a matrix: 111, fj 8, no. 5. Minors, complementary: 111, 3 8, no. 6. Mixed tensor: 111, 3 5, no. 6. Module, bigraded: 11, 3 11, no. 2. Module, divisible: 11, 3 7, Exercise 33. Module, dual: 11, 9 2, no. 3. Module, faithful: 11, 3 1, no. 12. Module, faithful, associated with a module: 11, fj 1, no. 12. Module, free: 11, 3 1, no. 11. Module, graded free: 11, 3 11, no. 2. Module, graded, graded module with positive degrees: 11, 9 11, no. 2. Module, graded quotient: 11, 3 11, no. 3. Module, indecomposable: 11, 3 2, Exercise 21. Module, injective: 11, 3 2, Exercise 11. Module, left, right module, A -module: 11, 9 1, no. 1. Module, monogeneous: 11, 3 1, no. 12. Module of finite length: 11, 3 1, no. 10. Module of formal linear combinations: 11, 4 1, no. 11. Module of linear relations: 11, Fj 1, no. 11. Module over an algebra: 111, 3 4, no. 3. Module, product: 11, 3 1, no. 5. Module, projective: 11, 3 2, no. 2. Module, quotient: 11, 3 1, no. 3. Module, reflexive: 11, 3 2, no. 7. Module, torsion -free, over an integral domain: 11, 3 7, no. 10. Module, torsion, over an integral domain: 11, 3 7, no. 10. Monogeneous group: I, 3 4, no. 10. Monogenous module: 11, 3 1, no. 2. Monoid : I, Q 2, no. 1. Monoid defined by generators and relations: I, 3 7, no. 2. Monoid, free, over a set: I, tj 7, no. 2. Monoid of differences: I, 8 2, no. 4. Monoid of fractions with denominators in S: I, 3 2, no. 4. Monoid operating faithfully on a set: I, 3 5, no. 1. 695 INDEX OF TERMINOLOGY Monoidal sum: I, 5 7, no. 3. Monomial: 111, 3 2, no. 9. Morphism, algebra: 111, 3 1, no. 1. Morphism, A-module: 11, 3 1, no. 2. Morphism, cogebra: 111, 4 11, no. 1. Morphism, graded bigebra, left graded bigebra morphism: 111, 3 11, no. 4. Morphism, graded cogebra : 111, 3 11, no. 1. Morphism, magma: I, 4 1, no. 1. Morphism, monoid : I, 3 2, no. 1. Morphism of extensions : I, fj 6, no. 1. Morphism, ring: I, 3 8, no. 4. Morphism, unital algebra: 111, 3 4, no. 5. Morphism, unital magma: I, 3 2, no. 1. +morphism (4 a homomorphism of one monoid of operators into another) : +morphism (4 a mapping of one set of operators into another): I, 3 3, M-morphism (M a monoid of operators) : I, 3 5, no. 1. a-morphism (R a set of operators) : I, 4 3, no. 1. Multiadditive, Z-multilinear, mapping: 11, 3 3, no. 9. Multidegree in the free algebra of a monoid: 111, 3 2, Exercise 13. Multiindex: I, $ 7, no. 7. Multilinear form: 11, $ 3, no. 9. C-multilinear mapping: 11, 3 3, no. 9. Multimodule: 11, $ 1, no. 14. Multimodule, quotient: 11, 3 1, no. 14. Multiple, left, right: I, 3 8, no. 1. Multiplication: I, fj 1, no. 1. Multiplication in an algebra. 111, 3 1, no. 1. Multiplication table: 111, 3 1, no. 7. Multiplicative group of a ring: I, 3 8, no. 1. I, $5, no. 1. no. 1. Negative of an element: I, 3 2, no. 3. Negative rational integer: I, 3 2, no. 5. Negative rational number: I, 3 9, no. 4. S-neighbouring element: I, 3 7, Exercise 18. Nielson -Schreier Theorem: I, 3 7, Exercise 20. Nilpotent group, nilpotent group of class n: I, 3 6, no. 3. Norm, Cayley: 111, 3 2, no. 4. Norm in a quadratic algebra: 111, 3 2, no. 3. Norm of an element in a K-algebra relative to K: 111, 3 9, no. 3. Norm of a scalar relative to a module: 111, 3 9, no. 1. Normal stable subgroup: I, 3 4, no. 4. 