Algebra I: Chapters 1-3 2705656758, 0201006391

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Elements of Mathematics

Algebra I

ACTUALITES SCIENTIFIQUES ET INDUSTRIELLES HERMANN ADIWES INTERNATIONAL SERIES IN MATHEMATICS A. J. Lohwater, Consulting Editor

NICOLAS BOURBAKI Elements of Mathematics English Translation: published: General Topology, Part I, 1966 General Topology, Part II, 1966 Theory of Sets, 1968

Commutative Algebra, 1972 Algebra, Part I, 1973

In preparation: Algebra, Part II

Functions of a Real Variable Integration Lie Groups and Lie Algebras Spectral Theories Topological Vector Spaces

NICOLAS BOURBAKI

Elements of Mathematics

Algebra I Chapters 1-3

db 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 © 1943, 1947, 1948, 1971 by Hermann, Paris

PRESTON _; POLYTECHNIC

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

© 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 knowledge of mathematics on the readers’ 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 indispensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands ofproofimpose 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

TO THE READER

volumes. This rule holds good within each Book, but for convenience of exposition these Books are no longer arranged in a consecutive order. At the beginning 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 axiom, and the theorem of the chapter. These are the parts that have mainly to be home 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 notations 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 1f.

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 abuse: oflanguage, 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 Hiytan'cal 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 importance in the evolution of the theory under discussion. It makes no sort of pre-

vi

TO THE READER

tence to completeness; in particular, references which serve only to determine

questions of priority are almost always omitted. As to 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) Ifreference 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 ofthe 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. The Summaries g“ Results are quoted by the letter R; thus Set Theory, R signifies “Summary ofResults of the Theory «J Se ”.

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES

I. THEORY or SETS

1. Description of formal mathematics. cardinals; natural numbers.

2. Theory of sets.

3. Ordered sets;

4-. Structures. II. ALGEBRA

l. Algebraic structures. 2. Linear algebra. 3. Tensor algebras, exterior algebras, symmetric algebras. 4. Polynomials and rational fractions. 5. Fields. 6. Ordered groups and fields. 7. Modules over principal ideal rings. 8. Semi—simple modules and rings. 9. Sesquilinear and quadratic forms. III. GENERAL TOPOLOGY

l. Topological structures. 2. Uniform structures. 3. Topological groups. 4-. Real numbers. 5. One-parameter groups. 6. Real number spaces, affine and projective spaces. 7. The additive groups R". 8. Complex numbers.

9. Use of real numbers in general topology.

10. Function

spaces. IV. FUNCTIONS or A REAL VARIABLE

l. Derivatives. 2. Primitives and integrals. 3. Elementary functions. 4-. Differential equations. 5. Local study of functions. 6. Generalized Taylor expansions. The Euler-Maclaurin summation formula. 7. The gamma function.

Dictionary. V. TOPOLOGICAL VECTOR SPACES

l. Topological vector spaces over a valued field. 2. Convex sets and locally convex spaces. 3. Spaces ofcontinuous linear mappings. 4. Duality in topological vector spaces. 5. Hilbert spaces: elementary theory. Dictionary. VI. INTEGRATION

l. Convexity inequalities. pact spaces.

2. Riesz spaces.

3. Measures on locally com-

4-. Extension of a measure. L” spaces.

sures. 6. Vectorial integration. representation.

7. Haar measure.

5. Integration of mea-

8. Convolution and ix

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES

LIE GROUPS AND LIE ALGEBRAS

l. 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.

logies.

2. Localization.

3. Graduations, filtrations, and topo-

4-. Associated prime ideals and primary decomposition.

6. Valuations.

7. Divisors. SPECTRAL THEORY

l. Normed algebras.

2. Locally compact groups.

DIFFERENTIAL AND ANALYTIG MANIFOLDS

Summary of results.

5. Integers.

CONTENTS

To THE READER ...............................................

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES ................ INTRODUCTION .................................................

mention ’00—-

CHAPTER I. ALGEBRAIG STRUCTURES .............................. § 1. Laws of composition; associativity; commutativity .......... . Laws of composition ............................... . Composition of an ordered sequence of elements ........ . Associative laws ....................................

. Stable subsets. Induced laws ......................... . Permutable elements. Commutative laws .............. . Quotient laws .....................................

§ 2. Identity element; cancellable elements; invertible elements . . @flmmfibfiM—I

. Identity element ................................... . Cancellable elements ............................... . Invertible elements ................................. . Monoid of fractions of a commutative monoid ..........

. . . .

Applications: I. Rational integers ..................... Applications: II. Multiplication of rational integers ..... Applications: III. Generalized powers ................ 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.

CONTENTS

SOQVOUIréOONf

§4. Groups and groups with operators ....................... .Groups ........................................... . Groups with operators .............................. . Subgroups ........................................ . Quotient groups ................................... . Decomposition of a homomorphism .................. . Subgroups of a quotient group ....................... . The Jordan-Holder theorem ......................... . Products and fibre products ..........................

. Restricted sums ................................... . . Monogenous groups ...............................

§ 5. Groups operating on a set ............................... l. Monoid operating on a set ........................... 2. Stabilizer, fixer ....................................

3. 4. 5. 6. 7.

Inner automorphisms ............................... Orbits ............................................ Homogeneous sets ................................. Homogeneous principal sets .......................... Permutation groups of a finite set ....................

§ 6. Extensions, solvable groups, nilpotent groups ............... 1. Extensions ........................................ 2. Commutators ...................................... 3. Lower central series, nilpotent groups .................

37 38 41 45 47 48

52 52 54 55 56 58 A 60 61 65 65 68

4. Derived series, solvable groups .......................

78 80

§ 7. Free monoids, free groups .......................... . Free magmas ...................................... . Free monoids ..................................... . Amalgamated sum of monoids ....................... . Application to free monoids ......................... . Free groups .......................................

81 81 82 84 88 89

cooougsmusoaww

5. p—groups .............................p ............. 6. Sylow subgroups ................................... 7. Finite nilpotent groups ..............................

Presentations of a group ............................ . Free commutative groups and monoids ...............

xii

E

5%"

9. ‘12

”0—!

00

GM

. Exponential notation ............................... . Relations between the various free objects ............. .............................................

nsequences of distributivity ........................

CONTENTS

Examples of rings .................................. Ring homomorphisms ............................... Subrings .......................................... Ideals ............. . .............................. Quotient rings .................................. . . . Subrings and ideals in a quotient ring ................. Multiplication of ideals ............................. Product of rings ................................... Direct decomposition of a ring ....................... Rings of fractions ..................................

101 102 103 103 105 106 107 108 110 112

§ 9. Fields ............................................... . 1. Fields ............................................ 2. Integral domains ...................................

114 114116

3. Prime ideals ......................................

116

3. 4-. 5. 6. 7. 8. 9. 10. 11. 12.

4-. The field of rational numbers ........................

117

............

118

l. Inverse systems of magmas ..........................

118

2. Inverse limits of actions ............................. 3. Direct systems of magmas ........................... 4. Direct limit of actions ............................... Excrms' es for § 1 ............................................

119 120 123

Exercises for § 2 ................ '............................

126

Exercises for § 3 ............................................

129

Exercises for §4 ............................................ Exercises for § 5 ...........................................

132

Exercises for § 6 ........................................... Exercises for § 7 ............ . ..............................

147

Exercises for § 8 ........................................... Exercises for § 9 ...........................................

171 174-

Exercises for § 10 ..........................................

179

Historical note ............................................

180

CHAPTER II. LINEAR ALGEBRA .................................. § 1. Modules .............................................. 1. Modules; vector spaces; linear combinations ........... 2. Linear mappings .................................. 3. Submodules; quotient modules ......................

191 191 191 194 196

§10. Inverseanddirectlimits

124-

140 159

xiii

4. Exact sequences ................................... 5. Products of modules ................................ 6 7 8. 9.

D1rect sum of modules .............................. Intersection and sum of submodules ................... D1rect sums of submodules ........................... Supplementary submodules .........................

Modules of finite length ............................ Freefamilies. Bases Annihilators. Faithful modules. Monogenous modules . . . Change of the ring of scalars ......................... Multimodules ...................................... § 2. Modules of linear mappings. Duality ..................... 1. Properties of HomA(E, F) relative to exact sequences. 2. Projective modules ................................. 10. ll. 12. 13. 14.

3. Linear forms; dual of a module ......................

4. 5. 6. 7. 8.

Orthogonality ..................................... Transpose of a linear mapping ....................... Dual ofa quotient module. Dual ofa direct sum. Dual bases Bidual ........................................... Linear equations ................................... § 3. Tensor products ....................................... l. Tensor product of two modules ....................... 2. Tensor product of two linear mappings ................

197 200 202 205 208 210

212 214 219 221 224

227 227 231 232

234 234 236 239 240 243 243 245

3. Change of ring ..................................... 246 4. Operators on a tensor product; tensor products as multimodules ......................................... 247 5. Tensor product of two modules over a commutative ring 249

6. Properties of E 63)., F relative to exact sequences ........

251

7. Tensor products of products and direct sums ........... 8. Associativity of the tensor product .................... 9. Tensor product of families of multimodules ............

254 258 259

§ 4. Relations between tensor products and homomorphism modules 1. The isomorphisms HomB(E ®A F, G) —> HomA(F, Hom3(E, G)) and HomG(E (8),, F, G) —> HomA(E, Homc(F, G)) 2. The homomorphism E* ®A F -—> HomA(E, F) .......... 3. Trace of an endomorphism .......................... 4. The homomorphism

267

267 268 273

Homc(E1, F1) ®c H0mo(E2: F2)

-—> Hom(,(E1 @c E2, F1 ®c F2)

274

CONTENTS

§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 .............................

282 283 284-

§6.

Inverse and direct limits of modules ...................... 1. Inverse limits of modules ............................ 2. Direct limits of modules ............................ 3. Tensor product of direct limits ......................

284284286 289

§7.

Vector spaces ......................................... l. Bases of a vector space .............................. 2. Dimension of vector spaces ......................... 3. Dimension and codimension ofa subspace ofa vector space 4. Rank of a linear mapping ........................... 5. Dual of a vector space .............................. 6. Linear equations in vector spaces .................... 7. Tensor product of vector spaces ...................... 8. Rank of an element of a tensor product ............... 9.‘ Extension of scalars for a vector space ................ 10. Modules over integral domains ....................... Restriction ofthe field ofscalars in vector spaces ............. 1. Definition of K’-struetures .......................... 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 ..............................

292 292 293 295 298 299 304 306 309 310 312

317 317 318 319 320 321 322 323

Affine spaces and projective spaces ....................... 1. Definition of affine spaces ........................... 2. Barycentric calculus ................................ 3. Affine linear varieties ............................... 4. Afline linear mappings ............................. 5. Definition of projective spaces ........................

325 325 326 327 329 331

6. Homogeneous coordinates ........................... 7. Projective linear varieties ............................ 8. Projective completion of an affine space ...............

331 332 333

9. Extension of rational functions .......................

334

§8.

§9.

277 277

280

XV

CONTENTS

10. Projective linear mappings ........................... 11. Projective space structure ............................

335 337

§ 10. Matrices ............................................. 1. Definition of matrices .............................. 2. Matrices over a commutative group ................... 3. Matrices over a ring ............................... 4. Matrices and linear mappings ....................... 5. Block products ..................................... 6. Matrix of a semi-linear mapping ..................... 7. Square matrices ............................... . . . . .

338 338 339 341 342 346 347 349

8. Change of bases ...................................

352

9. Equivalent matrices; similar matrices ................. 10. Tensor product of matrices over a commutative ring . . . .

354 356

ll. Trace of a matrix .................................. 12. Matrices over a field ................................

358 359

13. Equivalence of matrices over a field ..................

360

§ 11. Graded modules and rings ..............................

363

Graded commutative groups ........................ Graded rings and modules .......................... Graded submodules ................................ Case of an ordered group of degrees ..................

363 364 367 371

5. Graded tensor product of graded modules ............. 6. Graded modules of graded homomorphisms ...........

374 375

Appendix. Pseudomodules ................................. 1. Adjunction of a unit element to a pseudo-ring ......... 2. Pseudomodules ....................................

378 378 378

Exercises for § 1 ........................................... ..... Exercisesfor§2

380

386

Exercises for § 3 ...........................................

395

Exercises for §4 ........................................... Exercises for § 5 ...........................................

396

Exercises for § 6 ...................... , .................... Exercises for § 7 ...........................................

399 400

Exercises for § 8 ...........................................

409

Exercises for § 9 ...........................................

410

1. 2. 3. 4.

398

Exercises for § 10 ........................................... 4-17 xvi

CONTENTS

Exercises for § ll ..........................................

424

Exercise for the Appendix ..................................

425

CHAPTER III. TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG

ALGEBRAS ................................................... § 1. Algebras ............................................. 1. Definition of an algebra ............................ 2. Subalgebras. Ideals. Quotient algebras ................ 3. Diagrams expressing associativity and commutativity... 4. Products of algebras ................................ 5. Restriction and extension of scalars ................... 6. Inverse and direct limits of algebras .................. 7. Bases of an algebra. Multiplication table ...............

427 428 428 429 431 432 433 434 436

§ 2 Examples of algebras ................................... 1.Endomorphism algebras ............................

438 438

2. Matrix elements ...................................

3. 4. 5. 6. 7.

438

Quadratic algebras ................................. Cayley algebras ................................... Construction of Cayley algebras. Quaternions .......... Algebra of a magma, a monoid, a group ............... Free algebras ......................................

439 441 443 446 448

8. Definition of an algebra by generators and relations . . . .

450

10. Total algebra of a monoid ...........................

454

9. Polynomial algebras ................................

452

11. Formal power series over a commutative ring ..........

455

§ 3. Graded algebras ....................................... l. Graded algebras ...................................

457 457

2. Graded subalgebras, graded ideals of a graded algebra . .

459

3. Direct limits of graded algebras ......................

460

§4. Tensor products of algebras ............................. l. Tensor product of a finite family of algebras ........... 2. Universal characterization of tensor products of algebras 3. Modules and multimodules over tensor products ofalgebras 4. Tensor product of algebras over a field ................ 5. Tensor product of an infinite family of algebras ....... . 6. Commutation lemmas .............................. 7. Tensor product ofgraded algebras relative to commutation factors .........................................

460 460 463 465 468 470 472

8. Tensor product of graded algebras of the same types . . . .

9. Anticommutative algebras and alternating algebras .....

474 480

482 xvii

CONTENTS

§ 5. Tensor algebra. Tensors ................................ 010-5t—

. . . . .

Definition of the tensor algebra of a module ........... Functorial properties of the tensor algebra ............. Extension of the ring of scalars ....................... Direct limit of tensor algebras ........................ Tensor algebra of a direct sum. Tensor algebra of a free module. Tensor algebra of a graded module ..........

65

. Tensors and tensor notation .........................

§ 6 Symmetric algebras .................................... l. Symmetric algebra of a module ...................... 2. Functorial properties of the symmetric algebra ......... 3. n-th symmetric power of a module and symmetric multilinear mappings ................................. 4-. Extension of the ring of scalars ....................... 5. Direct limit of symmetric algebras ....................

484 484 485 489 490 491 492 497 497 498 500 502 503

6. Symmetric algebra of a direct sum. Symmetric algebra of a

free module. Symmetric algebra of a graded module . .

503

§ 7. Exterior algebras ...................................... l. Definition of the exterior algebra of a module .......... 2. Functorral properties of the exterior algebra ........... 3. Anticommutativity of the exterior algebra ............. 4. n-th exterior power of a module and alternating multilinear mappings .................................

507 507 508 510

\10501

. Extension of the ring of scalars ....................... . Direct limits of exterior algebras .....................

511 513 514

. Exterior algebra of a direct sum. Exterior algebra of a graded module ................................... 515 . Exterior algebra of a free module ..................... 517 . Criteria for linear independence ..................... 519

Mb—I

§ 8. Determinants .......................................... . Determinants of an endomorphism .................... . Characterization of automorphisms of a finite-dimensional free module

......................................

. Determinant of a square matrix ...................... Calculation of a determinant ......................... Minors of a matrix .................................

. Expansions of a determinant ........................

. Application to linear equations ....................... . Case of a commutative field ......................... . The unimodular group SL(n, A) ...................... xviii

522 522 524 526 528 530 534 536 537

CONTENTS

10. The A[X]-module associated with an A-module endomorphism ...................................... 538 ll. Characteristic polynomial of an endomorphism ......... 540 §9. Norms and traces ......................................

l. 2. 3. 4. 5.

541

Norms and traces relative to a module ................ Properties of norms and traces relative to a module ..... Norm and trace in an algebra ........................ Properties of norms and traces in an algebra ........... Discriminant of an algebra ..........................

541 542 543 545 549

§ 10. Derivations ........................................... l. Commutation factors ...............................

550 550

2. 3. 4. 5.

General definition of derivations ..................... Examples of derivations ............................. Composition of derivations .......................... Derivations of an algebra A into an A-module ..........

551 553 554 557

6. Derivations of an algebra ...........................

559

7. Functorial properties ............................... 560 8. Relations between derivations and algebra homomorphisms ......................................... 561 9. Extension of derivations ............................

562

10. Universal problem for derivations: non—commutative case 11. Universal problem for derivations: commutative case . . . . 12. Functorial properties of K—difl‘erenfials ................

567 568 570

§ 11. Cogebras, products of multilinear forms, inner products and duality ............................................ 574 1. Cogebras ..........................................

574

2. Coassociativity, cocommutativity, counit ..............

578

3. Properties of graded cogebras of type N ...............

584

4. 5. 6. 7. 8. 9.

585 587 594 597 600

Bigebras and skew-bigebras ......................... The graded duals T(M) *3”, S(M)*" and A (M)*'“‘ ..... Inner products: case of algebras ..................... Inner products: case of cogebras ...................... Inner products: case of bigebras ...................... Inner products betweenT(M) andT(M*), S(M) and S(M*),

/\ (M) and /\ (M*) ............................. 603

10. Explicit form of inner products in the case of a finitely generated free module ........................... 605 11. Isomorphisms between A? (M) and A“? (M*) for an ndimensional free module M ........................ 607 12. Application to the subspace associated with a p—vector. . . .

608

13. Pure p—vectors. Grassmannians .......................

609 xix

CONTENTS

Appendix. Alternative algebras. Octonions .................... 1. Alternative algebras ................................ 2. Alternative Cayley algebras ......................... 3. Octonions .........................................

611 611 613 615

Exercises for § 1 ...........................................

618

Exercises for § 2 ...........................................

618

Exercises for § 3 ...........................................

626

Exercises for § 4- ...........................................

626

Exercises for § 5 ...........................................

627

........................................

632

Exercises for § 7 ...........................................

633

Exercises for § 8 ...........................................

637

Exercisesfor§9..... ......................................

644-

Exercises for § 10 ..........................................

645

Exercises for § 11 ..........................................

647

Exercises for the Appendix ..................................

654

HISTORICAL NOTE ON CHAPTERS II AND III ........................

655

INDEX OF NOTATION ...........................................

669

INDEX OF TERMINOLOGY .........................................

677

Exercises for § 6 .

INTRODUCTION

Algebra is essentially concerned with calculating, that is, performing, on elements of a set, “algebraic operations”, the most well-known example ofwhich 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 the form 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 0 ofthe set E. In other words, there is nothing more in this notion than that offunction: 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 peculiarity 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 function we give the name law of composition.

Alongside these “internal laws”, mathematicians have been led (chiefly xxi

INTRODUCTION

under the influence of Geometry) to consider another type of “law of composition” ; these are the “laws of action”, where, besides the set E (which remains

as it were in the foreground), an auxiliary set 9 occurs, whose elements are called operators: the law this time associating with an ordered pair (a, a) consisting of an operator at 6 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, § 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 structure: and it is the study of these which constitutes Algebra. There are several species (Set Theory, IV, § 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 subjected. 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 associativity, 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 ofcomposition occurs) and that of a ring (with two laws of composition), of which afield 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 ofcomposition, there occur one or several laws of action. The most important groups with operators are modules, ofwhich 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 II. Similarly, the rings with operators which occur most often are called algebras (or hypercomplex systems). In Chapters III 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 fimdamental to the theory of algebraic equations. In Chapter V there is an exposition of the general theory of commutative fields 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 remains 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 xxii

INTRODUCTION

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 of groups; on the other, the theory of quadraticfirms and Hermitianforms, with the study of the groups associated with them. Finally, the elementary geometries (affine, projective,

euclidean, etc.) are studied in Chapters II, VI and IX from a purely algebraic point of view: here there is little more than a new language for expressing results of Algebra already obtained elsewhere, but it is a language which is particularly well adapted to the later developments ofAlgebraic Geometry and Differential Geometry, to which this chapter serves as an introduction.

CHAPTER I

Algebraic Structures §l. LAWS OF COMPOSITION; ASSOCIATIVITY; COMMUTATIVITY l. LAWS OF GONIPOSITION

DEFINITION 1. Let E be a set. A mappingfofE x E into E is calleda law ifcomposition on E. The valuef(x, y) offfor an orderedpair (x, y) e E x E is called the composition of x and 1/ under this law. A set with a law of composition is called a magma. The composition ofx and y is usually denoted by writing at 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 ofx andy 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 .74.}; = xy being called the product of x and y) and we say that it is written multiplicatioely. In the general arguments ofparagraphs l'to 3 ofthis chapter we shall generally use the symbols T and J. 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 defined on E. Examples. (1) The mappings (X, Y) |—> X U Y and (X, Y) I—> X n Y are laws of composition on the set of subsets of a set E. (2) 0n the set N of natural numbers, addition, multiplication and exponentiation 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 1:”) (Set Theory, III, § 3, no. 4).

(3) Let E be a set; the mapping (X, Y) +—> X 0 Y is a law of composition on the set of subsets of E x E (Set Theory, II, § 3, no. 3, Definition 6); the mapping (f, g) |—>f o g is a law of composition on the set of mappings of

E into E (Set Theory, II, §5, no. 2). (4-) Let E be a lattice (Set Theory, III, § 1, no. 11); if sup(x, 1/) denotes the least upper bound of the set {x, y}, the mapping (x, y) v-—> sup(x, y) is a law of

l

I

ALGEBRAIG STRUCTURES

composition on E. Similarly for the greatest lower bound inf(x, y). Example 1 above 15 a particular case of this with 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‘l') will denote 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) »—> x T g). 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) u—> X T Y is a law of composition on the set of subsets of E. DEFINITION 2. Let E be a magma and T denote its low of composition. The law of composition (x, y) 1+ y T x on E is called the opposite of the above. The set E with this

law is called the opposite magma 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, § 1, no. 5), a bijec-

tive mappingfof E onto E’ such that

(1)

. f(xTy) =f(x) Tf(y)

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’. More generally: DEFINITION 3. A mappingfof E into E’ such that relation (1) holdsfor 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 ofa magma E is a homomorphism, the composition of two homomorphism is a homomorphism. 1' 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) I—> A U B between subsets of a set E; a law of composition

(21, %) I—> F(2[, Q5) is derived between subsets of ENE), F(9[, $) being the set of A U B with A e 91, B e $ ; but F(Q[, Q3) should not be denoted by Q! U 523, as this notation already has a different meaning (the union of 91 and 93 considered as subsets of ENE».

2

COMPOSITION or AN ORDERED SEQUENCE or ELEMENTS

§ 1.2

For a mapping f of E into E' to be an isomorphism, it is necessary and -1

sufi'icient that it be a bijective homomorphism and f is then an isomorphism of E’ onto E. 2. COMPOSITION ORAN ORDERED SEQUENCE OF ELEMENTS

Recall that afamily of elements of a set E is a mapping L |—> x, of a set I (called an indexing set) into E; a family (“Oder is calledfinite if the indexing set isfinite. ’A finite family (x1)LeI 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 (x,),en, where H is a finite subset of the

set N ofnatural numbers, can be considered as an ordered sequence ifH is given the order relation induced by the relation m S n between natural numbers. Two ordered sequences (xi)ie1 and (yk) keg are called similar if there exists an

ordered set isomorphism 4) of I onto K such that 31“,, = x, for all i e I. Every ordered sequence (9:0,)CHE A is similar to a suitable finite sequence. For there exists an increasing bijection ofA onto an interval [0, n] ofN. DEFINITION 4. Let (x,),,e A be an ordered sequence of elements in a magma E whore indexing set A is non-empty. The composition (under the law T) of the ordered sequence

my”, denoted by «L x.” is the element of E defined by induction on the number of elements in A asfiillows:

(I) zfA = {a} then; xe = xe;

(2) ifAhasp > lelementr, flietheleastelementqandA’ = A—{p}, then Txa=xaT(Txu) aeA at e A -

‘

It follows immediately (by induction on the number of elements in the indexing sets) that the compositions of two similar ordered sequences are equal; in particular, the composition ofany ordered sequence is equal to the composition of a finite sequence. If A = {1, p, v} has three elements (A < y. < v) the composition “IA x“ is x; T (xll T xv). 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 (x5, x", xv) would be (xh T x") T xv.

As a matter of notation, the composition of an ordered sequence (rem)me A is

written “.41 for a. law denoted by _L; for a law written additively it is usually denoted by 0;" x, and called the sum of the ordered sequence Ora)“ A (the x“ being called the terms of the sum) ; for a law written multiplicatively it is usually

3

I

,

ALGEBRAIC STRUCTURES

denoted by all x, and called the product of the ordered sequence (x,) (the x, being called thefactors of the product) .T 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, ; x, instead of“EA x“; similarly for the other notations.

For a law denoted by T the composition of a sequence (ea) with indexing set a

non—empty interval [[1, q] ofN is denoted by 9 J“ x, or j; x,; 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 (:c,,)me A of elements of E

(2)

f(uIA)= T for.) aeA

3. ASSOCIATIVE LAWS

DEFINITION 5. A law if composition (x, y) 1—» x T g on a set E is called associative if, for all elements x, y, z in E,

(xTy) T I. =xT (yT z). 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, III, § 3, no. 3, Corollary to Proposition 5) (2) The laws cited in Examples (1), (3) and (4) of no. 1 are associative. THEOREM 1 (Associativity theorem). Let E be an associative magma whose law is denoted by T. Let A be a totally ordered non-empty finite set, which is the union if an ordered sequence of nan-empty subsets (B,),EI such that the relations at 6B,, fleB,, i < j imply a < (5 ; let (1%),” be an ordered sequence ofelements in E with A as indexing set. Then

x.) (“T Tx =T 161 534

(3)

aeA x“

1' The use of this terminology and the notation 1:1 x“ must be avoided if there is

any risk of confusion with the product of the sets at:6defined in the theory of sets (Set Theory, II, § 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. 11 x“) in the above notation is the cardinal sum (resp. cardinal proE

one

duct) of the family (away, (Set Theory, III, § 3, no. 3).

4-

ASSOCIATIVE LAWS

§ 1.3

We prove the theorem by induction on the cardinal n of A. Let p be the cardinal of I and hits least element; let] = I — {It}. Ifn = l, as the B, are nonempty, of necessity p = l 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) 13,, has a single element [5. Let C = 1L5}; B1. The left-hand side of (3) is equal, by definition, to x, T (“1-0 x“); the right-hand side is equal, by definition, to

x, T (:IJ (cl-B; 1(a)); the result follows from the fact that the theorem is assumed true for C and

(131)151-

(b) Otherwise, let 3 be the least element of A (and hence of Bk); let

A' = A— {{3} and letB,’ = A' n B,forie I; then B; = B,forieJ. The set A’ has n — 1 elements and the conditions of the theorem hold for A’ and its sub-

sets B,’ ; hence by hypothesis: Isl-V x“ = (ul‘nxm) T (IIJ(u-el-B. xa».

Forming the composition of 1:3 with each side, we have on the left-hand side side, by definition, “IA x“ and on the right-hand side, using the associativity,

(x, T («Jr-3". x,» T (Mala. x,» which is equal, by Definition 3, to the right-hand side of formula (3). For an associative law denoted by T the composition “T“ x, of a sequence (x0149. a] is also denoted (since no confusion can arise) by x, T - ~ - T x,.

A particular case of Theorem 1 is the formula

xoTxlT---Tx,. =(xoTx1T~-Tx..-1)Tx..Consider an ordered sequence of 7: terms all ofwhose terms are equal to the

same element x e E. The composition of this sequence is denoted by T x for a law denoted by T, .l. x for a law denoted by _L. For a law written multiplicatively the composition is denoted by x” and called the n-th power ofx. For a law

written additively the composition is usually denoted by nx. The associativity 5

.

ALGEBRAIC STRUCTURES

I

theorem applied to an ordered sequence all ofwhose terms are equal gives the equation

(HT (annex).

In particular, ifp = 2,

(4)

"i“x = (i x) T (i x)

andifn1=n,=---=n,=m,

EFHHR).

x T 1/ Of A x A into A is then called the law induced on A by the law T. The setA 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 suflicient that A T A C A. A stable subset of E and the corresponding submagma are often identified. The intersection Ofa family ofstable 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 set ofA. The corresponding submagma is also said to be generated by X. PROPOSITION 1. Let E and F be two magmas andf a homomorphism g“ E into F. (i) The image underf of a stable subset of E is a stable subset of F. (ii) The inverse image underf of a stable subset F is a stable subset of E. (iIi) Let X be a subset ofE. The image underqthe stable subset ofE generated by X

is the stable subset Qf F generated byf(X) (iv) [fgisasecondhomomorphismqfiomFthesetofelementsxqsuchthat f(x) = g(x) is a stable subset of E. 6

PERMUTABLE ELEMENTS. COMMUTATIVE LAWS

§ 1.5

Assertions (i), (ii) and (iv) are obvious; we prove (iii). Let X be the stable subset of E generated by X andf(X) the stable subset of F generated byf(X).

By (0E?) cm?) and by (ii) 2 c f), whencefO—Q cx“). PROPOSITION 2. Let E be an associative magma and X a subset Qf E. Let X' be the set

ofxlTxaT ---Txn, where n 2 land where x‘EXfor l S i S n. The stable subset generated by X is equal to X’.

It follows immediately by induction on n that the composition ofan 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 o are elements of X'theyareoftheformu = xoTxlT ---Tx,,_1,u = aaT ---Tx,,+,

withx, eXforO g i s n +p;then(Theorem l) u T u = x0 T x1 T - - -T x“, belongs to X’. Examples. (1) In the set N of natural numbers the stable subset under addition generated by {1} is the set of integers 2 l ; 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 stable subset generated by a 1!

set {a} consisting of a single element is the set of elements T a, where It 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 as andy of E are said to commute (or to be permutable) ify T x = x T y. DEFINITION 8. A law of cmnposition on a set E is called commutative if any two

elements ofE commute under this law. A magma whose law qfcompositian is commutative is called a commutative magma. A commutative law is equal to its opposite. Examples. (1) Addition and multiplication of natural numbers are commutative laws on N (Set Theory, III, § 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 n between subsets of a set E. (3) Let E be a set of cardinal > 1. The law (f, g) |—>f o g between mappings of E into E is not commutative as is seen by takingf and g to be distinct constant mappings, but the identity mapping is permutable with every mapping.

ALGEBRAIC STRUCTURES

I

(4) Let (x, y) |-> x T y be a commutative law on E; the law (X, Y) I—> X T Y between subsets of E is commutative.

DEFINITION 9. Let E be a magma and X a subset ofE. The set ofelements ofE which commute with each of the elements of X is called the centralizer g“ X.

Let X and Y be two subsets of E and X’ and Y’ their respective centralizers. IfX C Y, then Y' C X'. Let (X0,e I be a family ofsubsets ofE and for all i e I let X; be the centralizer

of X,. The centralizer of EU; X. is [11 x;. Let X be a subset of E and X’ the centralizer of X. The centralizer X” of X’ is called the bicentralizer ofX. 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. If an element at of E commutes with each of the elements y and z of E, it commutes with y T 2. For

xT(yTz) = (xTy)Tz= (yTx)Tz =yT(xTZ) =yT(s) = (yTz)Tx. COROLLARY. Let E be an associative magma. The centralizer 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 of E 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. If every element ofX commutes with every element ofY every element of the 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 ofX. They are stable subsets 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. If x and y are permutable under the associative law T so are "I"I x and "l" yfor all integers m > 0 and n > 0; in particular 'l'n x and “it x are permutablefor all x and all integers m > 0 and n > O.

8

PERMUTABLE ELEMENTS. COMMUTATIVE LAWS

§ 1.5

COROLLARY 2. Ifallpairs ofelements ofa subset X are permutable under 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 ofcomposition on E; let (1:33)“A be a non-emptyfinitefamily of elements of E which are pairwise permutable; let B and C be two totally ordered sets with A as underlying set. Then

T x, = T x“.

at e B

ueG

Since the theorem is true if A has a single element, we argue by induction on

the numberp ofelements 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 — l] in N; the composition of the ordered sequence p_1 Uta)” A defined by the natural order relation on A is in x(.

Let A be given another total ordering and let h be the least element of A under this ordering and A’ the set ofother elements ofA (totally ordered by the induced ordering). Suppose first 0 < h < p — l and let P = {0, l, . . ., h — l} and Q = {h + l, . . ., p —~ 1} ; the theorem being assumed true for A', applying the associativity theorem, we obtain (since A' = P U Q)

n— 1 p- 1 T xa=(T x1)T(T xi)

a e A’

\t= o

1=h+ 1

whence, composing xn with both sides and repeatedly applying the commutativity and associativity of T :

.1... = .. T (.1.~«> = .. T (.1. ..) T Li. a) h- 1

P-1

n—1

p— 1

p— 1

i=0

1=h+ 1

l=0

= (T x.) Tx.T( T x.) = T x,, which proves the theorem for this case. If h = 0 or h = p — l, 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 a finite

family (x1)as A of elements of E is by definition the common value of the composition of all the ordered sequences obtained by totally ordering A in all possible ways. This composition will still be denoted by “IA x“ under a law denoted by T; similarly for other notations. THEOREM 3. Let T be an associative law on E and Ora)“ A a non-emptyfinitefamily of elements of E which are pairwise permutable. [f A is a union of non-empty subsets (B,),E1 which are pairwise disjoint, then (6)

T xu= T(T x“).

«EA

tel men.

ADGEBRAIC STRUCTURES

I

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 mm)“, B, e A x B is a finite family of permutable elements ofan associative magma whose indexing set is the product of two non-empty finite sets A, B (a “double family”), then

(7)

(:1. 3;:A x B x0,“ = «1451-3 x“) = BIBLIA x“)

as follows from Theorem 3 by considering A x B as the union of the sets {at} x B on the one hand and of the sets A x {B} on the other. For example, if B has n elements and for each or E A all the x0“, have the same value x“, then

(8)

IL

I

“HT xv!) = T (elm)-

If B has two elements, we obtain the following results: let (11,)“ A, Lyn)” A be

two non-empty families of elements of E. If the x“ and the 3/3 are pairwise permutable, then (9)

«IA x“ T 1/“ = (ct-el-A x¢)T

(ml-A y“).

Because of formula (7) the composition of a double sequence (x.,) whose indexing set is the product of two intervals (,0, q] and (r, s] in N is often denoted for a commutative associative law written additively by q

2'; ix”

or I=rl t-p 1= and similarly for laws denoted by other symbols. 2 22x“

2. Let n be an integer > 0 and let A be the set of ordered pairs of integers (131) such that 0 g i < n, 0 S j < n and i < j; the composition of a family (9%)“, D e A (under a commutative associative law) is also denoted by 0 s L < n x”

(or simply {I x“, if no confusion arises); Theorem 3 here gives the formulae

(10)

"-1 i ) W >

“11-1q =11) 1=1+1xu = 1:1 ¢=ox” '

There are analogous formulae to (7) for a family whose indexing set is the product ofmore than two sets and analogous formulae to (10) for a family whose indexing set is the set S, of strictly increasing sequences (ik)1 g k“ ofp integers such

that 0 g ik g n (p S n + l): in the latter case the composition of the family (Irma . . J’)a1' ' ‘ . J”) E Sp is denoted by 0"1 a T x (resp.

x I—> x T a) of E into itself is called left 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, 5,, (or y(a), 8(a)) denote the left and right translations by a e E; then 7,,(x) = a T x,

14-

8,,(x) = x T a.

INVERTIBLE ELEMENTS

§ 2.3

PROPOSITION l.‘ If the law T is associative, thenfor all x e E and y e E Yx'rv = Yx°Yw

aan! = all °8x'

Forall zeE: YxTv(z) = (x Ty) T 7' = x T (I! T z) = Yx(Tv(z))

sxTu(Z) = Z T (xTy) = (Z T 9‘) Ty = 5u(5x(z)) In other words, the mapping x v—> y,‘ is a homomorphism from the magma E to the set E” of mappings of E into itself with the law (f, g) +—>f o g; the mapping x r—> fix is a homomorphism of E into the set El with the opposite law. If E is a monoid, these homomorphisms are unital.

DEFINITION 5. An element a ofa magma E is called left (resp. right) cancellable (or regular) if left (resp. right) translation by a is injective. A left 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 ofthe 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 7.; and 8e are then the identity mapping of E onto itself.

Examples. (1) Every natural number is cancellable under addition; every natural number 95 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, :25 is the only cancellable element under the law U and E the only cancellable element under the law 0.

PROPOSITION 2. The set of cancellable (resp. left cancellable, resp. right cancellable) elements Qf an associative magma is a submagma.

If 7,, and W are injective so is 7“,, = yx o 7,, (Proposition 1). Similarly for ll'

3. INVERTIBLE ELEMENTS

DEFINITION 6. Let E be a unital magma, T its law ofcomposition, e its identity element and x and 16' two elements of E. x’ is called a left inverse (resp. right inverse, resp. inverse) (y‘xifx'Tx = e (resp. xTx’ = e, resp.x'Tx = xTx’ = e).

An element x ofE is called left invertible (resp. right invertible, resp. invertible) if it has a left inverse (resp. right inverse, resp. inverse). A monoid all of whose elements are invertible is called a group.

I5

ALGEBRAIC STRUCTURES

I

Symmetric and symmetrizable are sometimes used instead of inverse and invertible. 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 element f is 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, II, § 3, no. 8, Definition 11). Forf to be invertible it is necessary and sufficient thatf be 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. If x’ 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 at an element Qf E. (i) For x to be left (resp. right) invertible it is necessary and suficient that the right

(resp. left) translation by x be surjective. (ii) For x to be invertible it is necessary and suflicient that it be left and right invertible. In that case x has a unique inverse, which is also its unique left (resp. right) inverse. If x’ is a left inverse of x then (Proposition 1) 8x °sx’ = s.n’Tx = 63 = Id]:

and 8,, is surjective. Conversely, if 8,, is surjective, there exists an element x’ of E such that 8,,(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’Te =x’T (xTvx") = (x’Tx)Tx" =eTx' =x”,

whence (ii). Remark. Let E be a monoid and it 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 = Y: = hill:

and y,‘ is injective. In particular, ifx is left invertible, the left and right translations by x are bijective. Conversely, suppose that Yx is bijective; there exists x’ EE such that xx' = 7,,(x’) = e; then 7,,(x’x) = (xx’) x = x = 7,,(e) and hence x'x = e, so that x is invertible. We see similarly that if 5,, is bijective, x is invertible.

16

MONOID OF FRACTIONS OF A GOMMUTATIVE MONOID

§ 2.4

PROPOSITION 4. Let E be a monoid and x andy invertible elements ofE with inverses x’ and y' respectively. Then y’ T x’ is the inverse ofx T y. This follows from the relation

(y’Tx') T (No) =y’T (x'Tx)Ty =y’Tv = 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 9a).,“ of elements of E has an inverse x‘}, the composition “IA x“ has inverse “I" x;, where A’ is the totally ordered set derivedfrom A by replacing the order on A by the opposite order. This corollary follows from Proposition 4 by induction on the number of elements in A. In particular, if x and x’ are mverses, T x and T x’ are mverses for every integer n 2 0. o

o

n

n

e

COROLLARY 2. In a monoid the set Qf invertible elements is stable. PROPOSITION 5. Ifin a monoid x and x’ are inverses and x commutes with 3/, so does x’. From xTy=yTx, we deduce x’T (xTy)Tx’=x’T(yTx)Tx’ and hence (x'Tx)T (yTx') = (x’Ty)T (xTx’),thatisyTx' =x’Ty. 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 the inverse qfa central invertible element is a central element. 4. MONOID 0F 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 Rix, yf defined by: “there exist a, beE and p, q,seS’ such that x = (a,p), y = (b, q) and

aqs— = bps” is 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 = (b, q)

and z = (c, r) be elements of E x S' such that Rgx, y§ and Rig, 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)

l7

ALGEBRAIG STRUCTURES

I

and hence Rise, 2% 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’, 4’) be elements

of E x S’ such that Rgx, 31% and Rix’, y’i hold. There exist 5 and s’ in S’ such that aqs = bps,

a’q’s’ = b’p's’,

whence it follows that (aa’)(qq’) (ss’) = (bb’) (pp’) (93’) and hence Rgxx’, yy’é for ss’ 6 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 Es and is called the monoid qffi'actionsf 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 1). Then by definition (a/p) . (a’/p’) = aa’IPp’. The fractions (1/!) and a'/p’ are equal ifand only iftherc exists s in S’ with spa’ = sp’a; if so, there exist a and o" in S’ with oo- = a’a’ and

[76 = p’c’. In particular, a/p = sa/sp for a e E and s, p in S’. The identity element of Es is the fraction e/e. Let a/e = e(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 5(1)) = p/e; every element of s(S’) is therefore invertible. Then

a/p = (ale)-(e/11).whcncc (1)

0/! = e(0)49?“

for a e E and p e S’, the monoid E8 is therefore generated by e(E) U s(S)*. PROPOSITION 6. The notation is that ofDefinition 7 and 2 denotes the canonical homamorphism of E into Es. (i) Let a and b be in E; in order that 5(a) = e(b), it is necessary and suflicient that there exist s e S’ with sa = sb. (ii) For 1-: to be iry'ective it is necessary andsuficient that every element ofS be cancellable. (iii) For a to be bfiectioe it is necessary and suficient that every element if S be invertible. Assertion (i) is clear and shows that s 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. T It is also called monoid of deference: if the law on E is written additively.

l8

MONOID or martons OF A COMMUTATIVE MONOID

§2.4-

If e is bijective every element of S is invertible, for e(S) is composed of invertible elements of Es. 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 2 is injective by (ii) and a/p = s(a.p*) by (1), hence e

is surjective. THEOREM 1. Let E be a commutative monoid, S a subset of E, E3 the monoid offractions associated with S and e: E ——> E,s the canonical homomorphism. Further letf be a homomorphism ofE 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,3 into F

such thatf = fo e.

Iffis a homomorphism of Es into F such thatf = fo s, then

f(0/1?) =f(€(a)=(P)*) =f(=(a))f(e(fi))* =f(a)f(10* for a e E and p E S', whence the uniqueness off Let g be the mapping of E x S' into F defined by g(a, p) = f(a) .f(p)*. We show that g is a homomorphism of E x S’ into F. First of all,

g(e,e) =f(0)f(e)* = e. Let (a, p) and (a’, p’) be two elements of E x S’; as a’ and p commute in E, f(a') andf(p) commute in F, whencef(a’)f(p)* = f(p)*f(a’) by no. 3, Pro-

Position 5- M0rc°V6ff(PP')* =f(P’P)* = (f(P')f(P))* =f(P)*f(1")* by 110-

3, Proposition 4-, whence

g(M':PP’) =f(M')f(PP’)* =f(a) f(a')f(1’)*f(1")* = f(a)f(1’)"ff(a')f(P')* = g(a:P)g(a': Pl)

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’ = sap’, whence f(s)f(p)f(a’) = f(s) f(a)f(p’). As f(s) is invertible, it follows that

f(10f(a’) =f(a)f(12') and then by left multiplication by f(e)* and fight multiplic ation byf(p’)*

g(a',fi') =f(a')f(!")* =f(h_)*f(a) =f(a)f(10* = 301,1!)Hence there exists ahomomorphismfofE5 into F such thatf(a/p) = g(a, p),

when06f(=(a)) =f(a/e) =f(d)f(e)* =f(a)- Henccf° 8 =fCOROLLARY. Let E and F be two commutative monoidr, S and T subsets of E and F respectively, f a homomorphism of E into F such that f(S) C T and e: E —> Es,

7]: F —> FT the canonical homomorphisms. There exists one and only one homomorphism g:E3—+F1.suchthatgoe = 1] of:

The homomorphism 1; of of E into F.r maps every element of S to an invertible element of FT.

19

I

ALGEBRAIC STRUCTURES

Remarks. (1) Theorem 1 can also be expressed by saying that (ES, 5) is the solution of the universal mapping problem for E, relative to monoids, monoid homomorphisms and homomorphisms of E into monoids which map the elements of S to invertible elements (Set Theory, IV, § 3, no. 1). It follows (loo. cit.) that every other solution of this problem is isomorphic in a unique way to (E3, 5).

(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, §3, no. 2 (cf. Exercise 17).

We mention two important special cases of monoids of fractions. (a) Let E = E3. As the monoid E is generated by the set s(E) U a(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

group. Moreover, by Theorem 1 every homomorphismf of E into a group G can be uniquely factorized in the form f = fo 3, where f: E —> G is a homomorphism. E is called the group offraetions of E (or group qfdrfi’erences ofE in the case of additive notation). (b) Let (I) = E2, where 2‘. consists of the cancellable elements of E. By Proposition 6, (ii), the canonical homomorphism of E into (I) is injective; it will be profitable to identify E with its image in (1). Hence E is a submonoid of (1), every cancellable element of E has an inverse in (I) and every element of (I) is of the form a/p = a.p* with aeE and p62; then a/p = a’/p’ if and only if op’ = pa’. It is easily seen that the invertible elements of (I) are the fractions a/p, where a and p are cancellable, and flu is the inverse of (1/12. Now let S be a set of cancellable elements of E and S’ be the submonoid of E generated by S. If (1/1) and a'/p’ are two elements of E5, then (1/1; = a’/p’ if and only if ap’ = pa’ (for sap’ = spa’ implies ap’ = pa’ for all s e S’). E,s may therefore be identified with the submonoid of (I) generated by E U 8*. Ifevery element of E is cancellable, then (I) = E and E is a submonoid ofthe commutative group (1). Conversely, if E is isomorphic to a submonoid of a group, every element of E is cancellable. 5. APPLICATIONS: I. RATIONAL INTEGERS

Consider the commutative monoid N of natural numbers with law ofcomposition addition; all the elements of N are cancellable under this law (Set Theory, III, § 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 qfrotional 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 (m1, n1) and (ma, n2) which is written m1 + 712 = m, + 711; an element m of N is identified with the class consisting of the elements (171 + n, n), where

20

APPLICATIONS: I. RA'I‘IONAL INTEGERS

§2.5

n e N; it admits as negative in Z the class of elements (71, m + n). Every element (1:, q) ofN x Nmaybewritten in theform (m + n, n) ifp ) qorintheform A (n, m + n) ifp S q; it follows that Z is the union of N and the set of negative: 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}. formeN,m = —mifandonlyifm = 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 m = n +p;

(b) if m < n, then m + (-n) = —p, where p is the element of N such that M+P=m

(C) (“"1)+(—n)= —(m + '1)-

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 :4 + (—y) is abbreviated to x — 3/ (cf. no. 8). The order relation S between natural numbers is characterized by the

following property: m s n if and only if there exists an integer p e N such that m + [J = n (Set Theory, III, § 3, no. 6, Proposition 13 and § 5, no. 2, Proposition 2). The relation1/ — x e N between rational integers x and y is a total order relation on Z which extends the order relation < on N. For, for all x 62,

x —- x = OeNgify — xeNand z —yeN, then

z—x=eN, forN is stable under addition; ify —- x EN and x — y e N, theny — x = O, for

0 is the only element of N whose negative belongs to N; for arbitrary rational integersxandy,y — x eNorx — 3/ EN, forZ = N U (——N);finally,ifxandy are natural numbers, then y — x e N if and only if there exists ,0 e N such that x + 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 2 0; they are also called positive in-

tegers; the integers S 0, negatives of the positive integers, are called negative integers; the integers > O (resp. < 0) are called strictly positive (resp. strictly negative); the set of integers > 0 is sometimes denoted by N*. Let x, y and z be three rational integers; then x < y if and only if 21

ALGEBRAIG STRUCTURES

I

x + z Sy + z. Forx —y = (x + z) — (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 exist: a unique homomorphism f of N into E with f(1) = x and

f(n) = i xforallneN. (ii) If x is invertible, there exist: a unique homomorphism g of Z into E such that

g(1) = x and g coincides withf on N. Writingf(n) = “l3 x for all n e N, the formulae 'Io'x=e

and

($x)T('?x)=m‘ltnx

(no. I) express the fact thatf is a homomorphism of N into E and obviously f(l) = x. Iff’ is a homomorphism of N into E such that f'(l) = x, then f = f’, by § 1, no. 4, Proposition 1, (iv). Suppose now that x is invertible. By no. 3, Corollary 2 to Proposition 4-, f(n) = '1't x is invertible for every integer n 2 0. By construction, Z is the group ofdifferences ofN and hence (no. 4, Theorem 1) fextends uniquely to a homomorphism g of Z into E. If g’ is a homomorphism of Z into E with g’(l) = x, the restriction f’ of g' to N is a homomorphism of N into E with f’(l) = 1:. Hencef’ = 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 endomorphismf," of Z characterized by fm(1) = m. If m is in N, the mapping n n—> mn ofN into N is an endomorphism of the magma N (Set Theory, III, § 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 = fm(n) for m, n in Z. We shall establish the formulae:

(2) (3)

xy = we (xy)z = m2)

(4)

x(y + z) = xy + xz (x + y)z = xz + 3/2

(5) (6)

(7) (8)

0.x = x.0 = O 1.x = x.l = x (—l).x=x.(—l) = —x

for x, y, z in Z. (*In other words, Z is a commutative ring. *) The formulae x(y + z) = xy + xz and x. 0 = O express the fact thatf,c is an endomorphism 22

NOTATION

§ 2.8

of the additive monoid Z and f,(l) = at my be written x.l = x. The endomorphismfx ofy of Z maps 1 to xy and hence is equal tof”, whence (3). Now f,‘( ——y) = —fx(y), that is x(-—y) = -xy; similarly, the endomorphism y 1—» —xy of Z maps 1 to —x, whence (—-x) .g = —xy and therefore

(-x)(-y) = -(x(-y)) = -(-xy) = xyFor m, n in N, mn = nm (Set Theory, III, § 3, no. 3, Corollary to Proposition 5), whence (—m).n = n(—m) and (—m)(-—n) = (—n)(—m);asZ = N U (—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. Ifx is invertible in E, let gx be the homomorphism ofZ into E mapping 1 to

x. Let gx(n) = "l" x for all n e Z; by Lemma 2 this notation is compatible for n e N with the notation introduced earlier. Then

(11)

ll

—|o

(10)

1':

"$5: = (i- x) T (t x)

(9)

tx

e x

for x invertible in E and m, n in Z. Further, ify = ‘T’ x, the mapping 7: 1—» gx(m") of Z into E is a homomorphism mapping 1 to y, whence gx(mn) = gy(n), that is

Ex = 1"- (¥ x).

(12) .

.

.

"1

.

.

1

.

.

As —-1 is the negative of 1 in Z, T x is the 1nve1se ofx = T x in E. If we write -—m

_

m

n = —m in (9), it is seen that T x is the inverse of 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 x + (-y) is abbreviated to x —- y and similarly

x+g—z, x—y—z, x—y+z—t, etc.... represent respectively

It +y + (-Z), x + (—y) + (—z), x + (-—y) + z + (—t), etc. . .. 23

I .

ALGEBRAIC STRUCTURES

For n e Z the notation II" x is replaced by nx. Formulae (9) to (12) then become (13) (14-) (15) (16)

(m + n).x 0.2: 1.x m.(n.x)

= = = =

m.x + 72.x 0 x (mn).x

where mandn belong toN or even to Z ifx admitsanegative. Also in the latter case the relation (— l) .x = —-x holds. We also note the formula (17)

n.(x + y) = 11.x + my.

‘

(b) Let E be a monoid written multiplicatively. For n e Z the notation 1!“ x is replaced by x”. We have the relations xm+n ___ xmlxn

x° = l x1 = x (xm)fl=

xmn

and also (xy)" = xny" if x and y commute.

When at has an inverse, this IS pretzisely x'l. The notation E 15 also used Instead of x‘l. Finally, when the monoid E is commutative, E or x/y is also used

for spy—1

§3. ACTIONS 1. ACTIONS

DEFINITION 1. Let Q and E be two sets. A mapping qf £2 into the set E“ of mappings qf E into itself is called an action of Q on E.

Let a: r—>f,, be an action of Q on E. The mapping (on, x) u—->f,(x) (resp. (x, a) v—>fa(x)) is called the law qflefl (resp. right) action qf Q on ET associated with the given action of Q on E. Given a mapping g of Q x E (resp. E X (2) into E, there exists one and only one action at Hf“ of Q on E such that the associated law of left (resp. right) action is g (Set Theory, II, §5, no. 2, Proposition 3). In this chapter we shall say, for the sake of abbreviation, “law of action”

1' Or sometimes the external law of comfiosition on E with Q as operating set.

_24

ACTIONS

§ 3.1

instead of “law of left action”. The elementfa(x) of E (for a e Q and x e E) is sometimes called the transform of x under at or the composition of at and :5. It IS often denoted by left (resp. right) multiplicative notation 1.x (resp. x. at), the dot may be omitted, the composition of at and x is then called the product of at and x (resp. x and at). The exponential notation x“ is also used. In the arguments of the following paragraphs we shall generally use the notation at _L x. The elements of (2 are often called operators. Examples. (1) Let E be an associative magma written multiplicatively. The mapping which associates with a strictly positive integer n the mapping 1: »-> 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 .9: 1—> x“ 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 1—» x T A of the set of subsets of E into

itself is an action of E on ENE). (3) Let E be a set. The identity mapping of EE is an action of E3 on E, called the canonical action. The corresponding law of action is the mapping (f, x) I—>f(x) ofEE x E into E. (4) Let (0‘),“ be a family ofsets. For all i e I, letfl: (2.» E” be an action

of Q, on E. Let Q be the sum of the 9, (Set Theory, II, §4, no. 8). The maping f of 0 onto E“, extending the fl, is an action of Q 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 Q. on E with law denoted by _L, a subset E of Q and a subset X of E, E .L Xdenotes the set ofot J. xwith a 5 Sand .7: e X; when E consists of a single element a, we generally write a: J. X instead of {at} _L X.

The mapping which associates with at 6 Q the mapping X I—> a J. X is an action of Q on ENE), which is said to be derived from the given action by extension to the set of subsets. (6) Let a: Hf“ be an action of Q on E. Let g be a mapping of 9’ into (2. Then the mapping B 1—)fig” is an action of .Q’ on E. (7) Let f: E x E —> E be a law of composition on a set E. The mapping 7: x 1-) yx (resp. 5:1: 1-> 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 lefl (resp. right) action of E on itself derived from the given law. Whenfis 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 8'18 f (resp. the opposite law to f)

Let Q, E, Fbesets, at HfianactionofflonEandaHgaanactionofflon F. An Q—morphitm ofE into F, or mapping ofE into F compatible with the action of Q, is a mapping 1: of E into F such that

ga(h(x)) = 11mm) ‘ ,

25

ALGEBRAIC STRUCTURES

I

for all a e Q and x e E. The composition of two Q—morphisms is an Q-morphism. Let Q, E, E, F be sets, a H]; an action of Q on E, B »—> g“ an action of E on

F and (b a mapping of (2 into 9’. A ¢-morphism of E into F is a mapping h of E into F such that forallaeflande.

g¢(a)(h(x)) = h(fm(x))

2. SUBSETS STABLE UNDER AN ACTION. INDUCED ACTION

DEFINITION 2. A subset A of a set E is called stable under an action a Hf“ of Q on E

iffa(A) C Afor all on 6 9. An element x of E is called invariant under an element at of

Qiffodx) = x 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 (fa: ofuzon-ofiMXx), where xeX, n 2 O, aglor all i. Remark. Let E be a magma with law denoted by T. It should be noted that a subset A ofE which is stable under the left action on E on itselfis 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 ifand only if A T A C A whereas A is stable under the left (resp. right) action on E on itselfifand only ifE T A C A (resp. A T E C A). 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 {l} is the whole of N.

DEFINITION 3. Let a ref“ be an action of Q on E and A a stable subset of E. The mapping which associates with an element at 6 Q the restriction off“ to A (considered as a mapping ofA into itself) is an action of Q on A said to be induced by the given action. 3. QUO'I'IEN’I‘ ACTION

DEFINITION 4-. Let a safe. be an action (y‘a set 0. on a set E. An equivalence relation R on E is said to be compatible with the given action if, for all elements x and y of E such that x E 3/ (mod R) and all a e Q, f;(::) E fa(y) (mod R). The mapping which

associates with an element a e Q the mapping ofE/R into itselfderivedfromfa 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 left (resp. right) compatible with the law on E if it is compatible with the left (resp.

26

DISTRIBUTIVITY

§ 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 Propositions 6, 7 and 8 of§ 1, no. 6. 4. DISTRIBUTIVITY

DEFINITION-5. LetE1,...,E,, andF besets andu a mapping q1 x x E,l 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 if the partial mapping xi H “(an - ' ': “1—1: xi: “HI: - - -: an)

is a homomorphism (f EI into Ffir allfixed a, in E, andj ye i. If T denotes the internal laws on E, and F, the distributivity of u is given by the equations ' (I)

"(an ' ° ., at-l: x! T xi: “1+1: an)

= ”(01, ‘ - -: at—1:xt: “HI: ' . ': an) T “(an - - '3 “1-1: xi: aH-l: - - -: an)

for i = 1, 2:---: 71: a1 GED-“:at-IEE1-1: 155E” xlEEiaat+16Et+1:°--: an 6 En.

Example. Let E be a monoid (resp. group) written multiplicatively. The mapping (1:, x) tax" of N x E (resp. Z x E) into E is distributive with respect to the first variable by the equation 76”" = xmx" (with addition as law on N). If E is commutative, this mapping is distributive with respect to the second variable by the equation (xy)" = xny". PROPOSITION 1. Let E, E2, . . ., E, and F be commutative monoid: written additively

and let u be a mapping of E1 x - - - >< En into F, which is distributive with respect to » all the variables. For i = l, 2, . . ., 11, let L. be a non-emptyfinite set and (4,701,514 a

family qfelementson. Lety‘ = 2.21. xmfizri = l,2,...,n. Then 5!

(2)

u(y1, . . ., y”) = Z u(x1',1, . . ., arm“)

the sum being taken over all sequences at = (a1, . . ., at“) belonging to L1 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)

“(gun'syn-byn)‘: one 2I... "(y1:--‘:yn—1:xn,un)

27

ALGEBRAIC STRUCTURES

I

for y" = an; xmun and the mapping 2 H u(y1, . . .,y,,_1, z) of En into F is a magma homomorphism. By the induction hypothesis applied to the distributive mappings (21, . . ., 2,.-1) r—>u(zl, . . ., 2,.-1, xmm) from E1 x --- x E,"1 to F, (4)

“(l/1: - ' Wyn—1: xnmn) = “I“.Za’Pl “(xlmp - ~ -: xii-1.09.4, xenon”):

the sum being taken over the sequences («1, . . ., Gin—1) belonging to x Ln=M x Ln;writing M=L1 xx L,,_1.NowL1 x team...“ = “(x1,a1: - - ‘) xn.an)9

we have (5)

MHZ-.1!" tu1.....a,. = 112,.(u1.;.antu1"“’a"'1'a")

by formula (7) of § 1, no. 5. (2) follows immediately from (3), (4) and (5). Remark. If u(a1,...,a,_1,0,al+1, ..., an) = 0 fori=1, 2, ...,n and ajeE,

(j 79 i), then formula (2) remains true for families (x,_,'),‘em offinite support. A special case of Definition 5 is that where u is the law of action associated with the action of a set 9 on a magma E. If u is distributive with respect to the second variable, it is also said that the action of 52 on the magma E is distributive. In other words: DEFINITION 6. An action a: Hf“ of a set 9 on a magma E is said to be distributive

if, for all at 6 Q, the mappingfa is an endanwrphism of the magma E. If T denotes the law of the magma E and J. the law of action associated with the action of Q on E, the distributivity of the latter is then expressed by the formula aJ.(xTy)=(ocJ_x)T(ocJ_y) (foraeQandx,yeE). (6) By an abuse of language, it is also said that the law _L is distributive (or tight distributive) with respect to the law T. Formula (2) of § 1, no. 2 shows that then, for every ordered sequence (x,_) MEL of elements of E and every at e Q, (7)

a. .L (ll-L x1) = A; (at J. xx).

If an action at Hf“ is distributive and an equivalence relation R on E is compatible with the law ofcomposition of E and the action at I—>f[,, the quotient action on E/R is distributive.

When the law on E is written multiplicatively, we often use the exponential notation x“ for a law ofaction which is distributive with respect to this multipli-

cation, so that distribu'civity is expressed by the identity (xy)°‘ = fly“. If the law

28

DISTRIBUTIVIT’Y OF ONE INTERNAL LAW WITH RESPECT TO ANOTHER

§ 3.5

on E is written additively, we often use left (resp. right) multiplicative notation onx (resp. x.ot) for a law of action which is distributive with respect to this addition, the distributivity being expressed by the identity «(x+y) =auc+ccy

(resp.(x+y)ac=xac+yot).

We may also consider the case where Q has an internal law, denoted by T, and the law of action is distributive with respect to the first variable, which means that

(8)

(ot'TB)J.x=(ot_Lx)T(flJ_x)

for all a, {3 e Q and x e E. Then, by formula (2) of§ 1, no. 2

(9) (iii. a”) 'L x = 1.11.. (0“ J‘ x) for every ordered sequence («QML of elements of Q and all x e E. 5. DISTRIBUTIVITY OF ONE INTERNAL LAW WITH RESPECT TO ANOTHER

DEFINITION 7. Let T and J. be two internal laws on a set E. The law _L 2'." said to be

distributive with respect to the law T if (10)

x_L(yTz)=(x.Ly)T(xJ.z)

(11)

(xTy)iZ=(xJ-Z)T(yJ-Z)

for all x,y, z in E. Note that (10) and (11) are equivalent if the law J. is commutative. In general, one of the laws is written additively and the other multiplicatively; if multiplication is distributive with respect to addition, then:

(12) (13)

x.(y + z) = any + x.z (x+y).z=x.z+y.z Examples. (1) In the set 33(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

An (BUC) = (An B) U(AnC) AU (BnC) = (Au B) n(AuC). (2) In Z (and more generally, in any totally ordered set) each of laws sup and inf is distributive with respect to the other and with respect to itself.

(3) In Z (*and more generally in any ring‘) multiplication is distributive with respect to addition. (4-) In N addition and multiplication are distributive with respect to the

laws sup and inf.

29

ALGEBRAIC STRUCTURES

I

§4. GROUPS AND GROUPS WITH OPERATORS l. GROUPS

Recall the following definition (§ 2, no. 3, Definition 6).

DEFINITION l. A set with an associative law of composition, possessing an identity element and under which every element is invertible, is called a group.

In other words, a group is a monoid (§ 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 homomorphismfmaps 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 multiplicatively, e.e = e, whencef(e) .f(e) = f(e) and, multiplying byf(e) ' 1,f(e) =e’. Hencef. is unital. It then follows from no. 3 of § 2 thatf(x‘l) = f(x) '1 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) Hf o g, called the symmetric group of the set F and denoted by 6“.

In this paragraph, unless otherwise indicated, the law of composition of a group will always be written multiplicatively and e will denote the identity element of such a group law. A group G is called finite if the underlying set of G is finite; otherwise it is

called infinite; 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 ofx is an isomorphism ofG onto the opposite group (§ 2, no. 3, Proposition 4). Following our general conventions (Set Theory, II, §3, no. 1), we shall denote by A"1 the image of a subset A of G under the mapping x I—->- it”. But it is important to note that, in spite of the analogy of notation, A’1 is definitely not the inverse element of A under the law of composition (X, Y) v—> 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 invertible elements of $(G) under this law are the sets A consisting of a single

element (such an A, moreover, certainly has inverse A”). The identity

30

GROUPS WITH OPERATORS

-§ 42

(AB) ‘1 = B‘IA‘1 holds for A C G, B C G. A is called a symmetric subset ofG ifA = A”. For all A C G, A U A”, A n A‘1 and AA'1 are symmetric.

2. GROUPS WITH OPERATORS

DEFINITION 2. Let .Q be a set. A group G together with an action of Q on G which is distributive with respect to the group law, is called a group with operators in Q. In what follows x“ will denote the composition of at 6 Q and x e G. Distributivity is then expressed by the identity (1:31)“ = fly“. 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 so obtained by giving G the unique action of g on G. This allows us to consider groups as special cases ofgroups with operators and to apply to them the definitions and results relating to the latter which we shall state. Example. In a commutative group G, written multiplicatively, (xy)" = x'fiy" for all n e Z (§ 2, no. 8, equation (1)); the action 71 r—> (x n—>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 Q. A homomorphism of groups with operators of G into G’ is a homomorphism of the group G into the group G’ such that fir all qandallG.

f(95“) = (f(1))“

An endomorphism of the group with operators G is an endomorphism of the group G which is permutable with all the homotheties of G. As two homotheties of a group with operators G are not necessarily permutable, a homathety of G is not in general an endomorphism of the group with operators G.

The identity mapping of a group with operators is a homomorphism of groups with operators; the composition of two homomorphism 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.

31

ALGEBRAIG STRUCTURES

I

More generally, let G (resp. G') be a group with operators in Q (resp. 9’). Let 4) be a mapping of 9 into 0’. A b-homamorphism of G into G’ is a homomorphism of the group G into the group G' such that

f(95“) = (f(x))°‘°" forall ae‘Qand allxeG. In the rest of this paragraph we shall be given a set 9. Unless otherwise mentioned the groups with operators considered will admit Q as set ofoperators. 3. SUBGROUPS

DEFINITION 4. Let G be a group with operators. A stable subgroup ofG is a subset H of G with thefollowing properties: (i) eeH;

g

(ii) x,yeHimpliesxyeH; (iii) e implies x'leH; (iv) eandoceQimplyxueH.

If H is a stable subgroup ofG, 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. Let G be a group. A stable subgroup of G with the action of o (no. 2), which

is a subset ofG satisfying conditions (i), (ii), (iii) of Definition 4, is called a subgroup ofG. When we speak ofa subgroup ofa group ofoperators 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. Example (1). Let 2 be a species of structure (Set Theory, IV, § 1, no. 4) and S a structure of species 2 on a set E (loo. cit.). The set of automorphisms of S is

a subgroup of 63,-.

PROPOSITION 1. Let G be a group with operators and H a subset of G which is stable under the homotheties of G. Thefollowing conditions are equivalent:

(a) H is a stable subgroup of G. (b) H is non-empty and the relations at e H, y e H imply xy 6 H and x‘1 e H. (c) H is non-empty and the relations x E H, y e H imply xy'1 6 H. (d) H is stable under the law on G and the law of composition 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 at“ e H and e = xx'1 6 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

SUBGROUPS

§ 4.3

xx" = e is an element ofH. For every element x ofH, x"1 = ex'1 belongs to H; hence the relations at e H, y e H imply :c(y‘1)‘1 == xye 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 .7: e H implies x‘ 1 e H (no. 1). Remarks. (1) Similarly it can be shown that condition (b) is equivalent to the condition (0’) H ;é a and the relations at e H and y E H imply y'lx e H. (2) For every subgroup H of G there are the following relations

(1)

H.H = H and H—1 = H.

ForH.H C HandH'1 C Hby (b).AseeH,H.H D e.H = Handtaking inverses transforms the inclusion H“1 C H into H C H“, whence

formulae (1).

If H 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) ofH. PROPOSITION 2. Let X be a non-empty subset of a group with operators G and X the stable subset under the action of Q 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 = X U X“.

The latter subset Z is the set of compositions of finite sequences all of whose terms are elements of X or inverse of elements of X: the inverse of such a composition is a composition of the same form (§ 2, no. 3, Corollary 1 to Proposition 5) and Z is stable under the action of Q, as is seen by applying § 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 Q. The subgroup generated by X and the stable subgroup generated by X coincide. COROLLARY 2. Let G be a group and X a subset g“ G consisting ofpairwise permutable elements. The subgroup of G generated by X is commutative.

The set Y = X U X‘ 1 consists of pairwise permutable elements (§ 2, no. 3, Proposition 5) and the law induced on the stable subset generated by Y is commutative (§ 1, no. 5, Corollary 2).

33

I

ALGEBRAIG STRUCTURES

If G is a group with operators, the stable subgroup generated by a subset of G consisting of pairwise permutablc elements is not necessarily commutativc.

COROLLARY 3. Letf: G —> G’ be a homomorphism ofgroups with operators and X a subset of G. The image underf of the stable subgroup of G generated by X is the stable subgroup of G’ generated byf(X).

Let X' =f(X). Then X' =f(X) and x'-1 = f(X-1). Hence

f(X u X-l) = X' I.) X'-1. The corollary then follows from § 1, no. 4, Proposition 1.

Example (2). Let G be a group and x an element of G. The subgroup generated 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 6 Ni". These two sets are always distinct if x ¢ 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 ofG and H a stable subgroup of G not meeting P, the set ofstable subgroups of G containing H and not meeting P, ordered by inclusion, is inductive (Set Theory, III, § 2, no. 4). Applying Zom’s Lemma (Set Theory, III, § 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 subgroup G not meeting P. The set ty'stable subgroups ofG 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; ifR is left (resp. right) compatible (§ 3, no. 3) with the group law on G and compatible with the action of Q, the equivalence class of e is a stable subgroup H g“ G and the relation R is equivalent to x‘ y e H (resp. gee—1 e H). Conversely, if H is a stable subgroup of G,

the relation x“ y e H (resp. yx‘l e H) is an equivalence relation which is left (resp. right) compatible with the group law on G and compatible with the action of Q 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 relation1/ .=_ x (mod. R) is equivalent to x‘ y E e (mod. R), for y E x implies x'ly E x'lx = e and conversely fly 5 e implies y = x(x‘ 1y) E x. If H denotes the equivalence class of e, the 34

QUO'I'IENT GROUPS

§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 at, the relation as E e implies x“ a e“ = e, hence H“ C H and H is stable under the action of Q. It suffices to establish (Proposition 1) thate andyeH imply x‘lyeH, that is x E e andy E eimply x .=.. g, which is a consequence of the transitivity of R.