696 INDEX OF TERMINOLOGY Normal subgroup: I, 3 4, no. 4. Normalizer: I, 3 5, no. 3. Normalizing (element, subset) a subset: I, 3 5, no. 3. Null element: I, 3 2, no. 1. Number, prime: I, 4 4, no. 10. Number, rational: I, 0 9, no. 4. Number (rational), negative, positive, strictly negative, strictly positive : I, Numerator: I, 3 2, no. 4. Octonions, Cayley: 111, Appendix, no. 3. Octonions of type (cr, p, y, 6) (algebra of) : 111, Appendix, no. 3. Odd permutation: I, 3 5, no. 7. Operation by left, right, translation: I, 3 5, no. 1. Operation, left, right (laws of) : I, 3 5, no. 1. Operation, left, right, of a monoid: I, 3 5, no. 1. Operation, simply transitive: I, 3 5, no. 6. Operation, transitive: I, 3 5, no. 5. Operation, trivial: I, 3 5, no. 2. Operator: I, $ 3, no. 1. Opposite algebra: 111, 3 1, no. 1. Opposite cogebra: 111, 3 11, no. 1. Opposite law: I, $ 1, no. 1. Opposite magma: I, 3 1, no. 1. Opposite to an M-set, Mo-set: I, 3 5, no. 1. Opposite ring: I, $ 8, no. 3. Orbit: I, $ 5, no. 4. Orbital mapping: I, 3 5, no. 4. Order, element of infinite: I, 3 4, no. 10. Order of a cycle: I, 3 5, no. 7. Order of a formal power series with respect to certain indeterminates: 111, Order of a group : I, 3 4, no. 1. Order of an element in a group: I, 3 4, no. 10. Order of a square matrix: 11, 3 10, no. 7. Order, total order, ofa formal power series: 111, $ 2, no. 11. Ordered sequence: I, 3 1, no. 2. Ordered sequences, similar: I, 3 1, no. 2. Origin: I, 3 2, no. 1. Origin, choice of, in an affine space: 11, 3 9, no. 1. Orthogonal elements, sets: 11, 3 2, no. 3. Orthogonal family of projectors: 11, 9 1, no. 8. Orthogonal, submodule, to a subset of E (resp. E*) : 11, 5 2, no. 4. 3 9, no. 4. 32, no. 11. 697 INDEX OF TERMINOLOGY Parallel linear varieties: 11, 3 9, no. 3. Parallelogram: 11, 3 9, Exercise 1. Parameters, direction, of an affine line: 11, 5 9, no. 3. Partial derivative: 111, 3 10, no. 11. Partial graduation: 11, 5 11, no. 1. Passage, matrix of: 11, 3 10, no. 8. Permutable elements: I, 3 1, no. 5. Permutation, even, odd: I, 3 5, no. 7. Plane, affine: 11, 3 9, nos. 1 and 3. Plane passing through 0 in a vector space: 11, 3 7, no. 3. Plane, projective: 11, 3 9, no. 5. Point of an afhe space: 11, 3 9, no. 1. Point of a projective space: 11, 5 9, no. 5. Points, afhely independent: 11, 4 9, no. 3. Points at infinity: 11, 3 9, no. 8. Polynomial, characteristic, of an element in a K-algebra: 111, 4 9, no. 3. Polynomial, characteristic, of an endomorphism: 111, 9 8, no. 11. Polynomial, characteristic, of a scalar with respect to a module: 111, 4 9, Polynomial containing no term in X": 111, 3 2, no. 9. Polynomial identities: 111, 3 2, no. 9. Polynomial of degree n: 111, 5 2, no. 9. Polynomial relators: 111, 3 2, no. 9. Polynomial with no constant term: 111, 3 2, no. 9. Polynomial with respect to a family of indeterminates, with coefficients in a Positive rational integer: I, 3 2, no. 5. Positive rational number: I, fj 9, no. 4. Power, exterior, of a linear mapping: 111, 3 7, no. 4. Power, exterior, of a matrix: 111, 3 8, no. 5. Power, exterior, of a module: 111, 3 7, no. 4. Power, n-th, under an associative law : I, 3 1, no. 3. Power, symmetric, of a linear mapping, of a module: 111, 3 6, no. 3. Power, tensorial, of a linear mapping: 111, 3 5, no. 2. Power, tensorial, of a module: 111, 3 5, no. 1. Presentation of a group: I, 3 7, no. 6. Presentation of an algebra: 111, 3 2, no. 8. Presented, finitely (group) : I, 3 7, Exercise 16. Preservation when passing to the quotient: I, 3 1, no. 6. Prime ideal: I, 3 9, no. 3. Prime number: I, 3 4, no. 10. Prime, relatively (integers) : I, 3 8, no. 10. Primitive element in a free group: I, 3 7, Exercise 26. 698 no. 1. ring: 111, 3 2, no. 9. INDEX OF TERMINOLOGY Primitive element in a free magma generated by a single element: I, 8 7, Primitive element of a graded bigebra: 111, fj 11, no. 8. Principal G-set, homogeneous: I, 3 5, no. 6. Principal ideal: I, 3 8, no. 6. Principal series : I, 3 4, Exercise 17. Principal set under G, homogeneous: I, 3 5, no. 6. Principle of extension by linearity: 11, 5 1, no. 7. Problem, linear: 11, 8 2, no. 8. Product algebra: 111, 3 1, no. 4. Product, block, of matrices: 11, 3 10, no. 5. Product, crossed: 111, 3 2, Exercise 11. Product, exterior, of ap-vector and a q-vector: 111, 3 7, no. 1. Product, external semi-direct, of G by F relative to 7: I, 3 6, no. 1. Product, fibre, of groups with operators: I, 3 4, no. 8. Product, fibre, of modules: 11, 1, Exercise 4. Product, free, of algebras: 111, 3 5, Exercise 6. Product, free, of groups: I, 4 7, no. 5. Product, graded tensor, of two graded modules: 11, 3 11, no. 5. Product, graded tensor, of graded algebras: 111, 3 4, nos. 7 and 9. Product group, internal product group, of a family of quotient groups: I, Product group with operators: I, 4 4, no. 8. Product, internal direct, direct product, product, of subgroups: I, 3 4, no. 9. Product magma: I, 3 1, no. 1. Product of K'-structures: 11, 3 8, no. 3. Product of laws of composition: I, 3 1, no. 1. Product of matrices calculated according to a mapping: 11, 3 10, no. 2. Product of matrices calculated in a ring: 11, 3 10, no. 3. Product of modules: 11, 3 1, no. 5. Product of multimodules: 11, 3 1, no. 14. Product of an operator and an element: I, 3 3, no. 1. Product of an ordered sequence: I, 4 1, no. 2. Product of two elements: I, 3 1, no. 1. Product of two-sided ideals: I, 9 8, no. 9. Product, (right, left) inner: 111, 3 11, nos. 6 and 7. Product ring: I, 3 8, no. 10. Product, symmetric, of multilinear mappings: 111, 3 11, no. 2. Product, tensor, of a family of Z-modules relative to a triple (c, p, q) : 11, 3 3, Product, tensor, of algebras: 111, 3 4, no. 1. Product, tensor, of an infinite family of algebras: 111, 3 4, no. 5. Product, tensor, of bases of algebras: 111, 3 4, no. 5. Exercise 7. 3 4, no. 8. no. 9. 699 INDEX OF TERMINOLOGY Product, tensor, of cogebras: 111, 3 11, no. 1. Product, tensor, of two bases: 11, 3 3, no. 7. Product, tensor, of two elements: 11, fj 3, no. 1. Product, tensor, of two linear mappings: 11, 3 3, no. 2. Product, tensor, of two matrices over a commutative ring: 11, 3 10, no. 10. Product, tensor, of two modules: 11, 3 3, no. 1. Product, tensor, of two multimodules: 11, 3 3, no. 4. Product, tensor, of two semi-linear mappings: 11, 3 3, no. 3. Projection homomorphism: I, 4 4, no. 8. Projective field: 11, 3 9, no. 9. Projective group: 11, 3 9, no. 10. Projective hyperplane, plane: 11, 3 9, no. 7. Projective line: 11, 3 9, no. 5. Projective mapping: 11, 3 9, no. 10. Projective module: 11, 3 2, no. 2. Projective space: 11, 3 9, nos. 5 and 11. Projectively free, projectively related, family: 11, 3 9, no. 7. Projector: 11, 3 1, no. 8. Pseudomodule, left, right: 11, Appendix, no. 2. Pseudo -ring: I, 3 8, no. 1. Pseudo-ring with zero square: I, 3 8, no. 3. Purep -vector: 111, 3 11, no. 13. Pure quaternion: 111, 3 2, Exercise 3. Quadratic algebra: 111, 8 2, no. 3. Quasi -group: I, 3 3, Exercise 6. Quaternion, pure: 111, Fj 2, Exercise 