Conversely, let H be a stable subgroup of G; the relation x‘ y E H is reflexive since x'lx = e e H; it is symmetric since x'ly e H implies y'lx = (.2c“1_1/)'1 EH; it is transitive, for x“ y e H and y‘lz e H imply x" 17. = (x'1y)(y‘ 1z) e H; it is left compatible with the law of composition on G, for x'ly = (zx)‘1(zg) for all z e G; finally, for every operator at, the relation y e xH implies y“ e x°‘H°‘ C x“H and hence the equivalence relation K1}; 6 H is compatible with the action of Q on G.

Let G be a group and H a subgroup of G; the relation x’ 3/ EH (resp. ‘ yx‘l e H) is also written in the equivalent form 1/ e xH (resp. y e Hx). Every subgroup H ofG 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 left cosets of H (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, II, §6, no. 4), we obtain respectively the sets AH and HA. The mapping x »—> x‘1 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 subgroup H with respect to G and is denoted by (GzH); 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 ofH contained in a left coset of K has cardinal independent of the latter. Hence (Set Theory, III, § 5, no. 8, Proposition 9): , PROPOSITION 4-. Let H and K be two subgroups of a group G such that H C K. Then (G :H) = (GzK) (K:H). (2) COROLLARY. If G is a finite group of order g and H is a subgroup of G of order h, then h. (GzH) = g (3) (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: if R is such a relation, it is both left and right compatible with the group law on G and with the action of (2. Hence, if H is the class of e (mod. R), H is a stable subgroup such that the relations 3/ exH and y eHx are equivalent (since both are equivalent to R); hence

xH = Hx for all x e G. Conversely, if this is so, one or other of the equivalent 35

I

ALGEBRAIC STRUCTURES

relations y 6 AH, y e Hx is compatible with the group law, since it is both left and right compatible with this law (§ 3, no. 4) and is compatible with the action of 9. Since the equation xH = Hx is equivalent to xHoc‘1 = 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 of G if xHx’ 1 = Hfor all x e G.

If Q = o, 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 .‘e‘1 C H for all xe G; for if so then x‘1Hx C: H for all xe G, that is H C xHx‘l, 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 Q, the quotient by R of the action of Q on G, is distributive with respect to the internal law on G/R (§ 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 9, it is necessary and sufl‘icient that it be of theform x‘ y e H, where H is a normal stable subgroup qf G (the relation x“1y e H being moreover equivalent to yx' 1 e Hfor such a subgroup). The law of composition on G/R the quotient of that on G and the action of Q on G/R the quotient of that of Q 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 qf a group with operators G by the equivalence relation defined by a normal subgroup H qf 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 ofgroups with Operators, it is necessary and suflicient that its composition with the canonical mapping of G onto G/H be one: this justifies the name “quotient group” (Set Theory, IV,

§2, no. 6). The equivalence relation defined by a normal stable subgroup ofG is denoted by x s y (mod. H) or x E y(H).

36

DECOMPOSITION OF A HOMOMORPHISM

§ 4.5

PROPOSITION 5. Letf: G —-> G’ be a homomorphism ofgroups with operators and H and H' normal stable subgroups ofG and G’ respectively such thatf(H) C H’. The mapping fis compatible with the equivalence relations defined by H and H'. Let 1:: G —> G/H and 1c': G’ —> G’/H’ be the canonical homomorphisms. The mapping f: G/H —-> G’/H' derivedfromf by passing to the quotients is a homomorphism. If x :- y (mod. H), then x‘ y e H, whence

f(x) ‘1f(y) =f(x'1)f(!/) =f(x‘1y)ef(H) C H’ and hencef(x) E f(y) (mod. H’). The second assertion follows from the universal property of quotient laws (§ 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 a 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 ifH is a normal subgroup ofa group G and K is a normal subgroup of H, K is not necessarily a normal subgroup of G (I, § 5, Exercise 10). Let G be a group with operators. The intersection of every family of normal stable subgroups ofG 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 if G ye {e} and there exists no normal stable subgroup of G other than G and {e}. 5. DECOMPOSITION OF A HOMOMORPHISM

PROPOSITION 6. Let G be a group with operators and G' a magma with an action by 9, written exponentially. Let f: G —>G’ be a homomorphism of the magna G into the magma G’ such that,for all one Q and all are G,f(x°‘) =f(x)°‘. Thenf(G) is a stable subset ofG’ under the law on G' and the action off); the setf(G) with the induced laws is a group with operators and the mapping x n—>f(x) of G into f(G) is a homomorphism ofgroups with operators.

By virtue of § 1, no. 4, Proposition 1, f(G) is a stable subset of G’ under the internal law on G’. For every element x e G and for every operator at, f(x)°‘ = f(x‘) ef(G) and therefore f(G) is stable under the action of Q on G’. Writing the internal law of G’ multiplicatively,

(f(x)f(y))f(Z) =f(xy)f(2) =f((xy)Z) =f(x(yZ)) =f(x)f(yZ) = f(x) (fWV(2)) for all elements x, y, z in G; therefore the induced law on f(G) is associative.

37

ALGEBRAIG STRUCTURES

I

Let e be the identity element of G. Its image f(e) is an identity element of f(G) (§ 2, no. 1). Every element of f(G) is invertible in f(G) (§ 2, no. 3). Therefore the law induced on f(G) by the internal law on G’ is a group law. For all elements at and y in G and every operator at,

(f(x)fM)“ = (f(5%))“ =f((xy)°‘) =f(x°‘y°‘) =f(WV(31“) = (f(x))“(f(31))“ which shows that the action of Q is distributive with respect to the group law onf(G). Thereforef(G) with the induced laws is a group with operators and clearly the mapping x n—>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 (J the identity element of G’ is called the kernel off. The kernel offis often denoted by Ker(f) and the imagef(G) offis sometimes denoted by Im(f).

-

THEOREM 3. Letf: G ——> G’ be a homomorphism ofgroups with operators. (8.) Ker(f) is a normal stable subgroup of G; (b) Im(f) is a stable subgroup of G’; (c) the mappingfis compatible with the equivalence relation defined on G by Ker(f) ;

(d) the mapping]; G/Ker(f) -——> Im(f) derivedfrom f by passing to the quotimt is an isomorphism ofgroups with operators; (e) f = L ofo 11:, where L is the canonical injection of Im(f) into G’ and 11: is the canonical homomorphism of G onto G/Ker(f). Assertion (b) follows from Proposition 6. The equivalence relation f(x) = f(3/) on G is compatible with the group with operators structure on G. By Theorem 2 (no. 4), it is therefore of the form 1/ 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, II, §6, no. 5). 6. SUBGROUPS OF A QUOTENT GROUP

PROPOSITION 7. Let G and H be two groups with operators, f a homomorphism of G into H and N the kernel off. -1 (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 subgroup of G’. Iff is surjectioe, then H' = f(G’) andf defines an isomorphism of G’/N onto H’ on passing to the quotient. (b) Let G’ be a stable subgroup of G. The image H’ = f(G’) is a stable subgroup -

—1

ofH and f1(H') = G’N = NG’. Inparticular, f (H') = G’ ifand only ifN C G'. Iff is swjectioe and G’ is normal in G, then H' is normal in H.

(a) Let x and y be in G' and at e Q; then f(x) e H’ and f(y) e H’, whence 38

sunonoups OF A QUOTIENT GROUP

§ 4.6

f(xi/'1) = f(x)f(y) ‘1 EH’, that is xy‘1 6 G’; hence G’ is a subgroup of G. Now f(xu) = f(x)°‘ e H’, whence x“ E G’ and therefore G’ is stable. Suppose H’ is normal in H and let x e G’, g e G; thenf(x) e H’ and

f(yxy‘l) =f(y)f(x)f(y)‘1 eH’ whence yxy‘1 E G'; hence G’ is normal in G. For all n EN, f(n) = e eH’, whence N C G’; as N is normal in G, it is normal in G’. Finally, iffis surjective, —1

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 at in Q; there exist 1:, y in G’ with a = f(x) and b = f(y), whence ab'1 = f(xy‘l) eH’, hence H’ is a subgroup of H

which is stable, for a“ = f(x‘x) e H’. Let x e G; then x e f(H’) if and only if f(x) 6 H' =f(G’), that is ifand only ifthere existsy in G’ withf(x) =f(y); the relation f(x) = f(y) is equivalent to the existence of n e N with x = yn; —1

finally, x e f (H') is equivalent to x e G’N = NG’. Clearly the relation G’ = G’N is equivalent to G’ D N. Suppose finally thatf is surjective and G’ isnormalinG; leta eH' and b eH;thereexistxeG’ andyeitha =f(x) and b =f(y), whence bab‘1 =f(yxy‘1) Ef(G') = H’. Hence H’ is normal in H.

COROLLARY 1. Suppose thatf is surjeetioe. Let 6 (resp. 6’) be the set ofstable (resp. normal stable) subgroups of G containing N and 5 (resp. 55’) the set of stable (resp. normal stable) subgroups (y’ H, these sets being ordered by inclusion. The mapping G’ .—>f(G’) is an ordered set isomorphism (I): 6 —> if); the inverse isomorphism —1

‘F: % —> 6 is the mapping H’» f (H’). Further (I) and ‘I’ induce isomorphisms (1726’ —>%’ and ‘1”: 5’ —> 5’.

»

COROLLARY 2. Let f: G ——> H be a homomorphism of groups with operators, N the kernel of], G’ a stable subgroup of G and L a normal stable subgroup of G’. Then LN, L. (G’ n N) andf(L) are normal stable subgroups of G’N, G’ andf(G’) respectively

and the three quotient groups with operators G’N/LN, G’/L. (G’ n N) andf(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

f’ (f'(L)) = L-(G' n N) is a normal stable subgroup of G’. Let 7. be the canonical homomorphism ofH' onto H’/f'(L) = f(G’) [f(L): as A of’ is surjective with kernel

fem» = L. (G' n N), 39

ALGEBRAIC STRUCTURES

I

it defines an isomorphism of G’IL. (G’ n N) ontof(G’)/f(L). By Proposition —1 7, (b), f (H’) = G’N ; iff” is the homomorphism of G'N onto H' which coincides withf on G’N, the homomorphism A of" of G'N ontof(G’)[f(L) is surjective with kernel }1(f(L)) = LN; this proves that LN is a normal stable subgroup of G'N and that A of” defines an isomorphism of G’N/LN onto

f(G')/f(L). COROLLARY 3. Let f: G —> H be a homomorphism of groups with operators, N its kernel, X a subset ifG such thatf(X) generates H and Y a subsetq which generates N. Then X U Y generates N.

Let G’ be the stable subgroup ofG generated by X U Y. As Y C G', N C G'. —1 Asf(X) Cf(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. If A and B are subsets of H andf is surjectioe, then

f(AB) = }1(A)-}1(B), Fox-1) = f(A) Obviously }1(A).}1(B) c f(Ah); on the other hand, if z e }1(A.B), there

exists a e A and b e B such thatf(z) = ab; asfis suljective, there exists or e G such that f(x) = a; writing y = x’lz, f(y) = a‘ff(z) = b and z = xy, _1 _. 1 -1 whence z e f (A) . f (B). The relation x e f (A‘l) is equivalent tof(x) e A'l, _1

-1

hence tof(x‘1)e A, that is to x“1 e f(A) and finally to x e f(A) '1. 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'1 e B *(in other words, A normalizes B) *. Then AB = BA is a stable subgroup if G, A n B is a normal stable subgroup ofA and B is a normal stable subgroup QfAB. The canonical injection ofA into AB defines on passing to the quotient an isomorphism of A/ (A n B) onto AB/B.

The formulae

(ab) (a'b') = aa'(a' ‘1ba’ . b') (ab) '1 = a'1(ab‘1a‘1) (ab)°‘ = club“ for a, a’ E A, b, b' e B and every operator at on G, show that AB is a stable subgroup of G. Let a e A and x 6A A B; then ana’1 e B by the hypotheses

made on A and B and clearly area" 1 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) ‘ 1 = a(bb’b ‘ 1)a' 1 shows that B is normal in AB. Let c]: be the restriction to A ofthe canonical homomorphism

40

THE JORDAN-HOLDER THEOREM

§4.7

of AB onto AB/B; then (Mo) = a3 and hence the kernel of (I) is equal to A n B. Clearly G’/N is a bijection of the set ofstable subgroups of G containing N onto the set ofstable subgroups of G/N. (b) Let G' be a stable subgroup ofG containing N. For G’/N to be normal in G/N, it is necessary and suflicient that G' be normal in G and the groups G/G’ and (G/N)/(G’/N) are then isomorphic. (e) Let G' be a stable subgroup of G. Then G’N is a stable subgroup of G and N is normal in G’N. Further G’ n N is normal in G' and the groups G’/(G’ n N) and G'N/N are isomorphic.

Let f denote the canonical homomorphism of G onto GIN. For all x E G, f(x) e xN; therefore,f(G’) = G'/N for every subgroup G’ of G containing N. As f is surjective, assertion (a) follows from Corollary 1 to Proposition 7; similarly for the equivalence “G' normal” “G’/N normal”. Suppose that G’ is a normal stable subgroup of G containing N. By no. 4, Proposition 5 applied to ma, 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, (0) 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 (GQO‘Kn of stable subgroups of G, with G0 = G and G" = {e} and such that Gm is a normal subgroup ofG,for O S i S n — l. The quotients G,/G‘+1 are called the quotients of the series. A composition series 2’ is said to befiner than a composition series 21 ifZ is a series takenfrom 2'. {f (G,)0(K,, and (H,)o‘,‘m are respectively composition series of two groups with operators G and H, they are said to be equivalent if m = n and there exists a permutation

i + 1, G, is not in general a normal subgroup of G,.

THEOREM 5 (Schreier). Given two composition series 21, 22 of a group with operators G, there exist two equivalent composition series 21, 2;, finer respectively than 21 and

22. Let 21 = (Hanan: and 22 = (K,)og, 0. The group Z/aZ ofrational integers modulo a is a group g" order a.

PROPOSITION 17. Let H be a subgroup ofZ. There exists one and only one integer a 2 0 such that H = a2.

If H = {0}, then H = OZ. Suppose that H 56 {O}. The subgroup H has an element x gé 0. Then at > 0 or —x > 0, and therefore H has elements >0.

Let a be the smallest element >0 in H. The subgroup aZ generated by a is contained in H; we show that H C (12. Let y e H. By Proposition 16, there exists an integer r such that y E r (mod. a) and 0 < r < a. A fortiori y _=. r

(mod. H), whence rEH. But this is only ‘possible if r = 0 and therefore y 6 oz. The integer a is unique: if H = {0}, then necessarily a = 0, and if H 7e {0}, the integer a is the order of Z/H. DEFINITION 15. A group is called monogenous ifit admits a system ofgenerators consisting of a single element. Afinite 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.

49

I

ALGEBRAIG STRUCTURES

PROPOSITION 18. A finite monogenous group of order a is isomorphic to Z/aZ.. An infinite monogenous group is isomorphic to Z. Let G be a monogenous group (written multiplicatively) and x a generator ofG. The identity xmx" = x'" +" (§ 1 , no. 3, formula (1)) shows that the mapping n H x" is a homomorphism ofZ into G. Its image is a subgroup of G containing at 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 > O, the group G is finite of order a and if a = O, the group G is isomorphic to Z. PROPOSITION 19. Let a be an integer > 0. Let H be a subgroup o/aZ, b the order Qf H and e its index in Z/aZ. Then a = be, H = eZ/aZ and H is isomorphic to Z/bZ. Conversely, let b and e be two integers >0 such that a = be. Then aZ C oz and eZ/aZ is a subgroup ofZ/aZ, of order b and index 0. a = be 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 6. By Proposition 17 and the Corollary to Proposition 16, H’ = oz and hence H is monogenous. Finally, H is isomorphic to Z/bZ by Proposition 18. Conversely, if a = be, then aZ C oz for a eoZ: the quotient group (Z/aZ)/(eZ/aZ) is isomorphic to Z/eZ (no. 7, Theorem 4) and hence of order e (no. 4, Corollary to Proposition 4) and index b (no. 4, Corollary to Proposition 4).

COROLLARY. Every subgroup of a monogenous group is monogenous. Let a and b be two integers $0. The relation b 6 oz is also written: b is a multiple ofa, and also a divides b or a is a divisor of b.

DEFINITION 16. An integerp > 0 is calledprime ifp aé l and it admits no divisor >l other than p. PROPOSITION 20. An integer p > 0 is prime if and only if the group Z/pZ is a simple group.

This follows from Proposition 19. COROLLARY. Every commutative simple group is cyclic (y‘prime order. Let G be such a group. Then G aé {e}; let a 75 e be an element ofG. The subgroup 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 at c.

50

MONOGENOUS GROUPS

§ 4.10

Lemma 2. Let a be an integer >0. By associating with every composition smes (Hl)0 out 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 operatefreely on E if, for all x e E, the orbital mapping defined by

x is injective or also if the mapping (g, x) I—av (gx, x) of G x E into E x E is injective. 56

onerrs

§ 5.4-

Examples. (1) Let G be a group and consider the operation of G on itself by inner automorphisms. Two elements of G which are conjugate under this operation are called conjugate under 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 aeG such that

H’ = «Jinx—1, that is if they are conjugate under the extension to ‘B(G) of the operation of G on itself by inner automorphisms. (2) 1*In the space R”, the orbit of a point at under the action of the orthogonal group 0(n, R) is the Euclidean sphere of radius “x“qt

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) v-> n = xgH; under this operation, H operates trivially, whence a right operation of G/H on E/H. Let ‘i’ be the canonical mapping of E/H onto E/G; the inverse images under 4) of the points of E/G are the orbits of G (or of G/H) in E/H. Hence (1) 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, e and hEH.

The action of H on E is also a left operation of the opposite group H° to H. It then follows from § 4-, no. 9, Proposition 12 that the mapping which associates with the element (g, h) e G x H° the mapping .7: H g.x.h of E into itself is a left operation of G x H" 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

57

ALGEBRAIG STRUCTURES

I

(where at, B, y, 8, 3 denote the canonical mappings of taking quotients), Y c at = 8 o B = e.

Let G be a group and H a subgroup of G. Consider the right operation ofH on G by right translation (no. 1, Example 2). The set of orbits G/H is the set of left cosets modulo H; note that G operates on G/H on the left by the law (g, xH) I—> n (cf. no. 5). Similarly, the set ofright 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 l" is a left (resp. right) coset modulo H, then I‘K (resp. KI‘) is a left (resp. right) coset modulo K. The mapping I‘ 0—) PK (resp. I‘ -—> KP) 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 left 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 of H\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 subgroup 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 if the operation of G 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 at and y of E, there exists an element at e G such that our = y. e 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 left by (g, xH) »—-> n. Let N be the normalizer of H. The group N operates on G/H on the right by (AH, n) r—> a = a. 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 (I): (N/H)° —> 65,3 be the homomorphism corresponding to this operation. PROPOSITION 5. With the above notation, G/H is a homogeneous G—set. The mapping

66,]; the homomorphism corresponding to the operation of G on G/H. The kernel of d) 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 operates 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 (of. 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-morphirm f q/K into E such thatf(e.K) = a. IfK = H,fis an isomorphism.

Iffis 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, ify = xk, k e K, then y.a = xk.a = x.a.

A mappingf 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 = H and let us show thatfis injective. Iff(x.H) =f(y.H), thenx.a = y.a,

whence x’ly.a = a and x‘ y EH, whence x.H = y.H. THEOREM 1. Let G be a group. (a) Every homogeneous G-set is isomorphic to a homogeneous G-set of thefimn G/H where H is a subgroup of G.

(b) Let H and H’ be two subgroups ofG. 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). Letf: 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

I

ALGEBRAIC STRUCTURES

element of G/H’. The subgroups H and H’ are therefore conjugate (no. 2, Proposition 2). If H’ = atHot‘l, H’ is the stabilizer of the element at.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 63 operates transitively

on E. If x and y are two elements of E, the mapping 1:: E ——> E such that 1(x) = y, 1(y) = x and 1(2) = z for z ¢ x, y, is a permutation of E. Let as E. The stabilizer of a is identified with GE, where F = E - {a}. The homogeneous G-set E is thus isomorphic to 63/61:. (2) Let E be a set of n elements and (14).“ a finite family of integers > 0

such that ‘21), = n. Let X be the set of partitions (F0151 of E such that Card(F,) = p, for all i. The group 6,; operates transitively on X. The stabilizer H of an element (F,),61 of X is canonically isomorphic to 11 6F,

and is hence of order n pd. Applying Theorem 1 and § 4, no. 4, Corollary to E Proposition 4, a new proof is obtained of the fact that Card(X) =

n!

‘13!“ In particular, take I = {1, 2, . . ., r}, E = {1, 2, . . ., n},

F: = {P1 +"'+P1—1 +l:"':p1 +"‘+P1} for l < i < 1. Let S be the set of r e 63 such that 1|“ is increasing for

l S i < r. If (G1, . . ., G,) e X there exists one and only one we S which maps (F1, . . ., F,) to (G1, . . . , G,). In other words, each left coset of GE modulo H meets S in one and only one point.

(3) *Let n be an integer 21. The orthogonal group 0(n, R) operates transitively on the unit sphere Sud in R". The stabilizer of the point (0, . . .,0, 1) is identified with the orthogonal group 0(n — 1, R). The homogeneous 0(n, R)—set 8,...1 is thus isomorphic to 0(n, R) /O(n —— l, R)...l 6. HOMOGENEOUS PRINCIPAL SETS

DEFINITION 7. Let G be a group. An operation if G on a set E is called simply transitive if there exists an element x qfE such that the orbital mapping defined by x is a bijection. A set E, together with a simply transitive lefl action qf G on E, is called 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 aeGsuchthatocx =3]. 60

PERMUTATION GROUPS or A FINI’I‘E SET

§ 5.7

Condition (ii) is also equivalent to the following condition: (iii) the mapping (at, x) I—> (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 G563.

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) n—>f o a 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 (a,j) I—> a 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 afiine space (cf. II, § 9, no. 1)....

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 G0 (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 4;, is derived of G0 onto Aut(E). It should be noted that 4),, in general depends on a; more precisely, for a e G, ‘i’aw = ‘i’a ° Int G°(a) = ‘i’a ° Int(a—1)'

(3)

For, writing 8,, for the translation x n—> xa on G, Qua = (i’ma ° set

and ‘l’a(x) = wa°sx°mu_la

xEG,

whence 41““) = (”a ° 80: ° 5:: ° 80:1 ° 00:1 = (“a ° sa’lxu ° 00:1 = ‘l’a(°‘-_lx¢)o 7. PERMUTATION GROUPS OF A FINITE SET

If E is a finite set with n elements, the symmetric group 6,, (§ 4, no. 1) is a

finite group of order 72!. When E is the interval [1, n] of the set N ofnatural

numbers, the corresponding symmetric group is denoted by 6,, ; the symmetric

group of any set with n elements is isomorphic to 6”. 61

I

ALGEBRAIC STRUCTURES

DEFINITION 8. Let E be afinite set, E 6 61., a permutation of E and Z the subgroup of GE generated by C. C is called a cycle if, under the operation of Z on E, there exists one and only one orbit which is not reduced to a single element. This orbit is called the support of t.

Let C be a cycle. The support supp(C) of C is the set of xEE such . that C(x) 9e 1:. The order of a cycle t is equal to the cardinal of its support. The subgroup Z: generated by t operates transitively and faithfully on supp(§). As I is commutative, supp(§) is a principal set under 5 (no. 6, Example 2) and hence Card(supp(§)) = Card(z). Lemma 1. Let ((0.51 be afamily qfeycles whose supports are pairwise disjoint. Then the C, are pairwise permutable. Let c = 1:; C, and E be the subgroup generated by 0. Then a(x) = Ci(x)forxeS‘, ieI, andc(x) = xforx¢ HUI S,. ThemappingiHS‘isa biiection of I onto the set of E-orbits not consisting of a single element. Let C and C’ be two cycles whose supports are disjoint. If

x ¢ SUPPK) U SHPPG’), then CC’(x) = CCU?) = x. If x belongs to the support of C, then C’ (x) = x and C(x) belongs to the support of 2;, whence CC’(x) = CCU!) 2: §(x). Similarly when x belongs to the support of C', then C’C(x) = {C(x) = C(x). Hence CC’ = CC. Therefore the C, are pairwise permutable and, for i e I and x e 3,,

o-(x) = Mac) 6 St. The mappings o- and C, coincide on Si, hence S‘ is stable under a and the subgroup of 63, generated by the restriction of 0' to S, operates transitively on 8‘; therefore 8‘ is a E—orbit. As the S; are non-empty and are

pairwise disjoint, the mapping i e» s‘ is injective. As Lg s. is the set ofx such that 60:) 9e x, every E-orbit not consisting of a single element is one of the S,.

PROPOSITION 7. Let E be afinite set and o- a permutation ofE. There exists one and only onefinite set C ofeyoles satisfying the twofillowing conditions: (a) the supports of the elements of C are pairwise disjoints;

(b) o- = 1;; 2; (the elements of C being pairwise permutable by Lemma 1). Let E be the subgroup generated by a and let S be the set of E-orbits not consisting of a single element. For s e S, write C,(x) = o-(x) ifx e s and C,(x) = x if x q? s. For all s e S, C, is a cycle whose support is s and a = .5 s C” as is seen by

applying the two sides to any element of E. The uniqueness of C follows from Lemma 1.

62

PERMUTATION GROUPS or A rmrrn SET

§ 5.7

DEFINITION 9. A cycle of order 2 is called a transposition. Let x and y be two distinct elements of E. Let 1-,“, denote the unique transposition with support {x, 3/}. For every permutation a of E the permutation a. 1,“. a' 1 is a transposition whose support is {6(x), c(y)}. Hence: (4)

6 - T”. 1! . 6- 1 = 41000.0(”?

Transpositions thus form a conjugacy class in the group 65.

PROPOSITION 8. Let E be a finite set. The group 6,; is generated by the transpositions. For every permutation a let Fa be the set of x e E such that a(x) = x. We show by descending induction on p that every permutation a such that Card(F,,) = p is a product of transpositions. Ifp 2 Card(E), the permutation c is the identity mapping of E; it is the product of the empty family of transpositions. If p < Card(E), suppose that the property is proved for every per-

mutation e’ such that Card(Fol) > [7. Now E —F‘, 96 fl ; let e — F, and y e 6(x). Then y ye x and y e E — Fa. Let 6' = rare. The set Fa. contains Fa and x and hence Card(Fo.) > Card(Fo) = [1. By the induction hypothesis 6’ is a product of transpositions and hence a = 2-,“ . a’ is a product of transpositions. PROPOSITION 9. Let n be an integer 20. The group 6” is generated by the transpositions (Tt.t+1)1 E' such that p' o u = p and u o i = i ’, or, in other words, such that the following diagram is commutative:

I

ALGEBRAIG STRUCTURES

PaomsmoN 1. Lee: E!» E3) Game”: F—"> E’ 1; G be extensions ofG by F. Ifu: E ——>E' isamorphism ofd’into 6", uisanisomorphism q onto E’ andu'1 isa morphism ofé'” into 6’. Let x e E be such that u(x) = e. Then p(x) = p’(u(x)) = e, whence it e i(F). Lety EF be such that x = i(y); then i’(y) = u(i (31)) = e. As i’ is injective, y = e and x = e. Therefore u is injective. By virtue of § 4, no. 6, Corollary 1 to

Proposition 7, u is surjective since u(i (F)) = i’(F). The last assertion is immediate.

In other words, the extensions 6 and 6” are isomorphic if and only if there exists a morphism of 6’ into 6’. Let F and G be two groups and write E0 = F x G; let i: F-—>Eo be the

canonical injection and p: E0 —> G be the canonical projection. Every extension of G by F isomorphic to the extension 6’0: F —‘-> E.0 —’> G is called a trivial extension.

Paomsmon 2. Let c: F —‘> E I» G be an extension of G by F. Thefollowing conditions are equivalent: (i) d’ is a trivial extension; (ii) 6’ has a retraction r; (iii) (3’ has a section 3 such that s(G) is contained in the centralizer of i (F). Clearly (i) implies (ii) and (iii). If (ii) holds, the mapping (r, p): E -> F x G is a morphism of d’ into 6’0, whence (i). If (iii) holds, the homomorphism of F x G into E corresponding to (i, s) (§ 4-, no. 9, Proposition 12) is a morphism of 6’0 into 6, whence (i). It may be that an extension J:F—>E—->G is not trivial and yet the group E is isomorphic to F x G (Exercise 6).

DEFINrrION 2. Let F and G be two groups and r a homomorphism of G into the automorphism group of F. Write 1'(g)(f) = {ffor geG and feF. The set F x G with the law of composition

(1)

((f, g), (f': g')) '-> (13 g) '1 (f', g') = (flf', gg’)

is called the external semi-direct product of G by F relative to 1. The external semi-direct product of G by F relative to 1- is denoted by F x 1 G.

PROPOSITION 3. The external semi-direct product F x1 G is a group. The mappings i:F-—->F x,G definedhy i(f) = (f, e),p:F x,G—>G deflnedbypU; g) = g, and s: G —> F > F x , G 1’) G. Let 6”: F—'—> E'i; G be an extension ofG by F and 5’: G—>E’ a section of 6’. We define an operation 1- of G on the group F by:

(2)

i'(‘r(g,f)) = S’(g)i’(f)5'(g)’1 = Int(3’(g))(i’(f))-

PROPOSITION 4-. With the above notation, there exists one and only one isomorphism u of 6’, onto 6” such that u o s = 5'.

(f; g) = (j; e) ., (e, g) = i(f) .1 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

wmnmwavwwvwm =yumuw¢ww4wwam =yUWManwnww) =Mrmmnmg) =nmnwwe» Therefore, u is a homomorphism of F x 1 G into E’. Obviously u oi = i’, p’ou =panduos = 5’. Remark. The definition of the operation 1 by formula (2) depends on the extension 5’ and the section a". When F is commutative, the operation 1' does not depend on 5’. For Int(s’(g)) I i ’(F) depends then only on the coset

ofs’(g) mod. i’(F). More generally, let of : F —> E -—> G be an extension of G by a commutative group F (it is not assumed that J 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 e admits a section, this operation is that defined by formula (2). COROLLARY. Let G be a group and H and K two subgroups g" G1 such that H is 67

ADCvEBRAIC STRUCTURES

I

normal, H n K = {e} and H.K = G. Let 1- be the operation q on H by inner automorphism: of G. The mapping (h, k) 1—> his is an isomorphism of H x 1 K onto G.

Under the hypotheses of this corollary, G is said to be the semi-directproduct 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 permutations f of E with the following property: There exists 7 e I‘ 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 e A there exists a unique Y e P such that f is a y-morphisrn, we shall denote it by p(f). Letf,f’ be in A, y = p(f), y' = p(f’). Then, for all b e E and all g e G,

(f’°f)(gb) =f'(Y(£)f(b)) = Y'(Y(g))f’(f(b)) which proves thatf’ ofe A and that p(f' of) = p(f’)p(f). On the other

hand,f(Y'1(g)f"(b)) = 35, whencef'1(gb) = Y"(g)f'1(b) andf“€A~

Thus A is a subgroup of 63 and p is a homomorphism of A into I‘. The kernel ofp is the set AutG(E) of automorphisms of the G-set E. We fix a e E. We have defined 1n §5, no. 6 an isomorphism d), of G°

onto Auta(E) such that 11;“ (x) (go) = gxa for all g, x in G. On the other hand, for Y e I‘, let sa(-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 1‘

—> I‘ is an extension of I‘ by G° into A such that p o s“ = Id. Thus G° 4 A” and sa is a section of thls extension. This extension and this section define an

operation of I‘ on G“, sa(I‘) acting on ¢4(G°) by inner automorphisms; we write this operation exponentially. We show that this operation is the natural operation (§ 3, no. 1, Example 3): for x, g in G and 'Y e I‘,

(¢a(”x))(gd) = (MY) ° 4120:) ° SAY) "mm = (MY) ° ¢a(x))(Y"(g)a) = 5a(Y)(Y'1(g)xa)

= gY(x)a = ¢a(r(x))ga

whence Vx= fix). Proposition 4 then shows that A is isomorphic to the semidirect product of

1" = Aut(G) by G° under the natural operation of Aut(G) on G°. 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 T1 (n, A)... 2. COMMUTATORS

DEFINITION 3. Let G be a group and x and1/ two elements ofG. The element x" 1y’ 1xy of G is called the commutator ofx and y. 68

COMMUTATORS

§ 6.2

(x, y) is used to denote the commutator of x and y. Then obviously (y: x). = (x: y) _ 1'

For x and y to commute it is necessary and suflicient that (x, y) = e. More generally,

xy = Ma y). On the other hand we write

(3)

x” = y'lxy = x(x,y) = (y, x‘1)x.

As the mapping 2: v—> x” is the inner automorphism Int(y'1), (x, y)? = (x’, y?) for all x, y, z e G. For x, y, z e G, we prove the following relations: (4) (4' bis)

(x: yz) = (x9 Z)'(x2 y): = (x: z)'(za (y: x»'(x’ 1/) (xy: 2) = (x, Z)"-(.% Z) = (x: Z)'((": Z):y)-(y, Z)

(5)

0‘”, (y, Z))-(!l‘, (ZaylHs (x,y)) = e

(6)

(x, yz)'(z: xy)-(3/’ zx) = e

(6 biS)

(x31: z).(yz, x)-(s!l) = 6'

Now

(“2) = x‘IZ‘w‘lxyz = (x, 2)2“x‘1y"xyz = (x, z)(x,y)z = (x, 1X2, (7" y)-1)(x3 3/)

by (3), which proves (4). Formula (4- bis) follows similarly. On the other hand, (xv: (.11: Z» = (xy)_1(za y)(xy)(!/’ 7')

= y“‘x"yz‘1y“zyy"xyy"2'1yz = (yzy'lxy) "(#24112)Then writing u = yzy‘lxy, v = zxz‘lxy and w = xyx’lzx, we obtain

(1", (1/: 2)) = 14‘1”By cyclically permuting x, y, 2, we deduce (y‘, (z, x)) = 0‘1w and (zx, (”:3”) = w-lus

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) = x‘lz‘ly'lxyz = (yzx) ‘1(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 ifand only ifB normalizes A. IfA and B are normal (resp. characteristic), so is (A, B). 1' We here reject the notational convention made in § 1, no 1 of extending a law

of composition to subsets.

69

> ALGEBRAIC STRUCTURES

I

PROPosrrION 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, 0)) is generated by the elements (a, (b, 0)) with a 6A, b e B and e e 0. (iii) If A, B and C are normal, then

(A, (B, 0)) C (C, (B, A))-(B, (C, A»By (4 bis), for a, a’ GA and b GB, (0, b)“' = (aa’, b).(a’, b)”,

whence (i). Suppose now that (B, C) normalizes A. For a e A, b e B, c e C and x e G, (4) implies (a, (be c)'x) = (a, x)'(xa ((b, 6‘), a))(aa (b: 5))

and ((b, o), a) E A since (B, C) normalizes A, whence by induction on p the fact 1’

that a, 1:; (b,, 04) , for b, e B, ci e C, belongs to the subgroup generated by the elements of the form (a, (b, 0)). 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

(a: (b: 0)) E (C, (B: A» - (B, (C: A» o e C. Now by (5), writing a"'1 = u and B e b A, for all a e

(a: (b: 0)) = ((19): (b: 9)) = (6": (“2 I’ll—145°, (‘2 and, whence (iii).

DEFINITION 4. Let G be a group. The subgroup generated by the oommutators ofelemeuts 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. Letf: G -—> G’ be a group homomorphism. Thenf(D(G)) C D(G’). Iff is surjective, the homomorphism q(G) into D(G’) the restriction off is surjeotive. The image under f of a commutator of elements of G is a commutator of elements of G’. Iff is suijective, the image underfof 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 l. The derived group of a group G is a characteristic subgroup of G. In particular it is a normal subgroup (If G. 70

LOWER CENTRAL SERIES, N’ILPOTENT GROUPS

§6.3

COROLLARY 2. Let G be a group. The quotient group G/D(G) is commutative. Let 1:: G —> G/D(G) be the canonical homomorphism. Every homomorphismf of G intoa commutative group G’ can be expressed uniquely in the form f = f 0 1:, where

f: G/D(G) —> G’ is a homomorphism. Now n(D(G)) = {e}. As 1: 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 ofG. Thefollowing conditions are equivalent:

(i) H 3 D(G);

(ii) H is a normal subgroup and G/H is commutative. (ii) => (i) by Corollary 2 and (i) => (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 elements of X and (b: G—-> G/H the canonical homomorphism. The set ¢(X) generates G/H. The elements of (X) are pairwise permutable and hence H is commutative (§ 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 rt, is a solution of the universal mapping problem for G, relative to commutative groups and homomorphisms from G to commutative 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 l5e).

Examples. (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 91". For 21,. is generated by the products of two transpositions; if

r = 1-,“ y and 1’ = 7‘“. are two transpositions, let a be a permutation such that 0(x’) = xand c(y’) = y. Then 1' = (Flea and r1" = 1‘11’ = r‘lc'lw is a commutator. Hence 9!" c D(én). As 6n/9In is commutative, 2L, 3 D(Gn)

(Corollary 3). 3. LOWER CENTRAL SERIES, NILPOTENT GROUPS

Let G be a group, H a subgroup of G and K a normal subgroup of G. The image ofH in G/K is contained in the centre of G/K ifand only if (G, H) C K. DEFINITION 5. Let G be a group. The lower central series of G is the sequence (C"(G)),,>1 of subgroups of G defined inductively by:

01(6) = G,

C”+1(G) = (G, one». 71

ALGEBRAIG STRUCTURES

I

Let f: G —> G' be a group homomorphism. It is seen, by induction on n, that f(C"(G)) C C“(G’) and that, if f is surjective, f(C”(G)) = C"(G’). In particular, for all n 2 l, C2 (G) is a characteristic (and hence normal) subgroup of G. For all n 2 l, C”(G)/C"+1(G) is contained in the centre of G/C"“(G). Let (G1, G2, . . .) be a decreasing sequence of normal subgroups of G such that (1) G1 = G; (2) for all 2', G,/G,+1 is contained in the centre of G/GHI. Then C‘(G) C G,, as is seen by induction on 2'. Now _ (C"'(G), C“(G)) C C"'+"(G). (7) For, if this relation is denoted by (Fm, n), it follows from (Fm, n), by no. 2, Proposition 5, that

(0“(G), C"(G)))-(C"(G), (G: C"'(G))) C (G, (0“(G), C”*1(G)) C Cm+n+1(G) _(Cm+1(G), Cn(G)). Hence ((Fm'n) and (Fm+1_,,)) 2, (mn). As (Fm, 1) and (171,») are obvious

(Fm...) follows by induction. DEFINITION 6. A group G is called nilpotent if there exists an integer n such that 0"“ 1(G) = {e}. The least integer n such that GM 1(G) = {e} is called the nilpotency class of a nilpotent group G.

If n e N, a group of nilpotency class 71 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 nilpotent. Examples. (1) A group is nilpotent of class 0 (resp. G' is the canonical homomorphism, then C“(G’) = 1t(C“(G)). (4) A finite product of nilpotent groups is nilpotent.

PROPOSITION 7. Let G be a group and 11 an integer. The following conditions are equivalent: (a) G is nilpotent 1y“ class G2 :...=Gn+1 ={e}

such that (G, G") c Gk+1forallke[l,n]. 72

LOWER CENTRAL SERIES, NILPOTENT GROUPS

§ 6.3

(c) There exists a subgroup A ofG contained in the centre ofG such that G/A is nilpotent (felass (b): it sufiices to take G" = C"(G). (b) :> (a): by induction on k, C"(G) C Gk. (a) =:> (c): it suflices to take A = C"(G). (c) a (a): let 1:: G —> G/A be the canonical homomorphism; then 7t(C"(G)) = C"(G/A) = {e} and hence C”(G) C A, whence 0"“(G) = {e}. More briefly: a group is nilpotent of class N2:>---D N"+1 ={e} such that (G,N’°) (a) (no. 5, Corollary to Theorem 1). Suppose (a) holds and let P be a Sylow p-subgroup of G. If N is the normalizer of P in G, Corollary 1 to Theorem 3 shows that N is its own normalizer. By §6, no. 3, Corollary to Proposition 8, this shows that N = G. Hence (a) => (c). Suppose (0) 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 are q, PF 0 P4 is reduced to e for it is both a p-group and a q—group, hence Pp and Pa centralize

one another (§ 4, no. 9, Proposition 15). Let (I) be the canonical homomorphism (§ 4, no. 9, Proposition 12) of 1;]; Pp into G. The homomor-

phism 4. is surjective by the Remark of no. 5. As Card([1 1),) = Card(G), it follows that (b is bijective. 80

FREE MAGMAS

.

§ 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 equivalent: (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 = ill G,, where G, is the unique Sylow p-group of G.

(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 63 is of order 6. It contains a normal Sylow 3-suhgroup of order 3: the group 913. It contains three Sylow 2-subgroups of order 2: the

groups {e, 1:}, where 1: is a transposition. The group 63 is thus not nilpotent.

§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

A sequence ofsets M,,(X) is defined by induction on the integer n 2 l as follows: writing M1 (X) = X, for n 2 2, M,,(X) is the set the sum of the sets M,(X) x Mn-,,(X) for l S p S n —- l. The set the sum of the family (Mn(X)),,>1 is denoted by M(X); each of the sets Mn(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 6 MAX); it is called the length of w and denoted by l(w). The set X consists of the elements in M(X) of length 1. Let w and w' be in M(X); write [I = 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 M,+a(X) is called the composition of w and w’ and is denoted by ww’ or w.w’. Then l(w.w’) = l (w) + l(w’) and every element of M(X) of length 22 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 theflee magma constructed on X (§ 1, no. 1, Definition 1). PROPOSITION 1. Let M be a magma. Every mappingfofX into M may be extended in a unique way to a morphism q(X) into M. By induction on n 2 , mappings fl: Mn(X) —> M are defined as follows: let

81

ALGEBRAIG STRUCTURES

I

f1 = f; for n > 2, the mapping)3, is defined byj},(w.w’) = f,,(w) .fn_,,(w') for p = l, 2,. . ., n - land (w, w’) in M,(X) x Mn_,,(X). Letg be the mapping of M(X) into M which inducesf,, on Mn(X) for every integer n 2 1. Clearly g is the unique morphism of M(X) into M which extendsf2 Let u 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) . If0 is a mapping of Y into a set Z, the homomorphism M(u) o M(u) of M(X) into M(Z) coincides with o o u on X, whence M(u) :2 M(u) = M(v o u).

PROPOSITION 2. Let at X —-> Y be a mapping. If u is injective (resp. smjeotive, hijective), so is M(u).

Suppose u is injective. When X is empty, M(X) is empty, hence M(u) is injective. If X is non-empty, there exists a mapping 0 of Y into X such that v o u is the identity mapping of X (Set Theory, II, §3, no. 8, Proposition 8); the mapping M(u) o M(u) = M(o o u) is the identity mapping of M(X) and hence M(u) is injective. When u is surjective, there exists a mapping w of Y into X such that u o w

is the identity mapping of Y (Set Theory, II, §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 extended 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 (um, 0a)”: 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 (11,, on) (§ 1, no. 6). The magma M(X)/R is called the magma defined by X and the relator: (um 0a),“ Let h be the canonical

morphism of M(X) onto M(X) /R. Then M(X) IR is generated by h(X). Let N be a magma and (155),,“ 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(uu) = k(v¢) for all at e I, there exists one and only one morphism f: M(X)/R ——> N such thatf(h(x)) = nx for all): e X (§ 1, no. 6, Proposition 9). 2. FREE MONOIDS

Any finite sequence w = (Minn of elements of X indexed by an interval

(l, 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 l(w). There is a unique word of 82

FREE MONOIDS

§ 7.2

length 0, namely the empty sequence e. X will be identified with the set of words of length I.

Let w = (24:1)1‘K"I and w’ = (14)“ in be two words. The composition of w and w’ is the word u = (yfluk‘mfi defined by

(l)

__{xk y"_

forl [(6) for every decomposition o" 96 a of x. The case n = 0 being trivial, suppose n > 0. Let 0' = (a;i1,...,in;p1,...,pn)

be a decomposition ofx oflength n. Ifthere existed an integer at with l S a: < n and pa = eh, the sequence (a; £1, - ' " itit-1: ia+15 ' ' 'a inipla ' - -)pa—1’Pu+13 - - at»)

86

AMALGAMATED SUM or MONOIDS '

§ 7.3

would be a decomposition of x of length n — 1, which is excluded. Suppose that there exists an integer at with l < at < n and i, = i“; and let

Push“ = h,“(a’) 4’; with a’ e A and p; e P“; by Lemma 1 there exists elements a” e A, Pi 5 P11: - - uPiL—i 5 P1P;

such that

:1(111)-- -4.-1(l’a—1)h(a’) = h(a")u(fii)- - -¢t.-1(P&-1) and the sequence (flan; i1: ' ' 'sia-1aiwiu+2: - ' u inipl: - - upgt—lapéspa+2) ' - 'spn)

is a decomposition of x of length n —- l, which is a contradiction. We have thus proved that o- is reduced.

COROLLARY. Under hypothesis (A) the homomorphism M’ inducingf, on M; for all i e I. 87

I

ALGEBRAIG STRUCTURES

Hypothesis (A) is satisfied in two important cases: (a) A = {e}. In this case, there is a family (Mi),GI of monoids and M is called the monoidal sum of this family. Each M, is identified with a submonoid of M and M is generated by ‘EUI M,; further, M; n M, = {e} for i aé j. Every element of M may be written uniquely in the form x,. . .x,I with 1'1t

—{e},...,xn€M;n

—{e}

and 1], 7E in,” for l S a < n. Finally, for every family of homomorphisms (fl: M, —+M’), there exists a unique homomorphism f: M» M’ whose restriction to M, is fl for all is I.

(b) There is a family of groups (G,),EI containing as subgroup the same group A and h, is the injection of A into G,. The sum Of the family (Cow: amalgamated by A is then a group G: the monoid G is generated by {EUI 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") =fx(n) foeEX and n EZ; in particular, f(x) =fi,(l) =f(x) for all x e X and hencef extendsf.

Let u: X ——> 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(v o u) = F(v) o F(u)

is established for every mapping 0: Y ——> Z and it is shown that F(u) is injective (resp. surjective, bijective) ifu is. For every subset S ofX, 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 2' in I with its

89

ALGEBRAIC STRUCTURES

I

canonical image (MI) in the free group F(I); the latter will be denoted by T, (01' TI, X” . . . as the case may be) and called the indeterminate of index i. The

free group F(I) is then denoted by F((TQ‘eI) or F(T1, . . ., T") if I = {l,2,...,n}. Let G be a group and t = (Qua: a family of elements of G. By Proposition 8

there exists a homomorphism j; of F((TQEI) into G characterized by f;(T,) = t‘ for all i e I. The image of an element w of F((T,),EI) under fl will be denoted by w(t) or w(t1,. . ., tn) if I = {1, 2, . . ., n}; w(t) is said to result from the substitution T,l .—-> t, in w. In particular, if we take G = F((T.).eI) and (t‘) = (T,) = T, fr 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 = (t1, . . . , tn) a finite sequence of elements in G. Let t' = (u(tl), . . . , u(tn)); the homomorphism u oft of F (T1, . . . , Tn) into G’ maps T, to u(t,) for 1 < i S n and hence is equal tofv; for w in F(T1,..., Tn), then

(8)

“(w(tu - - u t.0) = w("(tl)s - - -,u(tn))-

Let w be given in F(Tl, . . ., Tu) and elements 01, . . ., 0,, in the free group F (Ti, . . ., T,’,,). The substitution Ti I—> at defines an element w’ = w(vl, . . ., 0n) of F(Ti, . . ., T,’,.). Let G be a group, t1, . . ., t", elements of G and u the homo-

morphism ofF(Ti, . . ., T5,.) into G characterized by u(T}) = t, for l < j < m. Then u(v,) = v,(t1, . . ., tm) and u(w’) = w(tl, . . ., tm); formula (8) thus implies

(9)

w’(t1,...,tm) = w(vl(t1, . . ., rm), . . ., vn(t1,. . ., t».))-

This justifies the “functional notation” w(tl, . . ., tn). The reader is left to extend formulae (8) and (9) to the case of arbitrary indexing sets. 6. PRESENTATIONS OF A GROUP

Let G be a group and t = 0.),“ a family ofelements of G. Letft be the unique homomorphism of the free group F(I) into G which maps i to t,. The image of j}; is the subgroup generated by the elements A of G. The elements of the kernel ofj; are called the relators of the family 1;. t is called generating (resp. free, basic) ifj; is surjective (resp. injective, bijective). Let G be a group. A presentation of G is an ordered pair (1:, r) consisting of a generating family t = (a).eI and a family r = (rj),E J of relators such that the kernel Nt offt is generated by the elements grjg‘1 for g e F(I) and j EJ. It amounts to the same to say that Nt is the normal subgroup of F(I) generated by the r, for J eJ (in other words, the smallest normal subgroup of F(I) containing the elements r, (j eJ), cf. § 4, no. 4). By an abuse of language the generators t, and the relations r,(t) = e are said to constitute a presentation of the I

group G. 90

PRESENTATIONS or A GROUP

§7.6

Let I be a set and r = (13),“ a family of elements of the free group F(I). Let N(r) be the normal subgroup of F(I) generated by the r, for j ej. Let F(I, r) = F(I) /N(r) and 1, denote the class Ofi modulo N(r). The ordered pair (1, r) with 1 = (10,61 is a presentation of the group F(I, r) ; if G is a group and (t, r) is a presentation of G with t = (mien there exists a unique isomorphism

u of F(I, r) onto G such that u(1,) = tfl for all i e I. The group F(I, r) is said to be defined by the generators 1, and the relators 1,, or by an abuse of language that it is defined by the generators T: and the relations r,(1) = e. When I = [1, n] and J = [1, m], it is said that F(I, r) is defined by the presentation (b): Suppose that (t, r) is a presentation of G and let t’ = (130‘eI be a family of elements of a group G’ with r,(t’) = e for all j e]. Let f’ be the homomorphism of F(I) into G’ characterized by f’(i) = t" for all i e I. By hypothesis f'(r,) = e for all j 6"] and, as N is generated by the elements grjg‘l forj e] and g e F(I),f’(N) = {e}. As the homomorphismf: F(I) —> G

is surjective with kernel N, there exists a homomorphism u: G —> G’ such that

f’ = u of- Then um) = we» =f’(i) = tz. (b) => (c):

Suppose condition (b) holds. Let t = (tom be a generating family Ofa group G such that r,(t) = e for allj e] and let u be a homomorphism of G into G such that 00,) = t; for all i e I. As the family (4)“, generates G, the homomorphism v is smjeetive. By property (b) there exists a homomorphism

u: G—->G such that u(:,) = z, for all is I. Then u(v(t‘)) = 2‘ for all is I and 91

I

ALGEBRAIC STRUCTURES

hence u o v is the identity on C, which proves that u is infective. Hence 0 is an isomorphism and condition (c) holds. (c) => (a): Suppose condition (0) holds. Let t; be the canonical image of i in F(I, r) and t’ = (t;),el; then r,(t’) = e for allje]. As r,(t) = e for alljeJ, there exists one and only one homomorphism o of F(I, r) into G such that

21(3‘) = t; for all i e I. By (c), v is an isomorphism ofF(I, r) onto G which transfomls the presentation (t’, r) of F(I, r) into a presentation (1:, r) of G. 7. FREE GOMMUTATIVE GROUPS AND MONOIDS

The set Zx of all mappings of X into Z is a commutative group under the law defined by (at + [3) (x) = oc(x) + [3(x) (at, {3 62", x e X); the elements of Zx are sometimes called multiindices. The identity element, denoted by 0, is the constant mapping with value 0. For a 62", the set S“ of x EX such that «(26) 95 0 is called the support of at; then So = Q and Saw C S“ U S, for on, p in 2“. Therefore the set Z“) of mappings on: X —> Z of finite support is a subgroup of zx called thefiee commutative group constructed on X. For all x e X, let 8,, denote the element of 2"" defined by

(10)

ify = x ye x. My) _— 01 W

Also, for an e Z“), the integer [a], the length of at is defined by the formula (11)

lat] = ”2;: ot(x).

The following relations are immediately established: (12)

a = 2;: oc(x).8,,

(13) (14)

[0| = 0 |3x| = 1: Ian + Bl = M + IBI

foru,BinZ‘x’andxinX. The order relation on S fl is defined in Z“) by ac(x) < {3(x) for all x e X. The relationsoc < Band oc' Z; it is also shown that Z“) is injective (resp. surjective, bijective) if u is.

Let S be a subset of X; if i is the injection of S into X, the mappingf = Z“) is an isomorphism ofZ“) onto the subgroup H of2"" generated by the elements 8, for s e S. By (15),

(f( Z; further, N‘") is injective (resp. suljective, bijective) if u is. If S is a subset of X,

N(s> = 2(5) n Nan. Remark. Let M be the multiplicative monoid of strictly positive integers and let ‘3 be the set of prime numbers (§ 4, no. 10, Definition 15). By Proposition 10 there exists a homomorphism u of NW) into M characterized by u(8,,) = p for every prime number 1). Then u(aL) = 11 pm”) for at in NW) and PE

Theorem 7 of § 4, no. 10 shows that u is an isomorphism of NW) onto H.

8. EXPONENTIAL NOTATION

Let M be a monoid, written multiplicatively, and u = (uQxEx a family of

elements of M, commuting in pairs. Let at be in N“); the elements 142‘“) and uzw of M commute for x, y in X and there exists a finite subset S of X such that

a?” = l for x in X — S. We may therefore write:

(16)

u“ = H 11:”). xeX

Let M’ be the submonoid of M generated by the family (“x)xex; it is commutative (§ 1, no. 5, Corollary 2 to Proposition 4). There thus exists (no. 7, Proposition 10) a unique homomorphismfof N0” into M’ such thatf(8“) = ux

for all x E X and f(at) = u“ for all (X in N”. We deduce the following formulae

(17)

If” = u“‘.uB

(18)

11" = l

(19)

u“: = u,

for a, B in N0" and x in X. Let v = (vflxex be another family of elements of M; suppose that vxvy = vgvx and uxvy = vyux for x, y in X. Then there exists (§ 1, no. 5,

Corollary 2 to Proposition 4) a commutative submonoid L of M such that u, e L and v,c e L for all x e X. The mapping a »—> u“.v°‘ of NO“ into L is then a homomorphism (§ 1, no. 5, Proposition 5) mapping 8,, to uxwx. Thus we have the formula u“.v°‘ = (u.v)°‘, (20) where u.v is the family (ux.vx)xex.

94-

RELATIONS BETWEEN THE VARIOUS FREE OBJECTS '

§ 7.9

When.M is commutative, u“ can be defined for every family a 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. 1 shows the existence 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 a: Mo(X) —-> F(X) (no. 2, Proposition 3). By no. 4, Proposition 6 and no. 5, Proposition 7, y. is injective. Similarly Proposition 10 of no. 7 and Proposition 8 of no. 5 show the existence of homomorphisms v: Mo(X) —> N“) and n: F(X) —> 20" characterized by v(x) = 8x and 1t(x) = 8,. for all x e X. If r. is the injection of N“) into 20", the two homomorphisms I. o v and 11: o y. of Mo (X) into 2‘19 coincide on X, whence v. o v = 1: o u. The situation may be summarized by the following commutative diagram:

M(X) ——‘—> Mo(X) —"> N00 U.

‘

F(X) ——'i—> 200. The homomorphisms 1, pt, v and 1: will be called canonical. Let w be in M(X); it is immediately shown by induction on l(w) that the length of the word l(w) is equal to that of w. Moreover it

(21)

v(x1...x,,) = i=1 Z 8,“

for x1,...,x,, in X, whence |v(x1. ..x,,)| = n by (13) and (14). In other words,

(22)

lv(u)| = l(u)

(u eMo(X)).

PROPOSITION 11. The canonical homomorphinn v of Mo(X) into N00 is swjective. Let w = x1...x,, and w' = x1...x:,, be two elements of Mo(X); in order that

v(w) = v(w’), it is necessary and suficient that m = n and that there exist apermutation 0-66,, withx; = xmnforl < i S n.

The image of v is a submonoid I of N00 containing the elements 8,: (for x EX). Formula (12) (no. 7) shows that N00 is generated by the family (8,. ”x, where I = N“). Therefore v is surjective. Ifm = nandxfi =xc(,,forl g i < n,then n

n

'l

v(w’) = g a,“ = .21 em, = l; s,“ = v(w) by formula (21) and the commutativity theorem (§ 1, no. 5, Theorem 2).

95

I

ALGEBRAIC STRUCTURES

Conversely, suppose that v(w) and v(w’) are equal to the same element at of NC“; by formula (22), n = [on] = m. For all x e X, let I,‘7 (resp. IQ) be the set of integers i such that 1 < i < n and x, = x (resp. x; = x). Hence (Ix) fix and

(1;)see)! are partitions of the interval [1, n] of N; further, the formula 7‘

u = 2:1 8,“ shows that oc(x) is the cardinal of Ix; similarly the formula n

at = :21 8,.“ shows that u(x) is the cardinal of 13,. There thus exists a permutation

a of[1, n] such that o-(IQ) = I, for all xeX, that is x,’ = a“, fori = l, . . ., 71. 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 N05) with a submonoid of N“). Then

(23)

M(S) = 7."1(Mo(S)).

Clearly A(M(S)) C Mo(S). Let w e A'1(Mo(S)); we show by induction on l(w) thatweM(S). Itis obviousifl(w) = 1. Ifl(w) > l,wemaywritew = wlwg with w1, w e M(X), l(w1) < l (w), [(wz) < l(w). Then 1(w1)}\(w2) e Mo(S), hence )t(w1) e Mo(S) and 1(w2) e Mo(S), whence w1 e M(S) and wa e M(S) by the induction hypothesis and finally w e M(S). Also

(24)

Mo(S) = v'1(N‘S’).

ediately from formula (21). This follows unm' Further, N03) is the set of elements of N0“ whose support is contained in S; if (54):“ is a family of subsets of X of intersection S, then N‘s’ = Q NW and formulae (23) and (24) imply

(25)

M(S) = ,QIM(S.).

Mo(S) = Q Mo(s.).

§3. RINGS l. RINGS

DEFINITION l. A ring is a set A with two laws of composition called respectively addition and multiplication, satisfl/ing thefollowing axioms: (AN I) Under addition A is a commutative group. (AN II) Multiplication is associative and possesses an identity element.

(AN III) Multiplication is distributive with respect to addition. The ring A is said to be commutative ifits multiplication is commutative.

96

mas

§ 8.1

In what follows, (x, y) »—>x + y denotes addition and (x, y) r—>xy multiplication; 0 denotes the identity element for addition and 1 that for multiplication. Finally, —x denotes the negative of x under addition. The axioms of

a ring are therefore expressed by the following identities:

(l) (2) (3) (4-) (5) (6) 7 E8;

x + (y + z) = (x + y) + z (associativity of addition) x +y =y + x (commutativity of addition) 0+x=x+0=x (zero) x + (—x) = (—x) + x = 0 (negative) x(yz) = (xy)z (associativity of multiplication) x.l = 1.x = x (unit element) x+ ).z=xz+yz . . .. If. (3/ 4y- 2) = xy + xz }(d1str1but1v1ty)

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 ofA. For all x eA, we define the left homothety 7,, and right homothety 8,, by 7,,(y) = xy, 5,,(y) = yx. By formulae (7) and (8), 7,, and 8,, are endomorphisms of the additive group of A and thus map zero to zero and negative to negative. Therefore (9)

(10)

x.0 = 0.x = 0

x-(—y) = (-x)-y = -xy;

it follows that (—x)(-y) = —((—x).y) = —(—-xy), whence

(11)

(-x)(—y) = xy-

Formulae (10) and (11) constitute the sign rule. It follows that

—x = (—l)x = x(—l)

and (—l)(—l) = 1. From (1 1) it follows by induction on n that

(12)

x” ifn is even

('x)" = {—x" ifnis odd.

When we speak ofeancellable elements, invertible elements, Immutable elements, central elements, centralizer or centre of a ring A, all these notions will refer to the

multiplication on A. Ifx, y e A and y is invertible, the element xy‘ 1 ofA 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 group called the multiplicative group of A, sometimes denoted by A*. Let x, y be in A. x is said to be a left (resp. right) multiple of y if there exists 11’ EA such that x = 3/31 (resp. x = yy’); it is also said that y is a right (resp.

97

AIJGEBRAIG STRUCTURES

I

left) divisor ofx. When A is commutative, there is no need to distinguish between “left” and “right”. In conformity with the above terminology, every element 3/ 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. lefl) divisor qf 0” is reserved for elements 3/ such that there exists x 75 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) noncancellable elements. Let x e A. x is called nilpotent ifthere exists an integer n > 0 with x" = 0. The . element 1 — x is then invertible, with inverse equal to 1+x+x2 +---+x"'1. As A is a commutative group under addition, the element 12x for n e Z and x EA has been defined -(§ 2, no. 8). As 7,, and 8,, are endomorphisms of the additive group A, yx(ny) = 1:7,,(y) and 8,,(nx) = 725,,(x), whence

x-(ny) = (MN = n-(xy)-

.

In particular, 11:: = (n. l)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. CONSEQUENCFS 0F DISTRIBUTIVITY

Distributivity of multiplication with respect to addition allows us to apply Proposition 1 of § 3, no. 4-, which gives

(13)

x fl (2 x”) = a12 .....u,. i=1 "W hen.

where the sum extends over all sequences («1,. . ., at") belonging to L1 x - - - x L, and fori = 1,. . ., n the family (x,,;,),,em ofelements ofthe ring

A is of finite support.

PRoposmON I. Let A be a commutative ring and (xQML afinitefamily qfelements qfA. For everyfamily ofpositive integersZ[3 = (BA)AELS let [m = ’23:. (3,“ Then

(14)

—Exgt. (A: a) = I“:n fii—ilfl REL

We apply formula (13) with L, = L and x, 1. = x,, for l< i< n. Then (15)

£21.15)" = a 2m” 1:41.. .xan, 1v-..

the sum extending over all sequences at = (11, . . . , at") e L".

98

CONSEQUENCES OF DISTRIBUTIVITY

§ 8.2

Let a be in L"; for all A e L let U: denote the set of integers i such that l g i < n and a, = A and let (I>(a:) = (Uzbek It is immediate that (I) is a

bijection of L" onto the set of partitions of {1, 2, . . ., n} indexed by L. For all [3 e NI‘ such that ”3| = 1:, let LE denote the set ofac e L” such that Card U3 = B, for all A E L. It follows that the family (L3) 5 I =,, is a partition of L" and that Card L: = (§ 5, no. 5).

I n.

H at.I

Finally, for one L3, ecu-“x“, = Q

fixes” =11} ‘Ie—tlgxk =m,

whence (“1----o°‘n)xa1. I la.“ = lBl-n wells xnu- - 'xun

= 2 Z Hxa" IDl=n eel/9‘ 5'11 " 71!

3:5"

= lBl—n _ —_ ll (3;! A.

and formula (14) thus follows from (15). COROLLARY l (binomial formula). Let x and y be two elements of a commutative ring A. 77ml: n

(x + y)" = p20 (p)x”y 1’. n

—

Formula (14) applied to L = {1, 2}, x1 = x and x2 = y gives (x +y)n = ”+Enfixpyq:

the sum extending over ordered pairs of positive integers p, q with p + q = n. The binomial formula follows immediately from this (Set Theory, III, §5, no. 3).

COROLLARY 2. Let A be a commutative ring, X a set, I: = (ax) xex and v = (ox) “x

two families of elements of A. Let u + v denote the family (u,c + ox)”, For all

x e N00 we write A! = E m) 1. Thenfor all a: 6 N00, in the notation ¢f§ 7, no. 8, !

B be a ring homomorphism. The mappingfis a homomorphism of the additive group ofA into the additive group ofB; in particular, f(0) = 0 andf( —-x) = —f(x) for all x e A. The image underfof 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. Examples. (1) Let A be a ring. It is immediately seen that the mapping 1: |—> n .l 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 1—) n .1 of Z into A is an isomorphism of Z onto A by the very construction of multiplication in Z (§ 2, no. 6). (2) Let a be an invertible element of a ring A. The mapping x»—> axa'1 is an endomorphism of A for a(x + y)a‘1 = nna'1 + aya‘l,

WWW"1 = (axa'lXaya'UIt is bijective, for the relation x’ = am"1 is equivalent to x = a‘lx’a. It is

therefore an automorphism of the ring A, called the inner automorphism asso-

ciated with a.

102

IDEALS

§ 8.6

5. SUBRINGS

DEFINITION 3. Let A be a ring. A subring ofA is any subset B (jA which is a subgroup

ofA under addition, which is stable under multiplication and which contains the unit ofA. The above conditions may be written as follows 06B,

B+BCB,

—BCB,

B.BCB,

leB.

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. (2) Let A be a ring and (A‘) t e I a family ofsubrings ofA ; it is immediate that

‘01 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 ofA generated by X.

(3) Let X be a subset ofa ring A. The centralizer ofX in A is a subring ofA. In particular, the centre of A is a subring of A. (4) Let G be a commutative group with operators; let 9 denote the set of operators and ac Hf“ the action of Q on G. Let E be the endomorphism ring of the group without operators G and F the set ofendomorphisms of the group with operators G. By definition, F consists of the endomorphisms yx gives 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 called ideals.

103

ALGEBRAIG STRUCTURES

I

Examples. (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 sometimes 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 annihilatar 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 9% a, a 9E 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 qf the set qfleft ideals distinct from A.

In other words, a is maximal if a 75 A and the only left ideals ofA containing a are a and A.

THEOREM 1 (Krull). Let A be a ring and a a left ideal qf A distinct from A. There exists a maximal ideal m qfA containing a. Consider A as operating on the additive group A + ofA by left multiplication. Then the left ideals of A are the stable subgroups of A+. The theorem thus

follows from § 4, no. 3, Proposition 3 applied to the subset P = {l} of A+. PROPOSITION 3. Let A be a ring, (xx) ”L afamily Qfelements ofA and a (resp. b) the

set qfsums 2L ahx,‘ where ((5)151, is afamily withfinite support ofelements qfA (resp. a; ahxhb,‘ where (aQML, (b,.),~5L arefamilies with finite support of elements qf A). Then a (resp. b) is the left (resp. two-sided) ideal qfA generated by the elements xx. The formulae

(18)

0 = 72:110.“

(19)

72:1. aw" + x21. afixk = AZ]; (ah + aux;

(20)

a.AZL ahxh = A21. (aah)x,v

prove that a is a left ideal. Let a’ be a left ideal such that xx 6 a’ for all 7. e L and 104

QUOTIENT RINGS

§ 8.7

let ((1,) “L be a family with finite support in A. Then am e a’ for all A e L, whence ”L akxh 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 (0,015,, afamily of19‘? ideals ofA. The left ideal

generated by ALEJL a, consists ofthe sums ELy, where m),e]. is afamily withfinite support such that 3],, e a,for all A e L. Let a be the set of sums A21.” with yA e aA for all AeL. The formulae 2 x,‘ + 2 y, = 7;“ (xx + y,,) and a. 52111:, = gnu, shows that a is a left

AEL

Lela

ideal ofA. Let AeL and xeah; write y, = x and y“ = 0 for y, 9e A; then x = A; y» whence x e a and finally a, c a. If a left ideal a’ contains a, for all

A e L, it obviously contains a and hence a is generated by ALEJL an. The ideal a generated by ALEJL a, is called the sum of the left ideal: aA and is

denoted by A; a, (cf. 11,§ 1, no. 7). In particular, the sum a1 + a, ofthe two left ideals consists of the sums a1 + a; where a, 6 a1 and a3 6 a2.

7. QUOTIENT RINGS Let A be a ring. If a is a two—sided ideal of A, two elements .7: and y of A are said to be congruent modulo a, written at "=" 1] (mod. a) or x E y(a), if x — y e a.