3. Quaternion algebra: 111, 3 2, no. 5. Quaternionic group: I, 3 6, Exercise 4. Quotient algebra: 111, 3 1, no. 2. Quotient graduation: 11, 3 11, no. 3. Quotient group with operators: I, 3 4, no. 4. Quotient law: I, 3 1, no. 6. Quotient magma: I, 3 1, no. 11. Quotient module, vector space: 11, 3 1, no. 3. Quotient multimodule: 11, 3 1, no. 14. Quotient of an action: I, 3 3, no. 3. Quotient ring: I, 5 8, no. 7. Quotients of a composition series of a group with operators: I, 3 4, no. 7. Rank of a free group: I, 3 7, Exercise 14. Rank of a linear mapping of vector spaces: 11, 3 7, no. 4. Rank of a linear system over a field: 11, 3 7, no. 6. 700 INDEX OF TERMINOLOGY Rank of a matrix over a field: 11, 3 10, no. 12. Rank of an affine linear mapping: 11, 3 9, no. 4. Rank of an element in a tensor product of vector spaces: 11, 3 7, no. 8. Rank of a semi-linear mapping of vector spaces: 11, 3 7, no. 4. Rank of a subset of a module over an integral domain: 11, 3 7, no. 10. Rank of a subset of a vector space: 11, 3 7, no. 2. Ratio of a homothety : 11, 3 1, no. 1. Rational integer: I, fj 2, no. 5. Rational number: I, 3 9, no. 4. Rational numbers, field of : I, 3 9, no. 4. Rational over a subfield (linear form) : 11, 3 8, no. 4. Rational over a subfield (linear mapping) : 11, 9 8, no. 3. Rational over a subfield (subspace, vector) : 11, 3 8, no. 2. Reduced decomposition of an element of an amalgamated sum: I, 3 7, Reflexive module: 11, 3 2, no. 7. Regular, leit regular, right regular, element: I, 3 2, n3. 2. p-regular element: I, 3 6, Exercise 28. Related system, subset: 11, 3 1, no. 11, and 8 9, no. 7. Relation, equivalence, compatible with a law of composition: I, 3 1, no. 6. Relation, equivalence, compatible with an action: I, 4 3, no. 3. Relation, equivalence, generated by a family of ordered pairs: I, 3 1, no. 6. Relation, equivalence, left, right, compatible with a law of composition: I, Relations, commutativity, in a multiplication table: 111, 3 1, no. 7. Relations, Grassmann: 111, 3 11, no. 13. Relations, module of linear: 11, 3 1, no. 11. Relatively prime integers: I, 3 8, no. 11. Relator: I, 9 7, no. 6 and 111, 3 2, no. 8. Relator, universal: 111, 3 2, no. 8. Relators, ideal of: 111, 3 3, no. 8. Relators of a presentation: I, 3 7, no. 6 and 111, 3 2, no. 8. Relators, polynomial: 111, 3 2, no. 9. Residue of a semi-group: I, 3 2, Exercise 11. Residually finite group: I, 3 5, Exdrcise 5. Restricted algebra of a monoid: 111, 0 2, no. 10. Restricted sum, internal: I, 3 4, no. 9. Restricted sum of groups: I, 5 4, no. 9. Restricted sum of groups with respect to subgroups: I, 3 4, no. 9. Restriction of scalars (algebra obtained by) : 111, 3 1, no. 5. Resulting (element) from substituting elements for indeterminates : I, 9 7, Retraction of an extension : I, 3 6, no. 1. no. 3. 9 3, no. 3. no. 5. 701 INDEX OF TERMINOLOGY Right hand side of a linear equation, right hand sides of a linear system: 11, Ring: I, $ 8, no. 1. Ring, Boolean: I, $ 9, Exercise 8. Ring, commutative: I, $ 8, no. 1. Ring (graded, graded with positive degrees, bigraded) : 11, $ 11, no. 2. Ring, graded quotient: 11, $ 11, no. 3. Ring obtained by the adjunction of a unit element: 11, Appendix, no. 1. Ring of endomorphisms of a commutative group: I, $ 8, no. 4. Ring of fractions with denominators in S: I, $ 8, no. 12. Ring of integers modulo n : I, $ 8, no. 1 1. Ring of polynomials which are not commutative relative to an endomorphism and a derivation: 111, $ 10, Exercise 3. Ring, opposite: I, $ 8, no. 3. Ring, product: I, $ 8, no. 10. Ring, quotient: I, $ 8, no. 7. Ring, total, of fractions: I, $ 8, no. 2. Ring, zero: I, 3 8, no. 3. Row of a matrix: 11, $ 10, no. 1. Rule, sign: I, $ 8, no. 1. $2, no. 8. Scalar: 11, 3 1, no. 1. Scalar linear equation: 11, $ 2, no. 8. Scalar matrix: 11, $ 10, no. 7. Schreier's Theorem on composition series: I, $ 4, no. 7. Section of an extension: I, $ 6, no. 2. Semi-direct product of groups: I, 3 6, no. 1. Semi-group, left, right: I, $ 2, Exercise 11. Semi -homomorphism, algebra, algebra p-homomorphism: 111, $ 1, no. 5. Semi-linear mapping: 11, $ 1, no. 13. Sequence, exact: 11, $ 1, no. 4. Sequence, ordered: I, $ 1, no. 2. Sequence, split exact: 11, $ 1, no. 9. Sequences, similar ordered: I, $ 1, no. 2. Series, algebra of formal power: 111, Ej 2, no. 11. Series, composition: I, $ 4, no. 6. Series, derived: I, $ 6, no. 4. Series, equivalent decomposition: I, $ 4, no. 7. Series, finer composition: I, $ 4, no. 7. Series, formal power: 111, $ 2, no. 11. Series, Jordan -Holder: I, $ 4, no. 7. Series, lower central: I, $ 6, no. 3. Series, normal: I, 3 4, Exercise 17. 702 INDEX OF TERMINOLOGY Series, principal : I, $ 4, Exercise 17. G-set, homogeneous (G a group of operators) : I, $ 5, no. 5. G-set, homogeneous principal, homogeneous principal set under G: I, 5 5, Set, generating, of a group: I, $ 4, no. 3. Set, generating, of a magma: I, $ 1, no. 4. M-set (M a monoid of operators) : I, 4 5, no. 1. Mo-set opposite to an M-set: I, $ 5, no. 1. Set of degrees of a graded group: 11, $ 11, no. 1. Set of subgroups satisfying the maximal condition, the minimal condition : I, Sets, orthogonal: 11, $2, no. 4. G-sets, weakly equivalent: I, 3 5, Exercise 26. Shift, shifting: 11, $ 11, no. 2. Signature of a permutation: I, $ 5, no. 7. Sign of a rational number: I, $ 9, no. 4. Similar endomorphisms: 111, 4 8, Exercise 26. Similar matrices: 11, 9 10, no. 9. Similar ordered sequences: I, $ 1, no. 2. Simple group: I, $ 4, no. 4. Simply transitive operation: I, $ 5, no. 6. Skew field: I, $ 9, no. 1. Skew graded bigebra: 111, $ 11, no. 4. Skew-symmetric tensor: 111, $ 7, no. 4. Skew-symmetrized tensor: 111, $ 7, no. 4. Skew tensor product of graded algebras: 111, $ 4, nos. 7 and 8. Solution of a linear equation, of a linear system: 11, 4 2, no. 8. Solution, trivial, zero solution of a homogeneous linear equation: 11,s 2, no. 8. Solvable group, solvable group of class n: I, $ 6, no. 4. Space, affine, attached to a vector space: 11, 3 9, no. 1. Space, canonical projective, associated with a vector space: 11, $ 9, no. 8. Space, projective: 11, $ 9, nos. 5 and 11. Space, projective, derived from a vector space: 11, $ 9, no. 5. Space, quotient vector, quotient space: 11, 3 1, no. 3. Space, right, left, vector, over a field: 11, 3 1, no. 1. Space, vector, associated with a module over an integral domain: 11, $ 7, Space, vector, obtained by taking an origin in an affine space: 11, 3 9, no. 1. Space, vector, of translations of an affine space: 11, 3 9, no. 1. Split exact sequence: 11, $ 1, no. 10. Stabilizing (operator, set of operators) a subset: I, 3 5, no. 2. Stabilizing, strictly (operator, set of operators), a subset: I, $ 5, no. 2. Stable subgroup of a group with operators: I, $ 4, no. 3. no. 6. $ 4, Exercise 15. no. 10. 