This is an equivalence relation on A. The relations x E y(a) and x’ E y’(a) imply x + x’ E y + y'(a), xx’ E xy’(a) for a is a left ideal and xy’ E yy’(a) for a is a right ideal, whence xx' E yy’(a). Conversely, if R is an equivalence relation on A compatible with addition and multiplication, the set a of x E 0 mod. R is a two-sided ideal and x E y mod. R is equivalent to x E 3/ mod. a. Let A be a ring and a a two-sided ideal ofA. A/a denotes the quotient set of A by the equivalence relation x E y(a), with addition and multiplication the quotients ofthose on A (§ 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 of A by the subgroup a. (b) Under multiplication, A/a is a monoid (§ 2, no. 1). (e) Let E, 7;, C be in A/a and let 1:: A —> A/o. be the canonical mapping; we choose elements x, y, z inA such that 7t(x) = E, 1t(y) = 1) and 7t(z) = C. Then

5(1) + C) = 1:007:01 + Z) = “(My + 2)) = T=(xy + x2)

= w(x)rc(y) + 7r(x)1=(Z) = in + it

and the relation (E + 19C = it + 112: is established similarly. 105

ALGEBRAIC STRUCTURES

I

DEFINITION 6. Let A be a ring and a a two-sided ideal ifA. The quotient ring ofA by a, denoted by A/a, is the quotient set of A by the equivalence relation x E 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. THEOREM 2. Let A be a ring and a a two-sided ideal ofA. (a) The canonical mapping 7: ofA onto A/a is a ring homomorphism. (b) Let B be a ring andf a homomorphism of A into B. Iff(a) = {0}, there exists one and only one homomorphisqfA/a into B such thatf f o 1:. By construction, 1c(x +31) = 1t(x) + n(y) and 11(xy) = 7:(x)7:(y) for x, y in A; also 1r(l) is the unit a of A/a, whence (a). Let A+ be the additive group ofA and B+ that ofB ; asfis a homomorphism of A+ into B”, zero on the subgroup a of A“, there exists (§ 4, no. 4,

Proposition 5) one and only one homomorphismf of A+/a into B+ such that

=fo 7:. Let E, 1) be in A/a; choose x, y in A with 11:(x) = E and 1:(y) = n; then in = i=(xy), whence

flin) =f(7=(xy)) =f(x!/) =f(x) fly) =f(E)-f(71) andf(e) =f(-n:(l)) =f(l), hencefis a ring homomorphism. THEOREM 3. Let A and B be rings andf a homomorphism ofA into B. (a) The kernel 0 off ts a two-sided ideal of B. (b) The image B’ f(B) off is a suhring of B (e) Let 1:: A—> A/a and i: B’ —> B be the canonical morphisms. There exists one and only one morphismfofA/a into B’ such thatf — i ofo 7: andfis an isomorphism.

Asf15 a morphism of the additive group ofA into that of B, a is a subgroup ofA. IfxeaandaeA, thenf(ax) =f(a)f(x) = 0, hence axe aand similarly xa e a; hence a is a two-sided ideal ofA Assertion (b)'is obvious. Asf18 zero on a, there exists a morphismfofA/a Into B’ such thatf — i of o n- (Theorem 2). The uniqueness off and the fact that f is an isomorphism follow from Set Theory, II, § 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= f1(B’) is a subring of A containing a. B’ andf | B defines when passing to the quotient an isoIff is surjectioe, thenf(B) B’. onto B/a Qf morphism (b) Let b' be a left (resp. right, two-sided) ideal ofA'. Then I) = f1(b’) is a left (resp. right, two-sided) ideal of A containing a.

106

MULTIPLIGATION OF muss

§ 8.9

(c) [f b’ is a two-sided ideal ofA', the composite mapping ofthe canonical morphism A’ —> A’Ib’ andf: A -> A' defines, when passing to the quotient, an injectioe morphismf q/b into A’Ib'. [ff is surjectioe, fis an isomorphism q/b onto A’lb’. (d) Supposefis surjectioe. Let (I) be the set ofsubrings (resp. left ideals, right ideals, two-sided ideals) ofA containing a. Let (D' be the set qfsubrings (resp. left ideals, right ideals, two-sided ideals) ofA'. The mappings B —>f(B) and B’ I—> 3‘1(B’) are inverse bijeetions of (I) onto (D’ and (D’ onto (I). (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’Ib’ considered in (c) has kernel I) and hencefis an injective morphism ofAll) into A’Ib’ (§ 8, no. 7, Theorem 3). Iffis suijective, g is suijective and hencefis sutjective. Suppose f is surjective. By the above, the mapping 0: B' n—> ?(B’) is a. mapping of (D’ into (1). Clearly the mapping 1;: B »—>f(B) is a mapping of(I) into (1”. Then 9 o 1; = Ida, 1; o 0 = Id,” whence (d).

Remark. In the above notation, 0 and '4 are ordered set isomorphisms ( A/a —> (A/a)/(b/a) defines when passing to the quotient an isomorphism ofA/b onto (A/a)/(b/a). _ It suflices to apply Proposition 5 to the canonical morphism of A onto A/a. 9. MULTIPLIGATION OF IDEALS

Let A be a ring and a and b two-sided ideals of A. The set of elements of the

formsclyl + - - . + xnynwithn 2 0, x, e candy, 6 b for l < i S n, isobviously

a two-sided ideal ofA, 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 = be + ca. IfA is commutative, multiplication of ideals is commutative. ab C aA C aandab CAb C b, hence

(21)

I

ab C a n b.

PROPosmON 6. Let a, 61,. . ., b, be two-sided ideals of A. If A = a + b, fir all 1.,t = a + b1b2...bn = a +(b1nban"'nbn).

107

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ALGEBRAIG STRUCTURES

By (21) it suffices to prove that A = a + 61132. . .6". By induction, it suffices

to consider the case where n = 2. By hypothesis, there exist a, a’ e a, b, e 6,, 626b, such that l = a + b1 = a’ + 62. Then

1 = a' + (a + b1)b2 = (a’ + abz) + blbaea + b162, whence A = a + blbz.

PROPOSITION 7. Let [3,, . . ., b" be two-sided ideal: qf A such that b, + b, = A for

£99 j. Then b, n b, nmn 5,, = cabana,” . .b,(,,. In particular, if A is commutative, b, n b2 (1- . -n b,l = blba. . .6” (cf. Exercise 2).

If

Suppose first n = 2. There exist a1 6b,, (12 Eb, such that a1 + a2 = 1. x 6 b1 n 62, then x = x(a1 + a2) = xal + xaa e bab1 + b1b2. Hence

5, n b, = b,b, + b,b,.

Suppose now the equation of the proposition is established for all integers (d). Suppose that the el exist with the properties in (b). For 1' ;E j, 1—- e,eb,, e, = e,(1— e,) 66,, hence leb, + b, and A = b, + 1),. On the other hand, if I is given a total ordering and 0:0,“ is a family ofelements ofA, then, since the e, are central,

Ex,“ — e.) = (11:06:! (I — e,)) = (1—1 x,)(l — HQ) = 0 Hal tel tel

hence 1:; b, = {0}. (d) => (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 (a), the family of b; = A’(l — e,) is a direct decomposition ofA’. Then b. = A(1 — e,) = Ab: for all is I. Hence condition (e) holds. Finally, suppose condition (e) holds. By (a) => (b), there exists a family (eon.=1 of idempotents of A’ such that e,e, = O for iaé j, l = “521e, and

b,’ = A’(l - e,) fori e I. Then 6, = Abt’ = A(l —— e.) for i e I, hence condition (b) holds. Remark. Let A be a ring. Let («10ml be a finite family of subgroups of the additive group A+ of A such that A+ is the direct sum of the a,. Suppose any 6 ai

for i e I and ma, = {O} for 1' ye j. Then a, is for all i e I a two-sided ideal ofA. With addition and multiplication induced by those on A, a, is a ring with unit element the component of l e A in (1,. If b, = 1; a,, clearly the b, satisfy con-

dition (c) of Proposition 10 and hence (50m: is a direct decomposition of A, which is said to be defined by ((1,),61.

. Example: Ideals and quotient rings of Z

An ideal of Z is an additive subgroup of Z and hence of the form n.Z with n 2 0; conversely, for every integer n 2 O, 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 (I) is equal to Z, the ideal (0) consists of 0 and the ideals distinct from Z and {0} are therefore of the form 7:2 with n > I. If m 2 land n 2 l, mZ D nZ ifand only ifn em.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

ideal: of Z are the ideals Q)“ the form [JZ where p is a prime number (§ 4, no. 10, Definition 16). Let m and n be two integers 2 l. The ideal mZ + 712 is principal, whence there is an integer d 2 1 characterized by dZ = mZ + n2; for every integer lll

ALGEBRAIG STRUCTURES

I

r 2 l, the relation “r divides d” is equivalent to r2 3 dz and hence to “rZ 3 ml and r2 3 n2”, that is to “r divides m and n”. It is thus seen that the

common divisors of m and n are the divisors of d and that d is the greatest of the divisors 2 1 common to m and n; d is called the greatest common divisor (abbreviated to g.c.d.) of m and n. As dz = mZ + 112, there exist two integers x and 3/

such that d = mx + ny. m and n are said to be relatively prime 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 = l.

The intersection of the ideals ml and n2 is non-zero for it contains mn and hence is ofthe form rZ with r 2 l. Arguing as above, it is seen that the multiples ofr are the common multiples ofm 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 multiple (l.c.m.) of m and n. f

f

The product of the ideals m2 and n2 is the set of 1:21 mxmy, = mat; x,y,) for :61, . . ., y, e Z and hence is equal to n.

For every integer n 2 l, 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 = l, we obtain the zero ring. PROPosrrION 11. Let n1, . . . , nd be integers 21 which are relatively prime in pairs and f

n = n1. . .n,. The canonical homomorphism of Z into the product ring H Z/n,Z is 1’

surjective of kernel nZ and defines a ring isomorphism of Z/nZ onto £1 Z/niz. Let a, = n,Z for i = 1,. . ., r. By hypothesis, a, + a, = Z for i séj. The

proposition then follows from Proposition 9. The above results, as also those concerning decomposition into prime factors,

will be generalized in Chapter VII, § 1, which is devoted to the study ofprincipal ideal domains and in Commutative Algebra, Chapter VII, § 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 ofA. Let As be the monoid of fiactions ofA (provided only with multiplication) with denominators in S (§ 2, no. 4). Let c: A —> AS be the canonical morphism. There exists on As one and only one addition satisfying thefollowing 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 As satisfying conditions (a) and (b).

112

RINGS or FRAcrmNs

§ 8.12

i Let x, y e As. Let S’ be the stable multiplicative submonoid of A generated by S. There exist a, b e A and [1, q e S’ such that x = (1/11, y = b/q. Then

x = sWham)“,

y = s(bp)s(pq)'1,

whence

(23)

x + y = (=(aq) + 80211))8999)‘1 = 8(49 + owe-1 = (a4 + I’M/liq-

This proves the uniqueness of the addition. We now define an addition on As by setting x + y = (aq + b1») /pq. It is necessary to show that this definition does not depend on the choice ofa, b, p, 4.

Now, if a', b’ e A, p', q’ e S’ are such that x = a’/p’, y = b’Iq', there exist 3 and t in S' such that ap's = a’ps, bq’t = b’qt, whence

(44 + [’1’) (11'9’) (5‘) = (“'9' + b'P’XMXIt) and hence

(a? + I’M/M = (“'9' + WWI/4'. It is easily verified that addition in As is associative and commutative, that 0/1 is identity element for addition, that ( —a)/p is the negative of a/p and that x(y + z) = xy + xz for all x,y, zeAs. Ifa, b EA, then

e(a + b) = (a + b)/l = all + b/l = 3(a) + s(b) and hence s is a ring homomorphism.

DEFINITION 8. The ring defined in Theorem 4 is called the ring offractions associated with S, or with denominators in S, and is denoted by A[S‘1].

The zero ofA[S‘1] is 0/1, the unit ofA[S‘1] is 1/1. We shall return to the properties of A[S‘1] in Commutative Algebra, Chapter II, § 2. If S is the set of cancellable elements ofA, the ring A[S‘ 1] is called the total ring offractions ofA. A is then identified with a subring ofA[S " 1] by means of the mapping 5, which is then injective (I, § 2, no. 4-, Proposition 6). THEOREM 5. Let A be a commutative ring, S a subset ofA, B a ring andfa homomorphism ofA into B such that every element off(S) is invertible. There exists one and only onefofA[S“1] into B such thatf=fo e.

We know (§ 2, no. 4-, Theorem 1) that there exists one and only one morphism f of the multiplicative monoid A[S‘1] into the multiplicative monoid B such that f = fo 3. Let a, b e A, p, q E S’ (stable multiplicative submonoid

113

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ALGEBRAIC STRUCTURES

of A generated by S). As the elements off(A) commute in pairs,

flu/p + 6/9) =f((aq + I’M/M) =f(aq + 111))f(liq)'1 = (fGM(9) + f(b)f(17))f(II) ' Ef(q) '1 = f(a)f(1)) ‘_‘ + fGM(9) ‘1 =f(a/P) +f(b/q)Hencef is a ring homomorphism.

§9. FIELDS 1. FIELDS

DEFINITION l. A ring K is called afield ifit does not consist only qfO 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 (§ 8, no. 1). The opposite ring ofa field is a field. A field is called commutative if its multiplication is commutative; such a field is identified with its opposite. A non-commutative field is sometimes called a skewfiela'. 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, § 1, no. 3), also the field of complex numbers (General Topology, VIII,

§ 1, no. 1) and the field of quaternions (General Topology, VIII, § 1, no. 4).

These fields are commutative with the exception of the field ofquaternions...

(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 subfield of K and K is then called an extensionfield of L ; it amounts to the same to say that L is a subring of K and that x‘1 e L for every non-zero element x of L. If (L,),eI is a family of subfields of K, then 91 L, is a subfield of K; for every subset'X ofK there thus exists a smallest subfield of K containing X; it is said to be generated by X. PROPOSITION 1. Let K be afield. For every subset X of K, the centralizer (§ 8, no. 5, Example 3) X' of X is a subfield of K. We know (loo. cit.) that X’ is a subring of K. On the other hand, if x ye 0 is permutable with z e X, so is x'1 (I, §2, no. 3, Proposition 5) and hence X’ contains the inverse of every non-zero element Of X’.

COROLLARY. The centre of afield K is a (commutative) subfiela' of K. 114-

FIELDS

§ 9.1

THEOREM 1. Let A be a ring. Thefollowing conditions are equivalent: (a) A is afield; (b) A is not reduced to 0 and the only left ideals ofA 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 all x EA, x = (xa‘1)a e a; hence a = A. Suppose that A satisfies condition (b). Let x 5* 0 be in A. We need to prove that x is invertible. The left ideal Ax contains a: and hence is not zero, whence Ax = A. There thus exists x’ e A such that x’x = 1. Then x’ aé 0 since 1 ye 0.

Applying the above result to x’, it is seen that there exists x” EA such that x”x' = 1. Then x” = x".l = x”x’x = 1.x = x, hence xx’ = 1. Hence x' is the inverse of x. Remark. In Theorem 1, left ideals may be replaced by right ideals. In Chapter VIII, § 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 M3(Q) ofsquare matrices of order 2 with rational coefficients is quasi-simple but is not a field),;.

COROLLARY 1. Let A be a ring and a a two-sided ideal ofA. For the ring A/a to be a field, it is necessary and sufioient that a be a maximal lefl ideal ofA. The left ideals ofA/a are ofthe form [3/a where b is a left ideal ofA containing a (§ 8, no. 8, Corollary to Proposition 5). To say that A/a 9e 0 means that a at: A. Under this hypothesis, A/a is a field ifand only if the only left ideals ofA containing a are a and A (Theorem 1), whence the corollary. COROLLARY 2. Let A be a commutative ring which is not 0. There exists a homomorphism ofA onto a commutativefield.

By Krull’s Theorem (§ 8, no. 6, Theorem 1), there exists in A a maximal ideal a. Then A/a is a field (Corollary 1).

COROLLARY 3. Let a be an integer 20. For the ring Z/aZ to be afield, it is necessary and sufieient that a be prime. This follows from Corollary 1 and §8, no. 11.

For p prime the field Z/pZ is denoted by Fr THEOREM 2. Let K be afield and A a non-zero ring. Iff is a homomorphism of K

into A, then the subringf(K) ofA is afield andfdefines an isomorphism ofK ontof(K) . Letabe thekerneloff. Then 1 ¢aforf(l) = l 96 OinAand, asaisaleft ideal of K, a = {0} by Theorem 1. Therefore f is injective and hence an isomorphism of K onto the subringf(K) of A; the latter ring is therefore a field.

115

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ALGEBRAIC STRUCTURES

2. INTEGRAL DOMAINS

DEFINITION 2. A ring A is called an integral domain (or a domain If integrity) if it is commutative, non-zero, and the product of two non-zero elements ofA 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 of the form a or —a with a > 0 and (—a)b = —ab, (—a)(—b) = ab for a > 0, b > O, 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 (§ 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. [f A is an integral domain, the total ring offractions K of A is a commutative ring.

The ring K is commutative. It is non-zero since A 96 {0}. As A is an integral domain, every non-zero element of A is cancellable and K consists of the fractions a/b with b aé 0. Now a/b 56 0 implies a aé 0 and the fraction b/a is then inverse of a/b. The total ring of fractions of an integral domain is called itsfield offractions. 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 the field qffi'actions A. The canonical injection of A into B can be extended uniquely to an isomorphismf ofA' onto a subfield of B. (c) The elements off(A’) are the ny‘1 where x e A, y e A, y aé 0. Assertion (a) is obvious. The canonical injection ofA into B extends uniquely to a homomorphismfofA’ into B (§ 8, no. 12, Theorem 5). Assertion (b) then follows from no. 1, Theorem 2. The elements of A’ are the fractions x/y with :4 EA, y e A, y aé 0 andf(x/y) = xy'l, whence (c). 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 aé pand,ifxeA —pandyeA—p, thenxyeA —p. (c) p is the kernel of a homomorphism of A into afield. 116

THE FIELD or RATIONAL NUMBERS

§9.4

The implications (c) a (b) $ (a) are obvious. IfAlp is an integral domain, let f be the canonical injection of A/p into its field of fractions and g the canonical homomorphism of A onto A/p ; then p is the kernel off a g, whence the implication (a) a (c).

DEFINITION 3. In a commutative ring A, an ideal I: satisfying the conditions qroposition 4- i: called a prime ideal.

Examples. (1) Let A be a commutative ring. If m is a maximal ideal ofA, m is prime; the ring A/m is a field (no. 1, Corollary 1 to Theorem 1). (2) IfA is an integral domain, the ideal {0} ofA is prime (but not maximal in general, as the example of the ring Z proves). 4. THE FIELD OF RATIONAL NUMBERS

DEFINITION 4-. Thefield qffractions qf the ring Z qfrational integers is called thefield if rational numbers and is denoted by Q. The element: qare called rational numbers. Every rational number is thus of the form a/b where a and b are rational integers with 6 ye O (and we may even take b > 0 as the relation

“/17 = (-a)/(—b) proves). Q+ is used to denote the set of rational numbers of the form a/b with a e N and b e N*. We have the relations: (I)

Q+ + Q+ = Q+

(2)

Q+~Q+ = Q+

(3)

(1+ 0 (-9») = {0}

(4‘)

Q+ U ("Q+) = Q.

(5)

(1+ 0 z = N.

The first two follow from the formulae a/b + a'/h’ = (ab’ + a’b)/hb’, (a/b)(a’/b') = aa’/bb', 0 e Q,“ l EQ+ and the fact that N is stable under addition and multiplication and N* under multiplication. Obviously 0 e Q,“

whence Oe (—Q+); let x be in (1,, n (—Q+); then there exist positive integers a, b, a’, b’ with b 9E 0, 12' 9e 0 and x = a/b = —a’/h’; then ab’ + ba’ = 0, whence ab’ = 0 (for ab' 2 0 and ha’ 2 O) and therefore a = 0 since b’ sé 0; in other words, at = 0. Finally, obviously N C Z and N C Q+.

Conversely, if x belongs to Z n QH it is a rational integer; there exist two rational integers a and b with a 2 0, b > 0 and x = a/b, whence a = bx; if

x¢N, then —x > 0, whence —a = b(—x) > O and consequently a < 0 contrary to the hypothesis.

Given two rational numbers at and y, we write x < y if y — xe Q+. It is easily deduced from formulae (1), (3) and (4) that x < y is a total ordering on 117

ALGEBRAIG STRUCTURES

I

Q, from (5) that this relation induces the usual order relation on Z. Finally, from (I) it follows that the relations x < y and x’ < y’imply x + x’ < y + y and from (2) that the relation x < y implies x2: < yz for all z 2 0 and xz 2 yz forz < 0 (of. VI, §2, no. 1). >0 strictly positive if Let x be a rational number. x is said to be positive if x2 x > 0, negative 1fx < 0 and strictly negative ifx < 0.1‘ The set ofpos1t1ve rational numbers is Q+ and that of negative numbers is — Q+. If Q* denotes the set of

non-zero rational numbers, the set Qt of strictly positive numbers is equal to Q" n Q+ and —Q".‘, is the set of strictly negative numbers. The sets Qt and {1, — l} are subgroups of the multiplicative group Q*. Every rational number x aé 0 can be expressed in one and only one way in the form 1. y, (— l). y, where y > 0; hence the multiplicative group *Q“ is the product of the subgroups Q* and {1, -— l}, the component of x in Q* is called the absolute value of x and is denoted by |x|; the component of x in {—1,1} (equal to 1 ifx > O, to —1 ifx < 0) is called the sign ofx and denoted by sgn x. Usually these two functions are extended to the whole of Qby setting [0| = 0 and sgn 0 = 0.

§10. INVERSE AND DIRECT LIMITS Throughout this paragraph, I will denote a non-empty preordered set, a < [3 the preordering on I. The notion of inverse (resp. direct) system ofsets relative to the indexing set I is defined in Set Theory, III, § 7, no. 1 (resp. Set Theory, III, § 7, no. 5, under the hypothesis that I is right directed). l. INVERSE SYSTEMS OF MAGMAS

DEFINITION 1. An inverse system of magmas relative to the indexing set I is an inverse system of sets (ww) relative to I, each Eu having a magma structure and eaehfafl being a magma homomorphism. Let (EmfaB) be an inverse system of magmas whose laws are written multiplicatively. The inverse limit set E = lim Ea is a subset of the product magma

—fi,5(x5) for a E“ such that 110‘ = f,” o it“ for a: S [5. There exists one and only one homomorphism in F -—> E such that an =f‘,I o u for all cc 5 I (namely x1—-> u(x) = (ua(x))“I).

If the magmas E“ are associative (resp. commutative), so is E. Suppose that

each magma E,I admits an identity element e, and that the homomorphisms fl“, are unital. Then e = (5001151 belongs to E for em = fuB(eB) for at < B and it is an identity element of the magma E; with the above notation, the homomorphisms fa. are unital and if the u“ are unital then u is unital. Further, an

element x = (370”: ofE 15 invertible if and only if each of the x‘, is invertible 1n the corresponding magma 1,,E and x =(x; 1),“; this follows from the formulafiw(x” 1) = flown)“ = x; 1 for a < (3. From these remarks it can be deduced that if the magmas E, are monoids (resp. groups) and thefig are monoid homomorphisms, then the magma E is a monoid (resp. a group). In this case we speak ofan inverse system ofmonoids (resp. groups). The universal property goes over immediately to this case. It 15 left to the reader to define an inverse system of rings (wus) and to verify that E = lim E, is a subring of the product ring H Ea, called the inverse limit ring qf the rings IE“; it can be verified that the universal property extends to this case. Let (‘3 = (Emfw) and (2' = (E;,f.{3) be two inverse systems of magmas (resp. monoids, groups, rings) relative to the same indexing set. A homomorphism of (E into 6’ is an inverse system (11‘0“I of mappings :1“: Eu —> E;

such that each u“ is a homomorphism. Under these conditions, the mapping u = lim u“ of l(u_n- E“ into 131E; is a homomorphism (cf. Set Theory , III,

§7,£Z2). 2. INVERSE LIMITS OF ACTIONS

Suppose there are given two inverse systems of sets (Q1, 4)“) and (Emfua) relative to the same indexing set I. Suppose there is given forall or e I an action of Q, on E,I such that (l)

fas(0’oxn) = un(‘°a) fad-”9)

for ac< 3, x5 6 EB, (93 6 QB. Then the family of actions considered'1s said to be = (3512)”: an inverse system qf actions. Let Q: lien} Q and E = 111 E“ andj; the canonical mapping of E“ into E. Recall that

(2)

fit °fBu =f¢ for a < B,

(3)

E = glam“ -

By (2), also (4)

.fix(Ea) Cfa(Ea) for a < B-

If xwy, 6E“ are such that faced) = fawn), there exists a [3 2 at such that ffloz(xu) =jl3m(ya)'

PROPOSITION 1. There exists on E one and only one magma structure fbr which the mappings fa: E¢—> E are homomorphisms. If the magmas E“ are associative (resp. commutative), so is E. Ifthe magmas E“ and the homomoiphismfim are anital, so are the magma E and the homomorphismsfa.

The magmas E“ will be written multiplicatively. Let x, y be in E. There exist at in I and x“, ya in E, such that x = fa(x¢) and y = foxy“). If there exists a magma structure on E for which f“ is a homomorphism, then x.y = j;(xayu), whence the uniqueness of this magma structure. To prove the existence, we must prove that for at, L3 in I, x“, ya in E“ and

x3, yg in Ea: the relations

(5)

'

am) =fa(xé),

imply fit(xuyu) =fB(xlil/l!)'

120

For

Y 2 0t

My“) =fa(yé) and

Y 3 is:

we

sat

x7 =fw(xa):

DIRECT SYSTEMS OF MAGMAS

§ 10.3

g, = fw(ya), x; = f,3(x{,), g; = fya(y§). By the definition of direct limit, there existsyiaithy 2 any 2 [3,x, = x.’,,y,, =y.’,. Then

fidxayu) =f!(f!¢§xuya)) =fy(xm) =fl,(x;y4) =fy(fya(xéyé))

= fa (”51/11) Suppose the magmas E,,I are associative. Let x, y, 2 be in E. There exist as e I and elements xa, ya, 2“ in Ea such that x =fx(xa):

y =.f:1(.’/a):

z =jix(zu)-

Then xy =flx(xaya), when¢e (xy)z =fa((xuyu)zu)s similarly x(y2) =fa(xa(yuza)), whence (xy)z = x(yz) for (mug/(upm = x¢(yuzu). The case of commutative magmas is treated analogously. Suppose finally that each magma E“ has an identity element e, and that fade“) = eflforoc < (3. Fora,[3inI,thereexistsyiaith-y 2 dandy 2 [5, whence

fade.) =.fiv(fm(eu)) =fy(ev) =f,(fya(ea)) =fe(ee) and there thus exists an element e in E such thatf“ (ea) = e for all at e I. Let x be in E; choose a: e I and noI 6 Eu such that x = fa(x¢). Then

ex =fix(ea) -fu(xu) =fu(ea-xa) =fa(xa) = x and similarly x.e = x, hence e is the identity element of E. The magma E is called the direct limit of the magmas Eu. PROPOSITION 2. Let (wflu) be a direct system ofmagmas and let E be its direct limits fl: E¢»E the canonical homomorphism. Suppose a magma F and homomorphism u“: E“ —-> F are given such that u, = “a 0fan,for at S p. There exists one and only one homomorphism u: E —> F such that u“ = u of,for all or e I. [f the magmas E0, and F

and the homomorphismsf,m and u“ are unital, the homomorphism u is unital. We know (Set Theory, III, § 7, no. 6, Proposition 6) that there exists one and only one mapping in E —> F such that u“ = u of; for all at e I. We verify that u is a homomorphism: let x, y be in E, a: in I and x“, ya in E, such that x = fi(xu)

and y = fay“). Then xy = fl,(xayu), whence

“(31” = um(xaya)) = "maul/oz) = ua(xa)ua(ya)

= "(fu(ua))u(.fit(ya)) = u(x)u(y).

We consider now the unital case and let em denote the identity element of Ed, e that of E and e’ that of F. Choose a: e I, then e = fu(e¢), whence

I

“(5) = “bake” = uu(eu) = 3'

for u“ is unital. Hence u is unital.

121

ALGEBRAIC STRUCTURES

I

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 (Eu,f3a)a_ Be: is a monoid. We show that E is a group if the E, are groups: let x e E, o: e I and :6m E E0, be such that x = fact“) ; the element y = fa(x; 1) of E is the inverse of:4 (§ 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 (Awfw) be such a direct system; let A = firm A“ andfix: A, —> A the canonical homomorphisms. There exists (Proposing:>2) on A one and only one addition and multiplication characterized by x + y = fu(xu + ya), xy = fa(xaya) for a e I, x“, ya in A“l and x = fax“), y = fay“). Under addition A is a commutative group and multiplication is associative and unital. Finally, for x, y, 2: in A, choose at in I and x“, ya and z“ in A“ with x =fu(xu):

y =j,(cou) and x = j;(xa) (4)“: $2“ —> (I andf“: Ea —-> E denote the canonical mappings); then om: = fu(mu.xu). The action of Q on E is called the direct limit qf the actions of the Q“ on the E“. Ifthe Q“ are monoids and each action of (2.,I 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.

123

EXERCISES

§1

'

1. Let E be a set, A C E x E and (x,y) u—>xTy, (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 ‘13(E). If (X¢)ue A and (YQBEB are any two families of subsets of E, then («LeJA X“) T (BLEJB YB) = («.mLEJAxB 0"“ T Y3)‘

2. Let T be a law of composition not everywhere defined on a set E. Let E' be the subset of 913(E) consisting of the sets {x} where .7: runs through E and the empty subset a of E; show that E’ is a stable subset of ‘D(E) under the law (X, Y) »—> X T Y (Exercise 1); deduce that, if E denotes the set obtained by adjoining (S_et Th_eory, II, §4, no. 8) to E an element a), the law T can be extended 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) »—> 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 2) 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 I of no. 3 can be generalized as follows: ifone ofthe two sides offormula (3) is defined, the other is defined and is equal to it. 4. (a) Given a set E, let (I) be the set of mappings into E ofany subset of E; iff, g, h are three elements of (I), show that if the composition (f o g) o h is defined so isfo (g o h), but that the converse is not true; if these two compositions are defined they are equal.

124-

EXERCISES

(b) Let 3 be a family of non-empty subsets of E such that no two have an element in common and let ‘i’ be the subset of (I) consisting of the bijective mappings of a set of 8‘ onto a set of 8. Show that, under the law induced by the lawfo g on ‘1’, the condition of Exercise 3(a) holds.

5. Show that the only triples (m, n, [2) of natural numbers #0 such that (m")” = m” are: (1, n, p), n and ,0 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 consistingofthe elementsxsuchthatx T (y T z) = (x T y) T zforally 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 1; two elements of E, the sets

{a} T E, E T {b}, {a} T E T {b}, E T {a} T E are stable subsets of E under the law T.

8. Let T be an associative law on E and a an element of E; for all x e E,

y e E, we write x J. y = x T a T y: show that the law J. is associative. 9. On a set E the mappings (x, y) »—> x and (x, y) »-> y are opposite associative laws of composition. 10. LetXandYbeanytwosubsetsofasetEandletXTY = XUYif XnY = a, XTY = E iaY 9e 2‘; show thatthelawofcomposition thus defined on ‘1)(E) is associative and commutative. l 1. 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 79 0 which are permutable under the law (x, y) »——> x” are 2 and 4.

13. Show that under the law (X, Y) t—> X a Y between subsets of E x E the centre is the set consisting of 25 and the diagonal. 14. Show that under the law of composition fo g between mappings of E into E the centre consists of the identity mapping.

15. A law T on a set E is called idernpatent ifall the elements of E are idempotent (no. 4) under this law, that is ifx 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; ifwe write x < 1/, any two elements x, y of E admit a least upper bound (under this order relation) equal to x T 1/. Obtain the converse.

125

ALGEBRAIG STRUCTURES

I

16. Let T be an associative and commutative law of composition on a set E. Let n be an integer > 0. The mapping .7: I—-> T x of E into E is a morphism.

§2 1. Let T be a law of composition on a set E. Denoting by F the sum set (Set Theory, II, § 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 associative, the law T 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. (a) For T to be associative, it is necessary and sufficient that every left

translation 7,, be permutable with every right translation 8,, in the set of mappings of E into E (with the lawfo g). (5) Suppose that E has an identity element. For T to be associative and commutative, it is necessary and sufficient that the mapping (x, y) »—> x T y be a morphism of the magma E x E into the magma E.

3. In the set F ofmappings of E into E, for the relation f o g = fo h to imply g = h, it is necessary and sufficient that f be injective; for the relation g of = h of to imply g = h, it is necessary and sufficient thatf be surjective; forfto be cancellable (under the law 0), it is necessary and sufficient thatf be bijective. 4-. For there to exist on a set E a law ofcomposition such that every permutation of E is an isomorphism of E onto itself under this law, it is necessary and sufficient that E have 0, l or 3 elements. 5. For 2 S 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 multiplicatively; e denotes the identity when it exists, ya and 8,, the translations by a e E; for every subset X of E we write 7,,(X) = aX, 5a(X) = Xa.)

6. For an associative law on a finite set, every cancellable element is invertible (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 x“ for n e N*; if there is an identity element, let B be the set of x" for n e N;

if further .7: is invertible, let G be the set of x“ for a e Z. Show that, if A (resp. 126

EXERCISES

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 (§ 1, no. 4) (observe that if x” and x4 are idempotents, then x” = x“, x4 = x", hence x” = x4) ; if h = x", the set of x” 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 an = a, show that ux = x for all x e E; in particular, if xu = x for all x e E, u is identity element. (b) 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). 11 11. E is called a left semi-group if every element x of E is left cancellable. (a) If u is an idempotent (§ 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 Ev = a and the sets Eu and Ev (with the laws induced by that on E) are isomorphic.

(6) 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 9e R for all a e R (use Exercise 9(a) 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 9E R), for all x eREu, xEu aé 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). (f) 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 xa = mm for all x e E). For example, if E is the set of integers 2 l, with addition, E is a commutative semi-

group such that E = R. (g) If there exists a e E such that 5,, is suijective, either E has an identity element or E = R (examine separately the case where a e R and the case

where a 6 Eu for an idempotent u). 127

ALGEBRAIG STRUCTURES

I

1] 12. Under a multiplicative law on E, let a e E be such that ya is suijective. (a) Show that, if there exists u such that ua = a, then we = 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 suijective and that b be left cancellable. 11 13. Suppose that, for all x e E, vx is surjective. Show that if, for an element a e E, 7., is a bijective mapping, 7,, is a bijective mapping for all x e E (use Exercise 12(b)); this holds in particular if there exist two elements a, b ofE 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.

fl 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, ordered by inclusion). (a) IfM = aE is minimal, then xM = xE = M for all x e M; with the law induced by that on E, M is a left send-group (Exercise 11) in which every left translation is bijective (cf. Exercise 13). (b) IfM = aE and M' = a’E are minimal and distinct, thenM n M' = {25 ; for all b e M, the mapping x' l—> bx’ of M’ into M is a bijective mapping of M’ onto M. Deduce that there exist an idempotent u' e M’ such that bu’ = b and an idempotent u e M such that uu' = u (take u to be the idempotent such that bu = b). Show that u’u = u’ (consider uu’u) and that every 3/ e M’ such that y’u = y’ belongs to M’u’. (0) Show that the mapping x’ I——> ux’ of M’u’ into M is an isomorphism of M’u’ onto Mu; deduce that M and M’ are isomorphic left semi-groups. (d) Let M, (l g i g r) be the distinct minimal subsets of the form aE: deduce from (b) that the idempotents of each left semi-group M, can be arranged in a sequence (uu) (1 g j < 3) so that uu-uk, = u” for all i, j, k. If K is the union of the M,, show that Eu” C K (note that, for all x e E, xq is a

minimal subset of the form aE) ; deduce that Eu” is the union of the quk, for l < k < r (show that (Euu) n Mk = uk,Euk,) and that Eu,,E = K. Prove finally that every minimal subset of the form Eb is identical with one of the 5 sets Eu” (note, using (a), that Ebuub = Eb and deduce that Eb C K). 15. Let (x01 «:2: be a finite sequence of left cancellable elements. (a) Show that the relation xlxz. . .xn = e implies all the relations xi”. . .xnxlxz. . .x, = e which are obtained from one another by “cyclic

permutation” for l s i S n. (b) Deduce that if the composition of the sequence (x‘) is invertible each of the x, is invertible.

1} 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

128

EXER-Es

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 8,,(E) C X. (a) Show that the intersection of two sets of 3 belong to ‘3. (b) Let (I) be the set offunctionsfdefined on a set of 3 taking their values in E and such that, for X e 3, f1(X) belongs to 8. Let R denote the relation between elementsf and g of (D: “there exists a set X e 3 such that the restrictions offand g to X are identical”. Show that R is an equivalence relation; let ‘I’ = (DIR be the quotient set of (I) under this relation. (c) Letfand g be two elements of (I), A e 5 and B e 8 the sets wheref and g are respectively defined; there exists X C B and belonging to 3 such that g(X) c A; if gx is the restriction of g to X, show that the mapping f 0 3x belongs to (I) and that its class (mod. R) depends only on the classes offand g (and not on X); this class is called the composition of that off and that ofg; show that the law of composition thus defined on ‘1’ is associative and has an identity element. ((1') For all a e E, show that the left translation ya belongs to (I) ; let (1),, be its

class (mod. R). Show that the mapping x r—> ([2,, is an isomorphism of E onto a submonoid of ‘I’ and that, ifx e E*, x is invertible in ‘1’ (consider the inverse

mapping of Yx: show that it belongs to (I) and that its class (mod. R) is the inverse of (bx). 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 properties: (at) ForallseSandallaeE,thereexistteSandbeEsuchthatsb = at. ((3) Foralla, b eEand allseSsuch thatsa = sb,thereexisttESsuchthat at = bt. (a) In E x S let (a, 3) ~ (b, t) denote the relation: “there exist 5’ and t’ in __ S such that tt' = 55' and as’ = bt'.” Show that ~ is an equivalence relation. We write E = E x S/~ , denote by as‘1 the equivalence class of (a, 3) (mod ~) and by e: E —> E the mapping a v—> ae" 1. (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(:) is invertible. (a) Show that (E, s) has the universal property described in Theorem 1 (no. 4-). (d) Show that e is injective if and only if the elements of S are cancellable. §3 1. Let a '->f¢ be an action of!) on E. Let F be the image o in EE and G the stable subset of EE under the law (f; g) v—>f o g generated by F and the identity mapping of E.

(a) Show that every subset of E which is stable under the action of Q is also 129

I

ALGEBRAIC STRUCTURES

stable under the restriction to G of the canonical action of ER 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) wheref runs through G and x runs through X.

2. Let J. be an internal law on E which is doubly distributive with respect to the associative internal law T; show that, ifx J. x’ and 3/ J. y' are cancellable under the law T, x J. y’ is permutable with y J. x’ under the law T (calculate the composition (x T 3/) J. (x’ T y') in two different ways). In particular, if the law _L has an identity element, two cancellable elements under the law T are permumble under this law; if all the elements of E are cancellable under the law T, this law is commutative.

3. Let T and J. be two internal laws on a set E such that J. is right distributive with respect to T. (a) If the law T has an identity element a, x J. e is idempotent under the law T for all x e E; if further there exists 3/ such that x .L y is cancellable under thelawT,thenxJ_e = e.

(b) If the law .L has an identity element u and there exists z e E which is cancellable for both laws .L and T, u is cancellable under the law T.

1} 4. Let T and J. 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 _L, prove first that eJ.e ==‘e,notingthate = e.L (uTe)). 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 multiplication 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, 2 such that x T z, y T z and (x + y) T 2 are cancellable under multiplication, then e + e = e (use Exercise 3(a)).

1f 6. 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 7,, and 8,, 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 (L of rational numbers, with the law of composition (x, y) -—> fix + y), is a distributive quasi-group.* (c) In a distributive quasi-group E, every element is idempotent. Deduce 130

EXERCISES

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 21t

morphisms of E.

(e) If E is a finite 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“,

Ax = A, (note, using (2), that there exists y EA“ such that x = a T y) and when x runs through E the sets Ax are identical with the equivalence classes with respect to a relation compatible with the law T. (It) If E is a finite distributive quasi-group and the law T is commutative, the number ofelements ofE is odd (consider the ordered pairs (at, y) ofelements ofEsuchthaty =yTx = a,whereaisgiven).

7. Suppose given on a set E an (associative and commutative) addition under which all the elements of E are invertible and an (associative) multiplication which is doubly distributive with respect to addition *(in other words, a ring structure)“ write [x, y] = xy — yx; the law (x, y) I—> [x,y] is doubly distributive with respect to addition. For x and y to be permutable under multiplication, it is necessary and suflicient that [x, y] = 0; and we have the

identities

[1“, y] = —[y, ’5];

[“3 [ya 1]] + Lu: [25 xii] + [2: [’9 Id] = 0

(the second is known as the “Jacobi identity”). The second identity may also be written [3" [1/2 zIl _ [951/], z] = [[2, 31:31]

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‘a = (‘T‘ x) T (T x). (b) If we write [x,y, z] = (x T y) T 2 — x T (y T 2) (deviation from

associativity of the law T), prove the identities

[x5 .1]: Z] + [2: y, x] = 0

131

I

ALGEBRAIC STRUCTURES

m, z] + [y, z, x] + [2, w] = 0 [x T y, u, Z] = [to [(x T y): Z)]] (the notation [x, y] meaning the same as in Exercise 7)

[xTy,u,z] + [yT z,u,x] + [s,u,y] = 0.

1T 9. Suppose given on E an (associative and commutative) addition under which all the elements of E are invertible and a multiplication which is nonassociative, but commutative and doubly distributive with respect to addition. Suppose further that n e Z, n aé O and we = 0 imply x = 0 in E. Show that if, writing [x, y, z] = (xy)z — x(yz), the identity [W1 “2 z] + [1/2, "a z] + [2.76, “LI/1 = 0

holds, then x’”“‘ = xmx" for all x (show, by induction on [7, that the identity [xq,y, xp‘q] = 0 holds for l < 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 = O 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 T 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 S n < 6 (cf. §2, Exercise 5). Determine the subgroups and quotient groups of these groups and also their Jordon-Holder series.

{I 2. (a) An associative law (x, y) I—> xy on a set E is a group law if there exists 2 e E such that, for all x e E, ex = x and if, for all x e E, there exists 1."

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 ofE onto E and ifthere exists some a e E such that the right translation 8,, is a mapping of E onto E.

3. In a group G everyfinite non-empty stable subset H is a subgroup of G (cf. § 2, Exercise 6). 132

EXERCISES

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,)1‘,‘2,,+1 of an (arbitrary) odd number of elements such that x1 6 A for 1' odd and x‘ e B for i even.

(b) For AB to be a subgroup ofG (in which case it is the subgroup generated by A U B), it is necessary and suflicient that A and B be permutable, that is that AB = BA. (c) If A 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). 7 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 O G“ = {e}; show that G is isomorphic to a subgroup of the product group

U (Glen). 7. If G is the direct product of two subgroups A and B and H is a subgroup of G such that A C H, H is the direct product of A and H n B. 8. Let A and A’ be two groups and G a subgroup of A x A’. We write:

N = G n (A x {6}),

H = pr1(G)

N' = G n ({e} x A'),

H' = pr2(G).

(a) Show that N is normal in H and N' is normal in H'; define isomorphisms H/N —> G/(N x N’) —> H’IN’. (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 of A'. 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 (resp. of normal subgroups of G) distinct from G. Show that 92, 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). 133

ALGEBRAIC STRUCTURES

1

12. Let G be a group such that, for a fixed integer n > 1, (1:11)” = x"y” for all x e G, g e G. If gm denotes the set of x", where x runs through G, and G0,)

the set ofx e G such that x" = e, show that G”) and G0,) are normal subgroups of G; if G is finite, the order of G0” is equal to the index of G0,). Show that, for all x, y in G, also xl‘"y1‘" = (xy)1'" and deduce that x"‘1y" = ”x”‘1; conclude from this that the set of elements of G of the form x”"““ 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’IH is isomorphic to S. Let in(G) denote the set of isomorphism classes of simple groups occurring in G. (a) Show that in(G) = {a ¢> 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). (0) Let G1 and G; be two groups. Show the equivalence ofthe two following properties:

(i) in(Gl) n in(Gz) = (2‘;

(ii) Every subgroup of G1 x G2 is of the form H1 x Hz with H1 C G1 and (Use Exercise 8.) *When G1 and G2 are finite groups, show that (i) and (ii) are equivalent to: (iii) Card(G1) and Gard(G2) 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 representative system of the left cosets mod. H (cf. Set Theory, II, §6, no. 2); this condition is equivalent to the following: The mapping (x, y) s—> 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 its T, let x(g, t) ET and y(g, t) EH 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, ta T generate H.

(If H' denotes the subgroup of H generated by these elements, show that T.H’ is stable under the 7,, g e S, hence also under the 7,, g e G, and deduce that T.H' = G, whence H’ = H.) (12) Suppose that (GzH) is finite. Show that G can be generated by a finite subset if and only if H can.

1[ 15. Let 9;- be a set of stable subgroups of a group with operators G; E} is said to satisfy the maximal condition (resp. minimal condition) if every subset of 3, ordered by inclusion, has a maximal (resp. minimal) element. 134-

EXERCISES

Suppose that the set ofall stable subgroups ofa group G satisfies the minimal condition. (a) Prove that there exists no stable subgroup of G isomorphic to G and distinct from G (argue by reductia 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 9!! be a set of minimal normal stable subgroups of G and S the smallest stable subgroup of G containing all the subgroups belonging to 9R; show that S is the direct product of afinite number of minimal normal stable subgroups of G (let (Mn) be a sequence of minimal normal stable subgroups of G belonging to 912 and , such that Mn 4. 1 is not contained in the stable subgroup generated by the union of M1, M2, . . ., Mn; let 8,, be the stable subgroup generated by the union of the Mn of index n 2 k; show that S,” 1 = 5,, 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 subgroup M of G is the direct product of a finite number of simple groups isomorphic to one another (let N be a minimal normal subgroup qf M; show that M is the smallest subgroup of G containing all the subgroups aNa‘ 1 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).

1} 17. Let G be a group with operators; a composition series (G,) of G is called normal if all the G, are normal stable subgroups qf G; a normal series 2 is called principal if it is strictly decreasing and there exists no normal series distinct from Z, finer than 2 and strictly decreasing. (a) If (G,) 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 G,’, = G, n (G,+1H,) and H}, = H, n (H,+1G,) in the sequences (G1,) and (H,) respectively. (b) If G has a principal series, any two principal series of G are equivalent; for every strictly decreasing normal series 2 there exists a principal series finer than 2. 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 minimal conditions. (0) If G is a group without operators and possesses a principal series and if the set of subgroups of G satisfies the minimal condition, every quotient group

135

ALGEBRAIC STRUCTURES

I

G,/G‘ + 1 is the direct product of a finite number ofsimple subgroups isomorphic

to one another (use Exercise 15) . fi 18. (a) Let G be a group with operators generated by the union of a family (How! of simple normal stable subgroups of G. Show that there exists

a subsetJ of I such that G is the restricted sum of the family (H,),,SJ (apply Zom’s Lemma to the set of subsets K of I such that the subgroup generated by the union of the H, for i e K is the restricted sum of this family). (b) Let A be a normal stable subgroup of G. Show that there exists a subset L, of I such that G is the restricted sum of A and the H, for i EJA. Deduce that

A is isomorphic to the restricted sum of a subfamily of the HI. 1) 19. (a) 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 gé 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 aé {e}, show that x 9% 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 simple, whence a contradiction. Conclude with the aid ofExercise 18.) Obtain the converse (cf. Exercise 18). (b) In order that there exist in a group G a family (Nu) of normal simple subgroups such that the G/N‘, are simple and Q N“ = {e}, it is necessary and sufl'icient that, for every normal subgroup N ye {e} of G, there exist a normal subgroup N’ ye G such that NN’ = G. (To see that the condition is necessary, consider an x e N distinct from e and an Nu not containing x. To see that the condition is sufficient, for all x 75 e in G, consider the normal subgroup N generated by x and a normal subgroup N’ aé 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 ;éG and deduce the conclusion.) 20. (a) Let G be a group, the direct product of two subgroups A and B. Let CA be the centre of A and let N be a normal subgroup of G such that NnA = {l}.ShowthatN C CA x B. (b) Let (3,),eI be a finite family of non-commutative simple groups. Show

that every normal subgroup of 1:! S, is equal to one of the partial products 1;; S, where] is a subset of I.

21. Let G be a group of finite length. Show the equivalence ofthe following properties:

(a) G is a product of simple groups isomorphic to one another.

136

EXERCISES

(b) No subgroup of G distinct from {I} and G is stable under all the automorphisms of G. i

1122. Let E be a quasi-group (§ 3, Exercise 6) written multiplicatively, possessing an idempotent e and such that (xy) (zt) = (x2) (gt) for all x, y, z, t. Let u(x) denote the element of E such that u(x)e = x and 0(x) the element of E such that ev(x) = x; show that the law of composition (x, y) 1—» u(x)v(y) is a commutative group law on E under which e is identity element, that the mappings x r—>xe and 1/ Hey are permutable endomorphisms of this group structure and that xy = u(xe)v(ey) (start by establishing the identities e(xy) = (ex) (ey), (xy)e = (xe) (ye), e(xe) = (ex)e; then note that the relations at = y, ex = ey and xe = ye are equivalent). Obtain the converse. 1} 23. Consider on a set E an internal law, not everywhere defined, written multiplicatively and satisfying the following conditions: (1) ifone of the compositions (xy)z, x(yz) is defined, so is the other and they are equal;

(2) ifx, 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 9:4, e; and x‘1 such that 6,1 = x, are; = x, x'lx = e; ; ex is called the left unit of x, e; the right unit of x, x."1 the

inverse of x (by an abuse of language).

(a) Show that the compositions xx'l, x‘lex, egx‘l, exex, ege; are defined and that xx'1 = ex, x‘lex = e§,x'1 = x‘l, exe,‘ = ex, ege; = 12;.

(b) Every idempotent e of E (§ 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 e, = 11/”); if xy = 2, then x‘lz = y, 231-1 = x, y‘lx'l = 2‘1, z‘lx = y”, yz"1 = x‘1 (the compositions appearing on the left-hand sides of these relations being defined). ((1) For any two idempotents e, e' of E, let Ge, ,1 denote the set of x E E such that e is left unit and e’ is right unit of x. Show that, if G,, e 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 a 6 Ge, y, show that x v—> xa is a bijective mapping of GM, onto G,_ ,I, y I—-> ay a bijective mapping of G“ 3' onto Ge.e' and x 1-) a'lxa

an isomorphism of the group Ge, 3 onto the group Gel, y. (f) Show that the law defined in § 1, Exercise 4(b) determines a groupoid structure on the set 1I", 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

ALGEBRAIC STRUCTURES

I

of (x, y) and (y', z) is only defined ify' = y and in this case has the value (x, 2). Show that E x E, with this law of composition, is a groupoid (Exercise 23). (b) Let (x, y) E (x’, y') (R) be an equivalence relation compatible with the above composition (that is such that (x, y) E (x’, y’) and (y, z) E (y’, 2') imply (x, z) E (x’, z')); suppose further that the relation R satisfies the

following condition: for all x, y, 2 there exists one and only one t E E such that

(x, y) E (z, t) and there exists a u E E such that (x, y) E (u, 2). Show that under these conditions the quotient structure by R ofthe groupoid structure on E x E is a group structure (prove first that the quotient law is

everywhere defined, since, if :2, y, 2 are three classes such that kg] = :22, then

3) = 2; establish finally that, for all x e E, y e E, (x, x) E (y, y)). (0) 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, fl,(x) denotes the class mod. R of (a, x), show thatf, is a bijective mapping of E onto G and that the relation (17,31) 5 (x', y’) is equivalent to the relation

fa(x) (152M) '1 = fi(y)(f.z(y')) ' 1For G to be commutative, it is necessary and sufficient that the relation

(x, y) E (#3 y') imply (x, x') E (y, y')1f 25. Let E be a set and f a mapping of E'" into E; we write f(x,, x2, . . ., xm) = xlxz. . .xm;

suppose thatf satisfies the following conditions: (1)

(xlxa. . .xm)x,,,+1. . .x2m_1 = x1(x2x3. . .xm+1)x,,,+2. . .x2,,,_1

identically; (2) For all a1, a2, . . ., a,,,_1, the mappings

x r-—> xalag. . .a,,,_.1

x n—> alaa. . .am_1x

are bijections of E onto E. (a) Show that identically (xlxz- - ‘xm)xm+1' - 'm—l = x1":- - 'xl(xi+1' - -xl+m)xt+nl+1' - Jam-1

for every index i such that l < i < m — I (argue by induction on i, by considering the element ((xixs- - 'xm)xm+1' - 'x2m-1)ala2' - Jim—1)(b) For all (1,, a2, . . ., am-” there exists it e E such that for all x x = xa1a3. . .am_2u = M102. . .am_ax.

138

EXERCISES

(c) In the set E" of sequences (ul, 11,, . . ., uk) of k elements of E

(I < k < m - 1) consider the equivalence relation R, which states: for all x1, x2, . . ., xm_k, u,u,...u,,x1x2. . .xm_,, = 01112...v,,x1x2. . .xm_,,; let Ek denote

the quotient set Ek/Rk and G the sum set (Set Theory, II, §4, no. 8) of E1 = E, E3, . . ., Em_1. Let one E1, (3 e E,; if (111, uz, . . ., u,) is a sequence of

class a, (v1, . . ., 1),) a sequence of class (3, consider the sequence (ul, “2, . . ., u” 111, . . ., 0,)

in E‘” ifi +j < m, the sequence (“1: “a: - - '2 "tn—m: (u£+j-m+1- - -utvlv2- - '01))

in E”"'"+1ifi+j 2 m; show that the class ofthis sequence in E”, (resp. E” ,_,,,+ 1) depends only on the classes at and (-1; it is denoted by at. (5. Show that

a group law is thus defined on G, that H = E"...1 is a normal subgroup of G and that the quotient group G/H is cyclic oforder m — l ; prove finally that E is identical with a class (mod. H) which generates G/H and that xlxa. . . x", is just the composition, in the group G, of the sequence (x1, x2, . . ., arm). 1} 26. Let G, G’ be two groups, f: G —> G' a mapping such that, for two

arbitrary elements m ofG,f G/N i) G’, where g is injective. Attention may therefore be confined to the case wheref is injective, which will be assumed throughout what follows. (b) Show that if xy = yx then f(x)f(y) = f(y)f(x) (consider f(x231) and f(x2313), expressing them in several ways).

(a) 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 5e yx.) (d) Show thatf(xyx) = f(x)f(y)f(x). (Use (a) or (b) according to whether xy = yx or xy 96 yx; in the second case, considerf(xay) and f(#2).)

(e) LetAbe the set ofxeG such thatf(xy) =f(x)f(y) for allyeG and B the set of x E G such that f(xy) = f(y)f(x) for all y e G. Show that A 96 G, B 9e 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) 76 f(y)f(x) and u, v in B such that f(uv) = f(v)f(u) ye f(u)f(v). Deduce from this a contradiction by considering successively the two possibilities xu e A, xu E B; in the first case considerf(xuv) and in the secondf(yxu).) The rest of the proof consists of proving that A U B 9e G is impossible, arguing by reductia ad absurdum. IfA U B 75 G, there exist a, b, c in G such that

f(G’)) =f(a)f(b) ¢f(b)f(a) and f(at) =f(¢)f(a) #f(a)f(6)139

ALGEBRAIG STRUCTURES

I

(f) Show thatf(c)f(a)f(b) = f(b)f(a)f(c). (Consider two cases according to whether bc 9!: ob or be = cb. In the first case, considerf(bar); in the second, use (b) and (c) and considerf(abc), f(bra) andf(bac).) (g) Considerf(abac) and obtain a contradiction.

§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 a and is therefore a divisor of the order of G. 11 2. *If G is a finite group of order n, the number of automorphisms of G is f1(H) is a bijection 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,fis 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(xLJG xHxT'l) s 1 + (Card(H) — l)Card(G/H).) 7. Let p be a prime number. (a) Show that n" E 71 (mod. p) for all n e Z. (Reduce it to the case wheren is positive and argue by induction on n using the binomial formula.) (b) Let x, y be two elements of a group G. Suppose that

yxy'1 = x", withnez,

and that x" = I.

Show that y’xy" = x"; deduce that y" 1 commutes with x. (0) 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 l or 2. (Use (12) to prove that G is commutative.) 8. Let A and B be subgroups of a finite group G, NA and N3 the normalizers of A and B, VA and v3 the indices of NA and NE in G, 7A the number of conju-

gates ofA which contain B and r3 the number ofconjugates ofB which contain A; show that vArn = VBYA. 11 9. Let G be a group of permutations of a finite set E; for all s e G, let x(s) denote the cardinal of the set of fixed points of s.

(a) Show that Z} x(s) = Nt, where N = Card(G) and t is the number 0 orbits of G in E (evaluate in two ways the number of ordered pairs (3, x) e G x E such that s(x) = x). (b) Suppose that, for all a e E, the stabilizer H, of a is not reduced to a. Show that, ifx(s) is independent ofs for s 75 e and is equal to an integer k,then k 3hr“ 1) of 6,, into Aut(H) defines when passing to the quotient an isomorphism of 64H onto Aut(H) and that the latter group is isomorphic to 63. (a) Let K be a subgroup of H of order 2. Show that K is not normal in 914.

11. Let E be a finite set, C e 6,, a cycle of support E and 1.- a transposition. Show that 6,; is generated by C and ‘1". 1i 12. (a) Show that every permutation a e 91,, is a product of cycles of order 3 (which are not in general component cycles of a). (Prove it for a

product of two transpositions and use § 5, no. 7, Proposition 9.) (b) If (11, . . ., a, are p distinct elements of [1, 11], let (alag. . .ap) denote the cycle of order ,0 whose support is {a1, a2, . . ., up} and which maps a, to a” 1 for l S i < p—l and a, to al. Show that 9!, is generated by the n — 2 cycles (1 2 3), (l 2 4),. . ., (l 2 n). (Use (a).) (c) Deduce that, if n is odd, 9!" is generated by (l 2 3) and (1 2 . . . n) and, n). ifniseven,by (1 23) and (23 ((1) Show that, if a normal subgroup of 9L, contains a cycle of order 3, it is

identical with 9|” (prove that such a subgroup contains all the cycles (1 2 k), 3 < k < n).

11 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 properties: (a) Every subgroup H' of G containing H is equal to H or G. (1)) 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 = H’x, then gY = Y for all g e H’ and gY n Y = z 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 (b) and not consisting only of the identity element is called primitive.

14-2

EXERCISES

14. A group of permutations I‘ of a set E is called r-ply transitive if, for any two sequences (41, a2, . . ., a,), (b1, 1),, . . ., b,) of 7 distinct elements of E, there

exists a permutation o- e 1" such that o(a,) = b, for l < i < 1', this property not holding for at least one ordered pair ofsequences ofr + 1 distinct elements ofE. (a) Show that an r-ply transitive group is primitive if r > 1. (b) The order ofa permutation group P operating on n objects which is r-ply transitive is of the form n(n — l). . . (n —- r + l)d, where d is a divisor of (n — 1')! (consider a subgroup of the permutations of P leaving invariant r elements and calculate its index). 1T 15. Let I‘ be an r-ply transitive group of permutations of a set E with n elements; for a permutation a e I‘ distinct from the identity permutation, let n -— r be the number of elements of E invariant under a. If .r > r, show that

there exists a permutation r e 1‘ such that a" 11'0“: ' 1 is distinct from the identity permutation and the number of elements of E it leaves invariant is 2n — 2(s — r + 1) (use the decomposition of 0' into its component cycles). Ifs = r, show similarly that there exists 1' e 1" such that 6' 11-61." 1 is a cycle of order 3. Deduce that, ifr 2 3 and I‘ does not contain the alternating group 21", then s > 21 — 2 for every permutation in I‘ (use Exercise 10). Conclude finally that, if I‘ is not identical with 91,, or 6", then r g n/3 + 1.

1f 16. (a) Show that the alternating group 91,, is (n — 2)-ply transitive. (b) Show that ‘21,, is a non-commutative simple group for n 2 5. (Use (a), the method of Exercise 15 and Exercise 12(d) to prove that 21,, is simple if n > 6; examine analogously the cases 71 = 5 and n = 6.)

(c) Show that, for n 2 5, the only normal subgroups of 6,, are {e}, 91,, 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 P. Let PA and I‘B denote

the groups consisting of the restrictions of the permutations in I‘ to A and B respectively and AA and An the subgroups of P which leave invariant respectively every element of A and every element of B. Show that AA and A3 are normal subgroups of I‘ and that PA is isomorphic to F/AA and I“; to F/AB; if AA3 (resp. ABA) is the group consisting of the restrictions of the permutations in AA (resp. A3) to B (resp. A), show that the quotient groups FA/ABA, I",,/AAB and P/(AAAB) are isomorphic (use Exercise 8). 19. Let G be a group, H a subgroup of G and G/H the set of left cosets (mod. 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

ALGEIBRAIC STRUCTURES

I

(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 detemlines on this set a structure isomorphic to that determined on G/H by the law T. (b) 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 isomorphic to the structure of the quotient group G’[H' (to see that the condition ~ is sufficient, show first, With the aid of § 4, Exercise 2(a), that if T is associative

it determines on G/H a group structure; denoting by K the largest normal subgroup of G' contained in H’, show then, using the associativity condition on T, that (r(X T Y)) ‘1r(X)r(Y) E K for all X, Y; deduce that the mapping X n——> r(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 cosets 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 I‘ be the group of permutations of E generated by the left translations 7x and A the subgroup of permutations in l" leaving 2 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) = 7x, the mapping x I-—> 7,, is an isomorphism of the set E with the law T onto the set 1"]A with the law (X, Y) HT(X)7(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). (b) Show that G is commutative if n $ 4 (use Exercise 10). 21. Let p(n) be the number of conjugacy classes of the symmetric group 6”. Show that p(n) is equal to the number of families (x1, . . ., x”) of integers 20 n

such that “21 Lac. = n. no

*Deduce the identity "2014")?

117

= ”-11

1

— Tm.*

22. LetA = 2/22, B = 63 and G = A x B. Ifs is an element ofB oforder

2, let E —p> 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 'c be a homomorphism of G into the automorphism group Aut(A) of A ; ifg e G, a e A, we write ”a = T(g) (a). A crossed homomorphinn of G into A is any mapping 4): G ——> A such that

(gg') = 4%.?) + '4>(g'), the group A being written additively. The crossed homomorphisms of G into A form a group under addition denoted by Z(G, A). (a) If a e A, the mapping g »—> “a — a is denoted by 0“. Show that a :—> 0,, is a homomorphism 0 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 0 is denoted by B(G, A). (12) Let X = A x 1 G be the semi-direct product of G by A. If (I) is a mapping

of G into A, show that g »—> (¢(g), g) is a section of X if and only if 4) belongs to Z(G, A). For the sections corresponding to (1)1, (b3 6 Z(G, A) to be conjugate

by an element of A, it is necessary and sufficient that 4’1 E 4:2 mod. B(G, A). 148

EXERCISES

(6) Let F —'—> E 1) G be an extension of G by a group F whose centre is equal to A. Suppose that, ify E E and x = My), then

i("J”) =y-i(f)-y"1 for all fe A. If ad, is an isomorphism of Z(G, A) onto the automorphism group Aut(E) of the extension E. This isomorphism maps B(G, A) to the group of automorphisms IntE(x), where x runs through A. 11 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‘ g'». If 0' denotes the corresponding homomorphism of G into Aut(C(G, A)), the semi-direct product C(G, A) x a G is denoted by E0. (a) Let s: A —> C(G, A) be the mapping which associates with each a e A the constant mapping equal to a. Show that s is an injective homomorphism, compatible with the action of G. (b) Let A —'-> E 3> G be an extension ofG by A such that i('a) = xi(a):c‘1 if aeA,e andg =p(x). Let p beamappingG—>Esuch thatpo p = Id“. For all x e E, let (1),, be the mapping of G into A such that

Show that

19(11(x‘1)g) = i (¢x(g))9(g) for all g e G‘l’xu = 4):: + ”(WM

ifxnt/ 5F“

Deduce that the mapping (1): E —> E0 defined by

(1’06) = (4):” 1106)) is a homomorphism which makes the following diagram commutative:

A—‘—>E—’>G is

o

lldq

C(G, A) —> Eo ——> G (c) A G-mean on A is a homomorphism

m: C(G, A) —> A

satisfying the two following conditions: (c1) m o e = IdA

(62) m('¢i>) = 'm() ifg E G, ¢ 6 C(G, A)149

AMEBRAIG STRUCTURES

I

Let m be a G-mean on A. Show that, if ), then 4) = 6-“ (cf. Exercise 7). In particular,

Z(G, A) = B(G, A). Show that the mapping (4), g) I» (m(¢), g) is a homomorphism of E0 into A x , G. Deduce that every extension E of G by A satisfying the condition in (b) is isomorphic to A $< TG (use the composition E L”, E, —>A x1 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 at »—> not is an automorphism of A. For all 4) e C(G, A), let m() denote the unique element of A such that

n.m 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 3: 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 sectionT s: G —> E without assuming that F is solvable. (Argue by induction on the order of G. Ifp 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 are E, conclude by means of the induction hypothesis; if N = E, use (a) and the induction hypothesis applied to F/P —> E/P —> G.) 11 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 ofG whose order divides m is contained in a conjugate of H (“Hall ’3 Theorem”)! (Argue by induction on the order of G. If mn ye 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 CIA and to (m/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.) 1' Here again it can be shown that two such sections are conjugate, cf. Farr and

THOMPSON, P100. Nat. Acad. Sci, U.S.A., 48, 1962, 968—970.

150

EXERCISES

11. Show that the simple group Q5 of order 60 contains no subgroup of order 15. 1f 12. Let G be a nilpotent group and H the set of elements of G of finite order. (a) Show that H is a subgroup of G. (b) Show that every finite subset of H generates a finite subgroup. (6) 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 and only if G/D(G) is.

14. Let G be a finite group. Suppose that, for all x, y e G, the subgroup ofG generated by {x, y} is nilpotent. Show that G is nilpotent (apply the hypothesis to x and y of orders p1” and ’5’, where [21 and p2 are distinct prime numbers). 15. Let G be a group and X a subset of G generating G.

(a) Write X1 = X; for n 2 2, define X“, by induction on n, as the set of commutators (x, y), x e X, y e Xn_1. Show that X" = {e} if and only if G is

nilpotent of class 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 ((1).) (Cf. VII, § 4, no. 6.)

20. Let G be a finite commutative group written additively. Let x = age g and let G3 be the subgroup of G consisting of the elements g such that 2g = 0.

(a) Show that x = ugh g. (b) Show that x = 0 if Card(G2) ¢ 2. If Card(G2) = 2, show that x is the unique non-zero element of G2. (a) Show that Card(G2) = 2 if and only if the Sylow 2-subgroup of G is cyclic and #0.

(d) *Let p be a prime number. Show that

(p — l)! a —l

(modp).

(Apply (b) to the multiplicative group of the field Z/pz.)

21. Let p and q be two prime numbers such that p > g. (a) Show that every finite group G of order pg is an extension of Z/qZ by Z/pz (note that every Sylow p-subgroup of G is normal). If further p 7*. 1 (mod q), show that G is cyclic. (b) Deduce that the centre of a group cannot be of index 69. (6) Show that, ifp E 1 (mod 9), there exist a. group oforderpg whose centre is {a}.

(d) Show that every non-commutative group of order 6 is isomorphic to 63.

22. Let G be a finite group of order n > 1 and let p be the least prime

152

EXERCISES

number dividing 11. 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 order

of N/P is prime to the order of the automorphism group of P).

1} 23. Let a be an automorphism of a group G. (a) Show the equivalence of the following conditions: (i) a(x) = x implies x = 2. (ii) The mapping x 1-9 x" 16(x) is injective. When these conditions are satisfied, 0' is said (by an abuse of language) to be withoutfixed point. (b) Suppose G is finite and a without fixed point. The mapping x n—> x" 111(x) is then bijective. If H is a normal subgroup of G stable under a, show that the

automorphism of G/H defined by a is without fixed point. (c) Under the hypotheses of (b), let p be a prime number. Show that there exists a Sylow p—subgroup P of G which is stable under a. (If P0 is a Sylow [asubgroup, there exists y e G such that 6(P0) = yPoy‘1; write 1/” in the form x’la(x) and take P = xPox'l.) Show that such a subgroup is unique and contains every p—subgroup of G which is stable under 0'. (d) Suppose further that e is of order 2. Show that a(g) = g"1 for all g e G (write g in the form x'la(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 X1, X2 be two subsets of the centre of H and s e G such that 5X13'1 = X3. Show that there exists 11 EN such that mm’1 = sxs‘1 for all

x 6 X1 (apply the theorem on the conjugacy of Sylow subgroups to the centralizer of X1). Deduce that two central elements of H are conjugate in G if and only if they are so in N.

25. Let G’ be a surjective homomorphism of finite groups; let P be a Sylow p—subgroup of G. , (a) Let P1 be a Sylow p-subgroup of G such that (HP) = (P1). Show that there exists an element x of the kernel of (I) such that P1 = xPx‘l. (b) Let N (resp. N’) be the normalizer in G (resp. G’) of P (resp. (P)). Show that d>(N) = N’. In particular, if the order of G’ is not divisible by p, then MP) = {e} and (MN) = G’. 1i26. Let G be a finite group. G is called supersalvable if there exists a composition series (Gt)0 solvable and that the converse implications are false.

153

I

ALGEBRAIC STRUCTURES

(6) Suppose G is supersolvable. Show that, if G 9!: {I}, there exists a normal subgroup C of G which is cyclic of prime order. Show that the derived group D(G) of G is nilpotent. Show that every subgroup of G distinct from G which is maximal is of prime index. ((1) A finite group G is bicyclic if there exist two elements a, b of G such that, if Ca and 0,, denote the cyclic subgroups generated by a and b respectively,

then G = 0.10, = CbCa. Show that every bicyclic group is supersolvable. (Reduce it to proving that there exists in G a normal cyclic subgroup N 96 {1}. Attention may be confined to the case where 0,, n Cb = {1}; let m 2 n>l be the orders of a and b respectively; by considering the elements If 1a", where 0 S k < m — 1, show that there exists an integer s such that (1“1 ye l and

b‘la‘b 6 Ca; 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 simple 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 9!” is minimal simple if and only if n = 5.

(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. 1} 28. Let G be a finite g'roup and p a prime number. An element 3 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) ofelements 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 x.)

(b) Let P be a Sylow p—group of G, C its centralizer and E the set of 1)regular elements of G. Show that

Card(E) E Card(E n C)

(mod p).