703 INDEX OF TERMINOLOGY Stable subset: I, 3 1, no. 4. Strictly negative, strictly positive, rational integer: I, Fj 2, no. 5. Strictly negative, strictly positive, rational number: I, 3 9, no. 4. Strict stabilizer: I, fj 5, no. 2. Strictly stabilizing a subset: I, 3 5, no. 2. Strict transporter: I, 3 5, no. 2. Strongly associative subset: 111, Appendix, no. 1. Structures, module (multimodule), compatible: 11, 3 1, no. 14. Structure, projective space: 11, fj 9, no. 11. K‘-structure, induced: 11, 3 8, no. 2. K’-structure on a vector K -space: 11, 3 8, no. 1. K’-structure, product: 11, 3 8, no. 3. Subalgebra: 111, 5 1, no. 2. Subalgebra generated by a subset: 111, 5 1, no. 2. Subalgebra, graded: 111, 3 3, no. 2. Subfield: I, 3 9, no. 1. Subfield generated by a subset: I, 3 9, no. 1. Subgroup, characteristic: I, 3 5, no. 3. Subgroup, invariant stable, invariant subgroup: I, 3 4, no. 4. Subgroup, normal stable, normal subgroup: I, 5 4, no. 4. Subgroup, stable, generated by a subset: I, fj 4, no. 3. Subgroup, stable subgroup: 1, 3 4, no. 3. Subgroup, Sylow, Sylow p-subgroup: I, 5 6, no. 6. Submagma: I, 3 1, no. 4. Submagna generated by a subset: I, 5 1, no. 4. Submagma, unital : I, 4 2, no. 1. Submagma, unital, generated by a subset: I, 3 2, no. 1. Submatrix: 11, 3 10, no. 1. Submodule: 11, 3 1, no. 3. Submodule, component, of a direct sum of modules: 11, 5 1, no. 6. Submodule generated by a family: 11, 3 1, no. 7. Submodule, graded: 11, 3 11, no. 3. Submodule, irreducible: 11, 3 2, Exercise 16. Submodule, orthogonal (or totally orthogonal) to a subset of E (resp. E*): Submodule, torsion, of a module over an integral domain: 11, 5 7, no. 10. Submodule, zero: 11, 3 1, no. 3. Submodules, supplementary: 11, 3 1, no. 9. Submultimodule: 11, 3 1, nos. 14. Subring: I, 3 8, no. 5. Subring generated by a subset: I, Subring, graded: 11, 5 11, no. 3. Subset, affine: 11, 3 9, no. 3. 11, 3 2, no. 4. 8, no. 5. 704 INDEX OF TERMINOLOGY Subset centralizing a subset: I, 3 5, no. 3. Subset, free, related subset: 11, 3 1, no. 11. Subset, homogeneous, of degree p with respect to certain indeterminates in a formal power series: 111, 3 2, no. 11. Subset normalizing a subset: I, 3 5, no. 3. Subset, stable: I, 5 1, no. 4 and 3 3, no. 2. Subset, stable, generated by a subset: I, 5 1, no. 4 and 3 3, no. 2. Subset, strongly associative: 111, Appendix, no. 1. Subset, symmetric: I, fj 4, no. 1. Subsets, conjugate: I, 3 5, no. 4. Subspaces associated with a homogeneous element of the exterior algebra: 111, Subspace associated with a homogeneous element of the symmetric algebra: Subspace associated with a homogeneous element of the tensor algebra: 111, Subspaces associated with an element of a tensor product of vector spaces: 11, Subspace rational over a subfield: 11, 3 8, no. 2. Subspace, vector, subspace: 11, 5 1, no. 3. Sum, amalgamated, of modules: 11, fj 1, Exercise 5. Sum, amalgamated, of monoids: 11, fj 7, no. 3. Sum, direct: I, fj 4, no. 9. Sum, direct, of a family of submodules: 11, 4 1, no. 8. Sum, external direct, of a family of submodules: 11, 3 1, no. 6. Sum, internal restricted, of subgroups: I, 3 4, no. 9. Sum, monoidal: I, 5 7, no. 3. Sum of a family of elements of finite support: I, 3 2, no. 1. Sum of a family of left ideals, of right ideals: I, 3 8, no. 6. Sum of a family of submodules: 11, 3 1, no. 7. Sum of an ordered sequence: I, 3 1, no. 2. Sum of two elements : I, 5 1, no. 1. Sum of two matrices: 11, 4 10, no. 2. 3 7, no. 2. 111, 3 6, no. 2. 8 5, no. 2. 3 7, no. 8. Sum, restricted, of groups with respect to subgroups, restricted sum of groups: I, 6 4, no. 9. SupeEsolvable group: I, 3 6, Exercise 26. Supplementary submodules: 11, 5 1, no. 9. Support of a cycle: I, 3 5, no. 7. Support of a family: I, 3 2, no. 1. Suppress columns, rows, in a matrix: 11, 3 10, no. 1. Sylow subgroup: I, 3 6, no. 6. Symbol, Kronecker: 11, 3 1, no. 11. Symmetric algebra of a module: 111, 4 6, no. 1. 705 INDEX OF TERMINOLOGY Symmetric group: I, $ 4, no. 1. Symmetric multilinear mapping: 111, $ 6, no. 3. Symmetric power of a linear mapping: 111, $ 6, no. 3. Symmetric power of a module: 111, $ 6, no. 3. Symmetric product of two multilinear mapping: 111, 3 11, no. 2. Symmetric subset in a group: I, $4, no. 1. Symmetric tensor: 111, $ 6, no. 3. Symmetrization of a tensor: 111, $ 6, no. 3. System, direct, of graded algebras: 111, $ 3, no. 3. System, direct, of magmas: I, $ 10, no. 3. System, direct, of modules: 11, $ 6, no. 2. System, direct, of rings (groups, fields) : I, 5 10, no. 3. System, free, related system: 11, $ 1, no. 11. System, generating, of a group: I, $ 4, no. 3. System, generating, of a magma: I, $ 1, no. 4. System, generating, of a module: 11, $ 1, no. 7. System, generating, of a projective space: 11, $ 9, no. 8. System, generating, of a unital magma: I, $ 2, no. 1. System, generating, of an algebra: 111, $ 1, no. 2. System, inverse, of algebras: 111, $ 1, no. 6. System, inverse, of magmas: 11, 5 10, no. 1. System, inverse, of modules: 11, $ 6, no. 1. System, inverse, of rings (groups, fields) : I, $ 10, no. 1. System of equations of a vector subspace: 11, $ 7, no. 5. System of factors: 111, $ 2, Exercise 11. System of homogeneous coordinates of a point: 11, $ 9, no. 6. System of linear equations, linear system, homogeneous linear system : 11, System, trivial, of commutation factors: 111, $ 4, no. 7. Table, diagonal, lower triangle, upper triangular, of matrices: 11, 3 10, no. 7. Table, multiplication, of an algebra: 111, $ 1, no. 7. Table, square, of matrices: 11, 3 10, no. 5. Tensor algebra of a module: 111, $ 5, no. 1. Tensor, contravariant, covariant tensor, mixed tensor: 111, Tensor of type (I, J): 111, $ 5, no. 6. Tensor, skew -symmetric: 111, $ 7, no. 4. Tensor, symmetric: 111, 5 6, no. 3. Tensorial power of a linear mapping: 111, 5 5, no. 2. Tensorial power of a module: 111, 5 5, no. 1. Term, constant, of a formal power series: 111, 3 2, no. 11. Term, constant, of a polynomial: 111, $ 2, no. 9. Term in Xu in a polynomial: 111, fj 2, no. 9. $ 2, no. 8. 5, no. 6. 706 INDEX OF TERMINOLOGY Term of a formal power series: 111, $ 2, no. 11. Term of a polynomial: 111, $ 2, no. 9. Term of a sum: I, $ 1, no. 2. Term of degree p with respect to certain indeterminates in a formal power Term of total degreep in a formal power series: 111, 3 2, no. 11. Theorem, associativity: I, $ 1, no. 3. Theorem, Cayley -Hamilton: 111, $ 8, no. 11. Theorem, commutativity: I, $ 1, no. 5. Theorem, Desargues’s: 11, 5 9, Exercise 15. Theorem, Erdos-Kaplansky: 11, $ 7, Exercise 3. Theorem, fundamental, of projective geometry: 11, $ 9, Exercise 16. Theorem, Hall’s: I, $ 6, Exercise 7. Theorem, Jordan-Holder: I, $ 4, no. 7. Theorem, Kaplansky’s: 11, 5 2, Exercise 2. Theorem, Krull’s: I, $ 8, no. 6. Theorem, Nielsen -Schreier: I, $ 7, Exercise 20. Theorem of the complete quadrilaterial: 11, $9, Exercise 13. Theorem, Pappus’s: 11, 5 9, Exercise 