Deduce that Card(E) i 0 (mod p). (Argue by induction on Card (G) and reduce it to the case where C = G; then use (a) to show that Card (E) = (G: P).) 1) 29. Let G be a finite group ofeven order and let H be a Sylow 2-subgroup of G. For all s e G, let 5(5) be the signature of the permutation x »—> 3:: of G. (a) Let s e H. Show that 2(5) = —1 if and only if s generates H. (b) 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

EXERCISES

(d) Show that the order of a non-commutative simple group is either odd'l' or divisible by 4-. 11 30. Let p be a prime number, 1' and t integers 20, S a cyclic group of order p’“ and T the subgroup of S of order p‘. (a) If S operates on a finite set E, show that

Card(E) a Card(ET) (mod1f“), where E" denotes the set of elements of E invariant under T. (1;) Let m, n 2 0. Show that (9;; )

(9:17”)

(mod PM 1).

(Let M be a set with m elements and E the set of subsets of S x M with [in 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 (11).) 31. Let G be a finite group and S a Sylow p—subgroup of G. Let r, t be integers >0 such that Card(S) = 12'”. Let E be the set of subgroups of G of order p'. (a) Suppose that S is cyclic. Show that Card(E) E 1 (mod 11'“). (Let S operate on E by conjugation and use Exercise 30.) Deduce that, ifthe subgroup ofS oforderp‘ is not normal in G, there are at least 1 + [9’ + 1 Sylowp-subgroups in G. (b) Show that Card(E) E 1 (mod p) even if S is not cyclic. (Let G operate by translation on the set F of subsets of G with p‘ 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” 1. Apply Exercise 30.) 1i 32. Let p be a prime number and G a p-group. Let G* = G’D(G) be the subgroup of G generated by D(G) and the x", x e G.

(a) Show that G* is the intersection of the kernels of the homomorphisms of G into 2/12. (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 g l.

(c) Let u be an automorphism of G of order prime to p and let :2 be the corresponding automorphism of G/G*. Show that 12 = 1 implies u = l. 1[ 33. Let G be a p-group and A its automorphism group. (a) Let (Gi)oD such that, writing these groups additively:

Ma + [3,?) = New) + M5,?) Morafl + Y) = Ma. B) + Mm)

MOW) = -(B,a) Mu, 0‘) = 0

for all cc, B, Y e G/C. (b) Show that, for every integer n, (yx)fl = ynxu(x, y)n(n - 1)l2

for x, y e G. Deduce that if n is odd and d“ = 1 for all d e D, the mapping .7: H x“ induces a homomorphism 0: G/D —> C.

a

11 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 9 of the preceding exercise is non-zero. Show that G is the semi~direct product ofa group oforder p by a cyclic group of order ,0”. Show that there exist elements x, y generating G such that xpa = 1:

if = 1’

(x,_1/)= 3":

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. (0) For [J = 2, consider the same question as in (b) with the hypothesis

0 95 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 D4 (of. Exercise 4) and also to a Sylow 2-subgroup of 6,. 156

EXERCISES

(11) Suppose that p is odd and that the homomorphism 0 of the preceding exercise is zero. Show that there are elements x, y, z generating G such that

(w) = z;

17" =y" = Z’ = (x, z) = (3/, Z) = I,

that G is characterized by this property and that such a group exists. *Show that G is isomorphic to the multiplicative group of matrices of the form 1

a

b

0

l

c

0 0

l

with coefficients a, b, c in the field with [2 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, 3/ such that x2 =1/2 = (x,y);

(”fl/)2 = 1:

that G is characterized by this property and is isomorphic to the quaternionic group (cf. Exercise 4-). (f) *Is the group of matrices

where a, b, 0 run through the field with 2 elements, of type (c) or of type (c) 2’... 37. (a) Let G be a finite group, H a normal subgroup ofG,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 numbering p1, . . ., p, of distinct prime numbers dividing the order __01 5: n— 1...,

of G and Sylow p,-subgroups P1, . . ., P, of G such that, for l S i < s, the set

G, = PlPa . . .P, is a normal subgroup of G. Show that, if this is so, G, does not

depend on the choice ofthe P,; it is the unique subgroup ofG oforderp? . . . f‘; further, G, is the semi-direct product of P, by G‘_1. (a) Show that a finite group G has property (ST) ifand only ifthere exists a Sylow subgroup P of G which is normal in G and such that G/P has property ST). ( (:1) Show that every subgroup, every quotient group and every central extension of a group with property (ST) has property (ST). 157

I

ALGEBRAIG STRUCTURES '

TI 38. Let G be a finite group oforder 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 = (GzN). Deduce that 1 belongs to the set R of positive divisors of m which are congruent to 1 (mod p). In particular, if R is {l}, P is normal in G. (b) If a = I, 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; iffiirther m is a power ofa 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 oforder 210. Show that H contains a normal subgroup G of order 105 (cf. Exercise 29). Let p = 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 (5)). (e) Show that the group 64, of order 24, does not have property (ST). Construct analogous examples for n == 48, 60, 72, 96. (For n = 36, see the follow-

ing exercise.) 11 39. Let G be a finite group, n = Card (G), p a prime number, X the set of Sylow p-subgroups of G and r = Card (X). Then G operates transitively on X by conjugation; let 4): G —> 6x 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. (b) Show that if G has no subgroup of index 2, then MG) C 21,; and that if n = kr! (resp. gm), then (G) = 6x (resp. 91x). (0) 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 91,. If n = 24 and G does not have property (ST), then G z 64.

Ifn = 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 91., and that K is central.) Ifn = 48 and G does not have property (ST), then G contains a normal subgroup K of order 2 such that G/K z 6,. If n = 60 and G does not have property (ST), then G z 915. (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 66 and, arguing as in Exercise 25 of § 5, that G 2: 915.) 158

EXERCISES

If n = 72 and a Sylow 3-subgroup of G is not normal, then G contains a normal subgroup K such that G/K is isomorphic either to 6, or to 91‘. If n = 96 and a Sylow 2-subgroup of G is not normal, then G contains a normal subgroup K such that G/K z 63. (d) Show that a group G of order s 100 is solvable unless G z 915. 40. Let G1, G2 be two groups and G a subgroup of G; x G; such that prl G = G1, pra G = G2; identifying G1 and G2 with G1 x {oz} and {e1} x Ga respectively, H1 = G n G, and H2 = G n G; are normal subgroups of G such that G/H1 is isomorphic to G2 and G/Ha isomorphic to G1.

(a) Show that for G to be normal in G1 x G2, it is necessary and sufficient that Gcontain the commutator group D(G1 x G2) = D(G1) x D(G2) (prove

that G contains D(G1) by writing :urzx‘lz'1 e G for x 6 G1 and z e G). Deduce that Gl/Hl is commutative. (b) If G is normal in G1 x Ga, show that a homomorphism is defined of

G1 x G, onto G1/H1 by associating with each ordered pair (:1, :2) the coset of .rltf1 modulo H1, where :1 6 G1 is such that (£1, 32) e G. Deduce that (G; x Ga) [G is isomorphic to G/(H1 x H2). 41. Let x, y be two elements of a group G. (a) For there to exist a, b in G such that bay = xab, it is necessary and sufficient that xy' 1 be a commutator. (b) For there to exist 27: + 1 elements a1, a2, . . ., am,+1 in G such that x = alaa. . 412,,“

and y = a2n+1a21v . .a,,

it is necessary and sufficient that xy‘ 1 be a product ofn commutators (argue by induction on n).

§7 1. Enumerate the elements oflength < 4 in M(X) when X Consists ofa single element.

2. Let X be a set and M one ofthe magmas M(X), Mo(X) or N“). Show that every automorphism of M leaves X stable. Deduce an isomorphism of 6,; 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 9 denote the homomorphism ofM(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 w i—> (Mw), 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 a have the same length.

159

ALGEBRAIC STRUGI‘URES

I

1i 4-. Let X be a set and c an element not belonging to X. We write Y = X U {a}. Let M be the magma with underlying set Mo(Y) and law of composition given by (u, v) n—> cuv. Let L be the submagma of M generated by X and let a be the mapping from Y to Z equal to —l on X and to l at c. (a) Show that L is the subset of M consisting of the words w = y1. . .y,, y, 6 Y, satisfying the two following conditions:

(i) 5(y1) +~~+=(yp) = -1

_

(ii) 3(3/1) +---+e(y,) 2 0 forl Fx, ,, such that g((0, 1)) = x and g((y,,, 0)) = x"yx"‘ andfand g are inverses ofone another.) (b) Deduce that the normal subgroup of Fx, 1, generated by'y is a free group with basic family (x‘yx"‘)uez. Obtain this result by applying Exercise 20 tothe 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 96 y. Show that the

165

ALGEBRAIG STRUCTURES

I

derived group of F has as basic family the family ofcommutators (x‘, y’), i e Z,

jezn'aé 0J9é 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 1:, denote the corresponding element of F. If s, t e S, let rm denote the element of F defined by:

1M = firm-,1 ifst aé ein G

ifst = e in G. rm = x,x, Let 4: be the homomorphism of F onto G which maps x, to .r and let R be the kernel of (b. Show that the family (rm ,) for r, t 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 x,.) 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 G —> E. (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 ofthe free group F(X). Let g = 1.1 avg“) be the canonical decomposition of g as a product of powers of elements of X (cf. no. 5, Proposition 7). The integer l(g) = Z |m(a)| is called the length of g. - (a) g is called cyclically reduced ifeither at; 56 x,l or x1 = x” and m(l)m(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 = g”. Show that every element z 95 e of F(X) can be written uniquely in the form x" with n 2 l and x primitive. (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 ofx; it is the cyclic subgroup generated by x. 27. Let G = *A G, be the sum of a family (G012: of groups amalgamated by a common subgroup A. For all ie I, choose a subset P, of G, satisfying

condition (A) of no. 3. , (a) Let x e G and let (a; 1'1, . . ., in; [11, . . ., [7,.) be a reduced decomposition of x; then

x = 4:11;»... with men. —{e..}, i. .. i... 166

EXERCISES

Show that, if i, aé in, the order of x is infinite. Show that, if i, = in, x is conjugate to an element with a decomposition of length < n — l.

(b) Deduce that every element of G offinite order is conjugate to an element ofone ofthe G,. In particular, if the G, have no element offinite 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 ie I, the group N is normal in GP Show that N is normal in G = *A G, and that the canonical homomorphism of *NN Gi/N into G/N is an isomorphism. (b) For all ie I, let H, be a subgroup of G, and let B be a subgroup of A.

Suppose that H, n A = B for all 1'. Show that the canonical homomorphism of *3 H, into G is injective; its image is the subgroup of G generated by the ‘3

1i 29. LetG = G1 *A G; be the sum of two groups G, and G3 amalgamated by a subgroup A. (a) Show that G is finitely generated if G1 and G2 are finitely generated. (6) Suppose G1 and G2 are finitely presented. Show the equivalence of the two following properties: (I) 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 (A0,“ of subgroups of A with L): A, = A and A, gé A for all i. Let H, (resp. H) be the kernel of the canonical homomorphism of G1 all G2 onto G1 *A, G2 (resp. onto G). Show that H is the union of the H, and that H, 9e H for all i; then apply Exercise 16(d).) (c) Deduce an example of a finitely generated group which is not finitely presented. (Take G1 and G2 to be fi‘ee groups of rank 2 and A a free group of

finite rank, cf. Exercise 22.)

30. Let A and B be two groups and G = A 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 2 of the form biaibaaa - - - bu — 141: — lbm

with n 2 l, (1,6 A — {e}, 6, EB — {e}. Show that every element of Z can be written uniquely in the form aza’ with a, a’ e A.

(b) Let x e G — A. Letfx be the homomorphism of A at A into G whose restriction to the first (resp. second) factor of A * A is the identity (resp. the mapping a -—> xax‘l). Show that fx is injective. (Write x in the form aza’ as above and reduce it thus to the case x = 2; use the uniqueness of reduced decompositions in G.) In particular, A n xAx’1 = {e} and the normalizer ofA in G is A.

e 167

ALGEBRAIC STRUCTURES

I

(a) Let T be an element of A\G/B. Show that T contains one and only one element t of the form 510117202 - - - bn— lan— 1,

with n 2 l, (a e A — {e}, b. e B -— {e}. Show that every element of T can be written uniquely in the form at!) with a e A, b e B. (d) Let xeG and let h, be the unique endomorphism of G such that hx(a) = a if a e A and hx(b) = xbx‘1 if b e B. Show that fix is injective. (Same method as in (b), using the decomposition x = atb of (0).) In particular, A n xBx"1 = {e}. Give an example where It, is not surjective.

31. Let G be the free product of a family (Cd‘s: of groups. Let S be the set of subgroups of G which are conjugate to one of the G, and G) the union of the elements of S. (a) Let H, H’ e S, with H aé H’, and let x e H — {e}, x’ e H’ — {e}. Show that xx’ 5 G — (E). (Reduce it to the case where H is one of the G‘; write H' in , the form yG,y'1 and proceed as in the preceding exercise.) (b) Let C be a subgroup of G contained in 9. Show that there exists H e S such that C C H. (a) Let C be a subgroup of G all of whose elements are offinite order. Show that C is contained in ('3 (cf. Exercise 27); deduce that C is contained in a conjugate of one of the G;. fl 32. Let A and B be two groups and G = A II B their free product. Let X be the set of commutators (a, b) with a e A — {e} and b e B — {e} and S = X U K”. (a) Show that X n X‘ 1 = a 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 :1, . . ., s" be a sequence of elements of S such that as,” 9!: eforl < i < n;letan EA — {9}, bn EB — {e}, 5(n) = il be such that 3,. = (an, hm“). Let g = :1. . .3". Show, by induction on n, that the reduced

decomposition of g is of length 212 + 3 and terminates either with anb,l if e(n) = l or with bnan ife(n) = ——l. (0) Let R be the kernel of the canonical projection A * B —-> A 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 (1:) to prove that X is free; if Rx 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 ofG (resp. G’) satisfying condition (A) of no. 3.

Let n be an integer 21. Let L,I denote the set of x e E whose reduced de168

EXERCISES

composition is of length GI and 4),: A, —> G“,1 be two injective homomorphisms;

by these homomorphisms A, can be identified both with a subgroup of G, and with a subgroup of G”. (a) Let H, = G1,H2 = H1 *A: G2, . . ., H" = H”.1 *Ahl G". The group

H, is called the sum of the groups (G,) amalgamated by the subgroups (A,); it is denoted by G1 atA1 G2 “2- - - *A._1 G". If T is an arbitrary group, define a bijection between the homomorphisms of H“ into T and the families 169

ALGEBRAIC STRUCTURES

I

(t1, . . ., tn) of homomorphisms 13,: G, -—> T such that ti 0 Br such that fw(Z’) = Z maps B33 onto B; . The inverse limit of the BT, with T running of B through E, is denoted by B. Show that B = lim (— B; and that the image in ET is B; . The set B is called the set of ends of G; it is non-empty if and only if G is infinite. (0) Let T be a non-empty S-connected finite subset of G and let Z 6 Bi. Show that there exist g e G such that gT n T = 25 and gT A Z 96 2; and that then gT C Z. Show that there exists Z1 6 Br such that n contains T and

all the Z’ 6 BT such that 2’ 9E Z (observe that T U 2'92 2’ is S—connected and does not meet gT); deduce that, if Z’ 6 BT is different from 2,, then gZ’ C Z. Show that, writing T’ = T U gT, the inverse image of Z in B3 consists of at least 11 — 1 elements, where n = Card(BiE’).

((1) Let S’ be a symmetric finite subset of G generating G and B’ the set of

ends of G relative to 8'. 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 qf ends of G. (e) Show, using (c), that the number of ends of G is equal to 0, l, 2 or 230. (f) Show that the number of ends of Z is 2 and that that of Z" (n 2 2) is l. 1] 38. Let X be a finite set, F(X) the free group constructed on X and S = X U X“. (a) Let S,, be the set of products 31. . .sm, 5, e S, m s n. Show that every connected S-component (cf. Exercise 18) of F(X) — S,I contains one and only one element ofthe form :1. ...s',,+1 with 5, e S and as,“ aé e for 1 s 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 (31, . . ., 5n, . . .) of elements of S such that

as,” 75 e for all i. In particular, the number of ends of F(X) is equal to 2“!) 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

EXERCISES

(a) If X e 8‘ and n e N, let dn(X) denote the number of elements in G of the form x1. . .xn with x. e X. IfX, Ye 8, show that there exists an integer a 21 such that .

d,,(X) s (1“,,(Y) for all n e N. Deduce that lim sup W does not depend on the choice of X in 3; this 1 n» a: limit (finite or infinite) is denoted by e(G). (b) If X e 3 and G is infinite, show that dn(X) < dn+1(X) for all n. Deduce that e(G) is 2 l. _ (6) Let H be a finitely generated group. Show that e(H) g 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. ((1) Show that e(Z") = n. (e) Let X e 8. Consider the following property: (i) There exists a real number a > I such that (1,,(X) 2 a” for all sufficiently large 71. Show that, ifproperty (i) holds for one element of3, it holds for every element of 3. G is then said to be g” exponential type; this implies e(G) = +00. (f) Let G1 and G2 be two groups containing the same subgroup A and let H = G1 *A G, the corresponding amalgamated sum. Suppose G1 and G2 are

finitely generated (hence also H is) and G‘ 9e A for i = l, 2. Show that H is of exponential type if at least one of the indices (G1 :A), (nA) is 23. (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 71 over Z with all the diagonal terms equal to I. Let 8(Tn) be the invariant of the group Tn defined in Exercise 39. Show that

am) < ,2 i 2‘“) the canonical mapping. (a) Show that there exists a unique multiplication on 2“” such that 2‘“) is a ring and such that u is a monoid morphism of the monoid M into the multiplicative monoid of the ring 2‘“). (6) Let A be a ring and v: M —> A a monoid morphism of the monoid M into the multiplicative monoid of A. There exists a unique ring morphism f: 2‘“) —> A such that fou=v.

6. Let A be a commutative ring and M a submonoid of the additive group of A. Let n be an integer >0 such that m” = 0 for m e M. (a) If n! is not a divisor of zero in A, the product of a family ofn elements of M is zero.

(11) Show that the conclusion in (a) is not necessarily true if n! is a divisor of

zero in A.

172

EXERCISES

7. For all n e N*,

n-1

n! = 20 (—1)v(:) (n — v)” (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. 1} 11. In a ring A, a two-sided ideal a is said to be 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 aé 0 is inductive and apply Zorn’s Lemma). ((2) 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 («10,51 a finite family of left ideals ofA such that the additive group of A is the direct sum of the additive groups at. If 1: gr ei(e, 6 a1), then e? = e,, etc, = Oifz' #jand a, = A5, (writex = x. l for all x e A). Conversely, if (ta).eI is a finite family of idempotents of A such that

6,6, = o for 1‘ ¢ j and 1 = Z en then A is the direct sum of the left ideals All. For the ideals As, to be two-sided, it is necessary and sufficient that the e! be in the centre of A. TI 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 «I = An and the left annihilator b of e (note that, for all x e A, x — xe e b).

(12) Every right ideal b of A is the direct sum of b n a and {5 n b. (c) If A: = eA, a and b are two-sided ideals of A and define a direct decomposition of A. 14. Let A be a commutative ring in which there is only a finite number n of divisors of zero. Show that A has at most (71 + 1)2 elements. (Let (11,. . ., a,l be the divisors of zero. Show that the annihilator 81 of a1 has at most n + 1

elements and that the quotient n'ng A/81 has at most 11 + 1 elements.) 15. A ringaid is a set E with two laws of composition: (a) an associative

173

ALGEBRAIC STRUCTURES

I

multiplication xy; (b) a law written additively, not everywhere defined (§ 1, no. 1) and satisfying the following conditions: (1) it is commutative (in other words, if x + g is defined, so is y + x and x +y =y + x;xandyarethensaidtobeaddible); (2) if x and y are addible, for x + y and z to be addible it sufl‘ices that x and g 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-) if x and 2 on the one hand, y and z on the other, are addible and x+z=y+z,thenx=y; (5) if x, y are addible, so are x2 and yz (resp. 2:: and 23/) and (x+y)z=xz+yz (resp. z(x +y) = zx + zy) for all zeE. Every ring is a ringoid. (a) Examine how the definitions and results of § 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 addition and such that E. a C a). ((2) Let G be a group with operators andf and g two endomorphisms of G. For the mapping x I—>f(x) g(x) to be an endomorphism of G, it is necessary and suflicient that every element of the subgroupf(G) be permutable with every element of g(G); if this endomorphism is then denoted byf + g and fg is the composite endomorphism x v—>f(g(x)), show that the set E ofendomorphisms 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 that f(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 ifit 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.T 16. Give examples of ideals a, b, c in the ring Z such that ab ;6 a n b and

(a+b)(a+c)9éa+bc.

§9

1. Which are the field structures amongst the ring structures determined in ‘ §8, Exercise 1? 1' See H. Frmnc, Die Theorie der Automorphismenringe Abelscher Gruppen und ihre Analogen bei nicht kommutativen Gruppen, Math. Arm, 107 (1933), 514-.

174

EXERCISES

2. Afinite ring in which every non-zero element is cancellable is a field (cf. § 2, Exercise 6).

3. Let G be a commutative group with operators which is simple (§ 4, no. 4, Definition 7). Show that the endomorphism ring A of G is a field. 4. Let K be a field. Show that there exists a smallest subfield F ofK and that F is canonically isomorphic either to the field Qor to the field Z/pZ for some prime number p. In the first case (resp. second case) K is said to be of characteristic 0 (resp. p). 5. Let K be a commutative field of characteristic #2; let G be a subgroup of the 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, ifx and y are elements of G such that y 96 0, then xz/y e G; deduce that, if x, y, z are elements of G such that z ¢ 0, then xy/z e G).

6. Let K be a commutative field of characteristic #2; letf be a mapping of K into K such thatf(x + y) =f(x) +f(y) for all x and y and for all .1: ye 0 f(x)f(l/x) = 1. Show that f is an isomorphism of K onto a subfield of K (prove thatf(96") = (f (x))2).

7. Let A be a commutative ring. (a) Every prime ideal is irreducible (§ 8, Exercise 11). (b) The set of prime ideals is inductive for the relations C and D . (c) Let a be an ideal of A distinct from A; let I) be the set of x e A such that there exists an integer n 2 0 with x" e a (n depends on 5:). Show that b is an ideal and is equal to the intersection of the prime ideals of A containing a. fl 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 ‘13(E) of subsets of a set E,\show that a Boolean ring structure

is defined by setting AB = A03 and A + B = (AnflB) U (BnflA). This ring 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 function" Ff (so that F(_liin> X.)

can be identified with 13 F(X0). (c) State and prove analogous results for free magmas, free monoids, free commutative groups, free commutative monoids.

2. Show that a direct limit of simple groups is either a simple group or a group with one element.

179

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 Babylonians 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,1' 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 ([1], 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 operations 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 calculation with integers (for example, the commutativity of the product of two rational numbers). The most remarkable proofs of this nature are those T 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 coefficients; 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.

180

HISTORICAL NOTE

relating to the theory of magnitudes, the most original creation of Greek Mathematics (equivalent, as is known, to our theory of real numbers >0; see the Historical Note to General Topology, IV); here Euclid considers amongst other things the product oftwo ratios ofmagnitudes 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 § 1, no. 6) and that they commute ([2], Book V, Prop. 22—23).T It must not however be concealed that this progress towards rigour is accompanied in Euclid by a stagnation and even in some ways by a retrogression as far as the technique of algebraic calculations is concerned. The overwhelming preponderance ofGeometry (in the light ofwhich the theory ofmagnitudes was obviously conceived) paralyses any independent development of algebraic notation: the elements entering into the calculations must always be “represented” 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 representations 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 x'"“‘ = xmx" 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 negative numbersi; 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 problems; as for the axiomatic conception of laws of composition as begun by Euclid, this appears foreign to Diophantus’ thought as to that ofhis 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 1' 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, Definition 5) is considered to be interpolated: he has of course a perfectly clear conception of this operation and its properties.

2}: 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 allowing the “expansion” of such products as (a — b) (c — d).

181

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 >0 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, III), 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 dilference 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 calculations without investigating whether each step was well founded; a confidence moreover usually justified a posteriori, by the exact results to which the extension 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 1' 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 modem mathematics.

182

HISTORICAL NOTE

and Hindu mathematics spread to the West (via the Arabs), the manipulation of these numbers became more familiar and they took on other “representations” of a geometric or kinematic character. With the progressive improvement 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 numbers.T 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 calculus. In fact it was necessary to await the movement of ideas which, about 1800, led to the geometric representation of complex numbers (see the Historical 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 + 0y” with integer coefficients. Lagrange had defined on the set of forms with the 1' 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). ‘ i 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 relationT 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§ 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 generality 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 composition 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 numberi (ibid., p. 371), but proves also that the group of classes of quadratic forms of given 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. II, 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 ([6], (2), vol. I, p. 64) of the “product” of two permutations of a finite set,§

T Two forms are equivalent when one of them transforms into the other by a “change of variables” x’ = out + fly, 3/ = w + 81/, where a, (3, Y, 8 are integers

such that :28 —- By = l. i It is quite remarkable that Gauss writes composition of classes of quadratic

forms additioely, 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-

HISTORICAL NOTE

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 obtained 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 Galois'l' 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 ofAlgebra by applying this notion to a host of new mathematical entities: the algebra of Logic with Boole (see the Historical Note to Set Theory, IV), vectors and quaternions with Hamilton, general hypercomplex systems, matrices and non-associative laws with Cayley ([10], vol. I, pp. 127

and 301 and vol. II, pp. 185 and 475). A parallel evolution was pursued independently 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, III.§ 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 1' 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 discovers finite fields, which we shall study in Algebra, V. 1 Moreover those of Galois remained unknown until 1846 and those of Gauss

only exercised a direct influence on the Theory of Numbers. §The principal theories developed during this period are to be found in a remarkable contemporary exposition of H. Hankel [l l], 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 qfequations”. 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 respectively 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.‘]‘ 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 Commutative 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 ofafield (ofalgebraic 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 [l3] 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 for the more and more important role played by commutative groups and modules in the theory of algebraic fields; we shall retum to this in the Historical Notes to Algebra, III, V and VII. We shall also refer in the Historical Notes to Algebra, III 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, Frobenius, Molien) and in France (Laguerre, E. Cartan) independently of the Anglo—Saxons and using quite different methods. 1‘ 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 3 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

HISTORICAL NOTE

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 “Traité des Substitutions” of C. Jordan [15]. The latter there developed, greatly improving them, the works of his predecessors on the particular properties of permutation groups (transitivity, primitivity, etc.), obtaining 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 fundamental notion of a homomorphism of one group into another and also (a little later) that ofquotient group and who proved a part ofthe theorem known as the “Jordan-Holder Theorem”.1' Finally Jordan was the first to study infinite groups [16], which S. Lie on the one hand and F. Klein and H. Poincaré 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 [10], vol. II, pp. 123 and 131 and [14], vol. III, 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 consciously. 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 offinite 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 diflicult problems it raises, to the monographs of Burnside [20], Speiser [23], Zassenhaus [24] and Gorenstein [27]. Around 1880 too the systematic study of group presentation: was begun; previously only presentations of particular groups had been encountered, for example the alternating group 915 in a work of Hamilton [9 bis] or the monodromy groups of linear differential equations and Riemann surfaces (Schwarz, Klein, Schlafii). W. Dyck was the first [18] to define (without giving it a name) the free group generated by a finite number of generators, 1' 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.

'

I The existence of a subgroup of order [2" in a group whose order is divisible by p" is mentioned without proof in the papers of Galois [8].

187

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 described an interpretation ofa group ofgiven presentation, where each generator is represented by a product of two inversions with respect to tangent or intersecting circles (see also Burnside [20], Chapter XVIII). A little later, after the development of the theory of automorphic functions by Poincaré and his successors, then the introduction by Poincaré ofthe tools ofAlgebraic 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 subgroups [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 Dedekind 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, O. 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

l. O. NEUGEBAUER, Vorlesungen iiber Gesehiehte der antiken Mathematilc, Vol. I: Vorgriechische Mathematik, Berlin (Springer), 1934-. 2. Euclidis Elementa, 5 vols., ed. J. L. Heiberg, Lipsiae (Teubner), 1883—88. 2 bis. T. L. HEATH, The thirteen books qf Euclid’s Elements . . ., 3 vols., Cam-

bridge, 1908.

188

BIBLIOGRAPHY

3. Diaphanti Alexandrini Opera Omnia. . ., 2 vols., ed. P. Tannery, Lipsiae (Tcubner), 1893—95. 3 bis. Diophante d’Alexandrie, trad. P. Ver Eecke, Bruges (Desclée-de Brouwer),

1926. 4. G. W. LEIBNIZ, Mathematische Sehnfleu, ed. C. I. Gerhardt, vol. V, Halle (Schmidt), 1858. 5. C. F. GAuss, Werke, vols. I (Géttingen, 1870), II (ibid., 1863) and VIII ‘(ibid., 1900). 6. A. L. CAUCHY, Oeuvres camplétes (2), vol. I, Paris (Gauthier-Villas), 1905. 7. N. H. ABEL, Oeuvres, 2 vol., ed. Sylow and Lie, Christiania, 1881. 8. E. GALOIS, Ecrits et me’moires mathématiques, ed. R. Bourgne and J. Y. Azra, Paris (Gauthier-Villars), 1962. 9. W. R. HAMILTON, Lectures on Quatemions, Dublin, 1853.

9 bis. W. R. HAMILTON, Memorandum respecting a new system of roots of unity, Phil. Mag. (4), 12 (1856), p. 446. 10. A. CAYLEY, Collected mathematicalpapers, vols. I and II, Cambridge (University Press), 1889. 11. H. HANKEL, Vorlesungen fiber die complexen Zuhlen und ihre Functionen, lst part: Theorie der complexen Zahlcnsysteme, Leipzig (Voss), 1867. 12. J. A. SERRET, Cours d’Alge’bre supe’rieure, 3rd ed., Paris (Gauthier-Villas), 1866. 13. R. DEDEKIND and H. WEBER, Theorie der algebraischen Funktionen einer Verinderlichen, Crelle’s J., 92 (1882), pp. 181—290. 14-. R. DEDEKIND, Gesammelte mathematisohe Werke, 3 vols., Braunschweig

(Vieweg), 1932. 15. C. JORDAN, Troite’ des substitutions et des equations alge’lm'ques, Paris (Gauthier— Villars), 1870 (reprinted, Paris (A. Blanchard), 1957). 16. C. JORDAN, Mémoire sur les groups des mouvements, Arm. di Mat. (2), 2 (1868), pp. 167—215 and 322—345 (=0euvres, vol. IV, pp. 231—302, Paris (Gauthier-Villas), 1964). 17. L. SYLOW, Théorémes sur les groups de substitutions, Math. Ann., 5 (1872), pp. 584—594.

18. W. DYCK, Gruppentheoretische Studien, Math. Ann., 20 (1882), pp. 1—44.

19. G. FROBENIUS, Uber Gruppencharaktere, Berliner Sitzungsber., 1896, pp. 985—1021 (= Gesammelte Abhandlungen, ed. J. P. Serrc, Berlin— Heidelberg—New York (Springer), vol. III (1968), pp. 1—37). 20. W. BURNSIDE, Theory qroups offinite order, 2nd ed., Cambridge, 1911.

21. J. NIELSEN, Die Isomorphismengruppen der freien Gruppen, Math. Arm, 91 (1924), pp. 169—209. 22. O. SCHREIER, Die Untergruppen der freien Gruppe, Abh. Hamb., 5 (1927), pp. 161—185.

189

BIBLIOGRAPHY

23. A. SPEISER, Theorie der Crapper; van endlicher Ordnung, 3rd cd., Berlin (Springer), 1937. 24. H. ZASSENHAUS, Lehrbuch der Gmppentheorie, vol. I, Leipzig—Berlin (Teubncr), 1937. 25. B. L. VAN DER WAERDEN, Modeme Algebra, 2nd ed., vol. I, Berlin (Springer), 1937; vol. II (ibid.), 1940. 26. W. MAGNUS, A. KARRASS and D. SOLITAR, Combinatorial group theory, New York (Intersciencc), 1966. 27. D. GORENSTEIN, Finite groups, New York (Harper and Row), 1968.

190

CHAPTER 11

Linear Algebra

This chapter is essentially devoted to the study of a particular type of commutative groups with operators (I, §4, no. 2) called modules. Certain properties stated in §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 ofa commutative group with operators is always equivalent to that of a module suitably associated with the group with operators in question.

§ 1. MODULES l. MODULES; VECTOR SPACES; LINEAR COMBINATION!

DEFINITION 1. Given a ring A, a left module over A (or left A-module) is a set E with an algebraic structure defined by giving: (1) a commutative group law on E (written additively in whatfollows) ; (2) a law ty" action (a, x) 1—> a T x, whose domain of operator: is the ring A and which satisfies thefillowing axioms: (MI) «T (x +y) = («T x) + (ocTy)farallueA,e,yeE; (Mn) (at + [3) Tx = («T x) + (BTx)jbrallaeA,BeA,e; (Mm) uT (5T x) = («(5) T xforall aeA,BeA,e; (MN)1T x = xforalle. Axiom (M!) means that the external law of a left A-module E is distributive with respect to addition on E; a module is thus a. commutative group with operators.

If in Definition 1, axiom (Mm) is replaced by (Min) «T ([5 T x) = (Bu) T irall oceA, BeA,e,

191

II

LINEAR ALGEBRA

E with the algebraic structure thus defined is a right module over A or a right A-moa'ule. When speaking of A-modules (left or right), the elements of the ring A are often called 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 (Mm) is then written «(3:0 = (affix, condition (Min) is written (xfl)ar = x(Ba). If A0 denotes the opposite ring to A (I, §8, no. 3), every right module E over the ring A is a left module over the ring A°. It follows that an exposition can be given of the properties ofmodules whilst systematically confining attention either to left modules or to right modules; in §§ 1 and 2 we shall in general give this exposition for lefl 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: GA, the mapping x v—> our of an A-module E into itself is called the homothely of ratio at of E (I, §4, no. 2); by (MI) a homothety is an endomorphism of the commutative group structure (without operators) on E, but not in general of the module structure on E (I, §4, no. 2; cf. II, § 1, no. 2

and no. 13). Hence «0 = 0 and a( —x) = — (ax); if or is an invertible element of A, the homothety x »—> out is an automorphism ofthe commutative group structure (without operators) on E, for the relation 3/ = mx implies by virtue of (MW), :4 = a‘1(acx) = a'ly. Similarly, by virtue of (Mn), for all x e E, the mapping or +-> out is a homomorphism of the additive group A into the commutative group (without operators) E; hence 0x = 0 and (~—ac)x = —(arx) ; moreover, by (MN), for every integer n e Z, n.x = (n. l)x. When the ring A consists only of the element 0, every A-module E consists only of the element 0, for then 1 = 0 in A, whence, for all are E, x= l.x=0.x=0.

Examples. (1) Let 4) be a homomorphism of a ring A into a ring B; the mapping (a, x) »—> (a)x (resp. (a, x) n—> x¢(a)) of A x B into B defines on B a left

(resp. right) A-module structure. In particular if 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, 6’ the endomorphism 192

MODULES; VECTOR spAcrs; LINEAR COMBINATIONS

§ 1.1

ring of E (I, §8, no. 3: recall that the product fg of two endomorphisms is by definition the composite endomorphism f o g). The external law (f, x) v—>f(x) between operators fed" and elements e defines on E a canonical left 6-module structure. Consider now a ring A and suppose there is given on E a left (resp. right) A-m‘odule structure; for all aeA, the homothety have» out (resp. act—um) belongs to 6’; the mapping 42:0: aha is a homomorphism of the ring A (resp. the opposite ring A”) into the ring 6’ and by definition w: = (¢(a))(x) (resp. x0: = (¢(a))(x)). Conversely, giving a ring homomorphism ¢:A—-> 6 (resp. ¢:A° ——> 6’) defines a left (resp. right) A-modulc 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 —-> 6’ (resp. A° —> 6’). DEFINITION 2. A left (resp. right) vector space over afield K is a left (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 be of a real number i and a point x with coordinates x1, x3, x3 being the point with coordinates" txl, 1x2, txa. Similarly, the set of real-valued functions defined on an arbitrary set F is a vector space with respect to R, the product tf of a real num-

ber t and a function f being the real-valued function x I—> gf(10-.

For two families (mm, (yaw; of elements of an A-module E of finite sup-

port (I, § 2, no. 1), the following equations hold: g (”s + 11;) = 2x1. + gym

(I)

(2)

i

Z x. = glam) at. IEI

forall «EA;

the equations are immediately reduced to the analogous equations for finite sums by considering a finite subset H of I containing the supports of (x.) and '

(1h)-

DEFINITION 3. An element x of an A-module E is said to be a linear combination with cofioients in A of a family ((1.).91 of elements of E if there exists a family 0.).“ of Ml;elements of A, offinite support, such that x = 2 GI

In general there are several distinct families (7..) satisfying this condition (cf. no. 11).

193

I!

LINEAR ALGEBRA

Note that 0 Is the only linear combination of the emptyfamzly of elements of E (by the convention of I, § 2 no. 1). 2. LINEAR MAPPINGS

DEFINITION 4. Let E and F be two (left) module: with respect to the same ring A. A linear mapping (or A-linear mapping, or homomorphism, or A-homomorphism) q into F is any mapping u:E -> F such that:

“(x +y) = Wt) + "(w fire,y6E;

(3) (4G)

u()..x) = A.u(x) for AeA,e.

If E and F are two right A-modules, a linear mapping a : E -—> F is a mapping satisfying (3) and

(4D)

u(x.i) = u(x).7\ for A e A, x e E. Remark. When E and F are two commutative groups considered as modules over the ring Z (no. 1), 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

Z I—> M of the A~module A, into E is a linear mapping 0,, such that 0,,(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 realvalued functions defined on I. The mapping x:—> x’ which associates with every differentiable function x its derivative is a linear mapping from E to F...

Note that a homothety x x~—> out on an A-module E is not necessarily a linear mapping: in other words, the relation 104:) = Max) 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 I—> ax is then called a central homothety (cf. no. 13). If u:E —> F is a linear mapping, then, for every family (trawl of elements of E and every family (h),sI of elements of A such that the support of the family Dex)“ is finite,

(5)

4; Mo) = Z! hu(x,)

as follows immediately from (3) and (4G) by induction on the cardinal of the support of the family (Am).

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 a o u is linear.

194

LINEAR MAPPINGS

§ 1.2

PROPOSITION 2. Let E, F be two A-modules.

(l) [{q —->F and v:F —> E are two linear mappings such that v o u is the identity mapping on E anduov is the identitymapping on F, u isan isomorphism ofE onto F and v is the inverse isomorphism. (2) Every biieetive linear mapping u:E —> F is an isomorphism of E onto F. These propositions follow immediately from Definition 4.

Propositions l and 2 show that linear mappings can be taken as the morphisms for the species of A-module structures (Set Theory, IV, §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 HomA(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 commutative group F” of all mappings of E into F (I, §4, no. 8); recall that for two elements u, v of FE and for all x e E, (u + v)(x) = u(x) + v(x),

(—u)(x) = —u(x)

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, f1, f2 elements of Hom(E, F) and g, g1, g, elements of Hom(F, G), the following relations are immediately verified:

(6)

g°(f1+fa)=g°f1+g°fa

(7)

(g; + g2) °f= g1°f+ £2 °f

(3)

t” (-f) = (—g) °f= “(Pf)-

In particular, the law of composition (1; g) i—>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 1r. or Ida, is the identity mapping on E; the linear

mappings of E into itself are also called endomorphism Of the A-module E and the ring Hom(E, E) is also denoted by End (E) or EndA(E). The automorphisms of the A-module E are just the invertible elements of End(E) (Proposition 2); they form a multiplicative group, denoted by Aut(E) or GL(E) which is also called 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 module 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 fe Hom(E, F) is associated with the element 0 of o u e Hom(E’, F’), a mapping Hom(E, F) -—> Hom(E’, F’)

195

LINEAR ALGEBRA

II

is defined, which is Z-linear and which is denoted by Hom(u, v) or HomA(u, v). If u, ul, ua belong to Hom(E’, E) and v, 111, on to Hom(F, F’), then

(9) .

Hom(u1 + u2, v) = Hom(ul, v) + Hom(ua, v) Hom(u, 01 + 0,) = Hom(u, 01) + Hom(u, 03)

Let E', F” be two A-modules, u':E” —> E' and v’:F’ ., F" linear mappings. Then Hom(u o u’, o' o o) = Hom(u', o’) o Hom(u, o). (10) 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“, 0‘1) by (10). If h (resp. k) is an endomorphism of E (resp. F), Hom(h, 1r) (resp. Hom(lE, k)) is just the homothety with ratio It (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, §4, no. 3), for if this is so, the structure of a group with operators induced on M obviously satisfies axioms (Mn), (Mm) and (MW); 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 —> 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 O, by an abuse of notation). (2) Let A be a ring. The submodules of A, (resp. Ad) are just the left ideal: (resp. right ideals) of the 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 ocx, where on runs through a, is a submodule of E, denoted by ax. (4) In a commutative group G considered as a Z—module (no. 1), every subgroup of G is also a submodule. *(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 diferentiuble functions on I is a vector subspace of 0...,

Let E be an A-module. Every equivalence relation compatible (1, § 1, no. 6) with the module structure on E is of the form x — y e M, where M is a stable 196

EXACT SEQUENCES

§ 1.4

subgroup of E (I, §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, §4, no. 4) is an A-module structure, under which the canonical mapping E —> 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 AJa 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, §4, no. 5) (or directly from the definitions) that if u:E —> 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 E. In particular, the kernel N = u1(0) is a submodule of E and the image u(E) of E under a is a submodule of F (I, §4, no. 6, Proposition 7); by an abuse of

language u(E) is called the image of u. The quotient module E/N is also called the coimage of u and the quotient module F/u (E) the cokernel of u. In the canonical decomposition of u (I, §4-, no. 5)

(11)

u:E —’—> E/N —"—> u(E) —‘> F

o is an isomorphism 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 kernel 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 to be of the form v o d), where v is a linear mapping of E/M into F, it is necessary and sufficient that M C Ker(u); for, if this condition holds, the relation x — y e M implies u(x) = u(y), hence u is compatible with this equivalence relation and

clearly the mapping 0: E/M —-> F derived from u by taking quotients is linear. 4. EXACT SEQUENCES

DEFINITION 5. Let F, G, H be three A-modules; let f be a homomorphism of F into G and g a homomorphism of G into H. The orderedpair (f, g) is called an exact sequence if —1

g (0) =f(F)

in other words, if the kernel ofg is equal to the image off

197

LINEAR ALGEBRA

n‘

The diagram

F—f-—>G—5—>H

(12)

is also called an exact sequence. We consider similarly a diagram consisting of four A-modules and three homomorphism:

E —’> F .5.) G -"—> H.

(13)

This diagram is called exact at F if the diagram Ei> F-‘> G is exact; it is called exact at G if F -'-> G —h> 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 arbitrary number of terms are defined similarly. 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 homomorphism. 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 homomorphism a name in the

exact sequences where they appear. PRorosmon 3. (a) For

0—sE—’>F to be an exact sequence, it is necessary and suficient thatf be infective. (b) For

E—f—>F—>O to be an exact sequence, it is necessary and suflicient thatf be sutjective. (c) For 0——>E—f>F——>0

to be an exact sequence, it is necessary and suficient thatf be bifectt've (in other words (no. 2, Proposition 2) thatf be an isomorphism of E onto F). (d) [fF is a submoa'ule q and izF —> E is the canonical injection, [NE —> E/F the canonical homomorphism, the diagram

(14) is an exact sequence. 198

o——>F—">E—”—>E/F—>o

EXACT SEQUENCE

§ 1.4

(e) Iff : E —> F is a homomorphism, the diagram

0 —>}1(0) L» E —’—> F J» F/f(E)—>0 (where i is the canonical iry'ection and p the canonical surjection) is an exact sequence. The proposition follows immediately from the definitions and Proposition 2 of no. 2.

Remarks. (2) To say that there is an exact sequence

0—sE—f—sF—'>G——>o means that f is injective, g surjective and that the canonical bijection associated 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, §6, no. 7). (3) If there is an exact sequence with 4 terms

E—LF—‘sG—QH the cokernel offis F/f(E) = F/Tgl(0) and the kernel of h is g(F) ; the canonical bijection associated with g is therefore an isomorphism Cokerf—> Ker h.

(4) Consider an ordered pair of A-module homomorphisms

(15)

E —L> F —‘_> G.

For diagram (15) to be an exact sequence, it is necessary and sufficient that there exist two A-modules S, T and homomorphisms azE —> S, b:S —> F,

c:F -> T, d :T —> G such that the three sequences

E —‘—-> s ——> o

(16)

0——-—>S-—-b—->F—‘>T———>0 o ——> T —L> G

areexactandf: boaandg = doc.

If (15) is an exact sequence, then take 5 = f(E) = 2(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

conditions, then f(E) = b(a(E)) .—. b(S) and '5 (0) =—i(d1(0)) = 2E0), hence the exactness of (16) shows thatf(E) = 3(0). The use of explicit letters to denote homomorphism in an exact sequence is often dispensed with when it is not necessary for the arguments.

199

II

LINEAR ALGEBRA

Remark. (5) The definition of an exact sequence extends immediately to groups which are not necessarily commutative; in this case of course multiplicative notation is used with 0 replaced by l in the formulae (if no confusion arises). Parts (a), (b), (e) 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 that f(E) is a normal subgroup of F; Remarks 3 and 4 are valid without modification. 5. PRODUCTS 01" MODULES

Let (Ems; 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, §4, no. 8) is an A-module structure on the product set E = 1:! EL. With this structure the set E is called the product module of the modules E‘; if x = (x.), y = (y‘) are two elements of E, then -

x +l/ = (xi +%)

(17)

{1.x = (kg)

for all 16A.

Formulae (l7) express the fact that the projections prl —> El are linear mappings; these mappings are obviously surjective. Recall that if the indexing set I is empty, the product set i—‘i El then E

consists of a single element; the product module structure on this set is

then that under which this unique element is 0.

Paoposmon 4. Let E = g E‘ be the product of a family if A—modules (129.51. For every A-module F and every family of linear mappings fl:F —> E‘ there exists

one and only one mapping f of F into E such that pr,, 0f = j; for all t E I and this mapping is linear.

This follows directly from the definitions. Product of modules is “associative”: if (JmeL is a partition of I, the canoni-

cal mapping

11 E. —> AEL H (H E.) IEJ,‘ is an isomorphism. PROPOSITION 5. (i) Let (EJ‘EI, (F051 be two families of A-modules with the same indexing set I; for every family of linear mappings fizE. ——> F‘ (I. e I), the mapping

f: (x‘) —> (f;(x.)) q E. into E F. (sometimes denoted by E f.) is linear. (ii) Let (Go‘s; be a thirdfamily of A-modules with I as indexing set and, for all 200

PRODUCTS or MODULES

§ 1.5

LG I, let g,:FL —> GL be a linear mapping; let g = I: gP For each of the sequences E, 31> FI 3 G, to be exact, it is necessary and sufiicient that the sequence

UE,—L>UF,—'->HG. be exact.

Assertion (i) follows immediately from the definitions. 0n the other hand, to say that y = (y,) belongs to Ker(g) means that g.(y,) = 0 for all [61 and hence that y, e Ker(g,) for all I. e I; similarly, to say that 3/ belongs to Im(f) means that there exists x = (x.) e H E‘ such that y = f(x) , which is equivalent to saying that y, = f,(x,) for all r. e I or also that y, e Im(f.) for all I. e I; whence (ii).

COROLLARY. Under the conditions of Proposition 5, (i),

(18)

Kerm = H Kertfi),

1mm = U Im(.f.)

and there are canonical isomorphisms

Coim(f) -+ Q Coim(f,),

Coker(f) = H Coker(f,)

obtained by respectively associating with the class of an element x = (x,) of 1:]: Eu mod Ker(f), (resp. with the class ofan element y = (1].) t F“ mod. ImU)) the family of classes of the x, mod. Ker(fl) (resp. the family of classes of the y, mod. Im(f)). In particular, forf to be infective (resp. smjective, bijective, zero) it is necessary and suflicient that, fir all L e I, fl be injective (resp. surjectiae, bijective, zero). If, for all I. e I, we consider a submodule FI of E“ the module 1;! F, is a

submodule of 1:1: E, and by virtue of the Corollary to Proposition 5, there is a canonical isomorphism

FI mapping 1: e F to the constant function equal to x on I is linear. If (Ems, is a family of A-modules

201

LINEAR ALGEBRA

n

and, for all r. e I, f,:F —> El is a linear mapping, then the linear mapping

x ._> (me) of F into 11 E, is the composition of the mapping (x.) 9 (flag) of FI into E; E, and the diagonal mapping F —> F‘. 6. DIRECT SUM OF MODULES

Let (Ed‘s; be a family of A-modules and F = g E, 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 (E,) and denoted by EB: E, (I, §4, no. 9). When I is finite,

then 931 E, = g E.; ifI = (p, q] (interval o), we also write QEFEpeEpue-en. For all x e I, let jg be the mapping E,‘ —> F which associates with each x‘ 6 En the element of F such that pr,(j,‘(x,‘)) = 0 for t 96 K and pr,‘(jg(x.‘)) = xx; it is immediate that j,‘ is an injective linear mapping of E,‘ into the direct sum E of the E" which we shall call the canonical iry'ection; the

submodule j‘(EK) of E, isomorphic to Ex, is called the component submodule of E of index 1:. It is often identified with E: by means ofj‘. For all x e E = $91 Eu we have therefore

(20)

x = .2: Mr». x)-

PROPOSITION 6. Let (E.).E, be a family Qf A-modules, M an A-module and, for all I. e I, let f,:E, —-> M be a linear mapping. Then there exists one and only one linear

mapping g: ‘62 E, —> M such that,for all u. e I:

(21)

g °Jl =ft-

By virtue of (20), ifg exists, then necessarily, for all x 6 S131 Eu

g(x) = Z g(j. = Zmprxx», whence the uniqueness of g. Conversely, setting g(x) = Z f;(pr,(x)) for all

x e EDI Eu it is immediately verified that a linear mapping has been defined satisfying the conditions of the statement. 202

DIRECT sUM OF MODULES

§ 1.6

When no confusion can arise, we write g = 21 f, (which is contrary to the conventions of I, §2, no. I, when the family (f,) is not Of finite support). In particular, if J is any subset of I, the canonical injections j, for LEJ define a canonical linear mapping j;: g?! E, —> $91 Eu which associates with each (10,51 the element (x35, such that x{ = x, for v. e], x: = 0 for I. gt]; this mapping is obviously injective. Moreover, if (JQAGL is a partition of I, the

mappingi: £91.. (£9; E,) —-> EB: E, corresponding to the family (in) by Proposition 6 is an isomorphism called canonical (“associativity” of direct sums). COROLLARY 1. Let (Ema, (Foul, be twofamilies ofA-moa'ules. The mapping

(22)

Hom,( 62 E” E 17,) —> H”. HomA(E,, F,) (l. A) 61

which associates with each g e HomA( ‘63 E” 1;! FA) the family (pr, 0 g oj,), is a Z-module isomorphism (called canonical).

This follows from Proposition 6 and no. 5, Proposition 4-. COROLLARY 2. Let (E,), 51 be a family of A-modules, F an A-madule and, for each

tel, letfl:E,—->F be a linear mapping. Forf= Elf, to be an isomorphism of E = g)! E, onto F, it is necessary and :nfiieient that there existfor each IE I a linear mapping g,:F —> El with thefollowing properties: (1) g,of, = lE‘forall L51. (2) g,0f,, = Oforv. sex. (3) For all y e F, thefamily (g,(y)) has finite support and

(23)

y = gnaw».

Note that if I is finite, the last condition may also be written as-

(24)

2 r o g, = 1F.

tel . Obviously the conditions are necessary for they are satisfied by the

g, = pr, 0}? Conversely if they hold, for all yeF, g(y) = Z j,(g,(y)) is defined and it is immediate that g is a linear mapping of F into E. For all

3/ EF, f(g(y)) = gf,(g,(y)) = y by hypothesis. 0n the other hand, for all

x e E, gm») = gt(Zf.(pr. F, (v. e I), the restriction to Q E, of the linear mapping (x,) I—> (f,(x,)) is a linear mapping f : ‘6?! E, ——> g)! F,

denoted bog—)1}: or (of. (wheref=f,,(-Bf,,+1®

(-Bf, ifI = [[2,4] is an

interval in Z). (ii) Let (G,),e1 be a thirdfamily of A-modules with I as indexing set and, for all I. e I, let g,:F, —> G, be a linear mapping; we write g = ® g,. For each of the se-

quences E, i; F, L; G, to be exact, it is necessary and suflicient that the sequence

QB E, .19 (1:9 F, —i> (19 G, be exact.

Obviously, for all (x,) e $ E,, the family (fl(x,)) has finite support, whence (i). On the other hand, to say that an element 3/ = (y,) of ('P 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 i e I an x, e E, such that y, = j, (x,) and when y, = 0, it may be supposed that x, = 0; hence y 6 Im(f) and the converse is obvious.

COROLLARY 1. Under the conditions of Proposition 7, (i),

(25)

Kerm = 9+, Kerm),

Im(f) = $19, Im(f.)

and there are canonical isomorphisms

Coim(f) —> (131 Coim(f,),

Coker(f) —+ (l); Coker(f,)

defined as in no. 5, Corollary to Proposition 5. In particular, forf to be infective (reap. sudective, bijectioe, zero), it is necessary and suficient that each of thef, be injectioe (reap suriectioe, bijective, zero). If, for all L E I, we consider a submodule F, of E,, the module EDI F, is a sub-

module of g E, and, by virtue of Corollary 1 to Proposition 7, there is a canonical isomorphism

E, (resp. vfiF,‘ —> Fl) a linear mapping. Then the diagram

11046313311170 L

H Hom(E:.F;.)

(l. A.) e I x L

Ham( @ It, I} In.)

H Hom(ul,,n,)

Hom( 6') E, H FL) —> tel

1.6L

4:

(t,l.)eJxL

Hom(E,,, FA)

(where 4) and (b' are the canonical isomorphisms defined in Corollary 1 to Proposition 6) is commutative.

The verification follows immediately from the definitions. When all the E, are equal to the same A-module E, the direct sum $91 E.

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 Proposition 6, a linear mapping E“) —> E is obtained, called codiagonal, which associ-

ate: with every family (30.51 of elements of E, of finite support, its sum ‘62; xP Remark. Recall that the definition of direct sum extends immediately to a family (Ed‘s, of groups which are not necessarily commutative, multi-

plicative notation of course replacing the additive notation; we then say “restricted product” or “restricted sum” instead of “direct sum” (I, §4, no. 9). Note that E is a normal subgroup of the product F = I; E, and that each of the jg(EK) is a normal subgroup of F; moreover, for two distinct indices 7., u, every element of j,‘(E,.) is permutable with every element of ju(Eu). Proposition 6 extends to the general case with the hypothesis that, for two distinct indices 1, it, every element offA(E,,) is permutable in M with every element offi.(E,,) (I, §4, no. 9, Proposition 12). The property of “associa-

tivity” of the restricted sum follows immediately from this. Proposition 7 and its Corollaries l and 2 hold without modification. 7. INTERSECTION AND SUM OF SUBMODULES

For every family (M051 of submodules of an A-module E, the intersection Q E. is a submodule of E. If, for each I. e I, (b. denotes the canonical homo-

morphism E —> E/Mu ‘01 M, is the kernel of the homomorphism 4) :x v—> (¢,(x)) of E into I; (E/Ml), in other words, there is an exact sequence

(27)

o —> n M. ——> E is H (E/Mt). $91

lEI

205

LINEAR ALGEBRA

II

The linear mapping (I: and the mapping

WQW+QWW which is obtained by passing to the quotient, are called canonical. In particular:

PROPOSITION 8. [f afamily (Mme! of submodules of E has intersection reduced to 0

E is canonically isomorphic to a submodulc 0‘11 (E/M.). Given a subset X of an A-module E, the intersection F of the submodules

of E containing X is called the submodulc generated by X and X is called a generating set (or generating system) of F (I, §4, no. 3); for a family (“that of elements of E, the submodulc generated by the set of a. is called the submodulc generated by the family (a). An A-module is calledfinitely generated if it has afinite generating set. PROPOSITION 9. The submodulc generated by a family (a‘).51 of elements of an Amodule E is the set of linear combinations of thefamily (at). Every submodulc of E which contains all the aI 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 submodulc of E which obviously contains all the a. and is therefore the smallest submodulc containing them. COROLLARY 1. Let u:E—>F be a linear mapping, S a subset of E and M the

submodulc of E generated by S. Then u(M) is the submodulc of F generated by u(S). In particular, the image under u of any finitely generated submodulc of E is a finitely generated submodulc of F. Remark. If u(x) = 0 for all are S, then also u(x) = 0 for all are 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 —> F is of the form a a 4), where s/M —-> F is linear and ¢:E —> E/M is the canonical projec. tion, it suffices to verify that u(S) = 0.

COROLLARY 2. The submodule generated by the union of a family (Mme! of submodules of a module E is identical with the set of sums E! x" where (leI runs through the set offamilies of elements of E offinite support such that x; e Mtfor all r. e I.

Clearly every linear combination of elements of ‘EUI M. is of the above form; the converse is obvious.

206

mmcnon AND sUM or SUBMODULES

§ 1.7

The submodule of E generated by the union of a family (Mme! of sub-

modules of E is called the sum of the family (M,) and is denoted by ”21 Mr If for all I. e I, h, is the canonical injection M, -—> E and h: (x.) I-—> Z h,(x,) the

linear mapping of Q 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 sequence

(23)

'

t—LE—eE/(IEZIMJHO.

COROLLARY 3. If (Mflha‘ is a right directedfamily of submodules of an A-module

E, the sum A; M, is identical with the union ALE}L MA. xLe'JL M; C 7.21. MA is always true without hypothesis; on the other hand, for every finite subfamily (Ming of (MQML, there exists by hypothesis a p. e L such that M; C Mu for all A 6], hence ’21! MA C Mn and it thus fol-

lows ftom Corollary 2 that x21. M,“ C 1.91. ML. COROLLARY 4-. Let 0 -—> E —f> F 1) 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 subset of F such that g(T') = T, T’ Uf(S) is a generating system of F.

The submodulc F’ of F generated by T’ Uf(S) containsf(E) and, as g(F’) 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 arefinitely generated, so is F. PROPOSITION 10. Let M, N be two submodules of an A-module E. Then there are two exact sequences

(29)

0—>MnN—"—>M@N—':j->M+N—->0

(30) 0 ——-—> E/(MnN) —i-> (E/M) 6-) (E/N) ’—_q> E/(M + N) ——> 0 where i :M—> M + N, j :N ——> M + N are the canonical injections,

p:E/M—>E/(M + N)

and q:E/N—> E/(M + N)

the canonical surjections and where the homomorphism: u and v are defined as follow:

207

LINEAR ALGEBRA

H

ifsnN—>M—>M®N and g:M nN—>N—>M®N are the canonical injections, u =f + g, and ifrzE/(M n N) —> E/M -—> (E/M) GB (E/N) and szE/(M n N) -—> E/N ——> (E/M) 6-) (E/N)

are the canonical mappings, v = r + s. We prove the exactness of (29): obviously i — j is surjective and u is injective. On the other hand, to say that (i — j) (x,y) = 0, where xeM and yeN, means that i(x) ——j(y) = 0, hence i(x) =j(y) = 26M 0 N, whence by definition x = f(z), y = g(z), which proves that Ker(i — j) = Im a. We prove the exactness of (30): clearly p — q is surjective. On the other hand, to say that v(t) = 0 for teE/(M 0N) means that r(t) = s(t) = 0, hence t is the class mod. (M n N) of an element 2 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 e/M, y eE/N means that p(x) = q(y), or also that there exist two elements z’, z” 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” e N such that z’ —- z" = t’ — t”, whence

z'—t'=z”—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 2', that is x; similarly s(w) = g, which completes the proof that Ker([1 —- q) = Im v. 8. DIRECT SUMS 0F SUBMODULES

DEFINITION 6. An A-module E is said to be the direct sum of a family (Mom! of submodules of E if the canonieal mapping $131 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 = 2:14, where x, GB, for all LEI; the element x, thus

corresponding to x is called the component of x in E,; the mapping .2: tax. is linear. Remark. (1) Let (M,),,eI, (NJ,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 (NJ and that Nl C M, for all t 61. Then N| = M, for all L e I, as follows immediately from no. 6, Corollary 1 to Proposition 7

applied to the canonical injectionsfl: N, —> Mr PROPOSITION 11. Let (M,),eI be a family of submodules of an A-module E. The

following properties are equivalent:

(a) The submodule Z! M, is the direct sum ofthefamily (Mom. 208

DIRECT sums or SUBMODULES

§ 1.8

(b) The relation 21x, = 0, where x, e M, for all re I, implies x, = 0 for all

L e I.

(c) For all K e I, the intersection of MK and ‘2‘ M, reduces to 0. It is - immediate that (a) and (b) are equivalent, since the relation

2; x, = Z 3/, is equivalent to 2‘: (x, — 3],) = 0. On the other hand, by virtue of Definition 6, (a) implies (c) by the uniqueness of the expression of an element of g)! M, as a direct sum of elements x, E M,. Finally, the relation

2 x. = o, where 316 M. for all L, can be written, for all x e I, x, = .2: (-4); condition (c) then implies x: = 0 for all K e 1, 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 (60,51, ofprojectors is called orthogonal if e" o e“ = OfiJr A 95 y.. PROPOSITION 12. Let E be an A-module. (i) If E is the direct sum of a family (MDMH, of submoa'ules and, for all x E E, e,(x) is the component of x in MA, (eh) is an orthogonalfamily ofprojectors such that x = ”I. e,(x)for all x E E. (ii) Conversely, if (aha, is an orthogonal family of projectors in End(E) such that x = AZL e,,(x) fir all x e E, E is the direct sum of the family of suhmodules M}. = €A(E).

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 —> MA.

Note that when L is finite the condition x = 1.21. e,‘(x) for all x e E may also be written in End(E).

(31)

1E = 2 e,. AeL

COROLLARY. For every projector e of E, E is the direct sum of the image M = e(E) andthekernelN = ,‘1(0) qfe;forallx = x1 + x26E with x16M andxzeN, x1 = e(x) ; l — e is a projector of E of image N and kernel M.

(l —e)2= l —2e+e2= l —e in End(E) and hence I —e is a projector; as also e(l — e) = (l — e)e = e — e2 = O, E is the direct sum of the

images M and N of e and l —- e by Proposition 12. Finally, for all x e E, the relation x E M is equivalent to x = e(x); for x = e(x) implies by definition x E M and, conversely, if x = e(x’) with x' e E, then e(x) = e2(x') = e(x') = x;

209

LINEAR ALGEBRA

II

this shows therefore that M is the kernel of l — e and, exchanging the roles of e and l — o, 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 finite family (M01 “a. of submodules and F the direct sum of a finite family (N,)1‘,‘,. of submodules. Then it is known (no. 6, Corollary 1 to Proposition 6) that HomA(E, F) is canonically identified with the product 1;} HomA(M‘, N,); to be precise, to a family (111,), where u” e HomA(M, N,)

there corresponds the linear mapping u:E —> F defined as follows. It suflices to define the restriction of u to each of the LI, and for each age M‘, n

u(x,) = 121 u,,(x.). Now let G be a third A-module, the direct sum of a finite family (Pk)1‘k‘,, of submodules; let a be a linear mapping of F into G and let (om) e El HomA(N,, Pk) be the family corresponding to it canonically. For all x. 5 Mg,

vow) = ,2 auto.» = ,2; 31 Mum). Thus it is seen that if we write 1|

(32)

wt. = 21 at, o a. e HomMMu Pt)

the family (wm) corresponds canonically to the composite linear mapping w = voufromE to G (cf. § 10, no. 5). 9. SUPPLEMENTARY SUBMODULES

DEFINITION 8. In an A-module E, two submodules M1, M3 are said to be supple-

mentary if E is the direct sum of M1 and M2. Proposition ll of no. 8 shows that, for M1 and M2 to be supplementary, it is necessary and sufficient that M1 + M2 = E and M1 n M2 = {0} (cf. I,

§4, no. 9, Proposition 15).

PROPOSITION 13. Let M1, M2 be two supplementary submodules in an A-module E. The restriction to M1 of the canonical mapping E —> E/Ma is an isomorphism of M1 onto E/Mz. This linear mapping is surjective since M1 + M2 = E and it is injective since its kernel is the intersection of M1 and the kernel M3 of E —> E/Ma and hence reduces to {0}.

COROLLARY. [f M2 and M; are two supplements of the same submodule M1 of E, the set of ordered pairs (x, x') 6 M2 x M; such that x — x’ G M1 is the graph of an isomorphism of M2 onto M5. 210

SUPPLEMENTARY SUBMODULES

§ 1.9

It is immediate that it is the graph of the composite isomorphism M2—->E/M1—>M§. DEFINITION 9. A submodule M of an A-modale E is called a direct factor of E if 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 submodnle M of a module E to be a directfactor, it is necessary and suficient that there exist a projector of E whose image is M or a projector of E whose kernel is M. This follows immediately from no. 8, Proposition 12 and Corollary. PROPOSITION 15. Given an exact sequence q-modules

(33)

0——>E—{—>F—‘>G—>0

thefollowing propositions are equivalent:

(a) The submoa'ulef(E) (Jr is a directfactor.

(b) There exists a linear retraction r:F~> E associated with f (Set Theory, II, §3, no. 8,‘Definition 11). (c) There exists a linear section szG —> F associated with g (Set Theory, II, §3, no. 8, Definition ll). When thisisso,f+ s:E(—BG—>F isan isomorphism. If there exists a projector e in End(F) such that e(F) = f(E), the homomorphism flo e:F —> E is a linear retraction associated with f; conversely, if there exists such a retraction r, it is immediate that f o r is a projector in F whose image is f(E), hence (a) and (b) are equivalent (Proposition 14-). Iff(E) admits a supplement E’ in F and j :E’ —> F is the canonical injection, g 0j 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 o g is a projector in F whose kernel is f(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. Note that being given r (resp. s) is equivalent to being given a supplement off(E) in F, namely the kernel of r (resp. image of s).

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, § 6, no. 1). 211

LINEAR ALGEBRA

II

COROLLARY 1. Let q —> F be a linear mapping. For there to exist a linear mapping v:F —> E such that u o v = IF (the case where u is said to be right invertible and v is said to be the right inverse qf u), it is necessary and suflicient that u be surjective and that its kernel be a directfactor in E. The submodule Im(v) Qf E is then a supplement qf Kcr(u).

It is obviously necessary that u be suijective ; 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 mapping

s —> E such that v o u = 1E (the case where u is said to be left invertible and v is said to be the lefl inverse qf a), it is necessary and sufiieient that u be injective and that its image be a directfactor in F. The submodule Kcr(v) of F is then a supplement qf Im(u).

It is obviously necessary that u be injective; as v is then a retraction associated with u, the conclusion also follows from Proposition 15. Remarks. (1) Let M, N be two supplementary submodules in an A-modulc 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) I—> u a p + v o q is an isomorphism of HomA(M, F) ® HomA(N, F) onto HomA(E, F). The image of HomA(M, F) under this isomorphism is

the set of linear mappings s ——> 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 0F FINITE LENGTH

Recall (I, §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 of finite length if it has a Jordan-Holder series (Mooggn and the nmnber 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, §4, no. 7, Definition 11); we shall denote it by long(M) or longA(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 (y‘ M; for M to be qffinite length, it is necessary and suflicient that N and M/N be so, and then

long(N) + long(M/N) = long(M). The proof has been given in I, §4, no. 7, Proposition 10.

(34)

212

MODULES or 17mm: LENGTH

§ 1.10

COROLLARY 1. Let M be an A-module (Jfinite length; for a submodule N of M to be equal to M, it is necessary and suflicient that long(N) = long(M).

COROLLARY 2. Let u:M——>N be an A-moa'ule homomorphism. [f M or N is of finite length, so is Im(u). If M is offinite length, so is Ker(u) and

long(Im(u)) + long(Ker(u)) = long(M).

(35)

IfN is offinite length, so is Coker(u) and long(Im(u)) + long(Coker(u)) = long(N).

(36)

COROLLARY 3. Let (M,)oM1—u‘->M2——>

——>M,,_1—"l'—§Mn—>0

then 1:

(38)

“Zn (—1)'=1ong(Mk> = o.

The corollary is obvious for n = l and is just Proposition 16 for n = 2; we argue by induction on n. If M,’._1 = Im(u,,_2), then, by the induction

hypothesis, u-fl

Z, (—1>*Iong(Mk) + (-1)»-110ng(M;,_1) = o.

On the other hand, the exact sequence 0 —> Ml,_1 —> M,,_1 -—> Mn —> 0 gives

long(M;-1) + long(M") =Iong(Mn-1), whence relation (38). COROLLARY 4. Let M and N be two submodules offinite length of an A-module E; thenM + Nisqffinitelengthand

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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—>0

using the fact that long(M ® N) = long(M) + long(N) by (34).

COROLLARY 5. Let M be an A-moa'ule the sum of a family (N‘) of submodules of finite length. Then M is offinite length and

(40)

long(M) < Z long(N) 213

LINEAR ALGEBRA

Moreover, for the two sides of (40) to be equal, it is necessary and sufin'ent that M be the direct sum of the N..

It has been seen (no. 7, formula (28)) that there is a canonical surjective linear mapping h: (“PM —> M; hence the corollary follows from (35).

Conomv 6. Let M and N he two submodules ofan A-module B such that E/M and E/N are modules offinite length; then E/(M n N) is offinite length and (4-1)

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) G) (E/N) —> E/(M + N) —> 0 using the fact that

Ion8((E/M) 63 (E/N)) = long(E/M) + lo“gal/N)COROLLARY 7. Let (Mi) be afinitefamily ofsubmodules qfan A-module E such that

the E/M, are modules qffinite length. Then E/(O M.) is offinite length and

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long(E/(Q M.)) < Z long(E/M‘).

It has been seen (formula (27)) that there is a canonical injective linear

mapping E/(Q M) a 6?) (E/M«)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

x‘) = 2; out. for every operator at, and Corollary 3 follows from it. «(2 IeI As for Corollary 1, it suffices to note that ifN is a stable subgroup ofF contain-

ing u(S), 741(N) is a stable subgroup of E containing 5, hence 711(N) contains M and therefore u(M) C N. 11. FREE FAWLIES. BASES

Let A be a ring, T a set and consider the A-module AS”. By definition, it is the external direct sum of a family (Mata. of A-modules all equal to A, and

for all teT there is a canonical injection j,:A, —>A§T’ (no. 6). We write

214-

m2 FAMILIES. BASES

§ 1.11

j.(l) = a. so that a, = (swig, where 8“. is equal to o if t’ as t, to 1 if t' = t (“Kronecker symbol”; (t, t’) u—> 8“. is just the characteristic function of

the diagonal ofT x T) ; every x = (5,), ET e A?” may then be written uniquely:

x = gr E‘ep The mapping E determined by thisfamily be surjectioe.

‘

This is just another way of expressing Proposition 9 of no. 7.

’

215

LINEAR ALGEBRA

II

DEFINITION 10. A family (a,),eT qf elements of an A-module E is called a free

family (resp. a basis of E) if the linear mapping Afr) —> E determined by thisfamily is infective (resp. bijective). A module is calledfree if it has a basis. In particular, a commutative group G is calledfree if G (written additively) is afree Z-module (cf. I, § 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 aé {0}, T is often identified with

the set of e, by the canonical bijection tI—> e,; this amounts to writing t; £,.t. nstead of t; Eta. for the elements of AS”. When this convention is adopted,

the elements of Ag” are calledformal linear combinations (with coefficients in A) Qf the elements of T.

Definition 10 and Proposition 17 give immediately the following result: COROLLARY 3. Let E be a free A-module, (data a basis of E, F an A-module and (b,),e.r afamily ofelements ofF. There exists one and only one linear mappingf: E —> F such that f(at) = b, for all te T. (45) Forf to be injective (resp. surjeetive), it is necessary and sufiieient that (b;) be afiee family in F (resp. a generating system of F). When a family (at) ,eT is not free, it is called related. Definition 10 may also be expressed as follows: to say that the family (atheT is free means that the

relation 2; Mat = 0 (where the family (h) is of finite support) implies A, = 0 for all teT; to say that (a¢)teT is a basis ofE means that every x e E can be written in one and only one way in the form x = t; Ea; for all te T, E: is

then called the component (or coordinate) Qfx of index t with respect to the basis (at); the mapping x —-> E, from E to A8 is linear. Suppose A ;é {0}; then, in an A-module E, two elements of a free family (“Drew whose indices are distinct are themselves distinct: for if a, = a" for t’ aé t”, then 2; Mat = 0 with At. = l, A”. = —l and x, = 0 for the elements

of T distinct from t’ and t”. A subset S of E will be called afree 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. 216

FREE FAMILIES. BASES

§ 1.11

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 free and is a basis of the submodule {0} of E.

PROPOSITION 18. For a family (at),e.r qf a module E to be free, it is necessary and suflicient that everyfinite subfamily Q," (11,)“, befree. 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, III, § 2, no. 4); as it is non-empty (since 2‘ belongs to it), it has a maximal element (dots; by Zorn’s Lemma. It follows

(if A ¢ {0}) that for all xe E there exist an element p. 96 0 of A and a

family (5.) of element A such that px = 2 Km. (cf. § 7, no. 1). PROPOSITION 19. Let E be an A-module, the direct sum qf a family (M1),“ of submodules. If, for each A e L, S, is afree subset (resp. generating set, basis) of M,_, then S = ALGJL S, is afiee subset (resp. generating set, basis) qf E.

The proposition follows from the definitions and the relation Afis’ = Q A?" (associativity of direct sums, cf. no. 6). Remark. (1) By Definition 10, if A at {0} and (amel is a free family, no element a.‘ can be equal to a linear combination of the a, of index 1. 9E 31:. But conversely, a family (a,) 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-modulc, 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 = xa (cf. however § 7, no. 1, Remark).

An element x of a module E is calledfree if {x} is a free subset, that is if the relation ax = 0 implies a = 0. Every element of a free subset is free and in particular 0 cannot belong to any free subset when A aé {0}. Remarks. (2) A free module can have elements 960 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 a fortiori Z/ (n) is not a free module.

(4) It can happen that every element at 0 of an A-module is free without the module being free. For example, the field Q of rational numbers is a Z-module with this property, for two elements #0 of Q always form a

217

I!

LmEAR ALGEBRA

related system and a basis of Qcould therefore only contain a single element a; but the elements of Q are not all of the form M with n e Z (cf. VII, § 3).

PROPOSITION 20. Every A-madule E is isomorphic to a quotient module qf a free A—module. If T is a generating set of E, there exists a suijective linear mapping

A?” —> E (Corollary 2 to Proposition 17), and ifR is the kernel ofthis mapping, E is isomorphic to Ash/R. In particular we may take T = E; then there is a surjective linear map-

ping A?” —> E, called canonical.

In particular, to say that an A-module E isfinitely generated (no. 7) means that it is isomorphic to a quotient of a free A-module with a finite basis or also that there exists an exact sequence of the form A:->E—>0

naninte8er >0 .

Note that if A ;E {0} every basis of afinitely generated free module E is necessarily finite, 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 21. Every exact sequence qf A-modules

0—9G—5—>E—f>F——>0 in which F is a free A-module, splits (no; 9). To be precise, if (mm is a basis ofF

and, for each A E L, a,‘ is an element qf E such thatf(a) = bk, thefizmily (ah),_eL is free and generates a supplementary submodule ta g(G).

There exists one and only one linear mapping h : F ——> E such that h(b,‘) = a,_ for all A e L (Corollary 3 to Proposition 17). As h is a linear section associated with j; the proposition follows from I, §4, no. 9, Proposition 15.

Remark. (5) Let (a,)1 H0m3(E[B1: Fun)This mapping is not necessarily bijective; in other words a B—linear mapping Em —> Fm is not necessarily A-linear. For example, a sub-Bmodule of Em is not necessarily a sub-A-module of E: if A is a field and B

221

LINEAR ALGEBRA

II

a subfield of A, the vector subspace B, of the vector B-space (A,)m is not a vector sub-A-space if B 95 A. It is immediate that, for every family (Ema, of A-modules, the B-module

9.41:! E.) (resp. saga, E.)) is equal to 1:1 9*(E.) (resp. $9, 9*(E;))Every generating system of 9*(E) is a generating system of E but the converse is nOt necessarily true. PROPOSITION 24-. Let A, B be two rings and s ——> A a ring homomorphism.

(i) If p is surjeotive, the canonical mapping (47) is biiective. For every A-module E, every sub-B-module of 9*(E) is a sub-A-module of E; every generating system of E is a generating system of 9*(E). (ii) If p is injective, every flee family in the A-module E is a free family in the B-module 9*(E). The proposition follows immediately from the definitions. Note that even if p is injective, a free family in 9*(E) is not necessarily free in E. *For example, 1 and 1/2 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 1)... PROPOSITION 25. Let A, B be two rings, 9: B ->A a ring homomorphism and E an.A—module Let (mm, be a generating system (resp. free family of elements, basis)

ofA considered as a left B-module. Let (au)uEM be a generating system (resp.freefamily of elements, basis) of the A—module E. Then (09%)“ men” is a generating system (resp. (when p is injective) freefamily of elements, basis) of the B-module 9*(E). If x = “Zn yuan, where Y“ e A and (ax) is a generating system of A, we may write Y“ = 62L flaw)“, With [3,." 6B, for all p. EM, whencex— = 2 “BMW”... On the other hand, if (an) and (au) are free families? a relation 1.2“ 9(5tu)¢2.au = 0, With pm EB: may be written usZM (2 1.61. P(BAu)°‘A)au = 03

it thus implies A21. “(SM)“ = 0 for all p. eM and therefore B“, = 0 for all

A, y. if p is injective. COROLLARY. If A is a finitely generated left B-module and E a finitely generated left A-module, 9*(E) is a finitely generated left A-module. Let C be a third ring, 9’ : C —> B a ring homomorphism and p” = p o p' the

composite homomorphism. It follows immediately from the definitions that 222

CHANGE or RING or saunas

§ 1.13

p£(E) = p$(p*(E)) for every A-module E. In particular, if p is an isomorphism of B onto A, then B = p’...(p,,(E)), where 9' denotes the inverse isomorphism

of 9.

Remarks. (1) Let K be a field and A a subring of K with the following property: for every finite family (5,)1‘,“ of elements of K, there exists a y e A which is non-zero and such that ya, GA for l g i S 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 Em the A-module obtained by restricting the ring of scalars to A. Then, if a family (xx) ML isflee in Em it is alsofm in E.

Attention may be confined to the case where L = [1, n] ; if there were a rela-

tion “21 5.x, = 0 with E, e K, the 5, not all zero, it would follow that for all

(3 EA, 1; ([35,)x, = 0. By hypothesis we can suppose [-1 ye O in A such that [35, = cc, belongs to A for all i; but the relation “21 cox, = 0 is contrary to the hypothesis, the on, being not all zero. (2) If the ring homomorphism s —>A is surjective and b is its kernel (so that A is canonically identified with 3/1)), then, for every A-module E,

b is contained in the annihilator of 9*(E) and E is the A-module derived from

9*(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—-> 9*(E) (also called a B-limar 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, forxeF,yeF and BEB,

(48)

u(x + y) = u(x) + M) {u(fix) = p(fl)u(x)-

The set Hom3(F, p..(E)) of B-linear mappings of F into E is also written as Hom3(F, E) if no confusion can arise.

When 9 is an isomorphism of B onto A, the relation u(Bx) = p((3)u(x) for all [5 eB may also be written as u(p'(ac)x) = ax for all an EA, where 9’ 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 p;(F) into E.

Example. It has been seen (no. 1) that a homothety II“: x I—-> our on an A-module E is not necessarily a linear mapping. But if at is invertible, I!“ is a semi-linear mapping (which is moreover bijective) relative. to the inner automorphism

E 5—) «if 1 ofA, for «(NO = (ala‘1)(ocx). 223

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LINEAR ALGEBRA

Let C be a third ring, p’:C -—> B a homomorphism and G a C-module. If o:G-—> F is a semi-linear mapping relative to p’, the composition w = u o o

is a semi-linear mapping ofG into E relative to the homomorphism p' = p o 9'. If p is an isomorphism and u:F —> E is a bijectioe semi-linear mapping relative to p, the inverse mapping u’:E —-> F is a semi-linear mapping relative to the inverse isomorphism 9’:A —> B of 9. 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 dimorphisms (u, 4;) can be taken as morphisms (Set Theory, IV, §2, no. 1) , we shall always assume in what follows that this choice of morphisms has been made. Remark (3). Let A1, A; be two rings, A = A1 x A2 their product and let e1 = (l, 0), e2 = (O, l) in A, so that A1 and A2 are canonically identified with the two-sided ideals Ael and Ae, of A. For every A-module E, e1E and ezE are sub-A-modules E1, E2 of E, annihilated respectively by c2 and e1, so that, canonically identifying A/Ae2 with A1 and A/Ael with A2, E1 (resp.

E2) is given an Al-module (resp. Aa-module) structure. Moreover, E is the direct sum of E1 and E2, for every x e E can be written as x = elx + ezx and the relation elx = egg implies elx = egx = elezy = 0. Conversely, for every ordered pair consisting of an Al-module F1 and an A2~module F2, let E1 be

the A-module (pl) * (F1), E2 the A-module (p2) * (F2), [)1 and p2 being the projections of A onto A1 and A2 respectively; then in the A-module E = E, 6-) E2, E1 = elE, E2 = ezE. The study of A—modules is thus reduced to that of Al-modules and that of Az-modules. In particular, every submodule M of E is of the form M1 {-9 M3, where M1 = e1M and M2 = eaM. l4. 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 (3’ be the endomorphism ring of the additive group E and for all a GA (resp. B e B) let ha (resp. hg) denote the element x r—> our (resp. x »—> fix) of 6'. Clearly the three following properties are equivalent: (a) he, ohf, = hf, oh“ for all a and [3; (b) the image of A under the homomorphism a 1—» h. is contained in HomB(E, E); (c) the image of B under the homomorphism B» hf, is contained in HomA(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 A0 (resp. B“). The above properties can be expressed by saying that the two (left or right) module structures defined on E are compatible. DEFINITION 13. Let (Alba, (Bu),,eM be two families of rings; an ((A1), (Bu))-

multimodule (or multimoa'ule over the families of rings (Alba, (Bu)ueM) is a set E with, for each 7L EL, a left AA-module structure and, for each p. E M, a right Bumodule structure, all these module structures being compatible with one another. 224-

MULTIMODULES

§ 1.14

When the family (Bu) (resp. (AA)) is empty, E is called a left (resp. right) multimodule. When Card(L) + Card(M) = 2, we say “bimodule” 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 left 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

(49)

«(37(5) = (”)3 for x e E, at e A, [3 E B.

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. Examples. (1) On a ring A the module structures of A, and Ad are compatible and A can therefore be considered canonically as an (A, A)-birnodule. (2) A left A—module E has a canonical left module structure over the ring EndA(E) and the A-module and EndA(E)-module structures on E are compatible.

Clearly when E is a multimodule over two families (Ashen: (Bung, of rings, E is also a multimodule over any two subfamilies (Album (B ) EM': where the Ah-module and Bu-module structures for 7t 6 L’ and p. e li'II’I 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 ((AA), (Bu))-multimodules, a homomorphism u: E—> F is a mapping which is an AA-homomorphism for all MEL and a Bu-homomorphism for all p. e M. The stable subgroups ofan ((AA), (3,,»multimodule are ((AA), (Bu))-multimodules (called submultimodules), as also are the quotients by such subgroups (called quotient multimodules); similarly for products and direct sums. Let E be an ((AA), (BIN-multimodule and for each 7t 6L (resp. y. EM) let ot; -> A, (resp. 41“: Bf, —> B“) be a ring homomorphism; clearly the All-module structures associated with the (1),, and the AA-module structures

given on E and the BL-module structures associated with the 4;" and the Bh-module structures given on E are compatible with one another and hence define on E an ((Ag), (Em-multimodule structure, said to be associated with the given ((A1), (Bu))-multimodule structure and the 4),, and rpm. IfE, F are two ((A,_), (Bu))-multimodules, the additive group ofhomomorphisms of E into F is denoted by HommwmufiE, F) (or simply Hom(E, F)). Formulae (6) to (8) of no. 2 are obviously valid for ((AA), (Bun-multimodule homomorphisms and, in particular, Hom(E, E) = End(E) has a ring structure;

225

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moreover Hom(E, F) has a canonical left 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 p. e M) and on the other the rings of another family (Amway (resp. (Bh'ht'ew). Suppose similarly that F has a multimodule structure whose rings of left (resp. right) operators are on the one hand the AA for A e L (resp. the Bu for (LEM) and on the other the rings of another family (A1.) ”a, (resp.

(momma; to abbreviate we shall say that E is an ((A1), (AL); (Bu), (Bf, ))-

multimodule and F an ((AA), (A”»); (Bu), (Earn-multimodule. Consider E and

F as ((A5), (Bun-multimodules, thus restricting the operators to the subfamilies (AA) and (By). By what was said at the beginnings of this no., the multimodule structures given on E and F define canonically ring homomorphisms A3,: —> Endum, (BID) (E):

BL? ‘—) End(AA)n (3n) (E) ’

Ail _> Ent): (3“)(F),

Bf!” -—> End(A,.), (3“)(F) ;

moreover, two elements of End(A).).(Bu)(E) (resp. End(m,(nu,(F)) respective images of elements of two distinct rings among the A1. or the B13 (resp. the AZ» or the B3?) are permutable; it follows that the above homomorphism define on Hom(A,_,_ (3..)(E: F) a multimodule structure whose rings of lgfi‘ operators are the A”. (1” e L”) and the B;. (p! e M’) and whose rings of right operators are the AL. (2" e L’) and the B"~ (u' e M”). If now E' is an ((Ak), (AL); (Bu), (Wyn-multimodule and F’ an ((Ah), (A".); (Bu), (B;.))-multimodule, Hom(m,(nu,(E’, F’) is an

((A'»). (Bat): (Ab), (B:-))-mu1timodu1c; if u:E' —> E, v:F —> F’ are multimodule homomorphisms, H0m(u, v) :HOIIRAA’). (Bu)(E’ F) —> H0m(AA)' (Bu)(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 bimodal; structure whose rings of left operators are A and C. If E is another A-module, HomA(E, F)

has therefore a canonical C-module structure (where, for f e HomA(E, F) and

Y e C, yf is the homomorphism Jan—>7f(x)), if E’, F' are two A-modules,

u:E' —-> E, v: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)-bimodulc structure, so has the direct sum A”) for any indexing set T; by the above, 226

PROPERTIES or HomA(E, F) RELATIVE To EXACT SEQUENCES

§ 2.1

HomA( S”, E) has a canonical left A-module structure arising from the right

A-module structure on A3”: forf e HomA( S", E) and at e A, atf is the linear mapping x »-—>f(m). Corollary 2 to Pr0position 17 of no. 11 defines a canonical mappingjE. T from the product module E” to HomA(A§T’, E), the image under JE. T of a family (xt)te .1. being the linear mapping f: A?“ ——> E such that f(q) = x, for all te T (where (q) is the canonical basis of AS”); it is known (lac. cit.) thatjE_ T is bijective and it follows from the definition given above of the

A-module structure on HomA(Af,T’, E) thatjE, T is A-linear. Finally, if u: E —> F

is an A-module homomorphism, the diagram

ET —"——> Hom.(A§“, E) Hom(l,u)

u'r

(50)

FT -—-—> HomA(A:T), F) JF.’1‘

is commutative. Note that when T consists of a single element, 1'a —-> HomA(A,, E) is just the mapping x r—> 6,, defined in no. 2, Example 1.

§2. MODULES OF LINEAR MAPPINGS. DUALITY 1. PROPERTIES OF HomA(E, F) RELATIVE TO EXACT SEQUENCES

THEOREM 1. Let A be a ring, E', E, E” three A-modules and ME —> E, v:E —> E' two homomorphisms. For the sequence

(1)

E' 3—» E —"> E' —> o

to be exact, it is necessary and suflicient that, for every A-module F, the sequence

(2)

o ——> Hom(E', F) —"—> Hom(E, F) L Hom(E’, F)

(where :7 = Hom(u, IF), :7 = Hom(u, 13.)) be exact. Suppose that sequence (1) is exact. Ifw e Hom(E”, F) and 17(w) = w o v = 0, then w = 0 since 0 is suijective. Sequence (2) is therefore exact at Hom(E”, F). We show that it is exact at Hom(E, F). 12 o 5 = Hom(v o 14, 1,) (§ 1, no. 2, formula (10)) and you = 0 since sequence (1) is exact at E. Therefore 12 o 17 = 0, that is Im(fi) C Ker(fi). On the other hand, if w eKcr(11), then w o u = 0 and hence Ker(w) D Im(u). But as sequence (1) is exact at E, Im(u) = Ker(v) and hence Ker(w) D Ker(v); as v is surjective, it follows from §l, no. 3, Remark that there exists a w’ eHom(E", F) such that

227

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LINEAR ALGEBRA

w = w' a v = 17(w’). Therefore Ker(fi) C Im(5), which completes the proof that sequence (2) is exact. Conversely, supp05e that (2) is exact for every A-module F. As 51 o 17 = Hom(v 0 14,11.) = 0, w o v o u = Oforeveryhomomorphisms” —> F. Taking F = E" and w = 13., it is seen first that you = 0 and hence

u(E’) C Ker(v). Now take F = Coker(u) and let ¢:E—> F = E/u(E') be the canonical mapping. Then 12(4)) = o u = 0 by definition and hence there exists a i]; eHom(E”, F) such that F to be sutjective (resp. bijective, resp. zero), it is necessary and suficient that, for every A-module G, the mapping Hom(u, la) :Hom(F, G) —> Hom(E, G) be injective (resp. biiective, resp. zero).

It suffices to apply Theorem 1 to the case where E” = {0} (resp. E' = {0} resp. E" = Band 2/ =1“). Note that starting from an exact sequence

0 —> E’ —"> E —L> E' ——> o the corresponding sequence

0 ——> Hom(E”, F) L Hom(E, F) —i-> Hom(E’, F) ——> o is not necessarily exact, in other words, the homomorphism :7 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 qf linear mapping:

(3)

0—->E'—"——>E—’>E"——>O

splits (in other words, if u(E’) is a directfactor of E) the sequence

(4)

o —> Hom(E”, F) —‘> Hom(E, F) —"> Hom(E’, F) ——> o

is exact and splits. Conversely, ififiir every A-moa'ule 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 (§ 1, no. 9, Proposition 15); if 12’ = Hom(u', 1F) :Hom(E’, F) —> Hom(E, F),

228

PROPERTIES or HomA(E, F) RELATIVE TO EXACT SEQUENCES

§ 2.1

the fact that u’ o u is the identity implies that 11 o 12’ is the identity (§ 1, no. 2, formula (10)) and hence the first assertion follows from § 1, no. 9, Proposition 15. Conversely, suppose sequence (4-) is exact for F = E'. Then there exists an elementfe Hom(E, E’) such thatfo u = 1,3,, and the conclusion follows from § 1, no. 9, Proposition 15. Note that the first assertion of Proposition 1 can also be considered as a

special case of § 1, no. 6, Corollary 1 to Proposition 6, canonically identifying Hom(E’, F) 6 Hom(E”, F) with Hom(E’ Q E”, F) by means of

the

Z-linear

mapping

Hom(p’, 11.) + Hom(p”, 1].),

where

p’: E’ (B E” —> E’ and p”:E’ {-B E” —> E” are the canonical projections. THEOREM 2. Let A be a ring, F’, F, F' three A—modules and uzF’ —-> F, s —> F”

two homomorphism. For the sequence

(5)

o—)F'—1>F—“—>F'

to be exact, it is necessary and Witt that, for every A-module E, the sequence

(6)

o ——> Hom(E, F') —;—> Hom(E, F) —;-—> Hom(E,F”)

(where 12 = Hom(lE, u), 5 = Hom(ln, 0)) be exact. Suppose that the sequence (5) is exact. Note first that 5012 = Hom(lE,oou) = 0 (II, § 1, no. 2, formula (10)) since I) o u = 0. The image of Hom(E, F’) under 12 is therefore contained in the kernel N of 17; let f be the homomorphism of the Z-module Hom(E, F’) into N whose graph is equal to that of 17; it is necessary to prove that f is bijective and hence to define a mapping gzN -—> Hom(E, F’) such that f o g and g of are the identity mappings. For this, let w be an element of N, that is a linear mapping w:E —> F such that vow = 0. The latter relation is equivalent to w(E) C Ker(v) = u(F’) by hypothesis, hence, since u is injectivc, there exists one and only one linear mapping w’ : E —> F’ such that w = u o w’ and we take g(w) = w’; it is irmnediately verified that g satisfies the desired conditions. Conversely, suppose that sequence (6) is exact for every A-module E. As Hom(lE,oou) = 17012 = 0, then oouow = 0 for every homomorphism w:E—>F’. Taking E = F’ and w =13», it is seen first that vou = 0 and hence u(F’) C Ker(o). Now we take E = Ker(o) and let ozE—>F be the canonical injection. Then 17(4)) = v o (l) = 0 by definition and hence there

exists c]; e Hom(E, F’) such that d) = 17(4)) = u o 4», which obviously implies Ker(v) C u(F’) and completes the proof of the exactness of (5) at F. Finally, if 0 is the identity mapping of Ker u, then 12(6) = 0, hence 6 = O and Ker u = {O}, which proves the exactness of (5) at F’.

229

LINEAR ALGEBRA

II

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 (MA) of submodules of F

Hom(E, Q M,) = Q Hom(E, M,) for if ueHom(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 14 maps E into 0 M,"

COROLLARY. For an A-linear mapping u : E —> F to be injective, it is neeessmy and51505-

eient that,for every A-module G, the mapping Hom(l 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

0 ——> F’ —“—> F —-"-> F” —> o the corresponding sequence

0 —; Hom(E, F’) J» Hom(E, F) —”> Hom(E, F') —> o is not necessarily exact, in other words 17 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 1) o w, where w is a linear mapping of E into F. However:

PROPOSITION 2. If the exact sequence

(7)

_

0——>F'-—"——>F-—'—>F'—~>O

splits (in other words, if u(F’) is a. direetfaetor of F), the sequence

(8)

o ——> Hom(E, F’) —“—> Hom(E, F) l) Hom(E, F") ——> o

is exact and splits. Conversely, if sequence (8) is exact flJr every A-module E, the exact

sequence (7) splits. The first assertion follows from the fact that

Hom(E, F’) GE Hom(E, F”)

is canonically identified with Hom(E, F’ 6) F”) by means of the Z-linear

mapping Hom(lE,j') + Hom(lE, j”), j’:F' —> F’ {-9 F” and j”:F" —> F’ G F” being the canonical injections (§ 1, no. 6, Corollary 1 to Proposition 6). Con-

versely, ifsequence (8) is exact for E = F”, there is an element g e Hom(F", F) such that v o g = 1w and the conclusion follows from § 1, no. 9, Proposition 15.

230

PROJECTIVE MODULES

§ 2.2

Remark (2). The results of this no. are valid without modification for all commutative groups with operators. 2. PROJECTIVE MODULES

DEFINITION 1. An A-module P is called projective if, for every exact sequence F’ —> F —-> F" of A-linear mapping, the sequence

Hom(P, F’) ——> Hom(P, F) --> Hom(P, F”) is exact.

PROPOSITION 3. For an A-module P, the direct sum of a family of submodules (M. , to be pny'ective, it is necessary and sufiicient that each of the MI be projective. For every A-module homomorphism u : E -—> F,

Hom( 11,, u) :Hom(P, E) —-> Hom(P, F)

is identified with H Hom(lm, u) (§ 1, no. 6, Corollary 1 to Proposition 6) ; the conclusion thus follows from Definition 1 and § 1, no. 5, Proposition 5 (ii). COROLLARY. Everyfree A-module is projective. It suffices by Proposition 3 to show that A, is projective, which follows immediately from the commutativity of diagram (50) of § 1, no. 14. PROPOSITION 4. Let P be an A-module. Thefollowing properties are equivalent:

(a) P is projective. (b) For every exact sequence 0 —> F' —> F —> F" —> O of A-linear mappings, the sequence 0 —> Hom(P, F’) —> Hom(P, F) —> Hom(P, F”) —-> 0 is exact. (c) For every suriective A-module homomorphism u: E —> E' and every homomorphism f:P—>E", there exists a homomorphism g:P—>E such thatf= uog (it is said thatf can be “lifted” to a homomorphism of P into E). ((1) Every exact sequence 0 —> E' —> E —"> P —> 0 ofA-linear mappings splits (and therefore P is isomorphic to a directfactor of E). '

(e) P is isomorphic to a directfactor of a free A-module.

It is trivial that (a) implies (b). To see that (b) implies (c), it suflices to apply (b) to the exact sequence 0 —> E’ —-> E —"> E” —> 0, where E' = Ker(u), since (c) expresses the fact that Hom(lp, u) :Hom(P, E) —> Hom(P, E”)

is surjective. To see that (c) implies (d), it suffices to apply (c) to the suijective

23]

II

LINEAR ALGEBRA

homomorphism s —> P and the homomorphism lP —> P; the existence of a homomorphism g: P -—-> E such that IP = v o g implies that the sequence

0——>E’—->E—'—>P——>0 splits (§ 1, no. 9, Proposition 15). As for every A-module M there exist a free A-modulc L and an exact sequence 0 —-> R -—> L —> M -> 0 (§ 1, no. 11, Proposition 20), clearly (d) implies (e). Finally (6) implies (a) by virtue ofProposition 3 and its Corollary. COROLLARY 1. For an A-module to be projective andfinitely generated, it is necessary and suficient that it he a directfactor of aflee A-moa'ule with afinite basis. The condition is obviously sufficient; conversely, a finitely generated projective module E is isomorphic to a quotient of a free module F with a finite basis (§ 1, no. ll) and E is isomorphic to a direct factor of F by virtue of Proposition 4- (d). COROLLARY 2. Let C be a commutative ring and E, F two finitely generated projective C-modules; then Home (E, F) is afinitely generated projective C-module. It may be assumed that there are two finitely generated free C-modulcs M, N such that M = E (-9 E’, N = F G) F’; it follows from§ 1, no. 6, Corollary

1 to Proposition 6 that Homc(M, N) is finitely generated and free and on the

other hand that Homc(M, N) is isomorphic to

Home. F) ea Homcw'. F) ea Homes. F') 69 Hom(E'. F'), whence the corollary. 3. LINEAR FORMS; DUAL OF A MODULE

Let E be a left A-module. As A is an (A, A)-bimodulc, HomA(E, A,) has a canonical right A-module structure (§ 1, no. 14-).

DEFINITION 2. For every right A-module E, the right A-module HomA(E, A.) is called the dual module (f E (or simply the dual‘l‘ of E) and its elements are called the linearforms on E. If E is a right A-module, the set HomA(E, Ad) with its canonical left Amodule structure is likewise called the dual of E and its elements are called linearforms on E. 1' In Topological Vector Spaces, IV, we shall define, for vector spaces with a topalogy, a notion of “dual space” which will depend on this topology and will be distinct from the one defined here. The reader must beware of incautiously applying

to the “topologica ” dual space the properties of the “algebraic" dual which are established in this paragraph.

232

LINEAR FORMS; DUAL or A MODULE

§ 2.3

In this chapter, E* will be used to denote the dual of a (left or right) Amodule E.

Example. ‘On the vector space (with respect to the field R) of continuous real-valued functions on an interval (a, b] of R, the mapping a: I—> I: x(t)d¢ is a linear form...

Let E be a left A-module and E* its dual; for every ordered pair ofelements at e E, x‘" e E*, the element x* (x) of A is denoted by (x, 21*). Then the relations (9)

(10)

=+

(”’3‘+!I*>=+

(11)

(ex, 1*) = a(x, x‘)

(12)

(x,x*oc) = (x, x*>0t

hold for x, y in E, x*, 31* in E"I and a e A. The mapping (x, x‘") r—> (x, x*) of E x E* into A is called the canonical bilinearform on E x E* (the notion of bilinear form will be defined generally in IX, § 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 (ll) and (12) are written as

(16", x0!) = cc (ax‘, x) = atF 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‘l is the inverse image ‘u’1(M’). This follows immediately from (15). COROLLARY. The orthogonal of the image u(E) in F* is the kernel 'u‘1(0) of ‘u. The orthogonal of E in E* is 0.

If u:E —-> F is an isomorphism, l’uzF‘“ —> E"‘ is an isomorphism and if v:F —-> E is the inverse isomorphism of u, ‘o is the inverse isomorphism of 'u

(formulae (17) and (18)). "DEFINITION 6. Given an isomorphism u of an A-module E onto an A-module F, the transpose of the inverse isomorphism of u (equal to the inverse isomorphism of the transpose of u) is called the oontragredient isomorphism of u and denoted by 12'. The isomorphism a is thus characterized by the relation

(19)

(u(x), fi(y*)) = (x, x*) for x e E, x* e E“.

If v : F —> G is an isomorphism, the contragredient isomorphism of a o u is 15' o 11.

In particular, the mapping a +—> 12' is an isomorphism ofthe linear group GL (E) onto a subgroup of the linear group GL(E*).

235

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Let a: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 a semi-linear mapping (§ 1, no. 13) relative to a. Let a“ be the inverse isomorphism of o; for all y* e F*, the mapping x r—> "_1 of E into A is a linearform; if it is also denoted by ‘u(y*), a mapping ‘u:F* ——> E* is defined which is also called the transpose of the semilinear mapping a; it is thus characterized by the identity

(20)

("Mm") = (x, ‘u(y*)>°

for x e E, 31* e F‘. It is immediately verified that ‘u is a semi-linear mapping relative to 0'1. If 0 denotes the mapping u considered as an A-linear mapping of E into 5*(F) (§ 1, no. 13), we may write u = F, considered as a semi-linear mapping relative to c. It is immediate that ’u = ’v o ‘4) and (‘45 a“) is a di-isomorphism of F* onto (6* (F))* relative to the isomorphism o" 1; this relation allows us immediately to extend the properties of transposes of linear mappings to transposes of semi-linear mappings. 6. DUAL OF A QUOTlENT 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

E'—">E—'>E”——>0

(21)

an exact sequence of linear mappings. Then the sequence of transpose mappings

o—->E'*i>E*—‘“—>E'* is exact.

COROLLARY. Let M be a submodule of an A-module E and ¢:E —> E/M the canonical homomorphism. Then ‘4; is an isomorphism of the dual of E/M onto the submodule M' of E* orthogonal to M. Ifj:M —> E is the canonical injection. the kernel of ff is by definition the ‘ orthogonal of M in E“. Moreover, in the notation of the corollary, an injectioe homomorphism E*/M' —-> M' is Obtained from fj when passing to the quotient.

PROPOSITION 10. Let (Ema be a family of A-modules and for all I. e I let j.:E., —-> E = 91 E. be the canonical injection. Then the product mapping 5:" +-> (fj.(x*))

is an isomorphism ofthe dual E* ofE onto the product H E". 236

DUAL or A QUOTIENT MODULE. DUAL or A DIRECT sum. DUAL BASES

§ 2.6

This is a particular case of § 1, no. 6, Corollary 1 to Proposition 6, applied to the case where U F,_ = A,.

If, by means of the canonical injections j” the E. are identified with submodules of their direct sum E and if, by means of the product mapping x* +—> (fj.(x*)), E* is identified with 1:; ET, it can then be said that 1:; E?“ is the dual of £29: E” the canonical bilinear form being given by

(22)

«at or» = ; .

COROLLARY. Let M, N be two supplementary submodules in an A-module E and s -—> M, q —> N the corresponding projectors; then ‘p + ‘q:M* (—B N* —> E" is an isomorphism and ‘p (resp. ‘9) is an isomorphism of M* (resp. N*) onto the submoa'ule of E* orthogonal to N (resp. M). Moreover, if i:M —> E andj :N——> E are the canonical injections, ‘p o‘i and ‘qofj are the projectors E* —>‘p(M*), E* —> ‘q(N*) corresponding to the decomposition of E" as the direct sum of ‘p(M*) and ‘q(N*). [’01: = 1M: 4°j= 1N: 9°]. = q = 0: i°P +j°9 =1E: Whences by

transposition (no. 5, formulae (16), (17) and (18)), ‘i o ‘p = 1w: fjo ‘q = 1m, ff 0 ‘p = ‘i o ‘q = 0. ‘p o ‘i + ‘q o fj = In. and the proposition follows from § 1, ' no. 6, Corollary 2 to Proposition 6. Under the hypotheses of the Corollary, M“ (resp. N*) is often identified with the orthogonal ‘p(M*) (resp. ‘q(N*)) of N (resp. M) in E*, thus identifying every linear form u on M (resp. N) with the linear form on E attending u and which is zero on N (resp. M). When an A-module E admits a basis (e,),e.r, it has been seen that giving this

basis defines canonically an isomorphism u:A§T) -> E. By virtue of Proposition 10 and no. 3, Proposition 5, the dual of A?) is canonically identified with the product A3; consider the contragrcdient isomorphism i: A}? —> E*. If, for all t e T,fl is the element ofA3 all ofwhose projections are zero with the exception of that of index t, which is equal to l, and if we write e; = 1702), the elements e; of E" are, by (19) and (22), characterized by the relations (23)

*

_

(0‘, 8y) ""

0

for t’ aé t

l

for t, = t.

It amounts to the same to say that, for all x = 2; 5:8: EE, eflx) = 2,; also e?“ is called the coordinate farm of index t on E. It follows from (23) that (e?) is afree system in E“.

In particular, if T isfinite, the e?“ form a basis of E*, thef, then forming the canonical basis of Ag. Hence:

23.7

LINEAR ALGEBRA

II

PROPOSITION 11. The dual ofafree module with a basis ofn elements is afree 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. If E is afree module with afinite basis (e,), the basis (e?) of the dual E" of E defined by relations (23) is called the dual basis of (q). Relations (23) can also be written in the form (24')

(at: 4) = 8n'

where 8“. is the Kronecker symbol on T x T. Note that if T is finite and (e?) is the dual basis of (e,), then, for

x= ‘ZIEmeEnc" =2, «state-E", (25)

or, m = 2 as".

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-Inodule 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 isomorphic to a direct factor of A3, whence the corollary.

PROPOSITION 12. Let E be an A-module and (date-1. a generating system of E. The jbllowing conditions are equivalent: (a) E is a projective A-madule. (b) There exists a family (4‘),” of linear forms on E such that, for all x e E,

thefamily (a,.

There exists a sqectivc homomorphism u:L —> E, where L = A3”, such that if (e,), e .1. is the canonical basis of L then u(e,) = a. (§ 1, no. 11 , Proposition 17); for E to be projective, it is necessary and sufficient that there exist a linear

mapping v: E -—> L such that u o v = lE (no. 2, Proposition 4- and § 1, no. 9,

Proposition 15). If such a mapping exists and we write ‘v(e;") = of, then

(x, a?) = (x, ‘v(e;")> = e,isalinearmappingv: E—>Lsuch thatu ov =13. 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 lefl A-module (no. 3). For all x e E, it follows from no. 3, formulae (10) and (12), that the mapping x* I—> (x, 1*) is a linearform on the

right A-module E*, in other words an element of the bidual E**, which we shall denote by 1?; moreover, it follows immediately from (9) and (ll) (no. 3) that the mapping GE: 1: v—> 5 of E into E“ is linear; this mapping will be called eanonieal; in general, it is netither injective nor surjective, even when E is finitely generated (cf. Exercise 9(e) and §. 7, no. 5, Theorem 6). An A-module E is called reflexive ifthe canonical homomorphism an :E —> E" is biiective. Let F be a second left A-module; for every linear mapping q —-> F, the mapping ‘(‘u):E** —> P", which will also be denoted by “u, is linear and the diagram

E—"—>F (27)

:31

1:.-

E** —-—> F** “u

is commutative, as follows immediately from the definitions and formula (15) giving the transpose of a linear mapping. PROPOSITION 13. If E is afree module (resp. afree module with a finite basis), the canonieal mapping eEzE —> E" is injective (reap. bijective). Let (ethyl. be a basis of E and let (e?) be the family of corresponding co-

ordinate forms; by definition, if x e E is such that J? = 0, then (x, 4") = 0 for all t e T, in other words all the coordinates ofx are zero, hence x = 0. Suppose , flirther that T is finite; since {e}, e3) = 8“,, (it) is the dual basis of (e?) in E" and, as on transforms a basis of E into a basis of E", on is bijective (§ 1,

no. 11, Corollary 3 to Proposition 17). We have moreover proved: COROLLARY 1. Let E be afree A-module with afinite basis; for every basis (4) ofE, (en(e,)) is the dual basis of the basis (er) Qf E* dual to (e,). In this case it is said that (a) and (e?) are two dual bases of one another.

239

II

LINEAR ALGEBRA

COROLLARY 2. If E is a free A-module with a finite basis, every finite basis of E"l isthedualbasisqfabasisq. It suffices to consider in E“ the dual basis ofthe given basis and canonically to identify E with E**. COROLLARY 3. Let E, F be two A-modules each with a finite basis, E (resp. F) being canonically identified with its bidual E** (resp. F"). Then, for «my linear mapping u:E —> F, "u = a. This follows immediately from the commutativity of diagram (27). COROLLARY 4. If P is a projective module (rcsp. afinitely generatedprojective module) the canonical mapping cP:P —-> P“ is infective (resp. bijective). We shall use the following lemma: Lemma 1. Let M, N be two supplanentary submodules in an A-module E and i: M —-> E, j:N —> E the comical injections. Then the diagram

MeBN 131‘: M**(—BN** la. + n,

I'+J' 1

(28)

E —————-—> E** ‘5

is commutative. By definition, for x e M, y e N, 2* e E*,

(01:00:) +j(y)), 1*) = 606) +J'(y), 2*) = (1'04), Z*> + (6140‘), ‘i(2*)> + (My), 5(Z*)> = F. be a linear mapping. Every system of linear equations

(29)

v.06) = y. (. 6 I)

where the y, e F, are given, is equivalent to a single linear equation u(x) = y,

where u is the mapping x |—> (u.(x)) of E into F = 1:! F, and y = (3],). The system (29) is called homogeneous if y. = 0 for all . e I. (2) Suppose that E admits a basis (aflub; ifwe set u(a,() = b,‘ for all 7. e L, to say that 1: =72; aka; satisfies the equation u(x) = go is equivalent to saying that the family (of finite support) (5,),‘1, of elements of A satisfies the relation £2,152}; = yo-

(30)

Conversely, looking for families (55),,“ 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,_) = b,, for all A e L (§ 1, no. 11, Corollary 3 to Proposition 17). (3) A linear equation u(x) = yo is called scalar when F = A, and therefore a is a linear form on E and yo a scalar. If E admits a basis (afikfl, it follows

from Remark (2) that such an equation may also be written as

(31)

Lana. = y. e A

where the family of scalars (ax) is arbitrary and where it is understood that

the family (Er) must have finite support. In general, by the solution (in A) of a system of scalar linear equations

(32)

2 Esau = 7h

7.614

(a E I)

where on,u e A and 11, e A, is understood a family (mm, of elements of A of

finite support and satisfying (32) ; the a“ are called the caeficieats of the system of equations and the m the right hand sides. The solution of such a system is

equivalent to that of the equation u(x) = y, where y = (1].) and u:A‘,L) —> A: is the linear mapping (in) H (AZLEWM)

(4-) A linear system (32) is also called a system of left scalar linear equation: when it is necessary to avoid confusion. A system of equations (33)

242

LZLMMEA = 7h

(I E I)

mnson manner or Two MODULES

§ 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 EA, 1;. and MM as belonging to the opposite ring A0 to A.

§3. TENSOR PRODUCTS 1. TENSOR PRODUCT OF TWO MODULES

Let G1, G2 be two Z-modules; a mapping u of the set G = G1 x G2 into a Z-module is called biadditive (or Z-bilinear) if u(x1,x2) is “additive with respect to x1 and with respect to x2”; to be precise, this means that, for x1, 3/1 in G1: x3, 1/2 in G2:

u(x1 + 11:: x2) = "(11: J‘2) + “(.111: x2) “(x1: x3 + 1/2) = "(‘1’ x3) + “(xnyal-

Note that this implies in particular that u(0, x2) = u(x1,0) = 0 for all x1 E G1, x2 E G:-

Let A be a ring, E a right A-module and F a left A-module. We are going to consider the universal mapping problem (Set Theory, IV, §3, no. 1) where 2 is the species ofZ-module structure (the morphisms then being Z-linear mappings, in other words, additive group homomorphisms) and the zit-mappings are the mappings f of E x F into a Z-module G which are Z—bilz'near and further satisfy, foralle,yeF and 16A

(1)

f(x1, 1/) =f(x, 1y)-

We show that this problem admits a solution. For this we consider the Z-module C = 2“: "F’ of formal linear combinations of the elements of E x F with coefficients in Z (§ 1, no. 11), a basis ofwhich can be considered to consist of the ordered pairs (1:, y), where x e E and y e F. Let D be the sub-Z-module of C generated by the elements of one of the following types: (x1 + 162,11) _ (xlsy) _ (12d!)

(2)

(3,111 + .112) " (3:311) "" (x2312) (3951/) _' (x, All)

where x, x1, x, are in E, y, y1, y2 are in F and A is in A.

DEFINITION l. The tensor product of the right A-module E and the left A-module F, denoted by E § F or E ®A F (or simply E ® F if no confusion is to be feared)

is the quotient Z-madule C/D (the quotient of the Z-module C of formal linear 243

II

LINEAR ALGEBRA

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) of C = Z‘EXF’ is denoted by x ® y and called the tensor product ofx and y. The mapping (x, y) n—>x ®y of E x F into E ®A F is called canonical. It is a Z-bilinear mapping which satisfies conditions (I). We show that the tensor product E (8,, F and the above canonical mapping form a solution of the universal mapping problem posed earlier. To be preelse: PROPOSITION l. (a) Let g be a Z-linear mapping of E ®A F into a Z-module G. The mapping (1:, y) I»f(x, y) = g(x ®y) of E x F into G is Z-bilinear and satisfies conditions (1).

(b) Conversely, letf be a Z—bilinear mapping of E x F into a Z—module G satisfying conditions (1). Then there exists one and only one Z-linear mapping g of E ®AF into Gsuch thatf(x,y) = g(x ®y)fore,yeF.

If H a Z-bilinear mapping satfifi/ing 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 satisfl/ing (l)

244

TENSOR PRODUCT or TWO LINEAR MAPPINGS

§ 3.2

there exists a Z—linear mapping g:H—> G such that f = g o h. Then, if 1!) denotes the canonical mapping of E x F into E ®A F, there exists one and only one isomorphism GofE ®AFontoHsuehthath = 004).

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, no. 1). COROLLARY 2. Let E0 (resp. F°) denote the module E (resp. F) considered as a left (resp. right) module over the opposite ring A°; then there exists one and only one Zrnodule isomorphism a:E ®A F —>F° ®Ao E° such that o-(x ® 1]) = y (8 x for

x e E and y e F (“commutativity” of tensor products). By definition of the Ao-module structures on E0 and F9, the mapping (x, y) n—>y (8) x of E x F into F° ®Ao E0 is Z-bilinear and satisfies conditions (1), whence the existence and uniqueness of the Z—linear mapping 0-. Similarly a Z-linear mapping 1:F° ®A° E0 —>E ®A F is defined such that 1,-(y (8) x) = .7: (2) y and clearly a and r 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 By = 0 for all x e E and y e F; therefore, in E ®z F,

x®y = 3(x®y) - 2(x®y) = x® (3y) — (2x) SW = 0 for all x and 3/ (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 -> E’ and s -—> F’ two A-linear mappings. It is easily verified that the

mapping

(1‘, y) '-> "(’0 ® ”(y) of E x F into E’ (—9), F’ is Z-bilinear and satisfies conditions (1) of no. I. By Proposition 1 of no. 1 there thus exists one and only one Z-linear mapping w: E ®AF —> E’ ®A F’ such that

(3)

w(x SW) = 14x) ® v(y)

for x e E, y e F. This mapping is denoted by u ® 1) (when no confusion can arise) and is called the tensor product of the linear mappings u and o. It follows immediately from (3) that (u, v) »—-> n ® a is a Z—bilinear mapping called canonical HomA(E, E’) x HomA(F, F’) —> Homz(E ®A F, E’ ®A F’).

24-5

LINEAR ALGEBRA

II

There corresponds to it by Proposition 1 of no. 1 a Z-linear mapping called canonical HomA(E, E’) ®z HomA(F, F’) —> Homz(E (8., F, E’ ®A F’) (4)

which associates with every element n (8 v (y‘the tensorproduct the linear mapping u ® 02E (2),; F —> E’ ®A F’. Note that the canonical mapping (4) is not necessarily injectioe nor smjectioe. The notation u ® I) can therefore lead to confusion 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’ --> E”,

v’:F’ —> F” A-linear mappings; it follows from (3) that

(5)

0"”) ® (WW) = (u'®v') ° (“‘8’”)-

3. CHANGE O]? RING

PROPOSITION 2. Let A, B be two rings, p:B—>A a ring homomorphism and E (resp. F) a right (resp. left) A-module. Then there exists one and only one Z-linear

mapping ‘l’:9*(E) ®n P*(F) '—> E ®A F such that, for all e and yeF, the image under 4) of the element x®y of (6)

9*(E) ®3 p,,(F) is the element x (8) y of E ®A F; this Z-linear mapping is surfaclive. We consider the mapping (x, y) as: ® y of 9*(E) >< 9*(F) into E ®A F; it is Z-bilinear and, for all fleB, by definition (xp(B)) ® y = x ® (p((5)y), hence conditions (1) of no. 1 hold, whence the existence and uniqueness of 42 (no. 1, Proposition 1). The latter assertion follows from the fact that the elements at ® y generate the Z-module E ®A F.

The mapping (6) is called canonical.

COROLLARY. Let 8 be a two-sided ideal of A such that S is contained in the annihilator of E and in the annihilator of F, so that E (resp. F) has a canonical left (resp. right) (A/8)-module stmcture ( § 1, no. 12). Then the canonical homomorphism (6) ¢=E ®AF —> E @A/SF

corresponding to the canonical homomorphism pzA —-> AIS is the identity. For all EeA/S, all e and allyeF, x3 = xa (resp. a'zy = any) for all at such that 9(a) = E. If C = Z‘EXF’, the submodule of C generated by the elements (xot,y) — (x, any) is then equal to the submodule generated by the elements (xE,y) — (x, Ry).

With the hypotheses and notation of Proposition 2, let E’ be a right Bmodule, F’ a left B-module and consider two semi-linear mappings q’ -> E, 24-6

OPERATORS ON A 'I'ENSOR PRODUCT; 'I'ENSOR PRODUCTS AS MULTIMODULES

§ 3.4

v : F’ ——> F relative to the homomorphism s ~> A; u (resp. u) can be considered as a B-linear mapping E’ ——> 9*(E) (resp. F’ ——> p*(t")), whence a Z-linear mappingw:E' (3)3 F’ » 9*(E) (3)3 9*(F) such that w(x’ ®y') = u(x') ® v(y') for x’ e E', y' e F’; composing the canonical mapping (6) with this mapping,

a Z-linear mapping w’:E' ‘83 F' —> E ®A F is hence obtained such that w’(x’ ®y’) = u(x') ® 0(y') for x’ eE’, y' eF’; this is the mapping which will normally be denoted by u (8) v if no confusion can arise. Clearly (u, v) :—> u 8) v is a Z-bilinear mapping

HomB(E’, 9*(E)) x HomB(F’, 9*(F)) —> Homz(E’ 8),, F’, E ®A F). Moreover, ifC is a third ring, 0': C —> B a homomorphism, E” a right C-module, F" a left C-module, u’:E"—> E’ and v’:F' ——> F’ semi-linear mappings relative to c, then

(uou') ® (oov’) = (u®v)o (u'®v’). 4. OPERATORS ON A TENSOR PRODUCT; TENSOR PRODUCTS AS MULTIMODULES

With the hypotheses and notation of no. I, for every endomorphism u (resp. v) of the A-module E (resp. F), u ® 11, (resp. IE (8 v) is an endomorphism of the Z-module E ®A F; it follows immediately from (5) (no. 2) that the mapping u»—>u ® 11:- (resp. III—>11a ® 1)) is a ringlwmomorphism

EndA(E) —> Endz(E ®A F) (resp. EndA(F) —> Endz(E ®A F)); moreover,

(7)

(u®1r)°(1m®v)=(lm®v)°(u®1p)=u®v

and therefore (§ 1, no. 14) E (83‘, 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 ((B2);A, (C;))-multimodule structure

and on F a (A, (B7,); (C;))-multimodule structure (§ 1, no. 14); it amounts to the same to say that ring homomorphisms B; —> EndA(E), C;° —> EndA(E) are given with pairwise permutable images, and ring homomorphisms ,{—>- EndA(F), C;° —>EndA(F) with pairwise permutable images. If these homomorphisms are composed respectively with the canonical homomorphisms EndA(E) —> Endz(E ®A F) and EndA(F) —> Endz(E ®A F) defined above, it is seen (taking account of (7)) that ring homomorphisms

B; —> Endz(E @A F), B; —> Endz(E (g), F),

ego —> Endz(E op, F) 0,:0 —> Endz(E @, F)

are defined with pairwise Immutable images; in other words, there has been defined on E ®A F a ((B;), (Bi); (0;), (C;))-multimodule structure; it is this

multimodule which is also called the tensor product (relative to A) qf the 24-7

II

LINEAR ALGEBRA

((B1) ; A, (C;))-multimodule E and the (A, (B7,); (C;))-multimoa'ule F. This multimodule is the solution of a universal mapping problem analogous to that considered in no. 1; to be precise: PROPOSITION 3. Let G be a ((B1), (BZ); (Cj), (CZ))-multimodule. (a) Let g be a linear mapping of the multimodule E ®A F into G. The mapping f: (x, y) I—>g(x ®y) of E x F into G is Z-bilinear and satisfies relations (1) Qf no. 1 and the conditions

(8)

{ f(u1x,y) = uz’f(x,y),

f(“5.11) =f(x,y)V5

f(x, sly) = ulflmy),

f(x, M) =f(x,y)Vl

fir all x e E, y e F, u; e B,’, v; e C;, u}; e BZ, v,’; e CZ, i, j, h, k arbitrary.

(b) Conversely, letf be a Z-bilinear mapping qf E x F into G satigfi/ing conditions (1) (no. 1) and (8). Then there exists one and only one linear mapping g qf the multimodule E ®A F into the multimodule G such that f(x, y) _= g(x ® y) fir x E E, y e F.

Assertion (a) follows immediately from the definition of the multimodule structure on E (8,, F, since for example (x (8) y)v; = (xvj) ® 3/. 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 (8 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 (8) y generate the Z-module E ®A F, it suflices to verify the relations g(p.’(x (X) y)) = u{g(x ® 1/) and their analogues; but this follows immediately from the formula g(x ® 3/) = f(x, y) and relations (8).

SCHOLIUM. An element of E (8,, F may in general be written in several ways in the form Z (x, (8) 3],), where x, e E and y, e F; but to define a linear mapping g of the multimodule E ®A F into a multimodule G, there is no need to

verify that, HZ (x. a y.) = Z (x; o a), then 2m 69 y.) = 2w; on); it suffices to be given g(x ® y) for x e E and y e F and to verify that (x, y) n—> g(x ® y) is Z-bilinear and satisfies conditions (1) (no. 1) and (8). Let E’ be a ((B5), A, (Gm-multimodule, F’ an (A, (Bi); (Gm-multimodule and q -—> E’, s -—> F’ linear mappings of multimodules; it follows immediately from the definitions (no. 2) that u (8) v is a linear mapping of the multimodule E (8,, F into the multimodule E’ (8),, F'.

With E always denoting a right A-module, let ,Ad denote the ring A considered as an (A, A)-bimodule (§ 1, no. 14, Example 1); by the above, the tensor product E ®A (,Ad) has a canonical right A-moa’ule structure such that

(x (8) 70y. = x 63) (Au) for xe E, A EA, 9. e A. The mapping (x, A) 9x). of E X (8A,) into E is Z—bilinear and satisfies conditions (1) (no. 1) and (8) 24-8

TENSOR PRODUCT OF TWO MODULES OVER A COMMUTATIVE RING

§ 3.5

(where, in the latter, the Bi, C; and B; are absent and the family (0;) reduces to A); hence (Proposition 3), there exists an A—linear mapping g (called canonical) ofE ®A (,Ad) into E such that g(x ® A) = xA for x E E, A E A. PROPOSITION 4. If E is a right A-module, the mapping hzx r—->x ® 1 of E into E ®A (,Ad) is a right A-module isomorphism, whose inverse isomorphism g is such thatg(x® A) = fore, AeA. If g is the canonical mapping, g o h is the identity mapping 1E and h o g coincides with the identity mapping of E (8,, (,Ad) onto itself for elements of the form x (3 g, which generate the latter Z-module; hence the conclusion. We shall normally write E ®AA instead of E (8,, (,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 compatible with its right A-module structure, g and h are also isomorphisms for the B—module structures on E and E (8A A (and hence multimodule isomorphisms). Now let F be a left A-module; (,Ad) (8),, 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 ®A F onto F mapping A ® 1: to Ax, and its

inverse isomorphism is x I—> 1 ® x. In particular there exists a canonical isomorphism of the (A, A)-bimodule (,Ad) ®A (,Ad) onto ,A,, which maps A ® p. to An. 5. TENSOR PRODUCT OF TWO MODULES OVER A GOMMUTATIVE RING

Let C be a commutative ring; for every C-module E, the module structure on E is compatible with itseif(§ 1, no. 14-). IfE 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 ®c F, respectively such that 7(x @y) = (7x) ® 1/ and such that 7(x ® y) = x ® (yy) ; but as, by Definition 1 of no. 1, in this case (7x) (8) y = x 8) (yy), these two structures are the same. Henceforth when we speak of E ®c F as a C-moa'ule, we shall mean with the structure thus defined, unless otherwise stated. The canonical isomorphism. 61F ®cE—>E (80F

(no. 1, Corollary 2 to Proposition 1) is then a C—module isomorphism. It follows from this definition that, if (a) A61, (resp. (bu)uEM) is a generating system of the C-module E (resp. F), (a, 8) bu) is a generating system of the C-module E ®c F; in particular, if E and F are finitely generated C—modules, so is E ®c F.

For every C-module G, the Z-bilinear mappings f of E x F into. G for

which

(9)

firm) =f(x,ry) =Yf(x,y) fore,yeF,reC 24-9

II

LINEAR ALGEBRA

are then called C-bilinear and form a C-module denoted by $2(E, F; G);

Proposition 3 (no. 4) defines a canonical C-module isomorphism (cf. § 1, no. 14-, Remark 1).

(10)

2203, F; G) —> Homo(E @c F, G).

Let E’, F’ be two C-modules and u:E —> E’, ozF —> F’ two C-linear mappings; then (no. 4) a (X) o is a C—linear mapping of E (80 F into E’ ®c F’. Further, it is immediate that (u, o) n—> u ® I) is a C-bilinear mapping of Homc(E, E’) x HomC(F, F’) into Homc(E ®c F, E’ (8)0 F’); hence there

canonically corresponds to it a C-linear mapping, called canonical:

(ll)

Homc(E, E’) ®c Homc(F, F’) ——> Homc(E 8)., F, E’ (80 F’)

which associates with every element u (8) v of the tensor product

H0mc(E, E’) 69.: H0mc(F, F’) the linear mapping u ® a. Note that the canonical mapping (1 l) is not necessarily injective nor surjectioe (§ 4, Exercise 2).

Remarks. (1) Let A, B be two commutative rings, 9 :B —> A a ring homomorphism and E and F two A-modules; then the canonical mapping (6) of no. 3 is a B-linear mapping

(12)

9*(E) ® 9*(F) -> 9*(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 G into A such that 9(0) is contained in the centre of A (cf. III, § 1, no. 3). We may then consider the C-modules 9*(E) and 9*(F) and the hypothesis on 9 implies that these C—module structures are compatible respectively with the A-module structures on E and F (§ 1, no. 14-). The tensor product E ®A F is thus (by virtue of no. 4) given two C—module structures such that 7(x (8 y) = (xp(y)) (8) y and

7(x 6M) = x 69 (WW) respectively for y e C, x e E, 31 6F 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 q —>E’, o:F—> F’ are two A-linear mappings, then u (8 (KB ®AF—->E’ ®A F’ is C-linear for the G-module structures just defined; the mapping (u, v) »—> u (8) v:

HomA(E, E’) x HomA(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

PROPERTIES OF E ®A F RELATIVE TO EXACT SEQUENCES

§ 3.6

defined in§ 1, no. 14, Remark 1), whence we also derive a C—linear mapping, called canonical

(13)

HomA(E, E’) @o HomA(F, F’) —> Homc(E @A F, E’ @A F’).

(3) Let A be an integral domain and K its field of fractions. If E and F are two vector K-spaces, the canonical mapping

(Em) ®A (Fm) -> E 691: F (no. 3 and § 1, no. 13) is bijective. It suflices (no. 4) to prove that iff is an Abilinear mapping of E x F into a vector K-space G, f is also K-bilinear. Now, for all at 75 0 in A. whence

af(a'1x,y) =f(x:!/) = “f0”: 1—1.1!) fwd-1753.11) =f(x’ “-1.1” = “—1.11%!”

since G is a vector K-space. 6. PROPERTIES OF E ®A F RELATIVE TO EXACT SEQUENGES

PROPOSITION 5. Let E, E’, E” be right A-modules, F a 1m A-module and

(14)

E’——u—->E—'—>E”——>0

an exact sequence qf linear mappings. Writing 12 = n ® 1F, 17 = 0 8) IF, the sequence

(15)

E’ ®AF—E>E®AF—a—>E’ ®AF—>0

qfZ-homomorphisms is exact. By virtue of no. 2, formula (5), if 0 fi = (1)0 u) (81F = 0; the image H = 22(E’ ® F) is contained in the kernel L = Ker(17); by passing to the

quotient, we therefore derive from 5 a Z—linear mapping f of the cokernel =(E ® F) [H Of'u into E” (2) F; it must be proved that f 15 biiective and it will—hence suffice to define a Z-linear mapping g: E" ® F—> M such that of and”f o g are the identity mappings. Let x”eE”, yeF; by hypothesis there exists e such that v(x) = x'. We show that, if x1, x2 are two elements of E such that 0(x1) = v(x2) = x”

and ¢:E ® F —> M is the canonical mapping, then (Mn, ® y) = (Mac2 ® 3/). It suffices to prove that if 0(x) = 0 then 42(x (81/) = 0, which follows from the fact that x = u(x') with x' E E’, whence x ®y = u(x’) ®y = 12(x' ®y) e H. If (x', y) is mapped to the unique value of d>(x ® 3/) for all x E E such that 11(x) = x", a mapping is defined of E” x F into M; this mapping is Z—bilinear and satisfies conditions (1) (no. 1), since 0(xA) = x”A and (x1) (8) y = x (8 (kg) for x e E; hence there is a Z-linear mapping g of E” ® F into M such that

g(x" (8 y) = (Mac (81/) for y 6F, x e E and x' = v(x). This definition further proves that fog coincides with the identity mapping for the elements of 251

LINEAR ALGEBRA

II

E” 8) 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(41(x ® 31)) = 0(x) ® 3/ by definition, hence g(f(d>(x ® y))) = (x (8) y) and, as the elements of the form 42(x ® y)

generate M, g of is the identity mapping of M. COROLLARY. Let F, F’, F” be left A-modules, E a right A-module and

(16)

F’—’—>F-;>F'-——>O

an exact sequence qf linear mappings. Writing 5' = IE (8) s, i = In ® t, the sequence

q-homomorphisms

E®AF’—1>E®AF——f—>E®AF'——>0

(17)

is exact. When E (resp. F) is considered as a left (resp. right) Ao-module, F ®Ao E is identified with E ®A F and there are analogous identifications for F’ ®Ao E

and F” ®A° 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' —> E the canonical injection, the canonical mapping j® ls’®F—>E®F is not necessarily infective. In other words, for an exact sequence

0—-—>E'—"->E—'—>E'——>0

(13)

it cannot in general be concluded that the sequence

0——>E'®F—">E®F—">E'®F—->0

(19) is exact.

Take for example A = Z, E = Z, E’ = 22, F = 2/22. As E’ is isomorphic to E, E’ (8) F is isomorphic to E ® F which is itself isomorphic to F (no. 4, Proposition 4). But for all x’ = 2xEE’ (where e) and all

ye F, j(x’) ®y = (2x) ®y = 1: ® (2y) = 0, since 23/ = 0, and the canonical image of E’ 8) F in E (8 F reduces to 0. In other words, care must be taken to distinguish, for a submodule E’ of E and an element are E’, between the element :1: ® g “calculated in E’ ® F” and the element 7: ® g “calculated in E ® F” (in other words, the elementj(x) ® y).

We shall study later, under the name of flat modules, the modules F such that the sequence (19) is exact for every exact sequence (18) (Commutative Algebra, I, § 2).

PROPOSITION 6. Given two exact sequences (14) and (16), the homomorphism 0 ® t:E ®A F —> E” ®A F” is surjeetive and its kernel is equal to

Im(u ® 1,) + Im(1,, ® 5) 252

PROPERTIES OF E (8,, F RELATIVE TO EXACT SEQUENGES

§ 3.6

Now 0 (8) t = (mg) 1,.) o (1,, ® t) (no. 2, formula (5)) and 0 ® t is therefore suxjective, being the composition of two surjective homomorphisms by virtue of Proposition 5 and its Corollary. On the other hand, for z e E (8 F to be in the kernel of v ® t, it is necessary and sufficient that (lg, (3 t) (z) belong to the kernel of 0 (3) IF”, that is, by virtue of (15) to the image of

u® 11...: E’ ®F'—>E 8F”. But as the homomorphism t: F —> F” is suijective, so is 1E' ®ti E, ®F'—>E’®F'

by the Corollary to Proposition 5, hence the condition on 2 reduces to the existence of an a e E’ ® F such that

Us ® t)(Z) = ('1 t30(11)Let b = z -- (u ®1F.)(a); then (lE ®t)(b) = 0, and by virtue of (17), b belongs to the image of IE (8 s, which proves the proposition.

In other words: COROLLARY 1. Let E’ be a submaa'ule of a right A-moa'ule E, F' a submoa'ule of a left A-module F and Im(E’ ®A F) and Im(E ®A F') the sub-Z-modules ofE (8,, F, the respective images of the canonical mafipings E’ ®A F —> E (8)A F, E ®A F, —> E ®A F.

Then there is a canonical Z-module isomorphism

(20)

r: :(E/E’) ®A (F/F’) —> (E ®A F)/(Im(E’ ®A F) + Im(E ®A F’)) suchthat,forEEE/E’,neF,n-(E®n)istheclassofallelementsx®yeE ®AF suchthatxeiandyer]. Note that when E is a ((B;) ; A, (C;))-multimodule, F a (A, (BZ); (0%))multimodule and E' and F’ submultimodules of E and F respectively, the isomorphism (20) is an isomorphism for the ((33), (137,); (C;), (C;))-multimodule structures of the two sides (no. 3). COROLLARY 2. Let a be a right ideal of A, F a lefl A-module and aF the sub-Zmodule of F generated by the elements of the form M, where A e a and x e F. Then there is a canonical Z-module isomorphism

(21)

1:: (Ala) (2), F —> F/aF

such that, for all it eA/a and all x e F, 11(1 ® x) is the class mod. aF of Ne, 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

253

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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) (8“,, F are canonically isomorphic.

F and E ®A F are annihilated by a and hence have canonical (A/a)-module

structures (§ 1, no. 12) and if we write E’ = «B, then Im(E’ ®A F) = 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) ®Am F (no. 3, Corollary to Proposition 2). COROLLARY 4. Let a, b be two ideals in a commutative ring C; the C-module (Clo) ®c ((1/13) is then canonically isomorphic to C(a + b). 7. TENSOR PRODUCTS OF PRODUCTS AND DIRECT SUMS

Let (E0151, be a family of right A-modules, (F”MGM a family of left A-modules

and consider the product modules C = 1:1 Eh, D=1:1 F. The mapping ((xh), (yn)) v—> (xx (8 y“) of C x D into the product Z—module (MMSLXM

(EA. ®A Fu)

is Z-bilinear and obviously satisfies conditions (I) (no. 1). Thus there exists (no. 1, Proposition 1) a Z-linear mapping, called canonical

(22) halls.)®A(u1:,IJFu)—>MQXM(E. am.) such thatmx.) e (yo) = (xx m.)When C = RD, D = S“, R (resp. S) being a right (resp. left) A-module, the canonical mapping (22) associates with every tensor product u ® 0,

where u is a mapping of L into R and v a mapping of M into S, the mapping (7., y.) |—> 14(1) ® v(u) of L x M into R ®A S; even in this case the canonical mapping (22) is in general neither injective nor smjective (Exercise 3; cf. Corollary 3 to Proposition 7).

When the E; are ((Bfi) ; A, (Cm-multimodules and the F” (A, (B2) ; (0,1))multimodules, the homomorphism (22) is also a homomorphism for the ((B’), (Bi); (Cl), (Ck))-multimodule structures of the two sides. Consider now the submodule E = $91.13; (resp. “(a F“) of C (resp. D);

the canonical injections E ——> C, F —> D define canonically a Z-linear mapping E ®A F —> C ®A D which, composed with the mapping (22), gives a Z-

linear mapping g of E ®A F into E (EA (8“ F“) such that

sax») ® (m0) = (xi 8%); 254-

TENSOR PRODUCTS OF PRODUCTS AND DIRECT sums

§ 3.7

moreover, as the families (x1) and (ya) have finite support, so does (14',t ® 1/") and hence finally g is a canonical homomorphism

(E, @A F“), g:(£BL E.) @A ("C-BM F“) —> O» u)Gr) eLxM

(23)

which is a multimodule homomorphism under the same conditions as (22). PROPOSITION 7. The canonical mapping (23) is biiective. To prove this it suffices to define a Z-linear mapping h of the direct sum

G=

(43

(A..u)eLxM

(E; @A F") into E @A F such that g s h and h s g are the identity

mappings. But, to define a Z-linear mapping of G into E ®A F, it suffices (§ 1, no. 6, Proposition 6) to define a Z-linear mapping hs,‘ ®A F"—>E ®AF

for every ordered pair (1, pt), and we take hm = i,‘ ® j", where ih:E,.—>E and ju:Fu —> F are the canonical injections. Then clearly h o g coincides with

the identity mapping for the elements of the form (2 xx) (8 (211") which generate the Z-module E ®AF;similarly g o h coincides with the identity

mapping for the elements of the form 2 (xx (8 g“) which generate the Zmodule G, since for each ordered pair (1, p.) the products x1. ® ya

(”A 5 EM I’M 6 Fa)

generate the Z-module EA ®A F“. tnce the proposition. Let uA:E,_ —> E1, ou:FM —> F; be A-homomorphisms; clearly the diagram

(6;) Es) oi ( gr} (E.. or (Ft) (euh)®($vu)

$(I‘LQVH)

((19 E1) o. ( g2 (E; 8». Fa) is commutative. COROLLARY 1. If the left A-module F admits a basis (banal, every element of

E ®AF can be written uniquely in the form 2“: (xll ® bu), where x“ 6E and the family (xu) hasfinite support. The Z-module E ®A F is isomorphic to E“) considered as a Z-module.