14. Theorem, Schreier’s: I, $ 4, no. 7. Torsion element, module, submodule: 11, $ 7, no. 10. Torsion -free module: 11, $ 7, no. 10. Total algebra of a monoid: 111, $ 2, no. 10. Total graduation: 11, $ 11, no. 1. Totally orthogonal (submodule) to a subset; 11, $ 2, no. 4. Trace, Cayley: 111, $ 2, no. 4. Trace in a quadratic algebra: 111, $ 2, no. 3. Trace of a matrix: 11, $ 10, no. 11. Trace of an element in a K-algebra: 111, $ 9, no. 3. Trace of an endomorphism: 11, $ 4, no. 3. Trace of a scalar with respect to a module: 111, $ 9, no. 1. Transitive operation: I, $ 5, no. 5. Transitivity formulae: 111, 3 9, no. 4. Translation in an af€ine space, space of translations: 11, $ 9, no. 1. Translation, left, right: I, $ 2, no. 2. Translation, left, right (monoid operating on itself by) : I, $ 5, no. 1. Transporter, strict transporter: I, 5 5, no. 2. Transpose of a linear mapping, of a semi-linear mapping: 11, $ 2, no. 5. Transpose of a matrix: 11, $ 10, no. 1. Transposition: I, $ 5, no. 7. Transvection: 11, fj 10, Exercise 11. Triangular, lower, upper triangular, matrix: 11, fj 10, no. 7. Trivial extension: I, fj 6, no. 1. series: 111, $2, no. 11. 707 INDEX OF TERMINOLOGY Trivial graduation: 11, 4 11, no. 1. Trivial homomorphism: I, 3 2, no. 1. Trivial operation: I, 5 5, no. 2. Trivial solution of a homogeneous linear equation: 11, 3 2, no. 8. Trivial system of commutation factors: 111, 5 4, no. 7. Two-sided ideal: I, 4 8, no. 6 and 111, 5 1, no. 2. Type, exponential, group of: I, 3 7, Exercise 39. Type A, graduation of: 11, 5 11, no. 1. Unimodular endomorphism: 111, 3 8, no. 1. Unimodular group: 111, fj 8, no. 9. Unimodular matrix: 111, 3 8, no. 3. p-unipotent element: I, 3 6, Exercise 28. Unit element of an algebra: 111, 3 1, no. 1. Unit, left, right, in a groupoid: I, 5 4, Exercise 23. Unit, unit element of a magma: I, 3 2, no. 1. Unital algebra: 111, 3 1, no. 1. Unital algebra homomorphism, morphism: 111, 3 1, no. 1. Unital homomorphism: I, 3 2, no. 1. Unital magma : I, 3 2, no. 1. Units, matrix: 11, 3 10, no. 3. Universal algebra defined by a generating system related by a family of rela- Universal algebra subjected to identities: 111, 3 2, no. 8. Universal relator: 111, 5 2, no. 8. Unknowns of a linear system: 11, 3 2, no. 8. Value, absolute, of a rational number: I, 3 9, no. 4. Value of an element of a free algebra: 111, 3 2, no. 8. Vandermonde determinant: 111, 3 8, no. 6. Variety, affine linear, affine linear variety generated by a family: 11,s 9, no. 3. Variety, linear: 11, 3 9, nos. 3, 7 and 11. Variety, projective linear, projective linear variety generated by a family : Varieties, parallel linear: 11, 3 9, no. 3. Vector: 11, 3 1, no. 1. Vector, direction, of an affine line: 11, 3 9, no. 3. Vector, free, of an affine space: 11, 3 9, no. 1. p-vector: 111, 3 7, no. 1. p-vector, pure: 111, 3 11, no. 13. Vector rational over a subfield: 11, 3 8, no. 2. Vector space: 11, 5 1, no. 1. Word: I, 3 7, no. 2. 708 tors: 111, 3 2, no. 8. 11, 3 9, nos. 7 and 11. INDEX OF TERMINOLOGY Weight of a homogeneous element: 11, 3 11, no. 1. With no fixed point (automorphism) : I, 3 6, Exercise 23. Zassenhaus’s Lemma: I, 3 4, no. 7. Zero: I, 5 2, no. 1. Zero matrix: 11, 3 10, no. 22. Zero ring: I, 3 8, no. 3. Zero solution of a linear equation: 11, 3 2, no. 8. Zero submodule: 11, 3 1, no. 3. Made and printed in Great Britain by William Clowes & Sons, Limited, London, Beccles and Colchester