The basis (bu) defines an isomorphism of F onto “(59“ Ab“, whence there is an isomorphism E ®A F ni (E ®A Ab") by virtue of Proposition 7; as

255

II

LINEAR ALGEBRA

E 1—» 55,, is an isomorphism of A, onto Ab", x 1—» x ® bu is an isomorphism of E onto E ®A (Abu) by virtue of Proposition 4- of no. 4, whence the corollary. If E is a ((3,); A, (C,))-multimodule, the canonical isomorphism E ®A F —> E“) is a ((B’); (C’))-multimodule isomorphism. In particular, if also E admits a basis (ahheh, every 2 6E ®A F may be written in one and only one way in the form (ahfim) ® bu, where the 5m belong to A (and form a family of finite” support), the mapping 2 v—> (5,490“ men x M is an isomorphism of E (8,, F onto A‘L " M) for the Z-module structures (and even the module structures over the centre of A). More particularly: COROLLARY 2. If E and F are twofree modules over a commutative ring 0 and (a,_) (resp. (11,,» is a basis of the C-module E (resp. F), then (a; ® bu) is a basis of the C-module E ®c F. By an abuse of language, the basis (a, ® bu) is sometimes called the tensor product of the bases (ah) and (1),). Remark (1). Let E be a free right A-module, F a free left A-module, (“Omen a basis of E and (bung M a basis of F. Every element 2 e E 8),, F may be written uni‘1uelY as r. an ® 3/1., where y,‘ e F, and also uniquely as Z u xu (8) b u , where

x" e E. If we write y, = g: mubu, xu = Z ah, where the EM and 71m belong to A, then 5,“, = 7m- for all (A, u), for

Z (a. ® (. use.» = Z (on...) e b.) = Z ((2 am...) e bu). COROLLARY 3. Let (EQML be a family (J right A-moa'ules and F a free (resp. finitely generatedflee) left A-moa'ule. Then the canonical mapping (22)

(’15—; Ex) ®A F911;]; (E). ®A F) is injective (resp. bijective) .

If (bu) is a basis of F, every element of (“UL 13,) @A F can be written uniquely as 2 = 2 ((9:20) ® by.) (Corollary 1); to say that its canonical image is zero means that, for all 7\ e L, 2 (xi?) ® b“) = 0, hence xi“’— = O for all A e L and all (1. (Corollary 1) and therefore 7. = 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

256

TENSOR PRODUCTS 0]? PRODUCTS AND DIRECT SUMS ‘

§ 3.7

F = A,; but then the two sides are canonically identified with 711‘ E,_ (no. 4-,

Proposition 4) and after these identifications the canonical mapping (22) becomes the identity.

COROLLARY 4. Let A be a ring with no divisor of zero, E a fiee right A-module andFafreeleflA—module. Thentherelationx®y = OinE ®AF impliesx = O or y = 0.

Let ((1.) be a basis of E, (bu) a basis ofF and letx = 2 (1.5., y = Z nub”; then x (8 y = AZ“ ((ahgmu) ® 1),, and the relation x ® y = 0 implies Emu = 0 for every ordered pairs of indices (A, p.) (Corollary 1). Hence, if .1: 9E 0, that is 5,, aé 0 for at least one A, it follows that r)“ = 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 directfactor of E and N a directfactor of F, the canonical homomorphism M ®A N —> E ®A F is injective and the image of M ®A N

under this homomorphism is a directfactor of the Z-module E ®A F. This follows immediately from Proposition 7. Note that if E is a ((B1); A, (C;))-multimodule and F an (A, (Bi); ((3))multimodule and M and N direct factors in these multimodules, M ®N

is a direct factor of the ((B1), (B7,); (0;), (C;))-multimodule E ® F. COROLLARY 6. Let P be a projective left A-module and E, F two right A-modules. For every iry'ective homomorphism u:E —> F, the homomorphism u ® ls ®AP—>F ®AP

is infective. There exists a left A-module Q such that L = P 6-) Q is free (§ 2, no. 2, Proposition 4) and u ® 11, is identified (Proposition 7) with

(“® 11’) G (u ® 10); it therefore suffices to prove the corollary when P isfree (§ 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-. COROLLARY 7. Let C be a commutative ring. IfE and F are two prly'ectioe (ll-modules, E ®c 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 E; be a submodule of

EMFLasubmodule oer" and let E’ = gab 13;, F' = ”(29M Fa, Let Im(E’ @A F’) 257

LINEAR ALGEBRA

II

(resp. Im(E( ®A FD) denote the image of E' ®A F' (resp. E; ®A F;) in E ®A F (resp. E,‘ ®A F“) under the canonical mapping; then the isomorphism (23) identifies the sub-Z-modules

Im(E'®A F') and

er)

(A. u) e I x M

Im(E1®AFL);

this follows immediately from the commutativity of the diagram

(Q? 1Ei) gm (9;? 1FL) -—> ( (9(a)), a-(oo)) of 9‘ into L, x L;

for m e 9,, we write [23’ and qg’ instead of1;“, and on. Then for each i there is a “partial” tensor product F‘ = (confide); G”

We shall further make the following “permutability” hypothesis: (P) If a) e (2', pm (resp. q“) permutes with each of the endomorphism: p; and q,I

to" Gun) (“3513- Geo») such that £99 9', 71 ¢ 9' and 9(0)) = 9(5) = 0(7)) (resp6(0)) = 9(5) =00)»261

LINEAR ALGEBRA

[I

For each a) e (2’, let i be the index such that 9(0)) 6 Li; then consider the family (0,)”1“ where ”9(a) = pm and v,“ = 1G}. for A ye 9(a)); hypothesis (P) implies that the family (0,) satisfies conditions (27) (where p' and p must be replaced by p“), q’ and g by q“), a) by an element E, running through 9,); thus an endomorphismflgf) v,“ = r“, is derived of the Z—module F,. Similarly, an endomorphism so, is defined of the Z—module F, starting with q“, j being the index such that 6(a)) e L,; finally let d(w) = (i, j). Then we can define the tensor product (.189 F, and the corresponding canonical mapping 1}

(b‘d'n’): 1:11: F' 90183:) F" On the other hand, for each i, the canonical mapping

4:. = ¢(c“’.p“’,q“"> H, Gt —> F.; using the associativity of the product of sets, a Z—multilinear mapping 4; = ‘l’tdn. ,) o (42,) of G into “(3‘) F, is derived. We show that the ordered pair ( ® F‘, 4)) is a solution of the same universal problem as ((6850 GA, than”), our.» whence will follow the existence of a unique Z-module isomorphism 6:

®

Gh—>

®

F‘

(d. T. 5)

(c; P: G)

such that 4; = 6 0 them”) (Set Theory, IV, §3, no. 1). By induction on n, the proof is reduced to the case n = 2; for simplicity ® F, and ‘8 31,. Consider the we write F1 (8) F2 and 3/1 (8 y2 instead of (d.r.s) (11.1.8) (d)

(d)

mapping from G to F1 (gs) F2 ( h: (xii) —+ ((g) xh) 3 ((§) x1).

It is obviously Z-multilinear; we show that it satisfies conditions (26) for all a) 6 Q. This is obvious if to E 91 or o) e (In; otherwise, supposing, in order

to fix the ideas, that 9(0)) 6 L1 and 6(6)) 6 L2, the values of h o [7,, and h o in, at (xx) are respectively (rm((§) 961)) 9?) ((§) xx)

and

((g) xx) g (Sm((§) J(1'))

which are also equal by definition of F1 (8) F2. ( d)

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:F1 ® F2 —> H (a)

262

TENSOR PRODUCT or FAMILIES or MULTIMODULES

§ 3.9

such that u = u o h and that will prove our assertion (repeating the argument of no. 1, Corollary 1 to Proposition 1). For all 22 = (“New we consider the “partial” linear mapping of E GA into H 1

“(u 22) 3 (xh)heb1_ H “((xA)AEL1: £2) = “((xh)heh)'

(28)

Clearly it is Z-multilinear and satisfies conditions (26) for m e 91; by definition there thus exists a Z—linear mapping yl »—> w1(y1, x,) of F1 into H such that ® xx: 22) = "((xh)heh1a 22) w1( (0(1))

(29)

We next consider the mapping “23 (xxheba "’ w1(., (xA)LEL3)

of Q2 GA into Homz(F1, H); it is obviously Z-multilinear and satisfies

conditions (26) for to e (22, by virtue of the hypothesis on u and relations (28) and (29) and taking account of the fact that the elements of the form (@) x3, generate the Z—module F1. Hence there is a Z-linear mapping c

waz —> Homz(F1, H)

such that w2((§) xi.) = “2((xh)heba)

oralso x1.) = u(("A)>.eL)xz))(® (w2(® (9(2)) (cu);

(30)

We now consider, for 311 6 F1, yz 6 F2, the element of H w(y1,y2) = (w2(!/2)) (1/1)-

(31)

Clearly w is a Z-bilinear mapping of F1 x F2 into H. We show further that, for all to e (2’, (assuming, to fix the ideas, that 9(0)) 6 L1 and 6(0)) 6 L2) w(rw(y1),y2) = w(‘!l1, 54112))-

(32)

It sufl'ices to verify this relation when yl (resp. ya) is 'of the form (%) x,‘

(resp. ® xx), since these elements generate the Z-module F1 (resp. F3). But (cm)

I I ' ' = pm(x9(m,) and xxI = x" for where atom, mag) xi) = (cm) x“, by definition,

263

II

LINEAR ALGEBRA

A 75 9(0)) in L1; similarly so,( ®) x”) ((8 xi, WhCI‘C xam) = qm(xa(m)) and x1 = x,‘ for A 9e 6(0)) in L2; using (30) and (31), relation (32) then follows from (26). Hence there exists a Z-lz'near mapping 0 of F1 (8 F2 into H such

that 0(y1 @113) = w(y1,y2) and it then follows from (30) and (31) that v o h = u. The most important special case of the general tensor product defined above is the following: we start with a family (A1)1 T’(F) (8)

Tool

1m.)

T(F') 7+ "PM is commutative. The verification is immediate.

269

LINEAR ALGEBRA

II

Lemma 2. Let M, N be two supplementary submodules in F and izM —> F,j:N —>- F the canonical injections. The diagram m 63 m

T(M) 69 T(N) —> T'(M) C-B T(N) (9)

ire) +T'o)

mum)

T(F) ——> T’ (F) W

is commutative and the vertical arrows are bijective. The commutativity follows from Lemma 1, the other assertions from § 1, no. 6, Corollary 2 to Proposition 6 and § 3, no. 7, Proposition 7. Lemma 3. Under the hypotheses of Lemma 2, for VI; to be injective (resp. smjective), it is necessary and suflicient that vM and vN be so.

This follows from Lemma 2 and § 1, no. 6, Corollary 1 to Proposition 6. Then Lemma 3, together with § 2, no. 2, Proposition 4, shows that it suffices to

consider the case where F is a free module. But, if (bu) is a basis of F, every element of HomA(E, G) ®B F may then be written uniquely as 2 un ® bu, where u“ e HomA(E, G) (§ 3, no. 7, Corollary 1 to Proposition 7); the image of

this element under v is the A-linear mapping x 1—-> Z ul,(x) ® bu; it cannot be zero for all x e E unless uu(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 a finite basis, 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 HomA(E, G) (§ 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) = HomA(E, G ®B F) T(E) = HomA(E, G) (8)], F, and, for every left A-module homomorphism v:E --> E’, T'(v) = Hom(v, lG ® 11,); T(v) = Hom(v, 1G) ® 1F, on the other hand, we write vm instead of v. Then we have the two lemmas:

Lemma 4. For every homomorphism v:E —> E’, the diagram

T(E’) L T’(E’) (10)

T001

1'1" (a)

T(E) T T’(E) is commutative. 270

THE HOMOMORPHISM E* ®A F ——> HomA(E, F)

§4.2

Lemma 5. Let M and N be two supplementary submadules in E and p:E—>M, q: E —> N the canonical projections. The diagram

T ©T ”flit T'(M) @T'(N) TUI) +T(q)

1T1!) + T'(q)

T(E)

T’(E)

is commutative and the vertical arrows are bijective. They are proved as Lemmas l and 2, taking account of§ 1, no. 6, Corollary 2 to Proposition 6, §2, no. 1, Proposition 1 and § 3, no. 7, Proposition 7.

The remainder of the proof then proceeds as in (i) and is reduced to the case where E = As; the two sides of (7) are then canonically identified with G (833 F and v becomes the identity.

In particular take B = A and G the (A, A)-bimodule ,Ad (§ 3, no. 4), so that the right A—module HomA(E, ,Ad) is just the dual E* of E and (,Ad) ®A F is

canonically identified with F (§ 3, no. 4-, Proposition 4-). Homomorphism (7) then becomes a canonical Z-homomorphism

and 6(x* ® y) is the linear mapping of E into F x r-—> (x, x*>y.

Remark (1). The characterization of projective A-modules given in §2, no. 6, Proposition 12, can also be expressed as follows: for a left A-module E to be projective, it is necessary and sufiicient that the canonical homomorphism 0E:E* ®A E "9' HOmA(E, E) = EndA(E)

be such that 1,; belongs to the image of GE. COROLLARY. (i) When F is a projective (resp. finitely generated pray'ective) module, the canonical homomorphism (11) is injective (resp. biiective). (ii) When E is a finitely generated projective module, the canonical homomorphism (ll) is bijective. Even when E and F are both finitely generated, 0 is not necessarily surjective, as is shown by the example A = Z, E = F = Z/ZZ; the right hand side of (ll) is non-zero but E" = 0. On the other hand, examples

can be given where E is flu, but (11) is neither injective nor surjective

(Exercise 3(b)).

271

LINEAR ALGEBRA

n

When E admits a finite basis (e,), the inverse isomorphism 0‘1 of 0 can be found explicitly as follows. Let(e;") be the dual basis of (e,) (§ 2, no. 6);

for all u e Hom(E, F) and all x = 2 ac. with g, e A,

we = 2 Me.) = Z u HomA(E*, F).

On the other hand, there is a canonical A-homomorphism cEzE —> E**, whence there is a Z-homomorphism on ®1F:E ®AF—>E** 8)A F; com-

posing the latter with the homomorphism (14), we hence obtain a canonical Z—homomorphism

(l5)

0’:E ®A F —> HornA(E*, F)

such that 6'(x ® 3/) is the linear mapping x“ 1—» (x, x*>y.

If E and F are pny'eetive modules, the mapping (15) is infective. For ex is then injective (§ 2, no. 7, Corollary 4 to Proposition 13) and as F is projective, the Z—homomorphism eE ® 1F: E (8),, F —> E** (8,, F is also injective (§ 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 pny'ective andfinitely generated, the mapping (15) is bijeetive for the two mappings of which it is composed are then bijective (§ 2, no. 7, Corollary 4- to Proposition 13 and Proposition 2 above).

272

TRACE OF AN ENDOMORPHISM

§ 4.3

3. TRACE OF AN ENDOMORPHISM

Let C be a commutative ring and E a C-module. The mapping (x*, x) »—>(x, x*) of E* x E into C is then C-bilinear, since, for all 1 EC, (7x, x*> = fix, x*) and (x, x*y> = (x, x*>-y; we derive a canonical C-linear

mapping (16)

1-:E* ®cE——>C

such that 1(x* ® :4) = (x, x*) (§ 3, no. 5). Suppose now also that E is afinitely generated prty'eetive C—module; the canonical isomorphism (l 1) of no. 2 is then a C-moa'ule isomorphism and we can therefore define by transporting the structure a canonical linear form Tr = 1' 0 0g1 on the C-module Endc(E). For all u E- Endo(E) the scalar Tr(u) is called the trace of the endomorphism u; every n eEndc(E) can be written (in general in an infinity of ways) in the form x n—> Z (x, arr)!“ where xi" 6 E* and 31,6 E, by virtue of no. 2, Corollary to

Proposition 2; then

(17)

Tr(u) = 2 (y‘, :4") (cf. § 10, no. 11).

By definition, (18)

Tr(u + v) = Tr(u) + Tr(o)

(19)

Tree = mo)

for u, v in Endc(E) and Y e 0. Moreover: PROPOSITION 3. Let C be a commutative ring, E, F two finitely generated prty'ective C-modules and u:E —> F and 02F —> E two linear mappings; then

(20)

Tr(v o u) = Tr(u o a).

The two mappings (u, a) r——> Tr(u o a), (u, v) -—> Tr(o o u) of HomG(E, F) x Homc(F, E)

into C are C-bilinear; it therefore suffices to verify (20) when u is of the form x :—> (x, a*)b and v of the form yn—> (y, b*)a, with a e E, a* eE*, b 6F,

[7* e F*. But then u o a is the mapping x »——> (x, a*)(b, b*)a and u o a the mapping y r» (y, b*>(a, a*)b. Formula (17) shows that the values of the two sides of (20) are equal to (a, a*> Hom(E; ®o E2: F1 ®c F2)

Let C be a commutative ring and E1, E2, F1, F2 four C-modules; in § 3, no. 5,

formula (13) we defined a canonical C—module homomorphism

(21)

A:Hom(E1, F1) 99 Hom(Ez, F2) —> Hom(E1 (2; E2, F1 (8 F2).

PROPOSITION 4-. When one of the ordered pairs (E1, E2), (E1, F1), (E2, F2) consists offinitely generated projective C-moa’ules, the canonical homomorphism (21) is bijective. It is obviously sufficient to perform the proof for the ordered pairs (E1, F1) and (E1, E2). We consider first the case of the ordered pair (E1, F1); we fix E2, F1, F2

and write for every C-module T(E) = Hom(E, F1) ®c Hom(Ez, F3) and T’(E) = Hom(E ® E2, F1 (8 F2) and, for every C-homomorphism o:E —> E’,

T(”) = Hom(o, In) ‘8 lHODKEmFa) and

T’(v) = Hom(v ® 113,: lF1®Fa)' Then Lemma: 4 and 5 (no. 2) (where v is replaced by A) are valid and are proved by completely analogous methods. We next fix E2 and F2 and this time write, for every C-module F,

T(F) = Hom(C, F) ®c Hom(Ez, F2) and T’ (F) = Hom(C ® E2, F ® F2) and, for every C—homomorphism u: F —-> F', T(u) = Hom(lc, u) ® lHom (x1, xf)3/1 and ua of the form :c2 r—-> (x2, x§>yag then the image of x1 ® x2 under ul ® u; is by definition

(x1, xi>(y1 ® ya) x’f (8) x3 being canonically identified under (1. with an element of (E1 8) E2)*.

As (E®F ®G)*

is commutative, by virtue of formula (24). We note also that, without any hypothesis on the C-modules E, F, there are canonical isomorphirm

(28) (29)

'

(E 5;» F)* —-> Hom(E, F*) (E (3 F)* —> Hom(F, 13*)

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 bilinearform on E x F, the homomorphism: of E into F* and the homomorph-

ism: 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 (x, y) H (II, "(x)>

276

(resp' (563/) '4’ (x: ”(y)>)'

EXTENSION OF THE RING OF SCALARS OF A MODULE

§ 5.1

§5. EXTENSION OF THE RING OF SCALARS l. EXTENSION OF THE RING OF SCALARS OF A MODULE

Let A, B be two rings and p:A —> B a ring homomorphism; we consider the right A-module 9*(Bd) defined by this homomorphism (§ 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, b, b’ in B, these two module structures on B are compatible (§ 1, no. 14-). This allows us, for every left A-module E, to define a left B-module structure on the tensor product 9*(Bd) ®A E such that

B'(B ® x) = (3'3) (X) x for (3, {5’ in B and x e E (§ 3, no. 3). This left B-module is said to be derivedfrom E by extending the ring qfscalars to B by means of p and it is denoted by p*(E) or E03, if no confusion arises. PROPOSITION 1. For every left A-module E, the mapping (bun->1 8) x of E into the A-module 9*(p*(E)) is A-linear and the set ME) generates the B-module 9*(E). Further, for every left B-module F and every A-linear mapping f of E into the Amodule 9*(F), there exists one and only one B—linear mapping f of p*(E) into F such thatf(l®x) =f(x)firralle.

B can be considered as a (B, A)-bi.module by means of p; then there is a canonical Z-module isomorphism

(1)

HomB(B @A E, F) —> HomA(E, HomB(B,, 17))

as has been seen in § 4, no. 1 ,Proposition 1. But the left A-module HomB(B,, F) is canonically identified with 9* (F): for, by definition (§l, no. 14), there corresponds to an element y e F the homomorphism 9(y): B, ——> F such that (0(y))(l) = y; for all A e A, there thus corresponds to 9(7):; 6 F the homomorphism “ »-—> 51.9003] of B, into F, which is just 16(y) for the left A-module

structure on HomB(B,, F) (§ 1, no. 14). Using this identification, we obtain therefore a canonical Z-module isomrophism, the inverse of (l)

(2)

3 =H0mA(E, 9*(F)) —> H0mn(9*(E), F)

and it follows immediately from the definitions that if 3(f) =f then f(l ®x) =f(x) for all e. In particular, the mapping dun: xv—>l®x is just (3)

(la = 3'1(1m))-

Proposition 1 is therefore proved. The mapping (bEzE—> p,,(p*(E)) is called canonical.

277

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LINEAR ALGEBRA

Remarks. (1) Proposition 1 shows that the ordered pair consisting of Em and (be is a solution of the universal mapping problem (Set Theory, IV, §3, no. 1), where Z is the species of left B-module structures (the morphisms being B-linear mappings) and the ot-mappings are the A-linear mappings of E into a B-module. (2) If E is an (A, (Of); (D;))-multimodule and F a (B, (07,); (D;))multimodule, then the isomorphism (2) is linear with respect to the ((D;, (Ci) ; (Cf), (D;))-multimodule structures of the two sides (§ 1, no. 14 and § 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 § 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 (§ 1, no. 12); it is immediate that the canonical mapping 1:: 9*(E) —> E/aE defined in § 3, no. 6, Corollary 2 to Proposition 6 is an isomorphism for the (A/a) -module structures.

COROLLARY. Let E, E’ be two left A-modules;for every A-linear mapping u : E —> E’, v = lB ® a is the unique B-linear mapping which renders commutative the diagram E L E(B)

E’ —> EfB) 4W

where 4),: and 4m. are the canonical mappings.

It suffices to apply Proposition 1 to the A-homomorphism ¢[>E. o u: E —> EfB).

The mapping a defined in the above corollary is denoted by p*(u) or no”. If E” is a third left A-module and o:E’ —>E" an A-linear mapping, it is immediate that

(1’ ° ")(3) = ”(3) ° “(3)-

Extending the ring of operators Of a module is a transitive operation; to be precise:

PROPOSITION 2. Let p:.,A—>B a: B——> C be ring homomorphism. For every left A-module E, there exists one and only one C—homomorphism

(4)

6*(9*(E)) —> (6 ° P)*(E)

mapping 1 ® (1 ® x) to 1 (8 xfor all x e E and this homomorphism is bijeotioe. The underlying Z-modules of a*(p* (E)) and (a o p)*(E) are respectively C (8),, (B ®A E) and C. ®A E. There exists a canonical Z-isomorphism

C (83 (B 8),, E) —> (C ‘33 B) 8),, E (§ 3, no. 8, Proposition 8), which 1s also a C-isomorphism for the left C—module structures on both sides. Moreover, the 278

EXTENSION OF THE RING or SGALARS OF A MODULE

§ 5.1

C-module C ®n B is canonically identified with the C—module C, under the isomorphism which maps y (8) [3 to yo-(fi) (§ 3, no. 4, Proposition 4) and this iso— morphism is also an isomorphism for the right A-module structure on C ®B B defined by p and the right A—module structure on C defined by 0' o 9. Thus a canonical isomorphism (C ®BB) ®AE—>C ®AE

is obtained and, composing it with the isomorphism C ®n (B ®AE) —> (C 83B) ®AE defined earlier, the desired canonical isomorphism is obtained.

If:1), (b’ and 9*(E), 9* (E -—>) c*(p* (E)) and E —> (o- o p)*(E), 4/ a (1) is identified with ({2” under the canonical isomorphism 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

(5)

Eu» ®n Ein) —> (E ®A Ella)

mapping (1 ®x) ® (1 ®x’) to l® (x ®x’) for e, x'eE’, and this homomorphism is biiective. The left hand side of (5) may be written (B ®A E) ®B (B (8,, E’) and is identified with (E ®A B) (83 (B ®A E’) since A and B are commutative;

the latter product is identified successively with E ®A (B ®3 B) ®A E’, E ®A (B ®A E’), E ®A (E’ ®AB) and finally (E ®A E’) ®AB, using the associativity ofthe tensor product (§ 3, no. 8, Proposition 8), Proposition 4-, § 3, no. 4, and the commutativity ofA and B. The desired isomorphism is the composition of the successive canonical isomorphisms.

Clearly if S is a generating system of E, the image of S under the canonical mapping E —> E3) is a generating system of Em); in particular, if E is a finitely generated A-module, Em, is a finitely generated B-module. PROPOSITION 4. Let E be an A-module admitting a basis ((10,615 if ozx .—> 1 ® .7:

is the canonical mapping of E into 9*(E), then (¢b(a,()),,eL isa basis (y‘p*(E). If p is infective, so is (I). The first assertion follows immediately from § 3, no. 7, Corollary 1 to Proposition 7. Also, for every family (Emu, of elements of A of finite support,

¢(LZL 51.42») = 1.21.. p(€,_)d>(a,_) and the relation 4%,; EMA) = 0 is therefore equivalent to 9(5).) = 0 for all A e L, whence the second assertion. COROLLARY. For every [Irty'ectioe A-moa'ule E, the B-module p*(E) is prcy'ectioe. If firther p is injectioe, the canonical mapping of E into 9* (E) is injectioe.

279

LINEAR ALGEBRA

II V

By hypothesis there exists a free A-module M containing E and in which E admits a supplement F. It follows immediately from §3, no. 7, Proposition 7 that Mm, is identified with the direct sum of Em and F(B, and if 4) and ‘l‘ are the canonical mappings E ——> Em) and F —> F(B), the canonical mapping

M —> M03) is just x + y» ¢(x) + My). 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 9* (B,), B being considered this time as an (A, B)-bimodule and the right B—module structure on p*(E) being such that (x ® B)B' = x ® (69’) for BeB’ (3’ EB 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 lqfl A-module 9*(B,) defined by p and for every left A-module E, consider the Z-module

(5)

{5(E) = HomA(P*(Ba)a E)-

As 9*(B,) has a right B-module structure, a left B—module structure is derived on 5(E) (§ 1, no. 14) such that, if ue 5(E) and b’eB, Iz’u is the homomorphism bv—> u(bb') of p,,(B,) into E. We further define an A—linear mapping, called canonical

(7)

1r)=(>u:(t3(E)) ->E

associating with every homomorphism u e 5(E) the element 14(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

HomA(P*(Bs) ®n F: E) -> H0mn(F, HomA(P*(Ba)s 13)) (§ 1, no. 1, Proposition 1). As the left A-module p*(B,) (33 F is canonically identified with 9*(F) by virtue of § 3, no. 4, Proposition 4, we obtain a canonical Z-module isomorphism, the inverse of the above

(3)

H0mn(F, 5(E)) -> HomA(9*(F)2 E)

which associates with every B-linear mapping g of F into 5(E) the composite mapping 1} o g, considered as an A-linear mapping of 9*(F) into E. In particular, under the hypotheses of Proposition 2, if F is replaced by

0*(C,), we obtain a canonical C—isomorphism

(9)

3(5(E)) —> (6 ° P)”(E)-

2. RELATIONS BETWEEN RESTRICTION AND EXTENSION or THE RING or SCALARS Let p : A —> B be a ring homomorphism. For every left A-module E, a canonical A-linear mapping

(10)

¢n=E—> 9*(9*(E))

was defined in no. 1 such that (Inner) = 1 ® x. We consider now a left B280

RELATIONS BETWEEN RESTRICTION AND EXTENSION OF RING OF SCALARS

§ 5.2

module F and apply Proposition 1 (no. 1) to the A homomorphism 1,.m:p*(F) -—>p*(F): we obtain aB—linear mapping

44-: P*(P*(F)) -> F

(11)

equal to 30mm) and such therefore that, for all yeF and all [3613,

MB ®y) = ByPROPOSITION 5. Let E be a left A-module and F a left B—module; the composite mappings

H0mn(9*(E)a 9*(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 structure on B, cf. III, § 1, no. 3)*. Then the A-module structures of G’ and G” are identical and composing the homomorphisms (16) and (15) we thus obtain a canonical Z-homomorphism

(17)

(MB ®A H0mA(E: F) '9 Homn(E(n» Fan)

which is characterized by the fact that, for all u e HomA(E, F) and all b e B

(18)

(”(17 ® 11) = 71: ® u,

where r, denotes right multiplication by b in B. Moreover, the hypothesis that o) is a central homomorphism implies that (bb’) 9(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 (8,, HomA(E, F) a right B-module structure (§ 3, no. 4-) and also on

Fm, = B ®A F, and finally, as the left and right B-module structures on Fm) are compatible, a right B-module structure is also obtained on HomB(E(B,, Fm) (§ 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, B a ring, p:A —>B a central homomorphism and E, F two A-modules.

282

DUAL or A MODULE OBTAINED BY EXTENSION OF SCALARS

§5.4—

(i) If B is a [my'eetive (resp. finitely generated prcy'ective) A-module, the homomorphism (17) is injectioe (resp. bijeetioe). (ii) If E is a finitely generated prty'eotive A-module, the homomorphism (17) is bijeotive.

As (16) is‘bijective, the proposition follows from § 4-, no. 2. Proposition 2, applied to the canonical homomorphism (15). 4. DUAL OF A MODULE OBTAINED BY EXTENSION OF SGALARS

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 “m3 (E*) Ea be an Aa-linear mapping, these mappings forming an inverse

system; then u = lit—n u“ is an A-linear mapping of 1211 E; into hi1 E“. Moreover:

PROPOSITION 1. Let (Emfufl), (13;,fig), (E;,f;3) be three inverse systems of Aumodules and (ad), (on) two inverse systems qm-linear mappings such that the sequences

0 ——+ E; L E“ 11> E; are exactfor all ac. Then, wntmg u = l limEu E¢l>E;——>o it does not neeessarilyfbllow that the sequence limE; —-> 0 0 —-> limE; —u—> limEu —u—> —->

—>

u(li_In> E;) = li_n>1 uu(E;) and o‘1(0) = En) v;1(0) (Set Theory, III, § 7, no. 6,

Corollary to Proposition 7).

Loosely speaking, Proposition 3 can also be expressed by saying that passing to the direct limit preserves exactness. PROPOSITION 4-. Let (Emfm) be a direct system of Act-modules, E = EEO, its

direct limit and ouzAw—>A and fqa—> E the canonical mappings for all at e I.

g: for all n e I, s, is a generating system an“, then s = “lg! fuse!) is a generating system of E.

Every x e E is of the form fl,(x¢) for some a e I and some x“ e Em and by

hypothesis x = Z (”g/5,”, where A? EA“ and yfi? e S“; writing A“) = 420,09), yo) = j;(y§f’), we Obtain x = 2: Amy“). PROPOSITION 5. With the hypotheses and notation of Proposition 4, suppose that for all u e I, E“ is the direct sum of a family (MDHL of submodules (the indexing setL being independent of a) and that fm(M§) C Mzfir a < B andfor all A e L. Then

E is the direct sum of the submodules M7L = li_m> M: (A e L). It follows from Proposition 4 that E is the sum of the M". Let (11")151. be a family such that y" e M" for all A e L and whose support is finite and suppose

that z y’“ = 0. By virtue of Set Theory, III, § 7, no. 5, Lemma 1, there exist an at e I and a family (3:2)“,51, of finite support consisting of elements of E“

such that x2 6 M: and y’L = f,(x§) for all A e L. The relation 12(21‘ x2) = 0 implies the existence of a {5 2 at such that f”°‘(gn x2) = 0 (Set Theory, III,

287

LINEAR ALGEBRA

II

§ 7, no. 5, Lemma 1), which may be written as 2;, xf,‘ = 0, where x3 = fau(xfi) 6M3 by hypothesis; hence xi; = 0 for all AeL and therefore yh = fuses) = 0 for all A E L, which proves that the sum of the M" is direct.

COROLLARY. Let (Pa) be a direct system of subsets of Em and let P = li_>m P“. If, for all at e I, P“ is a free subset (resp. basis) of E“, then P is a free subset (resp. basis) of E. The second assertion follows immediately from the first and Proposition 4. It is therefore sufficient to prove that if the P, are free every subset {y“)}1 F, the family of v“ = v of“ 1s a direct system of A-linear mappings such that v = lim on. On the other hand we note that for a< [3 the mapping H°m(fflw 1F) = faflzHomA(EB’ F) _> HomA(Eou F)

is a Z-module homomorphism such that (HomA(E,, F),fin) is an inverse system of Z-moa'ules; as fl3(vfl) = on afim, the above remarks can be expressed as follows: PROPOSITION 6. For every direct system (Emfm) of A-modules and every A-module F, the canonical mapping u 1—» (u of“) is a Z-module isomorphism

(5)

domMEn> Ea, F) —> LimHomAEu, F).

COROLLARY 1. For every A—madule homomorphism v : F —> F’, the = Hom(lEu, v) :Hom(Eu, F) —> Hom(E“, F’)

288

§ 6.3

TENSOR PRODUCT or DIREGI‘ Lmrrs

form an inverse system of Z-linear mappings and the diagram

Hom(lim E“, F) i) lim Hom(E,,, F) .9

(—

“:50:

Hom(l;,v)l

(6)

Hom(h_n)1 E“, F’) 7—) 1(11_n_ Hom(E,,, F )

is commutative. For all u e Hom(£)n E“, F), dF,(v o u) = (v o u ofi) by definition and the commutativity of diagram (6) then follows immediately from the definitions. COROLLARY 2. If (Emfm) is a direct system of lefl A-modules and E = h_m) E“, (E3, ff”) is an inverse system of right A-moa'ules and 132 E: is canonically immm'phie to E*. Remark. Let E be an A-module and (Make; an increasing family of submodules of E such that E is the union of the M,; ifjguiMa ——> MB (for at < p) and j“: M“ —> E are the canonical injections, it is immediate that j = lii is an isomorphism of 11—11; Mm onto E (Set Henry, III, § 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 0F DIRECT LIMITS

Let (Au, pm) be a direct system of rings and (Emfflu) (resp. (Fa, gm» be a direct system of right (resp. left) Au-modules. For at S B, there is a Z-module homomorphism

fea ®gBu:Eu ®A¢ F, —> (EB)[A¢‘I ®A. (FB)EAu1 and on the other hand there is a canonical Z-module homomorphism (Brahma ®A. (FB)LA¢] —> EB ®Aa Fa

corresponding to the ring homomorphism pm (§ 3, no. 3, Proposition 2); whence by composition we obtain a Z-module homomorphism hs“ ®A¢ F“ —> EB ®A5 Fa

which maps the tensor product x, ® y“ tofan (eta) ® gnaw“). Clearly (Eat ®A¢ Fa, hm)

is a direct system of Z-modules. Let A = lim A“, E = lim Eu, F = lim F“ and

—>

—>

——>

289

II

LINEAR ALGEBRA

let puzAa —> A, fazEu ——> E, guzFu —> F be the canonical mappings. As above, a Z—linear mapping nqu ®Aa Fm -—> E 8)., F is defined, which maps the tensor product x“ (8 ya to f¢(xu) ® g,(y,), and it is immediate that these mappings form a direct system. Thus we obtain a Z—linear mapping

(7)

n = Hgnazgn (E. om.) +E (Em 8),,“ F“) and, for all «61, let h,:E,, ®A¢ Fu—->P be the canonical mapping. On the other hand, for all a: e I, let tqu x F, —> Eu 69A“ F“ be the canonical Z—bilinear mapping; for at < B, tfl(fflu(xoc)a g3u(ya)) =.ffla(xu) ® gfla(y¢x) = hB¢(ta(xaayu))

and hence (t!) is a direct system of mappings. Canonically identifying Luna x F“) with E x F (Set Theory, III, §7, no. 7, Proposition 10), we derive a mapping t = hm) tatE x F —-> P such that t(f;t(xa): gu(!/u)) = ha(tot(xocaya)) = ha(xot ®ya)'

Taking account of Set Theory, III, §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 (1 e I suCh that x =fit(xa)s y = goth/u), A = 90:00:) With MEAw qEa: yaEFu

(Set Theory, III, § 3, no. 7, Lemma 1); whence tot)», y) =

«((xuxot) ®ya) = have: ® (And/a» = K": )‘yh

Hence there exists one and only one Z—linear mapping rc':E (3,, F —> P such that 1c’(x ® y) = t(x,y) (§ 3, no. 1, Proposition 1). Moreover, by definition ”Kwanzaa @1100» = “IM(xa) ® gu(yu)) = 130:6”: @yot)

"(7:,(f':z(x¢) ® ga(y1A¢ by means of the canonical homomorphism Pu:Aot —->-A. Then (E;,fm ® 1A) is a direct system of right A-modules, whose direct limit is canonically isomorphic to l_1n_; E“.

It suffices to apply Proposition 7 with F, the ring A considered as an (A0,, A)-bimodule by means of pa. COROLLARY 3. Let A be a ring, (wfla) a direct system of right A-modules and F a left A-module. Then the Z-modules hm' (Ea ®A F) and (km E“) 8)A F are canonic, _ —> ~> ally isomorphic. It suffices to take A“ = A and F, = F for all a e I in Proposition 7. In particular, if p :A —> B is a ring homomorphism, lim 9*(Ea) and p*(h_m> E“) are canomcally 1somorph1c.

COROLLARY 4. Let M be a right A-module, N a left A-module, (mum afamily qfelements q, (y,)1‘.‘,, afamily qfelements q, such that: (x; ®y,) = 0 in M ®A N. Then there exists afinitely generated submodule M1 (resp. N1) of M (resp. N) containing the x, (resp. the y,) and such that Z (x, ® 3],) = 0 in M1 ®A N1.

M (resp. N) is canonically identified with the direct limit of the right directed family of its finitely generated submodules containing the x; (resp. the y,) and it suffices to apply Set Theory, III, § 7, no. 5, Lemma 1.

291

II

LINEAR AWEBRA

§7. VECTOR SPACES 1. BASES OF A VECTOR SPACE

THEOREM 1. Every vector spaee over afield K is afree K-module.

It must be proved that every vector space admits a basis; this will follow from the following more precise theorem: THEOREM 2. Given a generating system S of a vector space E over a field 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 = fl . To prove Theorem 2, we note that the set 52 of free subsets of E contained in S, ordered by inclusion, is an inductive set (Set Theory, III, §2, no. 4), by virtue of § 1, no. 11; so is the set 92} of free subsets containing L and contained in S. By Zorn’s Lemma, 9R admits a maximal element B and it sufiices 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 (a.),,EI be afleefamily 1y" elements of E; if b e E does not belong to the subspace F generated by (a,), the subset of E consisting of the a, and b isfree.

Suppose that there were a relation ab + 2 ha, = 0 with 96K and A. e K for all t. e I, the family (A) having finite support; if p. aé 0, it would

follow that b = —Z (a'lh)a, and hence b e F contrary to the hypothesis; !

hence p. = 0 and the relation becomes 2 ha, = 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 equivalent: (a) B is a basis Qf E. (b) B is a maximalfree subset (y‘ E. (c) B is a minimal generating system of E. This follows immediately from Theorem 2. Example. Given a ring A and a subfield K of A, A is a (right or left) vector space over K and therefore admits a basis; in particular, every extension field 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 basis,

292

DIMENSION 0F VECTOR SPACES

§ 7.2

Remark. For a family (mm; 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, a: belong to no subspace of E generated by the a, of index I ¢ 1:. 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 absurdum

and considering a minimal related subfamily of (a.). 2. DIMENSION OF VECTOR SPACES

THEOREM 3. Two bases of the same vector space E over afield K are equipotent. We note first that if E admits an infinite basis B, it follows from § 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, §4, no. 4, Definition 7), for it is gener-

ated by each of its elements #0, by virtue of the relation p.a = (y.).)(7t"1a) 1|

forueK, AeKand). se 0. Henceif(a,)1‘,‘,,isabasisofE, then E = @1191, K

up to isomorphism and the subspaces E, = (‘B Kal for 0 < k s n form a i=1 Jordan-Holder series of E, Ek/Ek.1 being isomorphic to Kak. Theorem 3 then

follows in this case from the Jordan-Holder Theorem (I, § 4, no. 7, Theorem 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 11 elements. The proposition is obvious for n = 0. If n 2 l, 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 (1 ¢ 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 (§ 1, no. 10, Proposition 13). As V admits a basis with at most It —- 1 elements, B’ — {a} has at most It — 1 elements by the induction hypothesis and hence B’ has at most n elements.

DEFINITION l. The dimension of a vector space E over afield K, denoted by dimx E or [EzK] (or simply dim E) is the cardinal ofany of the bases g" E. If M is a subset of E, the rank of M (over K), denoted by rg M or rgK 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 Kmodule offinite length and dimK E = longK E.

293

LINEAR ALGEBRA

II

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’ of V (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') S Card(B), whence the corollary. Theorems 2 and 3 immediately imply the following proposition: PROPOSITION l. (i) For a left vector space over K to be offinite dimension n, it is necessary and sufiicient that it be isomorphic to KS. (ii) For two vector spaces Kf," and K: to be isomorphic (m and n integers 20), it is necessary and sufiicient 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 offinite dimension n, everyfree subset has at most n elements; a free subset with n elements is a basis of E.

PROPOSITION 2. Let (Ema! be afamily of vector spaces over K. Then

(1)

dimx(gr)I E) = 21 dimK Ev

If the E. are canonically identified with subspaces of E = ”GB! E‘ and B‘ is a basis ofEl (v. e I), then B = ‘EUI B. is a basis ofE (§ 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 (§ 1, Exercise 16(c)). However:

PROPOSITION 3. Let A be a ring such that there exists a homomorphism p of A into afield D; thenfor everyfree A-module E, any two bases of E are equrpotent. Consider the vector space 9*(E) = D ®A E over D obtained by extending the ring of scalars to D (§ 5, no. 1) and let (a,()) is a basis of 9*(E) (§ 5, no. 1, Proposition 4); the proposition then follows from Theorem 3.

COROLLARY. If A is a commutative ring 790 and E a free A-module, any two bases of E are equipotent. There exists in A at least one maximal ideal m (I, §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. 294-

DIMENSION AND CODIMENSION or A SUBSPACE OF A VECTOR SPACE

§ 7.3

(3) Let A be a ring such that any two bases of a free A-module are equipotent and let K be a subfield of A, so that A can be considered as a left vector space over K by restricting the scalars. Every free A-module E can similarly be considered as a left vector space over K and it then follows from § 1, no. 13, Proposition 25 that dim]; E = dimx E.dimx A,.

(2)

(4-) In Chapter VIII we shall see examples of rings satisfying the conclusion of Proposition 3 but not the hypothesis. 3. DWSION AND CODIIKENSION OF A SUBSPACE OF A VECTOR SPACE

PROPOSITION 4. Every subspace F of a vector space E is a directfactor of E and (3)

dia + dim(E/F) = dimE.

As the quotient vector space E/F is a free module, we know (§ 1 , no. 11, Proposition 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 field K, every exact sequence of linear mappings 0 —> E —> F —-> G —> O splits. This is another way of expressing Proposition 4 (§ 1, no. 9). COROLLARY 2. Let (E,)°MnN—>M {—BN-—>M + N—>0

(§ 1, no. 8, Proposition 10) taking account of the fact that

dim(M @N) = dimM + dimN (no. 2, Proposition 2).

COROLLARY 4. For every subspace F of a vector space E, dimF S 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) = O by (3) and a vector space of dimension 0 reduces to O.

COROLLARY 5. If a vector space E is the sum of a family (F,) of vector subspaces, then

(8)

dimE < Z dim F,

[ffirther dim E is finite, the two sides of (8) are equal if and only if E is the direct sum of thefamily (E).

The inequality (8) follows from (3) and the fact that E is isomorphic to a quotient of 61') F, (§ 1, no. 7, formula (28)). The second assertion is a particular case of § 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 subspace F qf E, denoted by codirnE 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)

dimF+codimF=dimE.

PROPOSITION 5. Let F, F' be two subspaces Qf a vector space E, such that F C F’.

Then codimE F’ < codimEF S dim E. If codimEF is finite, the relation

coclimE F’ = codimm F implies F = F’. The inequality codimEF g dim E is obvious from (9) and if dimE is

296

DIMENSION AND CODIMENSION OF A SUBSPAGE OF A VECTOR SPACE

§ 7.3

finite the relation codimE F = dim E implies dim F = 0 and hence F = {0}. The remainder of the proposition follows from this, for codimE F’ = codirnE,F(F’/F), since E/F’ is canonically isomorphic to (E/F) /(F’/F) (I, § 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) = codimM + codim N.

It sufiices to apply Corollary 2 of Proposition 4 to the exact sequence

0 —> E/(M n N) —> (E/M) {-B (E/N) —> E/(M + N) —> 0

(§ 1, no. 7, Proposition 10) and use no. 2, Proposition 2. Note that if E is finite dimensional, (10) is a consequence of (7) and (9).

PROPOSITION 7. If (F,) is a finite family of subspaces Qf a vector space E, then

codizn(Q F.) < Z codim F,. If F = O F” E/F is isomorphic to a subspace of the direct sum of the E/F‘ (§ 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 (resp. 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 l. 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 correspondence between the subspaces of E containing a subspace H and the subspaces

of E/H (I, §4, no. 7, Theorem 4); if E is of dimension )1, 6 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.

In a vector space of finite dimension n 2 1, the hyperplanes are the subspaces qf dimension n -— l by formula (3). PROPOSITION 8. In a vector space E over a field K, every vector subspace F is the intersection of the hyperplanes which contain it.

It suffices to show that for all x (f: F there exists a hyperplane H containing F and not containing x. By hypothesis F n Kx = {0} and hence the sum M of F and Kx is direct. Let N be supplementary to M in E; E is then the direct sum of H = F + N and Kx and H is therefore a hyperplane with the desired property.

297

H

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 (§ 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 certain ideals are not principal (VII, § 1, no. 1) and any two elements of such a

ring are linearly dependent (§ 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 necessarily admit a supplement. 4. RANK 0]." 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 (ll)

rg(u) = codimE(Ker(u))

and therefore (12)

rg(u) + dim(Ker(u) = dim E.

Moreover, by formula (3) (l3)

rg(u) + dim(Coker(u)) = dim F.

PROPOSITION 9. Let E, F be two vector spaces over afield K and u:E —> F a linear mapping. (i) rg(u) < inf(dim E, dim F). (ii) Suppose that E is finite-dimensional; in order that rg(u) = dim E, it is p necessary and sufieient that u be injective. (iii) Suppose that F is finite-dimensional; in order that rg(u) = dim F, it is necessary and suflicient that u be surjective. This follows immediately from relations (12) and (13). COROLLARY. Let E be a vector space offinite dimension n and u an endomorphism of E. Thefollowing properties are equivalent:

(a) u is bijective; (b) u is injective; (c) u is surjective; (d) u is right invertible;

(e) u is left invertible; (f) u is of rank n.

298

§ 7.5

DUAL or 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, czK —>- K’ an isomorphism Of K onto K', E a vector

K-space, E’ a vector K’-space and uzK ——> K’ a semi-linear mapping relative to 6 (§ 1, no. 13); the dimension of the subspace u(E) of E’ is also called the rank ofa. It is also the rank of u considered as a linear mapping of E into a... (E’), for every basis of u(E) is also a basis Of 6* (u(E)). 5. DUAL OF A VECTOR SPACE

THEOREM 4-. The dimension of the dual E* of a vector space E is at least equal to the dimension of E. For E* to be finite-dimensional, it is necessary and sufiicient that E be so, and then dim E* = dim E.

If K is the field of scalars Of E, E is isomorphic to a space K9) and therefore

E* is isomorphic to Kf, (§ 2, no. 6, Proposition 10). As Kg” is a subspace of Ki, dimE = Card(I) < dim E* (no. 3, Corollary 4- to Proposition 4);

further, if I is finite, K; = K5,” (cf. Exercise 3(d)).

COROLLARY. For a vector space E, the relations E = {0} and E* = {0} are equivalent.

THEOREM 5. Given two exact sequences Qf vector spaces (over the same field K) and linear mappings 0 —> E' ——> E —> E” —> 0 0 —> F’ —> F —-> F" -—> 0 and two vector spaces G, H over K, the corresponding sequences

0 ——> Hom(E”, G) —> Hom(E, G) —> Hom(E', G) —> 0 0 —-> Hom(H, F’) —> Hom(H, F) -—> Hom(H, F”) —> 0 are exact and split. This follows from the fact that every vector subspace is a direct factor (no. 3, Proposition 4-) and from § 2, no. 1, Propositions l and 2. COROLLARY. For every exact sequence

0

E'

ll

E

0

E”

0

of vector spaces over the samefielcl K and linear mappings the sequence

O—-—->E"*-—-.3—>E*—."—>E’*——>O is exact and splits. 299

LINEAR ALGEBRA

II

It follows in particular that for every vector subspace M of E the canonical homomorphism E*/M’ —> M*, where M' is the subspace of E* orthogonal to M (§ 2, no. 4), is bijective. THEOREM 6. For every vector space E over a field K, the canonical mapping cE:E—>E** (§ 2, no. 7) is injective; for it to be bijective, it is necessary and suficient that E befinite-dimemional. The first assertion and the fact that if E is finite-dimensional cE is bijective are special cases of § 2, no. 7, Proposition 14-. Suppose that E is infinite-dimensional, so that we may assume that E = K9", where L is an infinite set and therefore E* = K‘g. Let (and, be the canonical basis of E and let (aha, be

the corresponding family ofcoordinate forms in E* (§ 2, no. 6) ; the vector subspace of E* generated by the ef is just the direct sum F’ = Ki,” and the hypothesis that L is infinite implies that F’ 9E 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** (§ 2, no. 6, Corollary to Proposition 9). But H” n cE(E) is contained in the image under cE of the orthogonal of F’ in E, 'which is by definition 0; hence cE(E) = E** is impossible. E will usually be identified 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. We shall define canonical isomorphisms: (1) 0f the dual ofIm(u) = u(E) onto Im(‘u) = ‘u(F*).

(2) 0f the dual QfKer(u) = 711(0) onto Coker(‘u) = E*/‘u(F*). (3) 0f the dual of Coker(u) = F/u(E) onto Ker(‘u) = ‘u'1(0). We write I = Im(u), N = Ker(u), C = Coker(u) ; from the exact sequences

(14)o

N

E ” I

o,

0——>I——J—>F—>C—>0

we derive, by transposition (Corollary to Theorem 5), the exact sequences

(15)

0——>I*—‘p—->E*——>N*—->O, 0—->C*-—>F*l>I*—>0.

Moreover, as u = j o p, ‘u = ‘p o fj; the exact sequences (15) thus define canonical isomorphisms of C“ onto Ker(‘u), of 1* onto Im(‘u) and of N* onto Coker(tu), since ‘1) is injective and fj surjective. To be precise, let y e Im(u), z e Ker(u), te Coker(u), y’ C Im(‘u), z’ e Coker(‘u), t’ e Ker('u); when 3/,

z’, t’ are canonically identified with linear forms on Im(u), Ker(u) and Coker(u) respectively, then (y,y’) = (x,y') for all x e E such that u(x) = y; (16)

300

DUAL or A vnc'ron SPACE

§ 7.5

(17)

(z, z’) = (z, x*) for all x* e E“ whose class mod. ‘u(F) is equal to z’;

(18)

(t, t’) = (s, t’)

for all s e F whose class mod. u(E) is equal to t.

In particular we derive from these results: PRorosmON 10. Let E, F be two vector spaces over the same field K and u:E —> F a linear mapping. (i) For u to be injectioe (resp. surjective), it is necessary and sufiicient that 'u be surjectioe (resp. infective).

(ii) rg(u) < rgcu) andrg = rgcu) ergo) isfim‘te.

The second assertion follows from the above and Theorem 4-. THEOREM 7. Let E be a vector space over afield K, F a subspace (J E and F' the orthogonal of F in E". (i) dim F’ 2 codimE F; for dim F’ to be finite, it is necessary and suficient that codirnE F befinite, and then dim F’ = codimm F. (ii) The orthogonal ofF’ in E is equal to F. (iii) Every finite-dimensional subspace G' of E* is the orthogonal of some subspace of E, necessarily equal to the orthogonal of G’ in E and offinite eodimension. (i) We know that F’ is isomorphic to the dual (E/F)* (§ 2, no. 6, Corollary to Proposition 9) and hence the assertion follows from Theorem 4-, since dim(E/F) = codimE F by definition. (ii) Let F1 be the orthogonal of F’ in E; clearly F C F1 and the orthogonal F1 of F1 is equal to F’ (§ 2, no. 4); the canonical linear mapping (E/F1)* —> (E/F)*, the transpose of E/F —> E/F1 is therefore bijective (§ 2, no. 6, Corollary to Proposition 10); it then follows from Proposition 10 that the canonical mapping E/F —> E/F1 is bijective, which implies F1 = F. (iii) Let G’ be a subspace of E* of finite dimension p and let F be its orthogonal in E; then codimE F < dim G’. For, if (ablag, is a basis of G', F is the kernel of the linear mapping x I» ((34, a?» from E to K: whose rank is at most p (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’ S dim G'; but on the other hand obviously G’ C F’, whence F’ = G’ (§ 2, no. 3, Corollary 4- to Proposition 4). Remark. An infinite—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)).1’

1' 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-

sional (cf. Topological vector spaces, II, §6).

301

LINEAR ALGEBRA

II

COROLLARY 1. Let 031‘“, be a finite sequence (9“ linear forms on E and let F be the subspace of E consisting of the x such that (x,x}")=0 flr1E’—>E-—>E"—>O

of right vector spaces over a field K and linear mappings and a left vector space F over K, the corresponding sequence of Z-linear mappings

0—>E’ ®KF——>E ®KF—>E' ®KF—>0 is exact and splits. As the sequence (23) splits, this is a particular case of § 3, no. 7, Corollary 5 to Proposition 7 and § 3, no. 6, Proposition 5. Because of Proposition 14, when E’ is a vector subspace of E, j :E’ -—> E the canonical injection, E’ ®KF is usually identified with a sub-Z-moa'ule of E (81: F by means of the injection j (8) IF. With this convention:

COROLLARY. Let K be a field, E a right vector space over K, F a left vector space over K, (MeowA a family of vector subspaces of E and (Ne)nsn a family of vector subspaces of P. Then (24)

aeA M“) ®K (303 N9) = (mmrslxn (M°‘ ®K NB)‘ (n

It is obviously sufficient to prove the particular case

Q”

QMM®J=£MW®JN

Clearly the left hand side of (25) is contained in the right hand side. To prove the converse, we consider a basis (f,),,eL of F. Every element of E ®x F

can then be expressed uniquely in the form A211 x, (8f,U where x, E E (§ 3, no. 7, Corollary 1 to Proposition 7); if E’ is a vector subspace of E, the relation 22:1. xA ®fi e E’ ®K F is equivalent, by Proposition 14, to x, e E’ for all

7k 6 L. To say that ,2; x, (8f, belongs to each of the Ma (8),; F thus means

that for all x e L and all or. e A, x, 6 Ma, that is x, 6 at; M, for all A e L, which proves that the right hand side of (25) is contained in the left hand side. PROPOSITION 15. If (E,,),,eL is a family of right vector spaces over afield K and (F”he,“ afamily of left vector spaces over K, the canonical mapping

m

(Qmafihm>

(§ 3, no. 7, formula (22)) is infective. 306

(MIDELXM

mam

TENSOR PRODUCTS or VECTOR SPACES

§ 7.7

We write F = g F“; the mapping (26) is the composition of the canonical mappings

(Ll Ea) ®x Fe U. (a ®xF)

and

1:1 (B; an: F) —> ELIE—la. seem); as F and the E; are vector spaces over K, this reduces to §3, no. 7, Corollary 3 to Proposition 7.

When the conditions of Proposition 15 are fulfilled, the tensor product

(E EA) (3,, (“ls—lld F“) is often identified with its canonical image in

1,] (EA ex F.)With this convention:

COROLLARY. Let F be a left vector space over K; fir every set X, the left vector space K3: ®K F is identified with the subspace qf the space Fx qf all mappings of X into F, consisting of the mappings u such that u(X) is offinite rank in F.

If (fl) is a basis of F, the element AZ]; 1),, ®f,_ of K? ®KF is identified under (26) with the mapping x +—> 2 01(x)fi. As I),~ = 0 except for the indices A 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 f, of indices A e H. Conversely, let u:X —> F be a mapping such that u(X) is contained in a finite-dimensional subspace G of F and let 001‘,“ be a basis it

of G. For all x e X, we may write u(x) = 2:1 v,(x)b‘, where the v,(x) are well determined elements of K; thus 72 mappings vuX —> K are defined and clearly u is then identified with the element ‘21 v, ® b,.

Similarly, for a right vector space E over K and a set Y, E ®K K3 is identi-

fied with a subspace of the space EY, consisting of the mappings v:Y—>E such that 0(Y) is offinite rank. More particularly, for every field K, K? ®K K3 is identified with a subspace of the space Kx” of mappings of X x Y

into K (K being considered as a (K, K)-bimodule); an element 2 u, (8 0,, where u, is a mapping of X into K and z)‘ a mapping of Y into K, is identified with the mapping (x, y) n—> 57: u,(x)v,(y) of X x Y into K.

307

LINEAR ALGEBRA

II

PROPOSITION 16. (i) Let K, L be two fields, E a [gt vector space over K, F a left vector space over L and G a (K, L) -bimodule. Then the canonical Z—homomarphism (27)

v : HomK(E, G) ®L F —> HomK(E, G ®L F)

(§ 4, no. 2, formula (7)) is iniective; it is bijective when one of the vector spaces E, F isfinite-dimensional.

(ii) Let E1, E2, F1, F2 be four vector spaces over a commutative field K; then the canonical K-homomorphism (28)

A:Hom(E1, F1) (2) Hom(E2, F2) —> Hom(E1 ® E2, F1 ® F2)

(§ 4, no. 4, formula (21)) is injective; it is bijective if one of the ordered pairs (E1, E2), (E1, F1), (E2, F2) consists affinite-dirnensional spaces.

Assertion (i) is a particular case of § 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 Ff‘ (i = l, 2) and that Hom(E1 ® E2: F1 ® F2)

is canonically identified with a vector subspace of the space (F1 ® F2)E1”E2

(II, §3, no. 1, Proposition 1); when these identifications are made and also

the left hand side of (28) is identified with a subspace of Ff‘ 8) FE“ (Proposition 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* ®x F —> HomK(E, F) (§ 4, no. 2, formula (11)) is infective; it is bijective when E or F is finite-dimensional.

This is a special case of Proposition 16 (i). COROLLARY 2. Let E be a right vector space and F a left vector space over the same field K; the canonical mapping (29)

E ‘81: F —> HomK(E*, F)

(§ 4, no. 2, formula (15)) is infective; it is bijective when E isfinite-dimensional.

This is a. special case of § 4,'no. 2, Remark 2. Remarks. (1) Let K be a commutative field, E, F two vector spaces over K,

308

§ 7.8

RANK or AN ELEMENT or A TENSOR PRODUCT

(ax) a basis of E and (bu) a basis of F; then (a,L (8 b“) is a basis of the vector K-space E ®K F (II, § 3, no. 7, Corollary 2 to Proposition 2) and therefore

(30)

dimK(E @K F) = dimK E.dim]; F.

(2) Let K be a commutative field, E1, E2, F1, F2 four vector spaces over K and u:E1 —> F1, ”3E2 —> F2 two linear mappings; then

(31)

rg(u ® 0) = 1PEG!) -rg(v)-

_

It is immediate that (u ® D)(E1 ® E2) is the canonical image of u(E1) ® 0(E2) in F1 (8) F2 and hence (Proposition 14-) is isomorphic to u(E1) ® 0(E2); the conclusion then follows from (30). (3) Under the same hypotheses as in Remark 1,

(32)

dimK(HomK(E, F)) 2 dimK E.dimK F.

If E is isomorphic to K“), Hom(E, F) is isomorphic to (Hom(K, F))I (§ 1, no. 6, Corollary 1 to Proposition 6) and hence to FI (§ 1, no. 14); as F“) is a subspace ofFI and dim(F A an injective central homomorphism and E, F two vector spaces over K. Then the canonical homomorphirm

m :A @K Hom(E, F) .9 HomA(E(A,, Fm)

(36)

(§ 5, no. 3, formula (17)) is injective; it is bijective if A or E is a finite-dimensional vector space over K.

This is a particular case of § 5, no. 3, Proposition 7. 10. MODULES OVER INTEGRAL DOMAINS

PROPOSITION 23. In a module E over an integral domain A, the ret T of non-flee element: is a submodule of E. If x and y are not free, there exist two non-zero elements 0!, 3 in A such that co: = 0 and By = 0. Then «[5 ye 0 since A is an integral domain and «(50.x + 9.31) = 0 for all A and p. in A since A is commutative, hence 1x + Ag 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

312

MODULES OVER IN'I'EGML DOMAINS

§ 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 2/62 considered as a module over itself,

3 and 4 are not free, but 3 + 4- = l is free.

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-free 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 #0 of A) E is called a torsion module. When the torsion submodule of E is reduced to 0 (that is every non-zero element of E isfree) E is called (by an abuse of language) a torsionfree module. Every submodule of a free A-module (and in particular every projective A-module) is torsion-free. The Z-module Qis torsion-free. PROPOSITION 24-. Let A be an integral domain. For every A-module E, let T(E) denote the torsion submodule of E. Letf:E —> E’ be an A-linear mapping, E and E’ being A-modules.

c T(E)(i)f(T(1_3)) (ii) Iff is zry'ective, f(T(E)) = T(E’) nf(E).

(iii) Iff is surjective and Ker(f) C T(E), thenf(T(E)) = T(E’). Assertions (i) and (ii) are obvious. On the other hand, iff is sqective and x’ E T(E’), then x’ = f(x), where x e E, and by hypothesis there exists a: 7': 0 in A such that f(ax) = ocx’ = 0; whence ax e Ker(f) and by the hypothesis there exists (5 ye 0 in A such that (3(a) = 0; as (in 7E 0, x eT(E).

COROLLARY 1. For every A-module E, E/T (E) is torsion-free. 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 ofA-modules and A-linear mappings

o —> E' -L> E —‘+ E" the sequence

0 ——> T(E’) A T(E) 1L T(E') is exact.

Ker(grr) = Ker(g) n T(E) =f(E') n T(E) =f(T(E’)) = ImUT)PROPOSITION 25. Let A be an integral domain and (E,) a family of A-modules;

then

(37)

T(@ E) = 63 T(E,). 313

II

LINEAR ALGEBRA

Let (x.) be an element of ("P E, such that x, ET(E.) for all L; then each of the x. is annihilated by an element a. aé O of A and it may be assumed that

atL = 1 when x, = 0; as the family (x.) has finite support, the element

a = U a. of A is defined and aeo; it obviously annihilates 69 x, and hence ‘

(P T(E,) C T(E? E,); the converse is immediate. If E and F are two A-rnodules, clearly T(E 83,; F) contains the canonical

NN

images of T(E) ®A F and E (8),, T(F); but examples can be given of torsionfree A-modules E, F such that T(E ®A F) ¢ 0 (Exercise 31). Note that an infinite product of torsion modules is not necessarily a torII (Z/p"Z) (,0 an integer > 1), sion module; for example, in the Z—module n: ' the element all of whose coordinates are 1 is free.

PROPOSITION 26. Let A be an integral domain, K its field offractions, E an Amodule and Em = K ®A E the vector space over K obtained by extending the ring of operators; let 1 (8 x of E into Em. (i) Every element qm is qftheform A'1¢(x)jbr A EA, 1 95 0 andxe E. (ii) The kernel of 4) is the torsion submodule T(E) of E. 1|»

Z E,¢(x¢) with £16K and (i) Every element of Em is of the form 2 = i=1 n

x, E E; for all i, there exists a, e A such that a; aé 0 and 0‘15: e A; ifa = 1:1 at, then a aé 0 and a5, = (5, e A for all i, whence, in Em,

z = a'1(acz) = at'l 1:1 35““) = “-1¢ (§i 3%)

since 4) is A—linear.

(ii) If x 79 0 is not free in E, there exists at aé 0 in A such that ax = O,

whence a¢(x) = Max) = 0 in Ex), which implies (Mac) = 0. Conversely, suppose that, for some at E E, 1 ® x = 0 in Em ; we show that x is a torsion element of E. We consider the set cm 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 = tit, B = C’l’q, Where E, n, C belong to A and Z sé 0, hence Am: C A.§’1 and A.[3 C A.§'1. Moreover K is the union of the modules

M e 921 and can therefore be considered as the direct limit ofthe direct system defined by the modules M e 922 and the canonical injections (§ 6, no. 2, Remark). Hence also, to within a canonical isomorphism, Em, = lain> (M ®A E) (§ 6, no. 3, Proposition 7) and the relation 1 (3) x = 0 in Eu!) implies that there exists an

314

MODULES OVER INTEGRAL DOMAINS

§ 7.10

M eQRsuch that 1 EM and 1 (8 x = 0 in the tensorproductM ®AE (Set Theory, III, § 7, no. 5, Lemma 1). It may fiirther be supposed (replacing if need be M by a monogenous submodule M’ :3 M of K) that M = A4”, where y E A and 7 ¢ 0. Now the mapping 8 r-—> yE is an isomorphism of M onto the AmOdule A; on the other hand, the canonical isomorphism A ®A E —> E (§ 3, no. 4-, Proposition 4-) maps E ® x to the element fix of E; thus there exists an isomorphism M ®A E ——> E which maps the tensor product E, ® x to the element

(15):: of E. The hypothesis 1 ® x = 0 in M ®A E thus implies yx = 0.

Remark. Let a‘1¢(x), B'1(y) be two elements of En!» with «EA, 36A, xe E, ye E, as ye 0. In order that on‘lcb(x) = B'1¢(y), it is necessary and sufficient that 13:: — any be a torsion element of E, for this relation is equivalent to {54>(x) = a¢(y), which may also be written (Mix — my) = 0.

COROLLARY I. If E is a torsionfree A-module, the canonical mapping ¢:E'—> Em is infective. Recall (§ 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: Ea!) —-> F such that

f = fo (b ; we shall say thatf is associated withf COROLLARY 2. Let f be an A-linear mapping of E into a vector space F over K; a" Ker(f) C T(E), the K-lz'near mappingfassociated withf is iry'ective. We write an element of Ker(f) in the form Floor), where A GA, A 9’: 0, x e E; the relation f(A‘1¢(x)) = 0 is equivalent to A'1f_((x)) = 0 in F and hence to f(x) = f(¢(x)) = 0. By hypothesis, this implies x eT(E) and hence (Mac) = 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 3 associated with g is an isomorphism (y‘ Ens) onto F.

g is injective by Corollary 2 and the hypothesis that g(E) generates F implies that g is surjective. For every A-module E, the vector space Em 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 Em: in other words (no. 2, Definition 1) the dimension over K of the vector subspace of Em generated by 42(5). When E is a torsionfree A-module, it is usually identified with its canonical image (ME) in Em. With this convention, every generating system of E contains a basis of Em (no. 1, Theorem 2). In particular:

COROLLARY 4-. Everyfinitely generated A-module is offinite rank. 315

I!

LINEAR ALGEBRA

Note that the converse of this corollary is not necessarily true; for example 9, is a Z-module of rank 1 but is not finitely generated over Z.

Recall (§ 5, no. 1) that for every linear mapping f:E -> E' (where E and E’ are A-modules),fix, denotes the K—linear mapping 1x ®f: Ea!) —-> Efx).

PROPoe'ION 27. For every exact sequence

E' -—f——> E —£-> E” q—linear mappings, the corresponding sequence of K-linear mappings fan can I Eat) "‘9 Ea!) _“>

I!

(x)

is exact.

Suppose that gang—1 (8) x) = 0, with A 6A, A 96 0, xe E; this is equivalent to 1'1 ® g(x) = O in E69 and hence also to

1 ®g(x) = M751 (81:06)) = 0; by Proposition 26, there exists a at: 0 in A such that ag(x) = 0 in E”, or also g(otx) = 0. By hypothesis, there is therefore an x' e E’ such that out = f(x’) and therefore )C‘1 18 x = fix, (oi-11‘ 1 (8 x’), which proves the proposition. COROLLARY 1. If E’ is a submodule of E, Ex, is canonically identified with a vector subspace (y’Em and (E/E’)(K, with E(K)/E(IK). It suffices to apply Proposition 27 to the exact sequence

0—>E'—>E—>E/E’—>O. COROLLARY 2. For every A-linear mapping f:E —-> F, Keem) = (Ker(f))(K,, Im(fim) = (Im(f))(K,, Coker (fin) = (Coker(f))(m to within canonical isomorphisms. In particular, fiir for) to be injective (resp. surjective, resp. zero), it is necessary and suflicient 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-moa'ule and (scold, a family of elements of E. For (xx) to be a free family, it is necessary and suflicient that in the vector K-space Em thefamily (1 ® xx) befree.

The family (xx) defines an A-linear mappingf: A‘L’ 4—) E such thatf (el) = x,‘ for all A e L ((eh) being the canonical basis of Am) and to say that (xx) is free means thatf is injective. It suffices to apply Corollary 2 to f, observing that A“) is torsion-free (Proposition 25). 316

DEFINITION or K’-s'mucrUREs

§ 8.1

§8. RESTRICTION OF THE FIELD OF SCALARS IN VECTOR SPACES K. On a set V, a Throughout this paragraph, K denotes a field and K’ a subfield of of scalars, a right on restricti right (resp. left) vector space structure over K defines, by (resp. lefl) vector space structure over K’. 1. DEFINITION OF K’-STRUGTURES

V which is PROPOSITION 1. Let V be a right vector space over K and V’ a subset g“ nt: equivale are s condition wing a vector subspace over K'. Thefollo that (a) The K—linear mapping A of van = V’ ®KI K into V such A(x’ 8) E) = x’Efor all x' e V’, 56 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 mappingf ofV into W. (e) Every basis «JV’ overK’ isa basis QfV overK. (d) There exists a basis of V’ over K’ which is also a basis ofV over K. (e) The vector K-space V is generated by V' and every subset of V’ free over K' isflee over K. We know (§ 5, no. 1) that m has a right vector K-space structure under which (x' (3) EM = 16’ ® (in) (E, 1; in K, x’ eV') 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 f’ of Van 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 A is bijective, then for every K’-linear mapping f ':V’ -> W, f’ o A"1 is the unique K—linear mapping of V into W extending f’; in other words, (3.) implies (b). Conversely, if (b) holds, there exists in particular a K-linear mapping p. of V into f, such that y.(x’) = x’ ®1 for all x’ e V'; it is immediate that y. o A = Ivan; on the other hand,

A(u(x’)) = x’ for all x’ e V’ and, as by hypothesis j :V' ——>V can be extended

uniquely to an endomorphism of V, of necessity A o p. = 1v, which completes

the proof that (a) and (b) are equivalent.

For every basis B’ of V’ over K’, the set B of elements v’ ® 1 of Van, where

12’ runs through B’, is a basis of Va!) over K (§ 5, no. 1, Proposition 4) and A(B) = B’. For A to be bijective, it is necessary that the image under A of every basis of Van be a basis of V over K and it is sufficient that it be so for a single basis ofVan (§ 1, no. 11, 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’ (§ 7, no. 1, Theorem 2), (c) implies (6). Finally, suppose that (e) holds;

if B’ is a basis ofV’ over K’, it is a free subset of V over K; on the other hand B'

317

II

LINEAR ALGEBRA

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 subfiela’ of K.

Every vector sub-K’-space V’ of V satisfi/ing 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-K’-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 K[X1, . . ., Xn], then, for every subfield K’ of K, K’[X1, . . ., Xn] 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 of V is said to be rational over K’ if it 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{)‘el 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: vii. of V to be rational over K’, it is necessary and sufficient that E, e K’ for all I. 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’-stmcture on W said to be induced by the K’-structure 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’-stmcture on V. The mapping W’ n-—>W'.K is a bijection (J the set of vector sub-K’-spaces cf V’ onto the set of vector sub-K-spaces of V which are rational over K’ and the inverse bijection is W +—> W n V'. Clearly the bijecu'on 1'1:V——>V’ ®K, 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’ o—> x’ ® 1 and W’ . K onto W’ (83K, K; the assertions of Proposition 2 are thus consequences of Definition 2 and § 7, no. 9, Proposition 19.

COROLLARY 1. Every sum and every intersection of vector sub-K-spaces of V which are rational over K’ is a rational subspace over K’. 318

RATIONALITY FOR A LINEAR MAPPING

§8.3

The assertion concerning sums is obvious. On the other hand, if (wow, is a family ofvector sub-K’-spa.ces of V’, then (‘0! WI) ®K, K = Q (W, ®r K) (§ 7, no. 7, Corollary to Proposition 14), which proves the corollary. A basis B of V over K is said to be rational over K’ if it consists of vectors which are rational over K’.

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 V1, V2 be two right vector spaces over K with K’-structures V3,

V; respectively. A K-linear mapping f:V1 —->V2 is said to be rational over K' if

f(Vi) C W:If Va is a third right vector space over K, with a K’-structure Vg’, and a K—linear mapping g:V2 ——>V3 is rational over K’, clearly g of:V1 —>V3 is rational over K’. PROPOSITION 3. Let V1, V2 be two right vector spaces over K and V1, V’g K’structures on V1, V2 respectively. V1 (resp. V2) is canonically identified with V; ®K. K (resp. V; ®K, K) (no. 1, Proposition 1). (i) The mapping f’ I—>f’ (8 1K = fig, is a hijection of HomK,(Vi, Vé) onto the set of K-linear mappings of V1 into V2 which are rational over K’; the inverse bijection associates with every K-linear mapping f:V1 -—>V2 rational over K’ the K’linear mappingf’ :Vi —> V; with the same graph as the restriction iff to V’. (ii) For every K-linear mappingf:V, —> V2 rational over K’,

f(Vi) =f(V1) n v; and five) = V; + Ker(f)(i) Clearly with the abovc identifications, if f’:V; -—>V§ is a K’-linear mapping, fig, = f’ ® lK is rational over K’ and f’ is the mapping with the same graph as the restriction offig) to V1. Conversely, iff:V1 —>V2 is a K-linear mapping rational over K’ and f’: i —->V§ has the same graph as the restriction off to V1, f and fig) coincide on V1, which is a generating system of V1 over K, hencef = fig).

(ii) If f =f' ‘8 1x, then f(V1) =f(Vi ®x' K) =f'(Vl) ®x' K and, as

f’(Vi) C V3, f(Vi) = f’(Vi) = f(V1) n V; (§ 7, no. 9, Proposition 19); the fomulaJ—"1(V;) = V; + Ker(f) follows immediately. 319

H

LINEAR ALGEBRA

COROLLARY I. In the notation ofProposition 3, Im(f) = (Im(f”on:

Kerm = (Kamila,

Cokerm = (COkerCf’Dmr

In particular, forf to be infective (resp. suriective, zero), it is necessary and suflicicnt thatf’ be so. Iff is bijective, its inverse mapping is rational over K’. This is a particular case of § 7, no. 9, Corollary to Proposition 18. COROLLARY 2. Let f V, —> V2 be a K-linear mapping rational over K’. For every

vector sub-K-space W1 of V1 (resp. W2 Qf V2) rational over K’ f(W1) (resp

f (W2)) is a vector sub-K-space of V2 (resp. V1) rational over K’. In the notation of Proposition 3, for every vector sub-K’-space W; of V5, fég,(W1 ®K, K) = f'(Wi) ®K1 K; whence the assertion relating to W1 no. 2, Proposition 2). On the other hand, let W; be a vector sub-K’-space of V’2 and let g’ be the canonical K’-linear mapping V; aVQ/Wg; then

_;”(W§) = Ker(g’ of’) ; hence, by Corollary 1, —1

—- 1

flin(Wé ‘81:, K) =f'(Wé) ®x' K, whence the assertion relating to W2. Let V1, V2 be two right vector K-spaces with K’-structures V1, V3 respectively. It is immediate that V; x V; is a K’-structure on V1 x V2, called the product of the K’-structures V’l and Vé.

PROPOSITION 4. For a K—linear mapping f:V, —>V2 to be rational over K’, it is necessary and sufl'icient that its graph I‘ be rational over K' for the product K'-structure on V1 x V2.

Let g be the mapping x1 n—>- (x1,f (x1)) from V1 to V1 x V2; this is a Klinear mapping such that I‘ = g(V1); iffis 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 V1 x V2; it follows immediately from the definitions that the restrictions p1, p2 to 1" of the projections prl, pra, 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 isf = pg 0 91. 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 linearforms x* e V*, rational over K’, as the linear mappings of V into Kd, rational over K' for the

320

APPLICATION TO LINEAR SYSTEMS

§ 8.5

K’-structures on V and Kd. 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

(1)

V’* is K®K.V’* —"—> V*

where d>(x’*) = 1 ® x’* and u(E ® 74”“) is the linear form 31* On V such that y* (x') = Ea’ + v(x —a) 329

II

LINEAR ALGEBRA

is an affine mapping of E into E' whose associated linear mapping is v. 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 u' the linear mapping of T’ into T” associated with u’. Clearly u' o u is an affine mapping of E into E"; moreover, for a e E and t e T,

u’(u(a + t)) = u’(u(a) + ”(0) = u’(u(a)) + WW» and hence v’ o v 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 1) be so, and u‘1 is then an affine mapping whose associated linear mapping is 0'1. In particular, the affine bijections of E onto itself form a group G, called the (fine group of E. The mapping which associates with u e G the linear mapping 0 associated with u is, by the above, a homomorphism of G onto the linear group GL(T). If u is a translation, 1) 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 »—> utu'1 of T (where t is identified with the translation 5: r—> x + t) is the linear mapping 11 associated with u. For x e E and t e T, by definition

.7: + utu‘1 = u(u‘1(x) + t) = "(u—1M) + ”(t) = x + ”(t) and hence utu‘1 = v(t). Let a e E and G, be the subgroup of G consisting of the u e G such that u(a) = a. If E is identified with T by taking a as origin, G, is identified with

GL(T). Every 14 e G can be expressed uniquely in the form u = tlu1 (resp. in the form u = uztz), where ul, u2 are in G, and 1:1, 1:2 in T: for, writing 1:, = u(a) — a, u'lt1 6 Ga, whence the existence of u1 and t1; the existence

of uz and t2 is obtained analogously. The uniqueness follows from the fact that Ga n T reduces to the identity element of G. Moreover t1u1 = “10‘1- 1t1u1) whence u2 = ul, t2 = u; 1tlul. Finally, the linear mappings associated with

u and ul are the same and hence, if as above G, is identified with GL(T), u1 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, §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 330

DEFINITION or PROJECTIVE SPACES

§ 9.5

exists an affine mapping a of E into E' such that u(V) = V’: taking as origins in E and E’ points ofV 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 § 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 afiine space of dimension 1. An affine mapping of an affine space D (over K) into the affine space K is also called an affine linearfimction (or an afiineficnction). If a point a is taken as origin in E, every affine function on E can then be written uniquely as xr—> at + v(x), where a e K and a is a linear form on the vector space E thus obtained; the affine fimctions 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 E satisfying the equation u(x) = A is a hyperplane; conversely, for every hyperplane H in E, there exists an affine function no on E such that H = 1:01(0)

and every affine function u such that H = 31(0) is of the form “all, where neK (§ 7, no. 5, Proposition 11). If u is an afiine function on E, the hyperplanes with equations u(x) = a: and u(x) = [3 are parallel. 5. DEFINITION OF PROJECTIVE SPACES

DEFINITION 4. Given a left (resp. right) vector space V over afield K, the left (resp. right) projective space derivedfiom V, denoted by P(V), is the quotient of the complement V — {0} of {0} in V by the equivalence relation A(V) “there exists‘ A aé 0 in K such that y = M (resp. y = xx) between x and y in V — {0}.

When V = K3“, we also write Pn(K) instead of P(K;‘”) and An(K) instead ofA(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 canonically 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 — l is called the dimension of the projective space P(V) if n is finite, and the cardinal n otherwise; this cardinal is denoted by dimx 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 projective space of dimension 1 (resp. 2) is called a projective line (resp. projective plane).

Henceforth we shall only consider left projective spaces. 6. HOMOGENEOUS COORDINATES

Let V be a vector space of finite dimension n + 1 over K, P(V) the projective space of dimension n derived from V and (masks, a basis of V. Let 1: denote the canonical mapping of V — {0} onto the quotient set P(V). For every

331

LINEAR ALGEBRA

II 1!

point x = ‘20 file; of V — {0}, (£0, £1, . . ., Em) is called a system qf homogeneous

coordinates of the point 1t(x) with respect to the basis (a) of V. Every system (E0 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 (er); for two such systems (54), (£1) to be systems of homogeneous coordinates of the same point of P(V) with respect to the same basis (e1), it is necessary and suflicient that there exists an element A # 0 of K such that E; = XE, for 0 S i S n.

This definition is immediately generalized to the case where V is infinitedimensional. 1!

Given another basis (3,) of V such that eI = ’20 1,15, (0 s i < n) and a system (5,) of homogeneous coordinates of r:(x) with respect to the basis

(e‘), for a system (it) of n + 1 elements of K to be a system of homogeneous coordinates of 7c(x) with respect to the basis (5‘), it is necessary and sufficient that there exist A aé 0 in K such that 7|

A21 = ’20 £11”

for 0 g 1. < n.

In particular, if e, = 7.2, with y, 9e 0 (o g i g n), then 5, = ham where y. 95 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 prqjective linear variety (or simply a linear variety 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 prqjective hyperplane (or simply a hyperplane); if P(V) is of finite dimension n 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 (e,)o“‘,, 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

(4')

332

,

:20 Eiau=0

(1 (x) = 1c(l, 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(Vo) called the hyperplane at iry‘inity ofP(K, x V) (or ofV, by an abuse oflanguage) ; its points are also called the “points at infinity” of P(Ks x V) (or of V). If (a,) is a basis of V and in K, x V the basis is taken consisting of the elements e, = (0, a.) and the element e, = (l, 0), the points at infinity in P(K, x V) are those whose homogeneous coordinate of index to 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 M = 1::(M2) of the vector subspace M2 of K, x V generated by the affine variety M1 = {l} x M of K, x V. More precisely, if (a) is an aflinely free system of M generating M, the elements (1, a,) form a basis of M2 and therefore M is just the projective linear variety generated by (M); if M is finitedimensional, M has the sa_n_ie dimension as M. The complement of MM) in M is the intersection of M and the hyperplane at infinity and is equal to . the canonical image 1:(Mo), where Mo = {0} x D. Conversely, let N be a projective linear variety not contained in the hyperplane at infinity and let R = n1(N) ; R n V1 is an affine linear variety of K x V of the form {1} x M, where M is an affine linear variety of V and it is immediately seen that N is the affine linear variety M generated by M . (MTIiere is therefore a one-to-one correspondence between the affine linear

varieties of V and the projective linear varieties of P(K, 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 afl'ine linear varieties in question have the same points at infinity). 9. EXTENSION OF RATIONAL FUNCTIONS

If the results of no. 8 are applied to the vector space V = K8 of dimension 1, it is seen that there exists a canonical injection ([2 of K, into the projective line P1 (K) = P(K, x K,); for all E e K, Mg) is the point with homogeneous

coordinates (l, E) with respect to the canonical basis (§ 1, no. 11) of K, x K,. The complement of MK) in P1(K) consists of the single point with homogeneous coordinates (0, l) with respect to the above basis; it is called the

“point at infinity”. P1(K) is also called the projective field associated with

K and denoted by K, the point at infinity in K being denoted by 00.

*Consider in particular the case where K is a commutative field and let

334

PROJEC'I'IVE LINEAR MAPPINGS

§ 9.10

fe K(X) be a rational function in one indeterminate over K (IV, § 4); if f 76 0, there is a unique expressionf = cap/q, where an e K* and p and q are two

relatively prime monic polynomials (VII, § 1); let m and n be their respective degrees and let 1 = sup(m, n). We write

91(TaX) = T'MX/T),

91(T, X) = T'q(X/T);

p1 and q1 are two homogeneous polynomials of degree r over K such that p(X) = mg, X), q(X) = q1(l, X). Hence, for every element 56K which is not a zero of q(X),f(E) = ap(E)/q(§) is defined and we may write

f(5) = 0471(1, «9/410, E) = «1110» 10/910» A5) for all A aé 0 in K. Consider then the mapping (7]: E) H (9103: E): P101: 5))

of K2 into itself; it is compatible with the equivalence relation A(K3) and therefore defines, when passing to the quotients, a mappingf of K into itself which coincides with E»f(E) at the points where this rational function is definléd; it is said, by an abuse of language, thatf is the canonical extension of f to . For example, iff= l/X, thenf'(0) = 00 andf(oo) = 0; if f= (aX + b)/(cX + d) with ad — bc ¢ 0, thenf(—d/c) = oo,f(oo) = a/c ifc ¢ 0,f(oo) = 00 if c=0. Iff=aoX"+ +4.. is a. polynomial of degree n> 0, then f(°°) =°°-# 10. PROJEGTIVE LINEAR MAPPINGS

Let V, V’ be two left vector spaces over a field K, f a linear mapping of V —1 into V’ and N = f(0) its kernel. It is immediate that the image under f 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 prcy'ectz've linear mapping (or, simply, a projective mapping); although it is defined on P(V) — P(N) and not on P(V) (when N 75 {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 (afimm (bu)ueM are given in V and V' respectively, a projective mapping of P(V) into P(V’) maps a point of P(V) with homogeneous

335

LINEAR ALGEBRA

II

coordinates in (AEL) to a point of P(V') with a system of homogeneous coordinates 1)" (p. e M) of the form

m = 2 arm

(6)

he];

(ameK).

The centre ofg is the linear variety defined by the equations

A; gm“ = o

(a e M).

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

dimg(M) + dim(M n C) + 1 = dimM

(7)

(§ 7, no. 4, formula (12)). If M’ is a linear variety of P(V’),:gl(M’) U C is a linear variety of P(V) and

dim(:g1(M’) u C) = dim C + dim(M’ n g(P(V))) + 1.

(a)

It is said, by an abuse of language, that g1(M’) U C is the inverse image of M’ under g. As the values taken by a linear mapping on a basis (2.) 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 11-(e‘). But (even when g

is everywhere defined) giving g(rc(et)) does not determine g uniquely (Exer-

cise 10).

The composition of two projective mappings which are bijections is a projective mapping; so is the inverse mapping of such a bijection. The bijective 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 PGLn(K) or PGL(n, K) instead of PGL(K;‘). 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 irgfinity in P(V) ; the complement of

H in P(V) is then identified with an afline space whose translation space is

336

PROJECTIVE SPACE STRUCTURE

§ 9.11

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 (I) ofbijection: ofsubsets ofthe projective space P(Kfi") onto E satisfying the following axioms:

(EPI) The set of definition of every mappingfe (I) is a linear variety ofP(Kgm). (EPn) For every ordered pair of elementsj; g of (I) defined respectively on the linear

varieties P(V) and P(W), the bijection h = glo f of P(V) onto P(W) is a projective mapping. (EPm) Conversely, if fe (I) is defined on the linear variety P(V) and h is a bijective projective mapping of P(V) onto a linear variety P(W) C P(Kfim’), then fo h'1 e (I). Let E be a set, (V,_),LEL a family of vector spaces over K and suppose given for each A e L a bijectioa of P(Vh) onto E such that, for every ordered pair —1

of indices 1, [1,, f,L of; is a projective mapping of P(Vu) onto P(VA). Then we can define on E a projective space structure with respect to K as follows: let (emu be a basis of a space V; and write a,, = f,‘(1c(e‘)); let [1‘ be the element

of index a. in the canonical basis of K9?” (§ 1, no. 11). The relation v. aé K

implies b. 96 b" because of the hypothesis that fl is bijective; hence the b‘ form a basis of a vector subspace Wo of K9”) and there therefore exists a bijective projective mapping h of P(Wo) onto P(VA) such that h(n(bl)) = n(e.) for all LG I. If (I) is taken to be the set of all bijective projective mappings

fi 0 h o g'l, where g runs through the set of all bijective projective mappings

P(W) C P(K§E)), it is immediately verified that (I) satisfies axioms (EPI),

(EPH) and (EPm). It is moreover immediate that (I) depends neither on the

choice of index A e L, nor on the choice of basis (a) in VA, 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 bijection f e (I) defined on P(V) C P(Kgm’), .. 1 .

f (M) is a linear variety in P(V) in the sense ofno. 7 (this property then holds for allfe (1)). It follows from the above that every linear variety in a projective space has canonically a projective space structure.

A projective space E is said to be Qf dimension n if, for all fe ¢,}1(IE) is a linear variety of dimension n (it suffices that this hold for one mapping fe (D). 337

II

LINEAR ALGEBRA

§ 10. MATRICES 1. DEFINITION OF MATRICES

DEFINmON 1. Let I, K, H be three sets; a matrix 9f type (I, K) with elements in H (or a matrix (y’ type (I, K) over H) is any family M = (mm)(,_,‘,51 xx of elements of H whose indexing set is the product I x K. For all 1. e I, the family (muggex is called the row of M if index i; for all x e K, thefamily (mm)“=.I 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 identified with the product HI " K. The names “row” and “column” arise from the fact that, in the case where I and K are intervals [1, 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 4 columns (arranged vertically): "111

mm

. . .

mm

"131

ӣ33

. - .

mm,

mp1

mp2

...

mm

When p and q are explicit integers sufliciently small for it to be practicable, 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 determined by its place in the array; for example, when we speak of the matrix a

b

d

e f

o

we mean the matrix (m,,)1 Iz’h” of H” x H’ to G; it follows immediately from the definitions that (4)

'

‘(M’M') = tM".‘M’

where the product on the left (resp. right) hand side is calculated viaf (resp. via f0). When H' and H" are themselves commutative groups (written additively) and f is Z—bilinear (§ 3, no. 1), the distributivity formulae (5)

(M’ + N’)M” = M’M' + N’M” MI(M”

+

N”)

=

MIMI

+

MIN”

are immediately verified, the indexing sets being such that the sums and products appearing are defined. Now let H1, H2, H3, H12, H23 and H be commutative groups (written addi-

tively),f12:H1 x H2 —> H12, f231H2 x H3 —> H23 mappings and fail X H‘s—9H,

fi:H1 X HaséH

Z-bilinear mappings; suppose further that, for all x‘ e H‘ (i = l, 2, 3) fa(f12(x1, x2), x3) =f1(x1:f23(x2: ’53))

(which may also be written as above (x1x2)x3 = x1(x2x3)) ; then, if M’ = (m;,), M" = (m3), M” = (mg) are matrices over H1, H2, H3 respectively,

(6)

(M’M')M"’ = M’(M'M")

when the products on the two sides (calculated respectively via f“, f3, fag and f1) are defined; for

Z (2 m:,m:.)m::. = Z Z (mammmr. = 2} ,2 m:.(mzmr.)

= Z "#4? "1%) 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 precise: (a) Suppose H = (“)5q G“ where each G“. is a commutative group

written additively; then the sum M + M' may be defined when, for each ordered pair (L, K), m“‘ e G“: and mi: 6 G“.

340

MATRICES OVER A RING

§ 10.3

(b) Let I, K, L be three sets with K assumed finite and let H’ = a “LEJI x x Hm: I

H” = (MDEJKHHQ, H = (LMUIxLHu be three sets; suppose that each H" is a commutative group written additively and for each triple (i, k, I) let flkl:Hlk X Hit—>111:

be a mapping. Then if M, = (mlk)u,k)e1xi xy in A (or, as is also said, will be “calculated in A”). Then we have (no. 2) the associativity and distributivity formulae

(7)

(XY)Z= X(YZ)

3 ()

X(Y+Z)=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 E", (resp. Eh, Efi) are the matrix units in A!” (resp. AK“, A1“) respectively, with I = [1,12], K = [1, g], L = [1, r], we obtain the formulae

EflcEllfl = Eil-

Let A° be the opposite ring of A and let a a: b (=ba) denote the product of a and b in A0 ; then, for two matrices X, Y over A whose product is defined,

(10)

'(XY) = We ‘X 341

LINEAR ALGEBRA

II

where on the right hand side ‘Y and ‘X are considered as matrices with elements in A°; when A is commutative, then

(11)

‘(XY) = ‘Y.‘X

PROPOSITION 1. Let A, B be two rings and M = Gum)“. k)EIXK and

M, = (mlk)(l,k)eIxK

two matrices with finite indexing sets over an (A, B)-bimodule G. Suppose that for every matrix unit L = (40.51 with one row and elements in A and every matrix unit C = (keK with one column and elements in B, L.M.C = L.M’.C (the products

being calculated via the external law: of the (A, B) -module G); then M = M’. IfL is taken to be the matrix unit ((1,) with a, =1, a, = 0 for: qé i, and C

the matrix unit (1),) with b,, = l, b, = 0 for t aé k, the products L.M.C and

L.M'.C are matrices with a single element respectively equal to m", and m,’,,. Let A, B be two rings and azA —> B a homomorphism.

For every matrix M = (mm) over A, we shall denote by o‘(M) the matrix

(c(m.,:)) over B; clearly 6(aM) = a(a)c(M), 6(Ma) = 6(M)cr(a) for aeA, also 60M) = ”(o-(MD and 6(M + M’) = 6(M) + c(M’) (12) 6(MM') = a(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 0' is denoted

by x I——> x", we write M" instead of 6(M).

Consider in particular an anti-endomorphism e- of A, that is a homomorphism

of A to the opposite ring A°, or a mapping of A into itself such that a(aa’) = c(a')c(a) 0(a + a’) = 6(a) + c(a’,),

for all a, a’ in A; then, for two matrices M, M ' over A whose product MM ' is defined,

(13)

6(MM’) = ’(6(‘M’)-°('M))

where the products on the two sides are calculated in A; this follows immediately from (10) and (12). 4. MATRICES AND LINEAR MAPPINGS

Let A be a ring and E a (right or left) A-module admitting a basis (90:51-

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 x, (i 61) of x with respect to (e;) (§ 1,

no. 11); in calculations it will sometimes be convenient, in order to re-

34-2

MATRICES AND LINEAR MAPPINGS

§ 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 (xm). We now consider two (left or right) A-modules E and F with bases (ems; and (f,,)keK respectively; let (ff) be the family of coordinateforms corresponding to (fit). For a linear mapping u of E into F, we shall define the matrix of u with respect to the bases (e‘), (3%) in each of the following cases:

(D) E and F are right A-modules, u is A-linear. (G) E and F are left A-modules, u is A-h'near. 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

bum (en), (fr) is the "Wm M(u) = (“Woman such that (14)

.

um =fx=*(u(ea))

which is written respectively as

(14 D)

um = 03,1400)

(14' G)

"m = ("(et)nflc*>'

The column of M (u) of index i is therefore equal to M (u(e,)).

Clearly if u, o are two linear mappings of E into F and M(u), M(0) their .matrices with respect to the same bases, then (15)

M(u + o) = M(u) + M(o)

and

(16)

MM) = YM(u)

for every element 7 of the centre I‘ of A. In other words, once the bases (e1), (fk) are fixed, the mapping u -—> M (u) is a F-module isomorphism of HomA(E, F) onto a subset of the set AK”, equal to AK"I if K isfinite. PROPOSITION 2. Suppose I and K are finite. For every element e, the matrix M (u(x)) with respect to the basis (fl) is given by theformula

(17 D) (17 G)

M(u(x)) = M(u) .M(x) ‘M(u(x)) = ‘M(x) .‘M(u).

We verify for example (17 G). Let x = me” u(x) = zykofk With 37:0 EA: yko EA; then “(7‘) = "(Z ”1091) = 2‘: xwu(‘i) = ‘21; xtoukiflc; whence

343

II

LINEAR ALGEBRA

3],“, = 2‘: xwum. In order to bring the two indices 1' along side one another, we consider the transpose matrices ‘M (x) = (xal), where x6, = x“, and

‘M(u) = ('41:):

where 14;, = a“; then gm, = Z xgwgk and the right hand side is the element of index k of the matrix with one row ‘M (x) .‘M (u), whence (17 G). 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, (e,),el, (flakeg, (gamb respective finite bases ty’ E, F, G, u:E ——> F, v:F ——> G two linear mappings, M (u) the matrix g" a relative to the bases (a), (fk), M (v) the matrix qf v relative to the bases (jg), (g,) and M (v o u) the matrix qf v o a relative to the bases (at): (g1); the"

(18D)

M(vou) = M(v)M(u)

(18 G)

’M(v o u)= tM (u)‘M (v)

We prove for example (18 G). For all x e E, by (17 G):

*M E mapping every an e A to Jean (§ 2, no. 1). It is immediate that the matrix M(0,,) with respect to the basis 1 of A4 and the basis (e,) of E is just the matrix M(x); similarly M(0am) = M(u(x)) and formula (17 D) can therefore be considered as a translation of the relation 61“.”) = u 0 0.7"

PROPOSITION 3. Let E, F be two right (resp. lefl) A-modules and (301:1: (fiche: finite bases qf E and F respectively. For every linear mapping a qf E into F, let M (u) be the matrix qfu with respect to the bases (a) and (fi). Then the matrix qf‘u : F* —> E* with respect to the dual bases (f,f) and (e?) is equal to ’M (u). E is canonically identified with its bidual E** and (a) with the dual basis of (ef); then (supposing for example that E and F are right modules) (adj-It): at) = (fit: “(30):

whence the proposition.

34-4-

MATRICES AND LINEAR MAPPINGS

§ 10.4

Remarks. (2) Let E and F be two left A-modules with bases (e016! and (flake; respectively. For every A-linear mapping u:E—> F, by (14- G), u(e,) = 2: umfk; these relations can also be interpreted by saying that the column matrix (u(e,)‘ex) with elements in F is equal to the product ‘M(u) . (fk),

where (fake; is considered as a column 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 o : A —> B a ring homomorphism.

In the notation of Proposition 3, (e, (8) l) and (j;‘ 69 l) are respective bases of Ea” = E ®AB and Fm = F ®AB (§ 5, no. 1, Proposition 4-); moreover, if (e?) and (fi') are respectively the dual bases of (e,) and (1;), then (ef ® 1) and (fl;I ® 1) are respectively the dual bases of (a. ® 1) and (fl. ® 1) (§ 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 (fl) and the matrix of the B-linear mapping u ® 1 with respect to (e, ® 1) and (f;‘ (8 1). It follows from § 5, no. 4, formula (20) that

M(u F is the linear mapping such that the matrix M (u) with respect to the bases (at) and (fk) are equal to A = (ammoexxp This

matrix is called the matrix of the system of linear equations (19). Recall (§ 2, no. 8, Remarks 2 and 3), that, writing c, = kak” the system (19) is equivalent to the unique vector equation

(20)

Z qx, = b,

and as at 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 = (k) with one column is a linear combination of the columns of the matrix A. We leave to the reader the task of formulating the analogous definitions and remarks for systems of lefi: linear equations.

345

LINEAR ALGEBRA

II

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 (Ema; of submodules. For all x e E, let x = “21x, with x, 6E, for all i e I; we shall say that the column matrix M (x) = (950451 with elements in E is the matrix qf x with respect to the decomposition (E0,eI of E as a direct sum.

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 (Fk)lceK of submodules. For all u e Hom(E, F) and all x, e E” let u(x,) = Z uk,(x,) with uk,(x,) e Fk for all k e K; then u,cl e Hom(E,, Fk); we shall say that the matrix

M (u) = (uk,)(k',)exx1 of type (K, I) with elements in the set H the sum of the Hom(E,, Fk) is the matrix qf u with respect to the decompositions (E,) and (Fk) Qf E and F as direct sums.

With these definitions, it is obvious that if u, v are two A-linear mappings of E into F then, for matrices with respect to the same decompositions as

direct sums

(21)

M + v) = me + Me),

MW) = me)

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(“(#0) = M(u) -M(x)

where M(u(x)) is the matrix of u(x) with respect to the decomposition (Fk), the product on the right hand side of (22) being calculated for the mappings (t, z) »—> 15(2) of Hom(E,, Fk) x E, into Fk (no. 2, Remark). Let G be a third A-module, the direct sum ofa family (Gl)leL ofsubmodules, so, that there corresponds to every A-linear mapping azF —-> G a matrix M(v) = (0",) with respect to the decompositions (Fk) and (G,). If I, K and L arefinite, then

(23)

M(v o u) = My) .M(u)

where the left hand side is the matrix (wu) of w = v o u with respect to the decomposition (E,) and (G,) and the product on the left hand side is calculated for the mappings (t, s) rates of Hom(Fk, G,) x Hom(E,, E) into Hom(E,, G,) (no. 2, Remark). This is just formula (32) of § 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 E’f (resp. Fz‘) (§ 2, no. 6, Proposition 10). Then it is immediately verified that the matrix of‘u with respect to the decompositions (Fg‘) and (E?) is just (511c 05K“. 34-6

MATRIX or A SEMI-LINEAR MAPPING

§ 10.6

Suppose now that I and K are finite and further that each of the E, (resp. Fk) admits afinite basis. It amounts to the same to say that E (resp. F) admits a basis (c,),en (resp. (fshes) and that R (resp. S) admits a partition (R9,eI (resp. (Sk)keK) such that for all ie I (resp. k GK), (e,),em is a basis of E, (resp. (f Leak IS a basis of FR). Then, if X: M (u) is the matrix of u with respect to the bases (4),.ER and (f ),e 5, the matrix X“: M (um) with respect to the bases ((3,)r E a. and (f) fish is just the submam'x of X obtained by suppress-

ing the rows of index 3 ¢ 3,, and the columns of index r ¢ R.. Thus we define a one-to-onc correspondence (24)

X "’ (Xm)(k. 06K x1

between the set of matrices of type (S, R) with elements in A and the set of matrices qatriccs (XM)(,,, Dex x I of type K x I, where each X,“ is a matrix over

A of type (Sk, R,). Suppose that further G admits a finite basis (£19m 'r and that T = (T0151, is a partition of T such that, for each 16 L, (gt)te'1‘g is a basis of

G,; let Y = M(v) be the matrix of v with respect to the bases (jibes and (gamma Y": = M (”m) that Of ”Us With TCSPCCt to the bases (.fs)sesk and (gown:

Z = M(w) the matrix of w = on u with respect to the bases (4),“ and (gown. and Zu= M (wu) that of w" with respect to the bases (4),“, and

(g,),e.n; then it follows from (23) that the submatrices Z" of Z: YX are given by

(25)

Zn = Z 1'.k

in other words, the one-to-one correspondence (24-) tramfirms 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 submatrices Z" of the product YX are calculated thus, this product is said to be

carried out “in bloc ’ This name arises from the fact that, when I = [1, p] and K = [1, q), the table representing the matrix X is envisaged as divided into “blocks” forming an “array of matrices” X11

X12

. o .

X1?

X21

X22

---

X21»

X“

a

...

X,"D

which is considered as a symbol denoting X when p and q are specific integers sufficiently small for this to be practicable. 6. MATRIX OF A SEW-LINEAR MAPPING

Let A, B be two rings, a:A —> B a homomorphism of A into B, E a right

(resp. left) A-module with basis ((3,),eI and F a right (resp. left) B—module with

347

II

LINEAR ALGEBRA

basis (faker Let u:E—->F be a semi-linear mapping relative to a and u(e,) = kaficu,“ (resp. u(e,) = 12;: umfic), where the uk, are therefore elements of B; by definition, the matrix M (u) = (um) of type K x I is also called the matrix qf u with respect to the bases (e,) and (fi). 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 arefinite,

M(u(x)) = M(u) .a(M(x))

(26 D) (resp.

*M(u(x)) = «we» -*M(u)>.

(26 G)

Let C be a third ring, 1:3 -—> C a homomorphism, G a right (resp. left) C-module with basis (g,),e1. and v a semi-linear mapping of F into G relative to 1-; if M(v) is the matrix ofz) with respect to (fk) and (g,) and M(vo u) the matrix of v o u relative to (e!) and (g,), then, if I, K and L are finite, M(a o u) = M(v) .1-(M(u))

(27 D)

(resp.

‘M(v o u) = 1(‘M(u)) .‘M(v)).

(27 G)

To show for example (27 D), note that for all x e E, by (26 D),

M 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: AZ —> A? is a linear mapping whose matrix is Xwith respect to the respective canonical bases ((2,) and (f,) ofA: and '5', then X’ is the matrix ofu with respect to the bases (e;) and (f;) such that Q is the matrix of passage from (q) to (of) and P'1 the matrix of passage from

(J?) to (fl)Exampler of equivalent matrices. (1) Two matrices X = (x,,) and X’ = (14,) with m rows and n columns “difer only in the order of their rows” if there exists a permutation a of the interval (1, m] of N, such that for every ordered pair of indices (i, j ), xi, = atom, (we also say that X’ is obtained by performing the permutation o’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 6‘1 (cf. no. 7, Example II). Similarly X and X’ are said to dilfir only in the order of their columns if there exists a permutation r of [1, n] such that xi, = arm“, for every ordered pair of indices (1', j), X and X’ are also equivalent, for X’ = XQ where Q

is the matrix of the permutation 1:. Note that in the above notation P is the matrix of passage from a basis (f,)1‘,‘,,, to the basis (fir'lm)1 A” are two graded homomorphisms of graded rings

oftype A, so is h’ o h: A——>A’; for a mapping h: A—>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 homomor366

GRADED SUBMODULES

§ 11.3

phisms can be taken as the morphisms of the species of graded ring structure of type A (Set Theory, IV, §2, no. 1). Similarly, if u: M —> M’ and u': M' —> M” are two graded homomorphisms ofgraded A-modules of type A, of respective degrees 8 and 8’, u’ o u: M —> M” is a graded homomorphism of degree 8 + 8'. If 8 admits an inverse —— 8 in A and u: M —> 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 morphisms of the species of graded A-module of type A. But a bijective graded homomorphism u: M—>N of degree #0 is not a graded A-module isomorphism if M and N are non-zero and the elements of A are cancellable. Examples. (6) If M is a graded A-module and M00) is a graded A-module obtained by shifting (no. 2, Example 3), the Z-linear mapping of M00) into M which coincides with the canonical injection on each M“ ,‘o is a graded homo-

morphism of degree 7&0 (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 1—) ax of any graded A-module M is a graded homomorphism of degree 8. Remark (3) A graded A-module M is called a gradedfree A-module if there exists a basis (mg,GI of M consisting of homogeneous elements. Suppose it is and A is a

commutative group; let x, be the degree of ml and consider for each I the shifted A-module A( — 1,) (no. 2, Example 3) ; ife, denotes the element 1 ofA considered

as an element qfdegree A, in A( — t), the A-linear mapping u: @I A( —A,) —> M such that u(e,) = m, for all I, is a graded A-module isomorphism; Assuming always that A is a commutative group, now let N be a graded A-module, (m), 91 a system of homogeneous generators of N and suppose that n, is

ofdegree up Then the A-linear mapping 0: .6591 A( — Pa) -—> N such that u(e,,) = n. for all L is a surjective 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 suljective homomorphism of the above type with I finite. 3. GRADED SUBMODULES

PROPOSITION 2. Let A be a graded ring of type A, M a graded A-module oftype A,

(MA) its graduation and N a sub-A-module of M. Thefollowing properties are equivalent: (a) N is the sum of thefamily (N n Mr)“; A. (b) The homogeneous eomponents qf every element of N belong to N.

(c) N is generated by homogeneous elements. 367.

II

LINEAR ALGEBRA

Every element of N can be written uniquely as a sum of elements of the MA, and hence it is immediate that (a) and (b) are equivalent and that (a) implies (c). We show that (c) implies (b). Then let (mm: be a family of homogeneous generators #0 of N and let 8(t) be the degree of x.. Every element ofN can be

written as Z atx. with a. e A; if at, A is the component of a, ofdegree A, the conclusion follows from the relation 2:1(uZA awn) = AZACtHg-k awn)’ Remark (1) In the above notation, the relation «Zr age. = 0 is therefore equivalent

to the system of relations u + «20.: a. awnL = 0. When A is a. group, these relations can be written WI a“ hamx, = 0.

When a submodule N of M has the equivalent properties stated in Proposition 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 l. [fN is a graded submodule of M and (sq) is a generating system ofN, the homogeneous components of the x,form a generating system of N. COROLLARY 2. IfN is afinitely generated submodule ofM, 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 750. A graded submodule of A, (resp. Ad) is called a graded left (resp. right) ideal ofthe graded ring A. For every subring B of A(B 0 AA) (B n A) C B n AM“; if B is a graded sub-Z-moa'ule 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 ifN (resp. B) is a graded sub-A-module of M (resp. a graded subring of A), the canonical injection N—>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 (MQLGA 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 NA = MA n N, (ML + N) /N is identified with MA/N,‘ and it follows from Proposition 2 and

§ 1, no. 6, formula (26) that M/N is their direct sum. Moreover, A,(Mu + N) c AnMu + N c MM“ + N and hence A,,((Mu + N) /N) C (MAW + N)/N, which establishes our asser-

368

GRADED SUBMODULES

§ 11.3

tion. The graduation «M; + N) [N)1“ 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 graduation.

If b is a graded two-sided ideal ofA, the quotient graduation on A/b is compatible with the ring structure on A/b; the ring A/b with this graduation is called the quotient graded ring ofA by b; the canonical homomorphism A —-> All» 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 qf degree 8. Then: (i) Im(u) is a graded submodule of N. (ii) If 8 is a regular element ofA, Ker(u) is a graded submodule of M. (iii) If 8 = 0, the bijection M/Ker(u) —-> Im(u) canonically associated with u is an isomorphism ofgraded modules. Assertion (i) follows immediately from the definitions and Proposition 2(c). If x is an element of M such that u(x) = O and x = z x; is its decomposition

into homogeneous components (where x,L is of degree A), then

2 um) = u(x) = o and u(x,.) is ofdegree A + 8; if 8 is regular the relation A + 8 = p. + 8implies A = (1., hence the u(x,v) are the homogeneous components ofu(x) and necessarily u(x,() = 0 for all A 6A, 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 ofthe quotient graduation; whence (iii) when 8 = O. COROLLARY. Let A, B be two graded rings (y‘type A and u: A ——> B a graded homo-

morphism ofgraded 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 ofgraded rings.

It suflices to apply Proposition 3 to u considered as a homomorphism of degree 0 of graded Z-modules.

PROPOSITION 4-. Let A be a graded ring oftype A and M a graded A-module qftype A. (i) Every sum and every intersection of graded submodules {If M is a graded submodule. (ii) If x is a homogeneous element of M of degree a which is cancellable in A, the annihilator of x is a graded left ideal of A. (iii) If all the elements of A are cancellable, the annihilator of a graded submodule

of M is a graded two-sided ideal if A. 369

LINEAR ALGEBRA

II

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 O N‘ belongs to O N.; whence (i). To prove (ii), it suffices to note that Ann(x) is the kernel of the homomorphism a n—> ax of the A-module Aa into M and that this homomorphism is graded of degree (1.; 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' ofM containing E; N’ is the set of x F:- 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, § 1, no. 5) is a graded subring of A. Suppose that a is of degree 8; let 6 = z b,” be an element permutable with a, 1),, being the homogeneous component of b of degree A for all it e A. Then by hypothesis 2 (ab,L — bAa) = 0 and ab; - bha is homogeneous of degree A + 8; as 8 is cancellable, it follows that ab,' = bad for all )L, 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 (Au, (ban) ofgraded rings oftype A (resp. a direct system (wm) of graded Ant-modules of type A) is a direct system of rings (resp. Aamodules) such that each A“ (resp. M“) is graded of type A and each 4’34: (resp. fan) is a homomorphism of graded rings (resp. an Aa-homomorphism of degree 0 of graded modules). If (AQMA (resp. (M2)“;A) be the graduation of Au (resp. M.) and we write

A—{ugnAwA—gA. _.

h._'

A.

(resp.M InM =.

113M.»

L___'

A

or

it follows from§ 6, no. 2, Proposition 5 that (Al) (resp. (M70) is a graduation 370

CASE or AN ORDERED GROUP OF DEGREES

§ 11.4

of A (resp. M) and it follows from I, § 10, nos. 3 and 4 that this graduation is compatible with the ring structure on A (resp. the A—module structure on M). The graded ring A (resp. graded A-module M) is called the direct limit ofthe direct system of graded rings (Au, (baa) (resp. graded modules (Ma, fag). If 4:“: A¢—>A (resp. fix: M“—>M) is the canonical mapping, (bu (resp. f,,) is a homomorphism of graded rings (resp. a homomorphism of degree 0 of graded Act-modules). 4. CASE OF AN ORDERED GROUP 01" DEGREES

An order structure (denoted by 0 implies A + p. > ufor all u, for it implies A + p. 2 p. by definition and the relation E, + p. = p. is equivalent to E = 0. Let A be an ordered commutative group, A a graded ring oftype A and (AA) its graduation and suppose that the relation A, # {0} implies A 2 0; then it follows from the definitions that 50 = 20 AL is a graded two-sided ideal of A, by

virtue of the remark made above.

PROPosrrION 6. Let A be an ordered commutative group, A a graded ring of type A, (AA) its graduation, M a graded A-module oftype A and (M,_) its graduation. Suppose that the relation AA 96 {0} implies A 2 0 and that there exists A0 such that MM aé {0}

andM, = {0mm < A0. Then, 1s = A20A,, 50M 95 M. Let x be a non-zero element of MM, ; suppose that x 6 80M. Then at = 2: am, where the a, are homogeneous elements #0 of So and the x‘ homogeneous

elements #0 of M with deg(x) = deg(a,) + deg(x,) for all i (no. 2). But, as deg(a;) > 0, A0 = deg(a,) + deg(x1) > deg(x,), which contradicts the hypothesis.

COROLLARY I. With the hypotheses on A and A of Proposition 6, if M is a finitely generated graded A-module such that 50M = M, then M = {0}.

Suppose M 96 {0}. Let A0 be a minimal element of the set of degrees of a finite generating system of M consisting of homogeneous elements 790; then the hypotheses of Proposition 6 would be fulfilled, which implies a contradiction.

COROLLARY 2. With the hypotheses on A and A of Proposition 6, let M be a finitely 371

II

LINEAR ALGEBRA

generated graded A-module and N a graded submodule ofM such that N + 80M = M; then N = M.

M/N is a finitely generated graded A-module and the hypothesis implies that 80. (M/N) = M/N; hence M/N = 0. COROLLARY 3. With the hypotheses on A and A ofProposition 6, let u: M —> N be a graded homomorphism of graded right A-modules, where N is assumed to be finitely generated. If the homomorphism

u ® 11M ®A (A/So) “’N ®A (AISO) is surjeetive, then u is surjective. u(M) is a graded submodule of N and the (A/80)-module

(N/u(M)) ®A (A/So) is isomorphic to (N ®A (A/8°))/Im(u ® 1) (§ 3, no. 6, Proposition 6). The hypothesis therefore implies (N/u(M)) ®A (A/So) = 0 and hence N = u(M) by Corollary 1. Remark. It follows from the proof of Corollary 1 that Corollaries l and 2 (resp. Corollary 3) are still valid when, instead of assuming that M (reap. N) is finitely generated, the following hypothesis is made: there exists a subset A+ of A satisfying the following conditions:

(1) for 7x¢A+s Ma. = {0} (reSP- N2. = {0});

(2) every non-empty subset of A+ has a least element. 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 hypotheses on A and M of Proposition 6, consider the graded Ao-module N = M/BOM and suppose the following conditions hold: (i) each of the N,L considered as an Ao-module admits a basis (ymheb‘ ;

(ii) the canonical homomorphism 3° ®A M ——-> M is injectioe. Then M is a gradedfree A-module (no. 2, Remark 3) and, to be precise, lfxm is an element ofM,v whose image in N,“ is gm thefamily (x000, 1,51 (where I is the set the sum

oftheIA) isabasisofM.

We know (no. 2, Remark 3) that there is a graded free A-module L (of graduation (LL)) and a surjective homomorphism p: L —¥> M of degree 0 such that p(e.,.) = x“ for all (L, A) e I ((3:A)(i,7t)e1 being a basis of L consisting of homogeneous elements e”L 6 LA). It follows from the above Remark that p is

surjee‘tiae. 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

CASE OF AN ORDERED GROUP OF DEGREES

§ 11.4

it will suffice to show that 30R = R. Consider the commutative diagram (§ 3, no. 6, Proposition 5)

so®Rfli>so®Lfl+so®M—->o 0

‘1

.1

“l

>R

AL f

>M

>0

9

where j is the canonical injection, (1, b, c deriving from the canonical injection 80 —> A (§ 3, no. 4, Proposition 4); it must be shown that a is surjective. Note that, as L is free, b is injective (§ 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/SOR; then there is an exact sequence (§ 3, no. 6, Proposition 5 and Corollary 2 to Proposition 6)

R/ESOR —’—> L/SOL —’—> M/EioM —-+ 0 where j and [7 derive from j and p when passing to the quotients and p is by hypothesis a bijection; then j (i‘) = 0, in other words j (t) 6 EL. Then there is an element 2 e ‘50 ® Lsuch thatj(t) = b(z) ; asp(b(z)) = 0, c((l ®p)(z)) = O and, as c is injective, (l (X) p) (z) = O. In other words, z is the image of an element t' e 80 ® Runder l ®j and thenj(a(t’)) = b(z) = j(t) ; asj is injective, this implies t = a(t’). We shall show later (Commutative Algebra, II, §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 ofA, it is necessary andsufiicient thatA be torsionfree.

If there exists such an order structure on A and if A > 0, then A + p. > 0 for all y. 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, ifA is torsion-free, A is a sub-Z-module ofa vector Q—space

(§ 7, no. 10, Corollary 1 to Proposition 26) which may be assumed of the form Q5”; if I is given a well-ordering (Set Theory, III, §2, no. 3, Theorem I)

and Q its usual ordering, the set Q“) with the lexicographical ordering is totally ordered (Set Theory, III, § 2, no. 6); it is immediate that this ordering is compatible with the additive group structure of Q5”. PROPOSITION 8. Let A be a torsionfree commutative group and A a graded ring oftype A. Ifthe product in A oftwo homogeneous elements 7‘: 0 is #0, the ring A has no divisor

qfO.

Let A be given a total ordering compatible with its group structure (Lemma 373

LINEAR ALGEBRA

'

n

l) and let x = 72; x,” y = LZA y,‘ be two non-zero elements of A (x, and y; being homogeneous of degree A for all A E A); let on (resp. B) be the greatest of the elements A EA such that x” 95 0 (rCSp- y». 95 0); it is immediate that if A 9:9 at or p. aé [3, either xkyu = 0 or deg(x,.yu) < a + {3; the homogeneous component of xy of degree at + p is therefore xayfl, which is non-zero by hypothesis; whence xy 9E 0. 5. GRADE!) TENSOR PRODUCT 01'" GRADE!) 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 ®z N of the Z-modules M and N is the direct sum of the M, ®z NI, (§ 3, no. 7, Proposition 7) and hence the latter form a. bigradua—

tion of types A, A on this Z—module. Consider on M ®z N the total graduation of type A associated with this bigraduation (no. 1, Example 4-); it consists of the

sub-Z-modules PA = “2:“ (MI, ®z NV). It is known that the Z-module M ®A N is the quotient of M ®z N by the sub-Z-module Qgenerated by the elements (xa) ®y — x ® (ay), where xeM, yeN and a EA (§ 3, no. 1); if, for all A e A, xx, ya, a; are the homogeneous components of degree A of x, y, a

respectively, clearly (am) (8) y — x (8) (23/) is the sum of the homogeneous elements (xhav) ® y“ — x,“ (8 (avyu), in other words Qis a graded sub-Z-module of M (81 N (no. 3, Proposition 2) and the quotient

M ®AN = (M ®zN)/Q. therefore has canonically a graded Z-module structure oftype A (no. 3). Moreover (no. 3, Proposition 5), the centre C of A is a graded subn'ng of A; the graduation which we have just defined on M ®A N is compatible with its module

structure over the graded ring 0. For M ®z N has canonically two C—module structures, for which respectively o(x 8) y) = (xe) (8) y and (x ® y)o = x (8) (oy)

forxeM,yeN, 060 (§3, no. 3);ifxeM,,,yeNu, 060 0A,, the two elements o(x (8 y) and (x ® y)e belong to (M ®z N) “M, and their difference

belongs to Q and hence their common image in M (8),, N belongs to (M (2),, N),L+,,+,,, which establishes our assertion. When we speak of M ®A N

as a graded C-module, we always mean with the structure thus defined, unless otherwise mentioned. Note that (M ®A N) A can be defined as the additive

group of M ®AN generated by the x" ®yw where xueMu, yveNv and + v = A.

‘1. Let M’ (resp. N’) be another graded right (resp. left) A-module and u: M —> M‘, v: N -—> N' graded homomorphism of respective degrees a and [3.

Then it follows immediately from the above remark that 14 ® 0 is a graded (C-module) homomorphism of degree a + {3. 374-

GRADED MODULES or GRADED HOMOMORPHISMS

§1 1.6

When A is commutative, a graduation (compatible with the 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 isomorphisms such as (M ® N) ® P —> M (8 (N 63) P) (§ 3, no. 8, Proposition 8) are isomorphisms ofgraded modules. Remark. When A has the trivial graduation (no. 1, Example 1), (M ®A N) n. is then simply the direct sum of the sub-Y-modules Mu ®A N, of M ®A N suchthatp +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, no. 1, and such moreover that f(x,(,yu) EPA-Pu

for xAEMMqN“, )3flinA-

Thenf(x,y) = g(x' ®y), where g: M ®AN—-> P is a Z-linear mapping (§ 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 9* (Bd) is a graded n'ght A-module oftype A. IfE is a graded left A-module of type A and 9*(Bd) ®A E is given the graded Zmodule structure of type A defined above, the canonical left B-module structure (§ 5, no. I) is compatible with the graduation of Em.) = 9*(E) = 9*(Ba) ®A E-

The graded B-module thus obtained is said to be obtained by extending the ring ofscalars to B by means of p and when we speak ofE03, or 9* (E) as a graded B-module, we always mean this structure, unless otherwise mentioned. 6. GRADE!) MODULES 0F GRADED HOMOMORPI-IISMS

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 HA ‘ denote the additive group of graded homomorphism qf degree 7. 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 AEA) of the HA is direct. For, if there is a relation 2 an = 0 with an e H, for all A, it follows that

2: u,,(x,,) = 0 for all p. and all x“ e M“. As the elements of A are cancellable, u,_(x,,) is the homogeneous component of ; u,,(x,,) of degree A + 9.; hence u,((x,,) = 0 for every ordered pair (11., 7.) and every x“ GM“, which implies u,_ = 0 for all A e A. We shall denote (in this paragraph) by HomgrA(M, N) the

additive subgroup of HomA(M, N) the sum of the HA and we shall call it the

additive group of graded A-module homomorphism: of M into N. Let C be the

375

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LINEAR ALGEBRA

centre of A, which is a graded subring (no. 3, Corollary toProposition 5); for the canonical C-module structure on HomA(M, N) (§ 1, no. 14, Remark 1), HomgrA(M, N) is a submodule and the graduation (HA) is compatible with the Cmodule structure: for, if 0, e C n Av, x“ 6 Nu and a,L 6 HA, then by definition (oval) (xu) = cv.u,,(xu) C NM.”v and hence ova,“ E HHV.

Let M' and N' be two graded left A-modules of type A and u': M’ -—> M, v’: N —>N’ graded homomorphisms of respective degrees a and [-3. Then it is immediate that Hom(u’, v'): w v-> v' o w o u’ maps HomgrA(M, N) into HomgrA(M’, N’) and that its restriction to HomgrA(M, N) is a graded homomorphism into HomgrA(M’, N’) qf degree at + [5. In particular HomgrA(M, M) is a graded subn'ng of EndA(M), which is denoted by EndgrA(M). Remark. If M and N are graded left A-modules, HomgrA(M, N) is in general

distinct from HomA(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 S i < a); let d(i) be the degree of x,; let u e HomA(M, N) and for all A e A let 2“ 2. denote the homogeneous component

of u(x,) of degree A + d (i). We show that there exists a homomorphism u“: M —> N such that u,_(x,) = 2,, ,_ for all i. It suffices to prove that if2: am = 0

with a, GA for l g i < n, then 2 dz“, ='= 0 for all A e A (§ 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 Z a‘u(x,) = 0; taking the homogeneous component of degree A + p. on the left-hand side, we obtain 2 aiz‘, r. = 0, whence the existence of the homomorphism uh; clearly moreover a; is graded of degree A. Finally, a,“ = 0 except for a finite number of values of A,

and u = z u,“ by definition, which proves our assertion. In particular, HomgrA(A,, M) = HomA(A,, M) for every graded left Amodule M; moreover HomA(A,, M) has a graded left 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) (§ 1, no. 14, Remark 2) is a graded A-module isomorphism.

Similarly, HomgrA(M, As) 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*", 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*“’ of ‘u =-‘- Hom(u, 1A.) is a graded homomorphism of the graded dual N*" into the graded dual M*", of degree 8, called the graded transpose of a.

376

GRADED MODULES or GRADED HOMOMORPHISMS

§l 1.6

We sometimes consider on the graded dual M*" the graduation derived from the above with the aid of the isomorphism 7t I—> — )r ofA (no. 1, Example 2) so that the homogeneous elements of degree A in M*" are the graded linear forms Qf degree —A on M (when A has the trivial graduation, these are the zero linear forms of the M“ of index [1. #3 1). Then, if u: M —> N is a graded homomorphism of degree 8, u’ becomes a graded homomorphism of degree — 8.

Suppose A is comutatz've and graded of type A and let M, N, P, Q be four graded A-modules of type A. Then there are canonical graded homomorphirm: of degree 0 (3)

HomgrA(M, HomgrA(N, P)) —> HomgrA(M ®A N, P)

H0mgrs(M, N) ®A P -> HomgrA(M, N ®A P) (4) (5) H0mgrs(M, P) ®A HomgrA(N, Q.) —> HomgrA(M ®A N, P ®A Q) (the tensor products being given the graduations defined in no. 5) obtained by restricting the canonical homomorphisms defined in §4, nos. 1, 2 and 4; for,

if u: M —> HomgrA(N, P) is graded of degree 8, then, for all x 6 MA, u(x) is a graded homomorphism N ——> P of degree 8 + A and hence, for y 6 Nu, u(x)(y) e FHA”; if a: M ®A N—> P corresponds canonically to u, it is then seen that v is a graded homomorphisms of degree 8, whence our assertion concerning (3); moreover it is seen that this homomorphism is bfiective. The argument is similar for (4) and (5).

If in particular P = Q = A in (5), then there is a canonical graded homomorphism of degree 0

(6)

M*" @A N*" —> (M @A N)*".

377

APPENDIX PSEUDOMODULES

1. ADJUNGTION OF A UNIT ELEMENT TO A PSEUDO-RING

Let A be a pseudo-ring (I, § 8, no. 1). On the set A’ = Z x A we define the following laws of composition:

(1)

“ + b) {(m,a) + (n,b) = (m + n, "a + ab). + (m, o)(n, b) = ("2", mb

It is immediately verified that A’ with these two laws of composition is a ring in which the element (1, 0) is the unit element. The set {0} x A is a two-

sided ideal ofA’ and L: x +—> (O, 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 derivedfrom the pseudo-ring A by adjoining a unit element. IfA already has an identity element a, the element a = (0, e) of A’ is an idempotent belonging to the centre of A’ and such that A = eA’ = A’e. Then (eA’, (l—e)A’) is a direct decomposition (I, § 8, no. 11) of A’ and the ring (1 — ¢)A’ is isomorphic to Z. 2. PSEUDOMODULES

Given a pseudo-ring A with or without a unit element, 3. lg? pseudomodule over A is a commutative group E (written additively) admitting A as set ofoperators and satisfying axioms (MI), (Mn) and (Mm) of § 1, no. 1, Definition 1. 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 lqfl A’-module structure on E is associated with it by writing, for all x e E and every element (n, a) e A’,

(2)

(n, a) .x = me + ax.

Axioms (M1) to (MW) of § 1, no. 1, Definition 1 are immediately verified;

378

APPENDIX

moreover, by restricting the set ofoperators 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 pseudomodule 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. If E, F are two pseudomodules over A, the homomorphisms E —> F of 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 (Ems; is a family of pseudomodules over A, the groups with operators

1;; E and .62 E. are pseudomodules over A and the associated A’-modules are respectively the product and direct sum of the associated A’-modules Ev There are analogous results for inverse and direct limits of pseudomodules. The theory of pseudomodules over A an thus be entirely reduced to that of A’modules.

379

EXERCISES

§l 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

g: F —> F/N the canonical surjection. Show that u(M) = Im(u o i) and

‘u‘(N) = Ker(qo u). 2. Let E be a left A-module; for every left ideal a ofA let aE denote the sum of the ax where x e E, which is therefore a submodule of E.

(a) Show that for every family (EA)A EL ofA-modules, a. “(BL E,\ is canonically

identified with AGDL aEA. (b) If a is a finitely generated left ideal of A, show that for every family

(EL)AEL ofA-modules, a. LIL E,‘ is canonically identified with 11 any. Give an example where a is not finitely generated and the above property fails to hold (take A commutative and all the EA 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 Amodule of A21 by the submodule of A2 generated by Xe1 — Ye2 (where (:1, ea)

is the canonical basis of A”), show that (m n n)E qé (mE) n (nE).* 3. Let E be an A-module. (a) Let (Lu)ueM be a family of right directed ordered sets and, for all u e M, let (tsh be an increasing family of submodules of E. Show that

n (2 F“) = Z (hogan).

HEM 1.61:“

05131:»

(1)) Take A to be a field and E the A-module A"; let G be the submodule A“) of E; on the other hand, for every finite subset H of N, let Fa be the sub380

EXERCISES

module 11 M, of E, where M, = A for 11 ¢ H and M, = o for n e H. Show that the family (FE), where H runs throughout the set (I) of finite subsets of N is right directed and that

(30o F3) + G 9e no.» (FE + G)' 4. Let E, F1, F2 be three A-modules and f1: F1-—> E, f2: F2 —> E two Alinear mappings. The fibre product of F1 and F2 relative to f1 and f2 is the submodule of the product F1 x F2 consisting of the ordered pairs (x1, x2) such thatf1(x1) = f2(x2); it is denoted by F1 x E Fa.

(a) For every ordered pair of A-linear mappings 141: G —> F1, us: G -->F2 such that f1 0 u1 = f,, o uz, there exists one and only one A-linear mapping u: G —> F1 XE F2 such that pl 0 u = 14,, pa 0 u = uz, where [11 and p3 denote the restrictions to F1 x E F2 of the projections pr1 and prz. (b) Let E',

1, F; be three A-modules,f1’: F’1—-> E’, f;: F; —> E’ A-linear

mappings and F1 XE. F; the fibre product relative to these mappings. For every system of A-linear mappings 111: F1 —> F1, 02: F2 —> F;, w: E —-> E’ such thatfl o 221 = w of1,f; o 02 = w ofz,letz)betheA-linearmappingofF1 x,,,Fa into F’1 x E. F; corresponding to the two linear mappings 01 o [)1 and 02 0122.

Show that if 01 and 02 are injective, so is 0; give an example where 01 and 122 are surjective but 0 is not surjective (take E’ = {0}). If w is injective and 01 and 02 suljective, show that v is suijective. 5. Let E, F1, F2 be three A-modules andf1: E —> F1, f1: E —> F2 two linear mappings. The amalgamated sum ofF1 and F2 relative tof1 andf2 is the quotient module of F1 x F2 by the submodule the image of E under the mapping 2 » (f1(z), -—f2(z)); it is denoted by F1 6-)]; F2.

(a) For every ordered pair of A-linear mappings ul: F1 —> G, 112: F2 —> G such that ul of1 = uz ofa, show that there exists one and only one A-linear mapping u: F1 @E F2 —> G such that u ojl = ul, u ojz = uz, where j1 and j,

denote the respective compositions of the canonical mapping F1 X F29F1®EF2

with the canonical injections F1 —> F1 x F2, F2 —> F1 x F2. (17) Let E’, F3, F; be three A-modules, f1’: E’—>Fi, f2]: E’ —>F§ two A-

linear mappings and F1 (-91.3, F; the amalgamated sum relative to these mappings. For every system of A-linear mappings v1: F; -—> F1, ()2: F; -—> F2, w: E’—->E such that 01 off =f1 ow, 02 of; =j§ow, let 0 be the A-linear

mapping of F’1 @E. F; into F1 (-BE F, corresponding to the two linear mappings jl o 01 and j2 0 v2. Show that if 01 and 02 are surjective, so is I); give an example where 01 and v, are injective but 1; is not injective. Show that if :0 is surjective and 01 and v; injective, then a is injective.

381

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6. Given an A-module E, let 7(E) denote the least of the cardinals of the generating systems of E. (a) If 0 —> E —> F —> G —-> 0 is an exact sequence of A—modules, then

‘Y(G) S Y(F) < Y(E) + ‘Y(G)Give an example ofa productF = E x Gsuch that 7(E) = y(F) = 7(G) = 1 (take E and G to be quotients of Z). (1)) IfE is the sum ofa family (Faber. ofsubmodules, then 7(E) S A; y(F,_). (c) If M and N are submodules of a module E, show that

suP(Y(M), Y(N)) < T(M n N) + Y(M + N)- i 7. (a) 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)(a’, x’) = (aa', ax’ + xa’) ; E (canonically identified with {0} x E) is then a two-sided ideal ofB such that E2 = {0}. (b) By suitably choosing A and E, show that E can be a left B—module with 7(E) (Exercise 6) an arbitrary cardinal, although y(B,) = l.

*8. Let A be a commutative ring, B = A[X,,],.,1 a polynomial ring over A in an infinity of indeterminates and C the quotient ring of B by the ideal generated by the polynomials (X1 -— X2)X, for i 2 3. If 51 and 52 are the classes in C of X1 and X2 respectively, show that the intersection in C of the

monogenous ideals Oil and 052 is not a finitely generated ideal.,,g

9. Show that in an A-module E, if (F,,),,>1 is a strictly increasing sequence of finitely generated submodules, the submodule F of E the union of the Ffl 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 7(F) = 80 (=Ca’rd (N)). 10. Let E be a Z-module the direct sum of two submodules M, N respectively isomorphic to 2/22 and 2/32; show that there exists no other decomposition of E as a direct sum of two non-zero submodules.

11. Let A be a ring admitting a field of left fractions (1, §9, Exercise 15). Show that in the A-module A8, a submodule distinct from A, and {0} admits no supplementary submodule (cf. I, §9, Exercise 16). 12. Let G be a module and E, F two submodules such that E C F.

(a) If F is a direct factor of G, F/E is a direct factor of G/E. If also E is a direct factor of F, E is a direct factor of G. (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 G/E, then F is a direct factor in G. 382

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/p“ (k e Z, n e N). Let E be the product Z-module M x N, where M and N are both isomorphic to U,, M and N being canonically identified with submodules of E. Consider the endomorphism u: (x, y) I—> (x, y + fix) of E; show that u is bijective. 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 U, has no direct factor other than itself and {0}). 1T 13. Let E be an A-module and B the ring EndA E. Show that for an element u e B the following three conditions are equivalent: (l).Bu is a direct factor of the left B-module B,; (2) uB is a direct factor of the right B-module Ba; (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, § 8, Exercise 12 and use II, § 1, no. 9, Corollaries l and 2 to Proposi-

tion l5.) 14. Let L be a well-ordered set, E an A-module and (E0161, an increasing family ofsubmodules of E such that: (1) there exists a A e L such that E = {0}; (2) E is the union ofthe E,‘ for AeL; (3) ifA eL is such that the set ofy. < A admits a greatest element A’, E,' is the direct sum EL. and a submodule FA;

(4) if A E L is the least upper bound of the set of p. < A, then E,L 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 EA is the direct sum of the FI, such that p. < A.)

1115. Let c be an infinite cardinal. (a) Let I be a set and Ria, bf a relation between elements of I such that

for all at e I the set of ('3 e I for which Riot, (32 holds has cardinal < c. Show that there exists a well-ordered set L and an increasing family (1,01% of subsets of I such that: (1) there exists a A e L such that I,_ = a; (2) I is the union ofthe I" for A e L; (3) if A e L is such that the set of p. < A has a greatest element A’, Card(I,, — 1,) < c; (4) ifA EL'15 the least upper bound ofthe set ofp. < A, I,’is the union ofthe I“ for p. < A; (5) ifa e I, and Riot, tolds, then 351,, (Note first that for all a e I the set of p e I for which there exists an integer n and a sequence (7,)1‘,‘,. 0, every basis of the A-module A: has n elements (cf. § 7, no. 2, Proposition 3).* 17. Let A be a ring. (a) Show that if the A-module A: is isomorphic to no A1” for m > n, then, for all p < n, A: is isomorphic to no A: for q > p.

(b) Show that if the A-module A: contains no free system ofn + 1 elements, the A-module A: contains no free system ofp + 1 elements for p < n.

18. Let A be a ring for which there exists an integer p with the following property: for every family (a,)1‘,‘, of [2 elements of A, there exists a family

(:01 ‘4‘, ofelements of A, one at least of which is not a right divisor of 0, such P

that ‘21 01a; = 0.

384.-

EXERCISES

(a) Show, by induction on n, that for every family (as) ofp" elements of A2, there exists a family (9) of[2" elements ofA, one at least ofwhich is not a right divisor of 0, such that Z cyx, = 0.

(b) Deduce from (a) and Exercise 16 that for all n > 0, the A-modulc A: contains no free system of n + 1 elements.

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 l, for the A-module A: to contain no free system of n + 1 elements, it is necessary that A admit a field of left fractions (I, §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, §9,

Exercise 16.) (0) Show that if every left ideal ofA is monogenous, A admits a field of left fractions (use (a) and I, §9, Exercise 16).

20. (a) Let A be a ring admitting a field ofleft fractions (I, § 9, Exercise 15). Let E be an A-module; show that, if in E (xl)1E,,,_1—->. ..—>E1-—>Eo—>0 be an exact sequence of A-modules such that E0, E1, . . ., Em_2 are projective. Show that the Amodules "690 E”, _ 2,, and “(‘1')” E...” + 1 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 u = ‘u its transpose. For every submodule N’ of F*, the submodule of E ortho-

gonal to v(N’) is i41(N), where N is the submodule of F orthogonal to N’. 6. (a) IfE is an A-module, every injective endomorphism u ofE is not a left divisor of zero in the ring EndA(E). Conversely, if for every sub-A-module F aé {0} of E there exists an endomorphism v aé 0 of E such that v(E) C F, an element which is not a left divisor of zero in EndA(E) is an injective endomorphism. 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 #0 ofthe Z-module Qis bijective, although there exists no linear form #0 on Q. (6) Let U, be the Z-module defined in § 1, Exercise 12(d). Show that the endomorphism x »—> [m of U, is not injective and is not a left divisor of zero in Endz(U,). 7. (a) If u is a surjective endomorphism of an A-module E, u is not a right divisor of zero in EndA(E). Conversely, if for every submodule F 9E E of E there exists a linear form x* e E* which is zero on F and suljective, every element of EndA(E) which is not a right divisor of zero is a suijective endomorphism. (b) If E is the Z-module defined in Exercise 6(a), show that every endomorphism 960 of E is surjective, although there exists no linear form 9e 0 on E.

387

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(6) Show that if E is a free Z—module 750, there exist non-surjective endomorphisms of E which are not right divisors of O in Endz(E). 8. (a) 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 mappingw:E—>Fsuch thatu=vow. ' (1)) Give an example of two non-zero endomorphisms u, v ofthe Z-module E defined in Exercise 6(a) such that there exists no endomorphism w of E for which it = v o w (although u(E) = 11(E) = E). (0) Give an example oftwo endomorphisms u, v of the Z-module Z such that

“141(0) = 31(0) = {0}, but such that there exists no endomorphism w of z for

which u = w o 1) (cf. § 7, Exercise 14).

9. Given an A-module E, for every submodule M of E (resp. every submodule 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: (A) The canonical homomorphism 63: E —+ E" is bijective. (B) For every submodule M of E, the canonical homomorphism E*/M° -—> M* is bijective. (C) For every submodule M of E, M°° = M. (D) For every ordered pair of submodules M, N of E,

(MnN)° = M° + N°. (a) 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 andy e N).

(b) 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)*. If we 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 § 7, no. 5, Theorem 6). (0) 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 K", where] C I).

((1) Suppose that for two submodulesM, N of E such that N C M and N 96 M, the dual (M/N)* is non-zero; then show that condition (B) implies (C). (a) The kernel of cm 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 O).

(f) Let M be a direct factor of E; show that M°° = M + (E*)°. 10. Give an example of an A-linear mapping u: E —-> F such that tu is bijective but u is neither injective nor suijective (cf. Exercise 6(b) and (0)). 388

EXERCISES

Deduce an example where yo 6 F is orthogonal to the kernel of‘u but where the equation u(x) = yo has no solution. 1f 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. (a) Show that the following properties are equivalent: (at) I is injective. ((5) For every exact sequence 0 —> E’ —> E —> E' —> 0 of A-linear mappings, the sequence

0 —> Hom(E', I) —> Hom(E, I) —> Hom(E’, I) —> o is exact. (7) For every A-module M and every submodule N of M, every A-linear mapping of N into I can be extended to an A—linear mapping of M into I. (8) For every left ideal a of A and every A-linear mappingf: a —-> I, there exists an element b e I such thatf(a) = ab for all a e a. (C) I is a direct factor of every A-module containing it. (6) For every A-module E which is the sum of I and a monogenous submodule, I is a direct factor of E.

(To prove that (8) implies (7), 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 Zom’s Lemma. To prove that (6) implies (8), consider the A-modulc A, x I = M, the submodule N of M consisting of the elements (a, —f(a)) for a e a, and apply (0) to the quotient module M/N.) (b) For a Z-module E to be injective, it is necessary and suflicient that for all x GE and every integer n 95 0, there exists 3/ GE such that ny = x. In particular, the Z—modules Qand Q/Z are injective.

12. (a) For a product E E; ofA-modules to be injective (Exercise 1 1), it is necessary and sufficient that each of the EA be injective. ((1) 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 (a)) 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.

11 13. (a) 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 N the set of u eHomA(M, I) such that u(x) = 0 in N. Show that for all

y gt N, there exists a u e N such that u(y) 96 0. 389

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(12) Let A be a ring. For every (A, A)-bimodule T and every left (resp. right) A-module M, HomA(M, T) has canonically a right (resp. left) A— module structure deriving from that on T, and there is a canonical A—homomorphism cM_T: M —> HornA(HomA(M, T), T) such that 6M.T(x) is the

A-homomorphism u »—> 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 homomorphism 011,: is injective for every left A-module M. (6) Suppose that I is an (A, A)-bimodule which is injective as a left Amodule and a right A-module and that for every (left or right) monogenous A-module E 96 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, HomA(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’ ofa right A-module N, HomA(N’, I) is isomorphic to a quotient module of HomA(N, I) and apply (1)) 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—modulc E 96 {0} there exists a non-zero C-homomorphism of E into I. For every A-module M, Homc(M, I) is an A-module; show that for every projective A-module P, Homc(P, I) is an injective A—module (same argument as in (6')). 14-. Let A be a ring. Show that for every A-module M there exists an injective A-module I such that M is isomorphic to a submodule of I. (Apply Exercise 13(d) with G = Z, using Exercise 11(1)) and the fact that every Amodule is the quotient of a free A-module.) 1[ 15. An A-module homomorphism u: E —> F is called essential if it is in-

jective and if, for every submodule P 79 {0} of F, u1(P) aé {0}; it suflices that this condition holds for every momogenous submodule P 9i {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, 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 v is injective, then u and v are essential. (b) Let E be a submodule of an A-module F and let (F,_),_fl, 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. (e) Let (141),“, be a family of essential homomorphism uh: E,_ —> FL. Show

that the homomorphism £31. uh: Q E, 9% FL is essential. (Prove it first when L has two elements, then use (6).) (d) The Z-module Qis an essential extension ofZ but the product Q" is not an essential extension of Z“. (e) Let E be a submodule ofa module F. Show that there exists a submodule 390

EXERCISES

Qof F such that Qn E = {0} and that the restriction to E of the canonical homomorphism F —> F/Qis essential.

16. A submodule E of an A-module F is called irreducible with respect to F (or in F) ifE 9e F and E is not the intersection oftwo submodules ofF distinct from E. (a) For an A-module F to be an essential extension ofeach of its submodules #{0} it is necessary and suflicient that {0} be irreducible with respect to F.

(b) Let (Eh) ML be a family of submodules of an A-module F. (EA),6L is called a reduced irreducible decomposition of {O} in F if each of the F; is irreducible in F, if the intersection of the EA reduces to 0 and if none of the EA contains the

intersection ofthe E, for y. aé A. When this is so, show that the canonical homo-

morphism of F into g (F/E,_) is essential (if E; = "QA Eu, observe that the image F; ofE; in F/EA is non-zero and that the image ofF in AGE-)L (F/EA) contains

1.6531. F;; then apply Exercise 15(6)). Conversely, if (EQML is a family of submodules ofF irreducible in F and the canonical homomorphin F —> ACE—BL (F/EA) is essential, the family (EA) is a reduced irreducible decomposition of {0} in F. 1[ 17. (a) Let M, N, M', N’ be four submodules of a module E such that

MnN = M’nN’ = P. Showthat

M= (M+ (NnM’))n(M+ (NnN’)). Deduce that if M is irreducible in E, then necessarily P = M' n N or P = N' n N. (b) Let (Ni)1 «a be a finite family of submodules of E irreducible in E and M its intersection. Show that if (P,)1 ‘1‘", is a finite family of submodules of E such that M = 0 P,, then for every index i such that l < i < 71 there exists an

index so) such that 1 s W) s m and such that, if we write N; = Q. Nk, then M = PM, n N; (use (a)). Deduce that if none of the P, contains the intersection of the Pk of index aéj, then m s 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). 1] 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 (3) of Exercise 11(a), arguing as for the proof that (6) implies (8) in that exercise). (b) Let I be an injective A-module. For a submodule E of I to be injective,

it is necessary and sufficient that E admit no essential extension distinct from

391

LINEAR ADGEBRA

II

itselfand contained in I (using Exercise 15(e)), show that this condition implies that E is a direct factor of I). (0) Let I be an injective A—modulc, E a submodule of I and h: E —> F an essential homomorphism. Show that there exists an injective homomorphism j: F —> I such that j o 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: (at) I is injective and is an essential extension of E. ([3) I is injective and for every injective homomorphism j of E into an injective A-module J, there exists an injective homomorphism of I into J extendingj. (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 h: 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 ofevery submodule ofI 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 (b)). In particular, every A-module admits an injective 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 1’ leaving invariant the elements of E. 11 19. (a) Let E, F be two A-modules and M an injective submodule of E (43 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 #{0}. Show that for every injective

submodule P of E, the image of P under the projection ofE onto M (under the decomposition of E as a direct sum M Q 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 transforming M into M’ and leaving invariant the elements of N. 1[ 20. (a) Let A be a ring admitting a field ofleft fractions K (I, § 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 (8) of Exercise 11(a) noting that iff: a —> 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 as 0 (the inverse being taken in K) does not depend on x e a.)

(1)) Let U, be the sub-Z-module of (1/2 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 (II, § 1, Exercise l2(d)). Show that U, is an injective envelope of each of the Z-modules p‘“Z/Z isomorphic to Z/pnz. Show that there exist automorphisms of U, distinct from the identity automorphism leaving invariant the elements of a submodule p'”Z/Z.

11 21. (a) Let (E,),,e J be a finite family of A-modules and for all j e] let I, be an injective envelope of E,; show that g I, is an injective envelope of

63 E,.

16J

(b) 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: (at) {0} is an irreducible submodule in I. ([3) I is indecomposable. (y) I is an injective envelope of each of its submodules 950. (Use (a) and Exercise 18 (e).) (8) 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 76 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 indecomposable, it is necessary and suflicient that {0} be an irreducible submodule of E. (0) Give an example of an indecomposable Z-module E such that its injective envelope I(E) is decomposable (cf. VII, § 3, Exercise 5). (d) Let E be an A-module, I an injective envelope of E and I = SBL I,_ a decomposition of I as a direct sum of a finite family of indecomposable injective submodules; for all 7x 6 L let J,L = Q Iu and let NA = J,_ n E. Show

that (N1))“; is a reduced irreducible decomposition of {0} in E (Exercise 16 (6)) and that IL 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)).

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 indecomposable, it is necessary and sufficient that the non-invertible elements of the ring End(I) form a two-sided ideal of that ring. 393

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LINEAR AMEBRA

1} 23. Let M be an A-module, the direct sum of a family (finite or otherwise) (MQAEL of submodules such that for all A e L the non-invertible elements of End(MA) form a two-sided ideal of this ring (which implies that M; is indecmnposable) . (a) Letf, g be two endomorphisms of M such thatf + g = In. Show that for every finite sequence (Ak)1 < k“ of distinct indices in L there exists a family of submodules N,c (l < k S s) of M such that for every k one of the endomorphismsf, g restricted to Mm is an isomorphism of this submodule onto Nk and that M is the direct sum of the N,c (1 S k S s) and the MA for A distinct

from Ak. (Reduce it to the case 5 = 1; if TEL is the canonical projection of M onto MA corresponding to the decomposition as a direct sum ofMA and G3 M“, [I

note that either 11:; of or 1t,L o 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 onto f(MA) ; moreover f(MA) is a direct factor of M. (If g = 1M — f, note that there exists a finite sequence ofindices (Ak)1 ‘k‘, such that the inter-

section of Ker(g) and the direct sum of the MM: is #{0} and use ((1).) (c) Deduce from (b) that every indecomposable direct factor of M is isomorphic to one of the MA. ((1) Let (Nam. 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 ofindecomposable submodules

of M (under the relation of isomorphism) such that an MA (or an N“) belongs to one of these classes. For all te T, let R(t) (resp. S(t)) be the set of A e L (resp. x e K) such that MA 6 t (resp. NK 6 1)). Show that, for all ts T, Card(R(t)) = Card(S(t)). (In the notation of (a), let J(K) be the set A EL such that the restriction of 75;. to N,‘ is an isomorphism of NK 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 M,“ for A ¢ R(t) and a subfamily of (Ngxesm of cardinal equal to that of R(t).) (e) Deduce from (d) and Exercises 22 and 21 (d) that if (EA)A.eL and (EQK6K are two reduced irreducible decompositions of {O} 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 z e I such that az = {O}; Q has a natural (A/ a)-module structure. Let N be the submodule of M consisting of the x e M such that ox = {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 394

EXERCISES

projective A-module P and every submodulc 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 suflicient that, for every injectiw A-module Qand every quotient module Q” ofQ,every homomorphism of P into Q” be ofthe form Q” is the canonical homomorphism and u is a homomorphism of P into Q (use the fact that every Amodule is a submodulc of an injective A-module).

26. For every Z-module G and every integer n > 0, let ”G denote the kernel ofthe endomorphism xn—>nx ofG. If0——>G’—>-G——>G'—>0 is an exact sequence, define a canonical homomorphism d:,,G'—> G'/nG’ such that the sequence

0 —> ,G' ———> ”G” —"—> G’/nG’ —> G/nG —> G'InG” ——> o is exact. If ”G and G/nG are finite, we write

MG) = Card(C-lnG) - Card(nG); then show that "G’, "G", G'lnG', G'/nG” are finite and that

MG) = MG') + 11"((9)§3

1. Let A be a ring, E a right A-module and F a left A-module. Letf be a function defined on the set S of all finite sequences «#133109 (”ED .112): - ' '3 (x11, 1/11»

(n arbitrarY)

of elements of E x F with values in a set G and such that: (I)

f((x1’y1)a - - u (”M yn» =f((xa(1):ya(1))5 ' ' " (xom)’ya(n)))

for every permutation a e 6"; (2)

f((x1 + xi, yl): (x2! yB): ' ' '3 (xiv 1/11»

(3)

f((x1:y1 + Iii), (372,312), - - '2 (xii: ya»

=f((x1!y1)’(x12y1)’(x2!y2)a ' ' -: (“Mil/13)); f((x1: !/1): (5:12.71): (x2: ya), ' ' ., (”ml/13)); (4)

f((x17\,!/1), - - ': ("ting/11)) =f((x1: 1.7/1): - ' u (xmyn))

for all A e A. Show that there exists one and only one mapping g of E ®A F into G

such thatf((x1, g1), . . ., (xn, m) = A; x. ® y,). (Note that if in @111 = 21:“; @152

395

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LINEAR ALGEBRA

the difference 2 (x,, 3],) — 2’: (x), y§) in the Z-module Z‘Exm is a linear combination with integer coefiicients of elements of the form (2) ofno. 1.)

*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 G @n G —> C ®¢ G is not injective. (b) Show that the two C-module structures on C ®n C arising from each of the factors are distinct...

3. (a) Let (Elfin, be a family of right A-modules and F a left A-module. Show that if F is finitely generated, the canonical mapping

(H En) ®AF—>l:,[, (E; at F) is surjective. (b) Give an example of a commutative ring A and an ideal m of A such that the canonical mapping AN ®A (A/m) —> (A/m)N is not injective (cf. § 1, Exercise 2 (b)). (c) Give an example of a commutative ring A such that the canonical mapping AN ®A AN —> ANXN is not suxjective (observe that for given n, the image of N under a mapping of the form m+—> :21 u,(m)v,(n), where the u; and vi belong to A", generate a finitely generated ideal of A). (d) Deduce from (b) and (c) an example where the canonical homomorphism (22) (no. 7) is neither injective nor suijective. 4-. Let E be a right A-module and F a flee left A-module. Show that if x is a free element ofE andy an element 9&0 ofF, then x ®y 96 0; ifalso A is

commutative and y is a free element of F, then x (8) y is free in the A-module E ®A F.

§4 1. Let A, B be two rings, E a left A-module, F a right A—module and G an

(A, B) -bimodule; HomB(F, G) then has a left A-module structure and HomA(E, G) a right B-module structure. Show that the Z-modules HomA(E, HomB(F, G)) and HomB(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) = af(x,y) and f(x, 31(3) = f(x, y)[3 for «6A, BeB,e,yeF.

2. Let A be the ring Z/4Z, m the ideal 2Z/4Z of A and E the A-module

396

EXERCISES

A/m. Show that the canonical mapping E"l ®A E —>HomA(E, 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* (8;; 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 suijective, although F, = C, E1 = C and E, and F1 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 (B, A)-bimodule. (a) Show that there exists one and only one Z—linear mapping a ®A F —-> HomB(HomA(E, G), G ®A F) which maps the tensor product x (8) y, where x e E and y e F, to the mapping vmfiu »—>u(x) ®yofHomA(E, G) into G ®A F. When B = A and G = ,A,, the homomorphism reduces to the canonical homomorphism (15) (no. 2). (b) Show that if F is a projective A-module and E and G are such that for x =7é O in E there exists u e HomA(E, G) such that u(x) aé 0, then the homomorphism 7) is injective.

11 6. Let A, B be two rings, E a lefi: A-module, F an (A, B)-bimodule and G a right B-module. (a) Show that there exists one and only one Z—linear mapping azHomB(F, G) ®A E —> HomB(HomA(E, F), G) such that for x e E and u e Hom3(F, G), c(u ® x) is the mapping 1) I—-> u(v(x)) of HomA(E, F) into G.

(b) If E is a finitely generated projective A-module, show that 0' is bijecfive. (6) Suppose that G is an injective B-module (§ 2, Exercise 11) and that E is the cokemel of an A-linear mapping AL" —> AS. Show that a is bijective (start from the exact sequence Af,” —>A;‘ —> E ——>0 and use (b), the definition of injective modules and Theorem 1 of §2, no. 1, and Proposition 5 of §3, no. 6). 7. Show that, if E1, E2, F1, F2 are four modules over a commutative ring 397

II

LINEAR ALGEBRA

C, the canonical homomorphism (21) (no. 4) is composed of the homomorphisms H0m(E1, F1)

® H0m(E2, F2) -—) H0m(E1, F2 ® H0m(E2, F2))

—> H0m(E1, H0m(E2, F1 ® Fg)) + H0m(E1 ® E2, F1 ® F2)

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. commutative ring C. Show that if u is an endomorphism of E and I) an endomorphism of F then Tr(u ® 0) = 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 E and F. An additive mapping f of E* ®c F* into A is called doubly linear iff(aw) = qf(w) and f(wa) = f(w)a for all w e E* ®° F* and all a e A. Show that there exists one and only one C—linear mapping (1) of E ®A F into the C—module L of doubly linear mappings of E“ ®c F* into A such that

(4>(x ®y))(x* ®y*) = B be a ring homomorphism. For every A-module E, show that the diagram

E L E** ————-> (E**)(B,

Ea» .—> 03(3))“ 7;» «mar Em)

is commutative.

398

EXERCISES

5. Let K be a field, A the product ring K", a the ideal K0“) of A and B

the ring A/ a. Show that a is a projective A-module which is not finitely generated, although am, = {O}.

6. With the rings A and B defined as in § 1, Exercise 7, let M be an Amodule; show that for M to be finitely generated (resp. free, projective), it is necessary and suflicient that M (8.; B be finitely generated (resp. free, projective).

7. Let p:A—>B be a ring homomorphism. For every left B-module F, define a canonical B-module homomorphism

F —> 5(P:(F)) and show that if E is a left A-module, the composite mappings

5(E) 9 5(P*(§(E))) 9 5(E) 9*(F) -> 9*(§(P*(F))) -> 9*(F) are the identity mappings.

§6 1. Let (Emfm) be the inverse system of Z-modules whose indexing set is N, such that E, = Z for all n and, for n < m, f,,,,. is the mapping x I—> 3’"'"x. For all 1:, let u" be the canonical mapping En —> Z/2Z. Show that (an) is an inverse system of surjective linear mappings but that kg an is not surjective. 2. Let (Emfm) be a direct system of A-modules. Show that the A-module lim E, is canonically isomorphic to the quotient of the direct sum (‘9 E“ = F

—>the by submodule N generated by the elements ofthe form“(fax (xu)) —ja(xa), for every ordered pair (cc, B) such that at < (5 and all x“ e E“, the j“: E -—> F being the canonical injections.

3. Let (mm) be the direct system of Z-modules such that F" is equal to Up (§ 1, Exercise 12 (d)) for all n 2 0 and fmI is the endomorphism xv—>p"""x of U, for n S m. Show that lia = {0} although the fm are surjective. _)

11 4-. Let (wm) be a direct system of A-modules. (a) For every A-module E, define a canonical Z-module homomorphism e:11—111> HomA(E, F“) —-> HomA(E, li_r)n F“).

(b) Show that if E is finitely generated, a is injective. Give an example where s is not injective (cf. Exercise 3, taking E = 113 Fa). 399

II

LINEAR ALGEBRA

(0) Show that if E is finitely generated and thefan, are injective, e is surjective. Give an example where E = 1i_m> F“ is free, the f“ are injective and e is not surjective. (d) Show that if E is projective and finitely generated, s is bijective. *(e) Let A be an integral domain and a an ideal of A which is not finitely generated. If (ad) is the family of finitely generated ideals contained in a, show that E = A/ a is canonically isomorphic to the direct limit of the F0, = A/ a,” but that the homomorphism e is not surjective.*

§7 1. Show that if E is a vector space #{0} over a field K with an infinity of elements, the set of generating systems of E is not inductive for the order relation :3 (form a decreasing sequence (8,) of generating systems of E such that the intersection of the S,I is empty). 1] 2. Let K be a field, L a subfield of K, I a set and (x,)1‘, dim(Coker(u)). it can be reduced to 402

EXERCISES

the case where Ker(u) c Im(u) ; this time take 31(Ker(u)) =_1l(Ker(u)) and use the results of the preceding cases and Exercise 9.) 11. Let E, F be two vector spaces over a field K and q —>F a linear mapping. Show that for every vector subspace V of F,

dim-(114(V)) = dim(V n Im(u)) + dim(Ker(u)). 12. Let E, F be two vector spaces and u, 2) two linear mappings of E into F. (a) Show that the following inequalities hold:

rg(v) < rg(u + v) + rg(u)

rg(u + v) < inf(dim(E), dim(F), rg(“) + rg(”))~

If E and F are finite-dimensional, show that rg(u + 1)) can take any integer value satisfying the inequalities

lrg(u) - rg('J)| < 1’30! + v) S inf(dim(E), dim(F);rg(u) + rg(v))o (b) Show that

dim(Ker(u + 27)) < dim(Ker(u) n Ker(v)) + dim(Im(u) n Im(v)). (c) Show that dim(Coker(u + 11)) < dim(Coker(u)) + rg(v). (If V is a supplementary subspace of Ker(v) in E, note that u(Ker(v)) C Im(u + v) and dim(u(V)) S rg(zl).) 13. Let E, F, G be three vector spaces and u:E —>F, v:F—> G two linear

mappings. (a) Show that there exists a decomposition of E as a direct sum of Ker(u) and two subspaces M, N such that Ker(v o u) = M 63 Ker(u) and Im(v o u) = v(u(N)). (b) dim(Ker(u)) S dim(Ker(v o u)) S dim(Ker(u)) + dim(Ker(v)); if Ker(u) and Ker(v) are finite-dimensional, show that dim(Ker(v o u)) can take every integer value satisfying the above inequalities. (c) The following equalities hold: rg(u) = rg(v o u) + dim(Im(u) n Ker(v)) rg(v) = rg(v o u) + codimF(Im(u) + Ker(v)). If F is finite-dimensional, show that rg(v o u) can take any integer value satisfying the inequality

SUP“), 1£0!) + rg(v) - n) < rg(v ° 1‘) < inf(rg(u), 11:01))14. Let E, F, G be three vector spaces and u:E -—-> F, w:E -—> G two linear

mappings. Show that if_u1(0) C 201(0), there exists a linear mapping a of F into G such that w = v o u (cf. §2, Exercise 8). 403

LINEAR ALGEBRA

II

15. Let E, F be two vector spaces over a field K and u:E —> F a linear mapping. (a) The following conditions are equivalent: (1) Ker(u) is finite-dimensional; (2) there exists a linear mapping z) ofF into E such that v a u = lE + 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) < +00 implies rg(f) < +00. (b) The following conditions are equivalent: (1) Coker(u) is finite-dimensional; (2) there exists a linear mapping 0 ofF into E such that u o v = l;- + w, where w is an endomorphism of F of finite rank; (3) for every vector space G over K and every linear mapping n —> G, the relation rg(g o u) < +00 implies rg(g) < +00.

16. Let E, F be two vector spaces over K and u:E —> F a linear mapping. u is said to be offinite index if Ker(u) and Coker(u) are finite-dimensional and the number (1 (u) = dim(Ker (u)) —- dim(Coker(u)) is then called the index of u. Let G be a third vector space over K and IMF —-> G a linear mapping. Show that if two of the three linear mappings u, v, v o u are of finite index, so is the third and d(v o u) = a'(u) + d(v) (use Exercise 13 ((1)). 17. Let E, F, G, H be four vector spaces over K and u:E —-> F, s —> G,

w:G —> H three linear mappings. Show that rg(v o u) + rg(w o v) S rg(v) + rg(w is v o u). 18. Let E, F be two vector spaces over a field K and u, 0 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(v), dim(Ker(u)) = dim(Ker(v)) and dim(Coker(u)) = dim(Coker(v)). 19. Let E be a vector space over a field K.

(a) Show that every mappingfof E into E which is permutable with every automorphism of E is of the form x +-> ax, where at e K (show first that for all x e E there exists p(x) GK 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 a, ll of K such that g(x, y) = out + fly on the set of ordered pairs

(x, y) of linearly independent elements of E; moreover there exists a mapping 4) ofK x K into K such that goat, ax) = (HA, p.)x for all x e E (same method). If further g(u(x), u(y)) = u(g(x, y)) for every endomorphism u of E, then

g(x, y) = out + fly for all x, y in E. Generalize to mappings of E” into E.

20. Let E be a vector space.

(a) Show that if M and N are two vector subspaces of E and M', N’ the

404-

EXERCISES

orthogonals of M and N respectively in E*, the orthogonal of M nN is M’ + N’ (cf. §2, Exercise 9 ((1)). (b) 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). (0) Show that if E is infinite-dimensional there exists an infinite family (V,) of subspaces of E such that, if V; is the subspace of E* orthogonal to V” the subspace of E* orthogonal to O V, is distinct from Z V{. (d) Deduce from (b) that if E is infinite-dimensional, there exists a decomposition V’ ® 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 ofE to be finite-dimensional, it is necessary and sufficient that its orthogonal in E“ be of finite codimension.

21. With the notation ofTheorem 8 ofno. 6, let n denote the linear mapping

x +—> ( HomK(E, F) to be bijective, it is necessary and suflicient that one of the spaces E, F be finite-dimensional. (12) Let E be a right vector space and F a left vector space over a field K. For the canonical homomorphism E ‘31: F —> HomK(E*, F) to be bijective,

it is necessary and suflicient that E be finite-dimensional (use Exercise 3 ((1)).

27. (a) Under the hypotheses ofProposition 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 HomK(E, G) ®L F maps E to a subspace of G ®L F contained in a subspace of the form G (8;. F’, where F’ is a finite-dimensional subspace of F). (b) Under the hypotheses of Proposition 16 (ii) (no. 7), for the canonical homomorphism (28) (no. 7) to be bijective, it is necessary and suflicient that one ofthe ordered pairs (E1, E2), (E1, F1) , (E2, F2) consist offinite-dimensional vector spaces (use (a) and Exercise 7 of § 4). 28. Let s —>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,—> (E(A,)* to be bijective, it is necessary and suflicient that E be finite-dimensional or that A, considered as a right vector K—space, be finitedimensional (use Exercise 25 (b)). When one of these two cases holds, there exists a canonical bijection (E"”")(,0 ——> (Em)**. 29. Let A be an integral domain, K its field of fractions, E an A-module and T(E) its torsion module. (a) Show that every linear form on E is zero on T(E), so that the duals E* and (E/T(E))* are canonically isomorphic. (b) Show that the canonical mapping (E*)m——> (Em)* is injective and that it is bijective when E is finitely generated. (6) Give an example of a torsion-free A-module E such that E“ = {0} and

Em) 9e {0}1} 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 ® 3/ 95 0 in E ®A F. (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 (b) to show that there is an A-bilinear mappingf of E x F into K such thatf(x, y) aé 0.)

4:06

EXERCISES

*31. Let K0 be a commutative field, A the polynomial ring K0[X, Y] in two indeterminates over K0, which is an integral domain, and E the ideal

AX + AY of A. Show that in the tensor product E ®AE the element X ®Y —Y ®X is $0 and that XY(X ®Y —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 afield qf lgfl fractions K (I, §9, Exercise 15). (Note that if E, (l < i S n) are elements ofK, there exists an at 9E 0 in A such that all the «E. belong to A.)

33. Let A be an integral domain. An A-module E is called divisible if for alleandallot aé 0inA,thereexistsyeEsuchthatoty = 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 ®A F = {0}. If E is a torsion Amodule and F is torsion-free, then HomA(E, F) = {O}.

(b) Show that every injective A-module (§ 2, Exercise 11) is divisible and conversely that every divisible torsion-free A-module is injective (use criterion (8) 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 ® lzP’ ®AE —>P ®A E is injective (reduce it to the case where E is finitely generated and embed E in a fiee A-module). 1[ 35. (a) 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 11. Show that there exists a family (on) of n elements of E, which is a basis for each of the two vector space structures on E. (Note that if (63199,, 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 (b,), either V + W aé E, or V n W and W n EV are non-empty; in the latter case, ify e V n EW, 2 eW n EV,

y + 2 forms with the b, a family of m + 1 elements which is free for each of the two vector space structures on E.)

(b) Let F be a subbirnodule of E, whose two vector space structures over K have the same dimension [1 < 11. Show that there exists a family 001‘,“ of

elements of E which is a basis of E for each of its vector space structures and such that (601(«1, is a basis of F for each of its two vector space structures

(same method). (0) Let (b,)1‘, [x] be the canonical injection of E into

the vector space K?” of formal linear combinations of elements of E and let N be the subspace of Kim) generated by the elements [t+x] — [x] — [t+y] + [y] [M + x] - [x] — A[t + x] + A[x] 4-12

EXERCISES

for A e K, t e T, x, y in E; finally, let V be the quotient space of K?” by N and (l) the mapping of E into V which associates with every x e E the class of [x] modulo N. Then (b 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. pinggofVintoLsuch thatf=go. (b) Let (1)0: T —-> V be the linear mapping associated with 4), so that ¢(t + x) — 4>(x) = (>00). Show that d) (and therefore also (1)0) is injective

(consider the mapping x »—>x - a of E into T for some as E); for every family (1‘).“ of elements of K with finite support and every family (mm! of elements of E,

2 MM) = M? m.)

if 2 A. = 0

Z more = mg rim.) if 2 A. = u 9‘ o. Deduce that ¢0(T) is a hyperplane passing through 0 in V and ¢(E) a hyperplane parallel to (MT).

9. Let K be a finite field with 9 elements and V an n-dimensional vector space over K. (a) Show that the set of sequences (xi, x2, . . ., xm) of m < n vectors of V

forming a free system has cardinal equal to

(4" —1)(q" - q)---(q" - qm‘l) (argue by induction on n). (b) Deduce from (a) that the cardinal of the set of m—dimensional linear varieties in an n-dimensional projective space over K is equal to

(4"+1 - l)(¢1"+1 - q)---(q"+1 - 9’”) (q"'” -1)(q"'+1 — q).--(q""‘1 - 4'") 10. In an n-dimensional projective space P(V) over a field K, a pny'ective basis is a set S ofn + 2 points any two ofwhich form a projectively free system. IfS = MOM,“+1 and S' = (a;)o E’, let f0" denote the Z‘m-module homomorphism f ® 1: E09 —> E'W. Similarly, if A’ is another additive monoid with its identity element denoted by 0 and if at: A ——>A’ is a monoid homomorphism, ot(E) denotes the homomorphism E“) —> H“ such that at(E) (2.., (8 X“) = z,, ® X“). (E‘A’)‘A" is identified with Emu" by associating with "’62” (02A 2,”. ® X“) ® X°' the element 02" 2,”. ® X‘°'°". Finally,

let e(E) denote the homomorphism E‘“ ——> E mapping 2 2,, ® X“ to Z 2,. (a) Let E be a Z-module, (Ea),e A a graduation on E of type A and, for all

a e A, let p, be the projector E —> E, corresponding to the decomposition of E

as the direct sum of the E0. For all x e E, let 4W) = 021200;) 59 X”. Show that the homomorphism (in: E —> EW thus defined has the following properties:

(1)

=(E) ° m = 1n

(2)

8(E) a 4m = tr e «a,

where 8: A —> A x A is the diagonal mapping. Show that a bijection of the set ofgraduations on E oftype A is thus defined on

to the set of homomorphisms of E into E“) satisfying conditions (1) and (2). (6) Let E, F be two graded Z-modules of type A. For a homomorphism f: E —> F to be graded of degree 0, it is necessary and sufficient that 471» of = f(A) 0 4’2- For a submodule E’ of E to be graded, it is necessary and sufficient that ¢E(E’) C E’“’. (a) Let A be a ring; AW = A ®z Z“) is given a ring structure such that (a ® X“) (b ® X‘) = (ab) ® X°+'. Let (A0)“ Abea graduation ofthe additive 4-24-

APPENDIX

group A; for this graduation to be compatible with the ring structure on A and l 6 A0, it is necessary and sufficient that 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 aé 0 in M, z eAz. Show that if x, y are two elements of M such that

Ann(x) 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 = ay).

4-25

CHAPTER III

Tensor Algebras, Exterior Algebras, Symmetric Algebras

Recall the exponential notation introduced in Chapter I, of which we shall makefrequent use (I, §7, no. 8) : Let (xA)}.eL be a family of pairwise permutable elements of a ring A; for every mapping azL —> N offinite support we shall write

x“ = 7»[1 e L xi“). If B is another mapping of L into N offinite support, a + B denotes the mapping

1» 0‘(70 + 30») of L into N; with this law ofeomposition the set N“) of mappings of L into N offinite support is thefiee commutative monoid derivedfrom L and fix“ = eta”.

For all aeN‘L), we write [oil = ghee) eN; then [a + (5| = [nil + m,- [at] is called the order of the “multiindex” a. For all A e L, let 8,, denote the element Qf Na“) such that 8,0) = I, 8gp.) = Ofor (.1. ye A (Kronecker index); the 8, for 16L are the only elements ofNW of order 1. Na”) is given the ordering induced by the product ordering on N“, so that the relation on g B is equivalent to “‘10) S BO) for all 16 L”; then the multiindex 1» [3(1) — a0) is denoted by B — a, so that it is the unique multiindex such that a + (B — a) = B. For all aeN‘L’, there are only a finite number rfmultiindieesB S a; the 8,, are the minimal elements ofthe set Na" — {0}; the relation ac < B implies [a] S [BI and ifboth a: < B and M = [BL then at = B.

Finally, we write a! = E (am) 1, which is meaningfiel since 0! = 1. From §4 to §8 inclusive, A denotes a commutative ring and, unless otherwise stated, the algebras considered are assumed to be associative and unital and the algebra homomorphisms are assumed to be unital.

427

III

'I’ENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

§ 1. ALGEBRAS 1. DEFINITION or AN'ALGEBRA DEFINITION 1. Let A be a commutative ring. An algebra over A (or an A-algebra, or simply an algebra, when no confision is to be feared) is a set E with a structure defined by giving thefollowing: (1) an A-module structure on E; (2) an A-bilinear mapping (II, §3, no. 5) of E 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) »——> x.y,

or simply (x, y) 1—) xy. Let (09),“ and ((3,015,I be two families of elements of A, offinite support (I, § 2, no. 1). Then, for all families (x,),,_.1 and (1/1),“ of elements of E, the

general distributivity formula (I, § 3, no. 4)

(I)

(,2, max; m) = Z («451)(xiuy) (l. I) E I X J

holds; in particular (2)

(ouc)y = x(ocg) = a(xy)

foraeA, e andyeE.

The bilinear mapping (x, y) n——>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 E“. 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 E“ is also called an anti— automorphism of the algebra E.

When 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, §2, no. 1)), 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, §8, no. 1). Multiplication on E and the unique Z-module structure on E define on E an associative Z-algebra structure.

(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, no. 10) and the product 428

SUBALGEBRAS. IDEALS. QUOTIENT ALGEBRAs

'

§ 1.2

A-module structure (II, §l, no. 5) is an associative and commutative Aalgebra. (4) Let E be an A-algebra; the internal laws (x, y) v—>xy + yx and (x, y) »—>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

mwww+w is always commutative. DEFINITION 2. Given two algebras E, E’ over a commutative ring A, a homomorphism ofE into E’ itamappingf: E—>E’suchthat (l) f is an A-module homomorphism;

(2) f(xy) =f(x)f(y)f0r aux 613 My 5 E. 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 morphisms 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_m(E, E’) denote the set of A—algebra homomorphisms 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 subalgehra of the A-algebra E. Every intersection of subalgebras of E is a subalgebra of E. For every family (354)“: of elements of E, the intersection of the subalgebras

of E containing all the x, is called the subalgebra of E generated by the family (x,).eI and (Mia 13 called a generating system (or generating family) of this subalgebra. If u: E —> E’ is an A-algcbra homomorphism, the image a(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, § 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, III, § 1, no. 7, Proposition 2). If F is a subalgebra of E, the centre of F is the intersection F n F’ of F and its centralizer F’ in E. Note that if

429

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

F is commutative, then F C F' and hence F' D 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, § 1, no. 6).

A subset a of an A-algebra E is called a lefl 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 6 a). It amounts to the same to say that a is a left ideal of E or a right ideal of the opposite algebra E". 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 at e A and x e E, ax = (ate)x = x(rxe) by virtue of (2) (no. 1) and hence the (right, left, two-sided) ideals of the ring E (I, §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 E x' (mod. b) and y E 31’ (mod. b), then x(y -— y’) Eb and (x — x’)y’eb and hence xy E x’y’ (mod. b). Hence an internal law can be defined on the quotient A-module E/b, which is the quotient of the multiplication law (x, y) n—>xg of E by the equivalence relation x E x’ (mod. 1)) (I, § 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 ,0: E a 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 =_ul(0) is a two-sided ideal of E; further, in the canonical decomposition of a:

E ——"—> E/b —"—> u(E) —’—> E’ v is an algebra isomorphism. More generally, all the results of Chapter I, §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: (A3x) + (Pay) = (A + 9-2" +3!)

(Amway) = (Macy + w: + Ky) 7‘0": x) = 0‘”: M)-

4-30

DIAGRAMS EXPRESSING ASSOCIATIVITY AND commnvrrv

§ 1.3

It is immediately verified that E, with these laws of composition, is an algebra over A and (l, 0 is a unit element of this algebra. The set {0} x E is a two-sided ideal of and x» (0, x) is an isomorphism of the algebra E onto the subalgebra {O} x E, by means of which E and {0} x E are identified. E is called the algebra derivedfrom E by adjoining a unit element; it is associative (resp. commutative) if and only if E is. 3. DIAGRAMS EXPRESSING ASSOCIATIVITY AND COMMUTATIVITY

Let A be a commutative ring and E an A-module; being given a bilinear mapping of E x E into E is equivalent to being given an A—lz'near mapping:

s ®AE—>E (II, §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 if f is a mapping rendering commutative the diagram

E @A E £4» E' @A E'

~1

E

——f~—>

1..

E,

For an A-algebra E to be associative, it is necessary and suflicient (taking account of the associativity of tensor products, cf. II, §3, no. 8) that the diagram of A-linear mappings

E®AE®AE 1911'» E®AE 118ml

1»!

E®AE ——m——> E be commutative. Similarly, for the A-algebra E to be commutative, it is necessary and sufficient that the diagram of A-linear mappings

E®AE —°_> E®AE E

be commutative, where 0 denotes the canonical A-linear mapping defined by 4-31

In

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

a(x ®y) = 3/ ® x for x e E, y e E (II, §3, no. 1, Corollary 2 to Proposition 1 .

For all c e E, let 71c denote the A-linear mapping of A into E defined by the condition ma) = c. For I: to be a unit element of E, it is necessary and ' sufficient that the two diagrams 11:01:: A¢&._‘/E®AE

Imam E®AE E®AA-——>

E

E

I'

be commutative (i and i’ denoting the canonical isomorphisms (II, '§ 3, no. 4, Proposition 4-)). Let E be an A-algebra with unit element e and let 1; = '4, (also denoted by

7m); then 71%) = n(¢)n(fl) = “W”: for by (2) (n0- 1),

(«6)(86) = («W = «(66); hence n is an A-algebra homomorphism. Observe that the A—module structure on E can be defined using 1;, for (3)

ax=n(oc).x foraeA,e

(where, on the right hand side, multiplication is in E). The image of the homomorphism 1; is a subalgebra of E whose elements commute with all those of E. The kernel of the homomorphism v; is the annihilator of the element a of the A-module E; by (3), it is also the annihilator of the A-module E (II,

§ 1, no. 12). When the algebra E is unital and associative, 1; is a ring homomorphism. Conversely, let pzA —> B be a ring homomorphism such that the image p(A) 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))

M: p().).x forAeA,e. 4. PRODUCTS OF ALGEBRAS

Let (Ethel be a family of algebras over the same commutative ring A. It is immediately verified that on the product set E = g E“ the product A-

module structure (II, § 1, no. 5) and the multiplication

(4) 4-32

((xil, (W) '-> (Mt)

Rasmcnon AND nx'mnsion or SCALARS

§ 1.5

define an A-algebra structure; with this structure, the set E is called the product algebra of the family of algebras (E0,E :-

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

Let A0 and A be two commutative rings and p : A0 —> A a ring homomorphism. If E is an A-algebra, we denote (conforming with II, § 1, no. 13) by 9* (E) the Ao-module defined by addition on E and the external law 1.x = p().)x for all A e and all xe E. Multiplication in E and the Ao-module structure on 9,,(E) define an A0algebra structure on 9* (E). When A0 is a subring of A and p the canonical injection, the algebra 9* (E) is said to be obtained from E by restricting the ring A of scalars to A0. By an abuse of language, this is also sometimes said when p is arbitrary.

Let F be an Ao-algebra. An Ao-algebra homomorphism F——> 9*(E) is called a semi-homomorphism (relative to p) or a p-lwmomerplzirm of F into the Aalgebra E; it is also called an Ao-homomorphism if no confusion arises. If

E, E’ are two A-algebras, every A-algebra homomorphism E —> E' is also an Ao-algebra homomorphism 9* (E) —> 9*(E').

Consider now two commutative rings A and B and a ring homomorphism pzA —>B. For every A—module E, the B-module 9*(E) = E ®A B, obtained from E by extending the ring A of scalars to B, has been defined (II, § 5, no. 1). If E is.also an A-algebra, we shall define on 9*(E) a B-algebra structure. For this, observe that (E 8),, B) 8’3 (E ®A B) is canonically isomorphic to (E ®A E) ®AB (II, § 5, no. 1, Proposition 3). If m:E ®A E —> E defines the multiplication on E, the mapping m ® 13: (E ®A E) ®A B —> E ®A B

is therefore canonically identified with a B-linear mapping

"1': 9*(E) ®n 9*(E) —> 9*(E) which defines the desired B-algebra structure on 9*(E). Hence

(5)

(m8 WI" ® (5') = (906') ® (33')

for x, x’ in E, (3 and [5' 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 9). It is also denoted by EB, or E (8,, B. When E is associative (resp. commutative, resp. unital), so is p“ (E).

PROPOSITION]. For every A-algebra E, the canonical mapping ¢sn—>x (81 of E into E03, is an A-homomorphism of algebras. Moreover, for every B-algebra F 433

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

and every A-homomorphism s —> F, there exists one and only one B-homomorphis‘m

f:E(B)-—>F such thatf(x® l) =f(x)jbr alle. The first assertion follows immediately from the definition of multiplication in Em), which gives (x® l) (x’® l) = (xx’) ® 1 for at GE and x’ 6E. The existence and uniqueness of the B-linear mappingfof E03, into F satisfying

the relation f(x (8) l) = f(x) for all x e E follow from II, §5, no. 1, Proposition 1; here it all amounts to verifying that f(yy’) = f(y)f(y’) for y and y' in Em); as the elements of the form x (8) l (with x e E) generate the B-module Em), attention may be confined to the case where y = x® 1, y’ = x’ ® 1

with e, x’ GE; as yy’ = (xx’) ® 1, the relation f(yy’) =f(y)f(y’) then follows fromf(xx’) = f(x)f(37’).

It can also be said thatf,._,fis a canonical bijection (6)

HomA—alg.(El 9* (F)) "> H0mn—a13.(P* (E): F) The ordered pair consisting of Em) and (1),; is therefore a solution of the universal mapping problem (Set Theoty, IV, §3, no. 1) where E is the species of B-algebra structure and the «ac-mappings the A-homomorpln'sms from

E to a B—algebra.

COROLLARY. Let E, E’ be two A-algebras; for every A-homomorphirm of algebra: u: E —> E’, u (8 1B is the uniqueB—homomorphirm Qfalgebmsv: E (8A B -—> E’ ®A B rendering commutative the diagram

Eli» E®AB E’ -—> E’ ®A B 4512'

Let C be a third commutative ring and c:B —> C a ring homomorphism; it is immediate that the canonical C-homomorphism

6*(9*(E)) -> (6 ° 9)*(E) mapping (x® 1) ® 1 to x® l for all e (II, §5, no. 1, Proposition 2) is an algebra isomorphism. 6. INVERSE AND DIRECT LIMITS OF ALGEBRAS

Let I be a preordered set and (A,, 4)”) an inverse system of commutative rings with I as indexing set. Let (E,,fi,) be an inverse system of Armodules with I as indexing set (II, §6, no. 1) and suppose further that each Ei has an A1— algebra structure and that, for i S j, f” is an Arhomomorphism of algebras

(relative to 4)”) (no. 5). Let A = 13A, and E = Eli—n E1, which has an A434-

INVERSE AND DIRECT LIMITS OF ALGEBRAS

§ 1.6

module structure, the inverse limit of the structure of the Armodules El (11,

§6, no. 1); it is immediately verified that the law of composition on E, con-

sidered as the inverse limit of the E, considered as magmas under multiplication

(I, § 10, no. 1), with the A-module structure on E, defines on E an A-algebra structure; (E,,fi,) is called an inverse system ofAralgebras and the A-algebra E is called its inverse limit. IffizE —> E” d),:A—> Ai are the canonical mappings, j; is an A-hamomorphism of algebras (relative to (1),). If the E, are associative (resp. conm'iutative), so is E; if each E, admits a unit element e, andfi,(e,) = e, for i < j, e = (e,) is a unit element of the algebra E. Let (E,’,f,9) be another inverse system of Aralgebras and for all i let

u,:E, —-> E; be an Aralgebra homomorphism, these mappings forming an inverse system; then u = l E:_be an A-algebra homomorphism such that (11,) is an inverse system of mappings; then u = lim u, is a homomorphism of the algebra F into the algebra E. Conversely, for every A-algebra homomorphism s —> E, the family of I), = j} o v is an inverse system of A-algebra homomorphisms such that v = lim 1),. As moreover, writingfl, = Hom(lp,fi,), (Homkflz. (F, E5),f-u) is clearlyf—arn inverse system of sets, it is seen that the above remarks can also be expressed by saying that the canonical mapping vu—> (fiov) isa bijectian IF:HomA_m., (F, ligEa -—> ligHomAM, (F, E,).

Moreover, for every A-algebra homomorphism w:F —>- F', the if), = Hom(w, 1m) :HomiF', E.) —-> HomlHug, (F, E,)

form an inverse system of mappings and the diagram ' 1' ' Hem“, (F I , 11 T(vfi) is a bilinear form on this module, *it follows from (18) and (20) that N is a quadratic form (cf. IX, §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

(22)

(x,y)(x’,y’) = (xx’ + W’s/w? + y’x);

clearly (e, 0) is unit element of F and E x {0} is a subalgebra of F isomorphic to E; we shall identify 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(0611)) = (1?, —y)

(x E E, y E E)-

PROPOSITION 5. (i) The ordered pair (F, t) is a Cayley algebra over A. (ii) Letj = (0, e), so that (x, y) = xe + yjfor xEE, yeE. The Cayley trace and norm TF and NF of F are given by theformulae

(24)

TM“ + yj) = T(x),

NEW + yj) = N(x) - “(NW)-

(iii) For F to be associative, it is necessary and suflioient that E be associative and commutative.

443

III

'I‘ENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

For (x, y) e F,

(25) (26)

(Jay) + t(0%)) = (x + E, 0) = T006 (x,!l)‘((x,!l)) = (Ml/XE: —y) = (x5 "' neyi '_ yx)

= (N(x) - YN(y), 0) = (N(x) - YN(y))¢-

To prove both (i) and (ii), it therefore sufiices to show that t is an antiautomorphism of F. Clearly t is an A-linear bijection. On the other hand,

t((x,y) - (x’,y’)) = t((xx’ + Way? + W» = (:79? + vay’, -y7c - y’x) = (’7’: —y')(3, “'11) = t((xley,))t((xay))

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

_,

_

u(u'u'> = (x u((x,),ex) is a homomorphism of A[(X),EI] into F. The elements of the kernel of h are the relators of the family (xi) in A[(X,),eI],

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,),el] (even when F is not commutative); if a is the ideal of polynomial relators between the x‘, then there is an exact sequence of A-modules

0—6w9MWMLLMWM—9Q PnoposmON 8. Let A[(X,)“I] be a polynomial algebra, J a subset of I and K the

complement of] in I. Writing A’ = A[(X,),EJ] and denoting by X}, (k e K) the 4-53

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

indeterminate: in the polynomial algebra LibascA.(K) = A’[(X,'¢)kex], there exists

a unique ring isomorphism of A'[(X,’,)kex] onto A[(X,),EI] which coincides with the identity on A’ and maps Xi: to Xkfor all k e K. Clearly A[(X),ex] is an A’-algebra generated by the X, for k e K. On the other hand, as a polynomial relator between the Xk (k E K) with coefficients in A‘ can be written uniquely as Z hv((X,),eJ)Xv where v runs through a finite subset of Na” and where the hv are elements of A[(X,),EJ], 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.),el] with polynomials with coeflicients in A’ = A[(X,),e J]. If u is an element #0 of A[(X,),EI], its total degree considered as an element

of A’[(Xk),¢eK] is also called its degree with respect to the X of index i e K.

Remark. Let I and] be two sets and (P,),EJ afamily of elements of Z[(X‘)‘fl] , if Q is an element of Z[(X,),eJ] such that Q((P,),EJ)B= 0, then, for every family (bi)leI of pairwise permutable elements of a ringB,

Q((Pj((bl)te1))lEJ) = 0The relations which hold of the form Q((P,),EJ) = 0 are sometimes called polynomial identities. For example (X1 + X2)2 — Xi — X; -— 2X1X3 = 0 with

Q=Yi—Ya.

P1=X1+Xm

P2=Xi+x2+2x1X2

Xi "' X; "' (X1 '— X2)(X’i_1 + xii—2X2 + ' ‘ ' + X3") = 0 with Q= Y1 '— Y2Ya:

P1 = Xi "

P3 =X1"1 + Xi'axa +

3:

P2 = X1 —' X2,

+ X3'1

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 (ashes of finite support; the multiplication in this algebra is defined by the relations (0(3) ((3,) = (7,), where, for all s e S,

(38) 454-

Z “tflu y, = tu=s

FORMAL POWER SERIES OVER A COMMUTATIVE RING

§ 2.11

(cf. no. 6, formula (35)). The sum on the right hand side of (38) is meaningful because (as) and ([3,) are families of finite support and so therefore is the double family 0145“)“ ”53,3. But the right hand side of (38) is also meaningful for arbitrary elements (cg), [3,) of AIs when the monoid S satisfies the following condition:

(D) For all s e S, there exists only a finite number of ordered pairs (t, u) in S x S such that ta = 3.

Suppose then that S satisfies condition (D); a multiplication law can then be defined on the product A-module A5 by formula (38). It is immediate that the multiplication thus defined on Asis A-bilinear; also it is associative, since,

for at, (3, y in A5,

2 «.45o = m=t 2 ((uv=r 2 my)?» = Z,a( (,2 10-: Bun,»

uvw=t

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 A5, 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 (35,)” s of the total algebra of S over A is also denoted by the same notation 8; Eg, (or even 2:5 firs) 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 #0 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 p“: A3 —> B8 is an A-homomorphism of the total algebra of S over A into the total algebra of S over B, which extends the canonical homomorphism A‘s’ —> E”. 11. FORMAL POWER SERIES OVER A COMMUTATIVE RING

For every set I, the additive monoid N“) satisfies condition (D) of no. 10; for, if s = (nous; with n, = 0 except for the indices i in a finite subset H of I, the relation s = t + u with t = (1),),61 and u = (Me: is equivalent to

p, + q. = n,foralli;butthisimpliesp‘ = q, = Oforie‘Handfit S ”:29: < rt. for is H; there are therefore 1:; (n, + I) ordered pairs (t, u) in N“) such that t + u = s.

455

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

III

We can therefore consider the total algebra of the monoid N“) over A, which contains the (restricted) algebra A[X,], e 1 ofthis monoid. It is a unital commutative associative algebra called the algebra offormal power series in the indeterminate: X, (i e I) with coefiez'enty in A and denoted by A[[X,]],eI ; its elements are called

formal power series in the indeterminates X (i e I) with coefl‘icients in A. Such an element (av)veNu> is also denoted, following the convention made in no. 10, by ”N“ , 0‘vX"; the «V are the we ient: of the formal P0wer series and the atVX" its terms ; a polynomial in the X1 is therefore a formal power series with only afinite number of terms #0. Clearly an algebra isomorphism A[[X,]],,311 —-> A[[X¢]],EI2 is canonically

derived from every bijection a: 11 —> I; by mapping the formal power series (in) am). Lil XI" to the formal power series (2,) «(my 1611

":0.

Let J be a subset of I; the algebra A[[X,]]‘EJ can be identified with a sub-

algebra of A[[X‘]]‘eI consisting of the formal power series (2) «WI—l X1" is! m where am, = 0 for every element (12.) e N“) such that n, 9:9 0 for at least one index i e I —J. Further, if K = I —J, A[[X,]],EI is canonically identified

with (A[[X,]],EJ) [[Xk]]kex, by identifying the formal power series n

in

'th 12116 fcml

(m) “(“0 1151 X1 W

' Wer SCnCS

P0

(”I219 60"")

.

I Ik

kex

k ’

Wh C

re

H x‘;1

‘3‘“) = (m) Your,“ 'with 70,!) = “(no for the sequence (11,) such that n, = p, fori EJ and n, = m‘ for i e K.

Given a formal power series a = Z avX", the terms avX" such that M = 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 homogenous part of u ofdegree p; when I isfinite, up is a polynomial for all p; no is identified with an element of A (also called the constant term of u). If u and v are two formal power series and w = at), then P

w, = Z are...

(39)

r=0

for every integer p 2 0. For every formal power series u ye O, the least integerp 2 0 such that u, 75 0

is called the total order (or simply the order) of u. If this order is denoted by (0(a) and u and v are two formal power series 990, then

(40)

w(u + v) 2 inf(¢n(u), (9(0))

(41)

@010) 2 «9(a) + (0(1))

456

ifu + 0 ye 0 ifuv 75 0.

GRADED ALGEBRAS

§ 3.1

Further, if m(u) aé m(v), then necessarily u + v 9E 0 and the two sides of (40) are equal. Note that the order of 0 is not defined. By an abuse of language, it is a

convention to say that “f is a formal power series of order 21) (resp. >9)” ‘ if the homogeneous part off of degree n is zero for all n < p (resp. n s p); 0 is therefore a “formal power series of order >12” for every integer p 2 0

Let J be a subset of I and let A[[X1]],el be identified as above with B[[Xk]]kex, where K = I —J and B = A[[X,]],EJ; corresponding to the

above definitions applied to B[[X,,]]he 1: there are new definitions for the formal

power series u EAHXJLa; a term “(no'g X" is said to be qfdegreep in the X ofindex i e K if“;K n; = p and the formal power series ofB[[Xk]]keK with the same terms of degree p as u and the others zero is called the homogenous part ofdegreep in the )Q of i e K. Ifu aé 0, the order mx(u) with respect to the x ofindex i e K is the smallest ofthe integersp > 0 such that the homogeneous

part of u of degree p in the X of mdex z e K is 91:0. Inequalities (40) and (4-1) still hold when a) is replaced by cox.

§3. GRADED ALGEBRAS The graduations considered in this paragraph will have as set of degrees a 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 type A (II, § 11, no. 2), (ADA: A it: graduation and E an A-algebra. A graduation (E,“),‘6A qftype A

on the additive group E is said to be compatible with the A-algebra structure on E if it is compatible both with the A-module and with the ring structure on E, in other words, if, for all 7t, p. in A,

(1)

AtEu c Em

(2)

ELEV. c EA+IP

The A-algebra E, with this graduation, is then called a graded algebra (J‘type A over the graded ring A. When the graduation onAis trivial (thatis (II, § 11, no. 1) A0 = A, A,L = {O} for A aé 0), condition (1) means that the E,_ are sub-A-module: of E. This leads to the definition of the notion of graded algebra of type A over a nan-graded 457

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conmiutative 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 [7, its

inverse x"1 is homogeneous and of degree —p: it suffices to decompose x'1 as a sum of homogeneous elements in the relations x'lx = xx‘1 = 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 homomorphism ifu(E,') C E; for all )t e A (where (EA) and (Eg) 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 (II, § 11, no. 2). Let E be a graded A-algebra of type N. E is identified with a graded A-algebra of type Z by writing E, = {O} for n < 0.

Remark. Definition 1 can also be interpreted by saying that E is a graded Amodule and that the A-linear mapping s®AE-—>E

defining the multiplication on E (§ 1, no. 3), is hmnogeneous of degree 0 when E ®A E is given its graduation of type A (II, § 11, no. 5). 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 (1, p.) of elements of A a Z-bilinear mapping

mm: Eh x Eu_>E;.+u such that for every triple of indices (7., p, v) and for a 6AA, x e E", y 5 EV, “'muv(xsy) = mk+u.v(°‘x:y) = mu,z,+v(x: 13/)-

Examples. (1) Let B be a graded ring of type A; if B is given its canonical Zalgebra structure (§ 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 Amodule oftype A. Suppose that all the elements of the monoid A are oancellable, which allows (II, § 11, no. 6) us to define on HomgrA(M, M) = EndgrA(M) a graded A-module structure oftype A; as this graduation is compatible with the ring structure on EndgrA(M) (II, § 11, no. 6), it defines a unital graded A-algebra structure on the A—algebra EndgrA(M). (3) Algebra of a magma. Let S be a magma and d): S —> A a homomorphism.

For all 16A, we write SA = (1:0); then SASu C SM“. Let A be a graded commutative ring of type A and (AQAEL‘ its graduation; we shall define a

graded A-algebra structure on the algebra E = A‘s’ of the magma S (§ 2, 4-58

GRADED SUBADGEBRAS, GRADED IDEALS or A GRADED ALGEBRA

§ 3.2

no. 6). To this end, let E, denote the additive subgroup of E generated by the elements ofthe form «.3 such that at EA“, 3 e 3., and y. + v = 7L. As the S,

are pairwise disjoint, E is the direct sum of the A,,Sv and hence also the direct sum of the EA and it is immediate that the E,L satisfy conditions (1) and (2) and 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 EA is the sub-Amodule of E generated by 8,. More particularly, if we take S = N“), A = N

and (I) the mapping such that ¢((n,)) = :21 n,, the ring A having the trivial graduation, a graduation is thus obtained on the polynomial algebra A[X,],eI,

for which the degree ofa homogeneous polynomial aé 0 is the total degree defined in § 2, no. 9 (cf. §6, no. 6). We now take S to be thefree monaid Mo(B) ofa set B (I, § 7, no. 2) and N which associates with each word its length. Thus a graded A—algebra structure is obtained on thefree associative algebra of the set B (§ 2, no. 7; cf. §5, no. 5). 2. GRADE!) SUBALGEBRAS, GRADE!) IDEALS OF A GRADE!) ALGEBRA

LetEbeagraded algebra oftypeveragraded rinnftypeA. IfFisa sub-A—algebra of E which is a graded .mb-A-module, then the graduation (FL) on F is compatible with its A-algebra structure, since F, = F n EL; 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-modale, then Elau C ah“ (resp. aAEu C ah”), since a,L = a n Eh; 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 subalgebra of E’, Ker(u) is a graded two-sided ideal of E and the bijection E/Ker(u) —> Im(u) canonically associated with a is a graded algebra isomorphism. PROPOSITION 1. Let A be a graded commutative ring qftype A, E a graded A—algebra qftype A and S a set qfhomogeneous element: a” E. Then the sub-A-algebra (resp. left 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 (resp. right) ideal generated by S is the sub-A-module generated by the ele-

ments of the form u1(u2(. . . (uns)) . . .) (resp. (. . .((su,,)u,,_1) . . .)u2)u1) where 459

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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 II, §ll, no. 3,

Proposition 2); finally the two-sided ideal generated by S is the union of the sequence (3,091, where 81 is the left ideal generated by S and 83,, (resp. 82“,) the right (resp. left) ideal generated by 5211—1 (resp. 82“), which completes the proof.

3. DIRECT LINIITS OF GRADED ALGEBRAS

Let (Am, (113,) be a directed direct system ofgraded commutative rings oftype A (II, § 11, no. 3, Remark 3) and for each a let E“ be a graded Aa-algebra of type A; for at s a letf»: E“ —> EB be an Aa-homomorphism of graded algebras and suppose that f7, = fig of” for a: s [5 S 7; then we shall call (weu) a directed direct system of graded algebras of type A over the directed direct system (A,, (113,) ofgraded commutative rings oftype A. Then we know (II, § 11, no. 3) that E = 1131> Ea has canonically a graded module structure of type A over the graded ring A = bin> A“ and a multiplication such that EAE" C EH" (where (E‘) denotes the graduation on E); then this multiplication and the graded A-module structure on E define on E a graded A-algebra structure of type A; E, with this structure, is called the direct limit of the direct system (wm) of graded algebras. The canonical homomorphisms E“ —> E are taken Aa-homomorphisms of graded algebras. Moreover, if F is a graded A-algebra of type A and (an) a direct system of Aa-homomorphisms an: E“ —-> F, u = li_£1>u,, is an A-homomorphism of graded algebras.

§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 homomorphism are assumed to be unital. l. TENSOR PRODUCT OF A FINITE FANIILY 0F ALGEBRAS

A always denotes a commutative ring with unit element. Let (E.).Ex be a finite family of A-algebras and let E = 2% E, be the tensor product A-module ofthe A-modules E, (II, § 3, no. 9). We shall define an A-algebra structure on E. Let m,: E, ®A E, ——> E, be the A-linear mapping defining the multiplication on

E, (§ 1, no. 3). Consider the A-linear mapping m = @m‘:@ (E1®AE,)—>@E, = E;

460

TENSOR PRODUCT or A FINITE FAMILY or ALGEBRAS

§4.1

the composite mapping

((8 E.) at (Q E.) 4> ® (E. on E.) #3 ® E. teI

where z- is the associativity isomorphism (II, § 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

(1)

(Q x.) (Q g.) = Q (m) for x,, y, in E. and is I.

It is therefore seen already, by linearity, that if e, is the unit element of E”

e = g e, is unit element of E. On the other hand, the associativity of each of' the E, implies the relation

((§xt)(;9iw))(§ 2:) = @ (w) = (§ x‘)((@w)(§2 21)) whence, by linearity, the relation x(yz) = (xy)z for all x, y, z in E. DEFINITION 1. Given a family (E0,GI of algebra: over A, the tensor product of this family, denoted by g E, (or, when I is the interval [1, n] of N, E1 «8,, E; Q - -- ®A E", or simply E1 ® E2 8) - - - ® E") is the algebra obtained

by giving the tensor product of the A-modules E, the multzflieation dqined by (1) . Relation (1) shows that the tensor product g E? of the opposite algebras to

the E, is the opposite algebra to Q E,; in particular, if the E, are commutative, so is 3% E1.

Let (Ems, and (E), 51 be two families of A-algebras with the same finite indexing set I. For each i e I, letf,: E. —> F, be an A-algebra homomorphism. Then the A-linear mapping f: ®fi1§Ei—>§F‘ tel

is an A-algebra homomorphism, as follows from (1). For every partition (1,),EJ of I, the associativity isomorphisms

£93 (23;, E:) -*§ E, (II, §3, no. 9) are also algebra isomorphism, as follows from (1) and their definitions. 461

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

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 @ E, is also denoted by B“. 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 (8) b) of a ®Ab in E ®AF is a left ideal of E ®AF; there are analogous results when “left ideal” is replaced by “right ideal” or “two-sided ideal”. Moreover:

PRoposmON 1. Let E, F be two A-algebras and a (resp. b) a two-sided ideal of E (resp. F). Then the canonical A-moa'ule isomorphism

(Ela) ® (E/b) -> (E ® F)/(Im(a ® F) + Im(E ® I’)) (II, § 3, no. 6, Corollary 1 to Proposition 6) is an algebra isomorphism. This follows from (1) and the definition given lac. cit.

COROLLARY 1. Let E be an A-algebra and a an ideal of A. Then the A-moa'ule aE is a two-sided ideal of E and the canonical (AM-module isomorphism (A/a) ®A E —-> E/aE

is an (A/a)-algebra isomorphism. COROLLARY 2. If a, b are two ideals of A, (A/a) ®A (A/b) is canonically isomorphic to A/fia + b).

COROLLARY 3. Let E, F be two A-algebras and a an ideal of A contained in the annihilator of F. Then the (Ala)-algebra E ®AF is canonically isomorphic to

(E/aE) on F.

PROPOSITION 2. Let (Elna, and (Fu)ueM be two families of A-algebras. The canonical mapping (II, § 3, no. 7)

( A"3 ”7’. The elements 2, ® e, (resp. e“, t), where 5 runs through S and t runs through

T, form a basis ofA“) ®A A”) by virtue ofII, § 3, no. 7, Corollary 2 to Proposition 7 (resp. of A“5 x'1'); the desired isomorphism is obtained by mapping e, (8) e, to em” and it follows from the definitions that this is indeed an algebra isomorphism. 2. UNIVERSAL CHARACTERIZATION OF TENSOR PRODUCTS OF ALGEBRAS

PROPOSITION 5. Let (E0,EI be a finite family of A-algebras and, for each is I, let

e. be the unit element q,. For each is I, let u.: E, —> E = @E, be the A-linear mapping dqinea' by u,(x.) = $.14 with x: = x. andx; = ejforj aé i. 463

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

(i) The n, are A—algebra isomorphism; further, for i 9!: j, the elements u.(x,) and u,(x,) are permutable in Efor all x, e E1 and x, e E, and E is generated by the union of the subalgebras u,(E,). (ii) Let F be an A—algebra and, for all i e I, let 0,: E, ——> F be an A—algebra homomorphism, where the o, are such that, for i =;é j, v,(x,) and v,(x,) are Immutable in F for all x, e E, and x, E E1. Then there exists one and only one A-algebra homomorphism w: E —> F such that

(4)

v,=wou, foralliel.

(i) The mapping u, is an algebra homomorphism by definition of the multiplication on E. If 1' 9E j, x, e E” x, e E1, then

u.(x,) = (fix; with x; = x4, x; = ek for k as i, u,(x,) = @142 with x} = 16,, x7, = ek fork aéj. Clearly 1451’, = x”,,x,', for all k e I and hence 14,09) and u,(x,) commute in E by formula (1) (no. 1) defining the multiplication in E. The last assertion follows from the relation g x, = 1;; u.(x,). (ii) For each i e I, let x, be an element of E.. The product 1:; v,(x‘) is then

defined in F independently of any ordering on I since the algebra F is associative and the elements v,(x,) are pairwise permutable. The mapping (13),“ —-> 1:! o,(x,) of 1:; E, into F is obviously A-multilinear and there there-

fore exists one and only one A-linear mapping w: E —> F such that

momento-

Now, the desired A-algebra homomorphism w: E—>F must satisfy (5), u,(x,). This proves the which follows from (4) and the fact that (g) x, = 1—1 leI 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, for j ye i and we obtain w(u, (x,)) = v,(x,). Finally, w is an algebra homomorphism, for

”(@xtxg’w» = w(® (”i“) = U ”(W = U (v.(x.)v:(y.)) = (H w») - (U mo) 464-

MODULES AND MULTIMODU‘LES OVER TENSOR PRODUCTS or ALGEBRAS §4.3

since o,(x,) commutes with 0,0,) forj aé i; hence

ww((®x«)(®w)) = 100$) xI)-w( m) which, by linearity, completes the proof.

The ordered pair consisting of E and the canonical mapping 4): (x5) '-> g x‘ of 1:]: E into E is a solution of the universal mapping problem (Set Theory, IV, § 3,

no. 1) where Z is the species of A-algebra structure, the morphisms being the

A-algebra homomorphisms and the cat-mappings the mappings 1:1 u, of U E‘ into an A-algebra such that the u, are A-algebra homomorphisms and u‘(xi) and 14,09) commute for i 9!: j, for all x‘ e E. and x, E E,. COROLLARY. Let (Ema, (F0421 be two finite families of A-algebras and, for all is I, letfi: E, —> F, be an algebra homomorphism. [frat EI —> 9&8; E,, 0,: F‘ —> 3% F,

are the canonical honwmorphisms, the mappingf = (8]: (cf. no. 1) is the unique Aalgebra homomorphism such thatfo u, = v, Ofifbr all i e I. It suffices to note that the homomorphisms g. = v, of. are such that g,(x,) = u,(j}(x,)) and g,(x,) = v,(f,(x,)) commute fori aé 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 v,(x.) and v,(x,) are permutable for i aé j is automatically satisfied. Hence, when F is commutative, there is a canonical bijection

(6)

HomA.m,(® 13,, F) _> U Hommm, (13,, F),

namely the one which associates with every homomorphism w of g E, into F the family of w o 14,. Note that if E is a commutative A-algebra, the ring structure of E ®A F is the same as that om (§ 1, no. 5). 3. MODULES AND MULTIMODULES OVER TENSOR PRODUCTS 01" ALGEBRAS

DEFINITION 2. Let E be a (unilal) A-algebra. A left (resp. right) E-module is a lefl (resp. right) module over the underlying ring of E. Unless otherwise mentioned, all the modules and multimodules considered in this no. are left modules and multimodules. If M is an E-module, the homomorphism n: A—>E (§ 1, no. 4-) then

465

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

defines on M an A—module structure, said to be underlying the E-module structure on M; foraeA,seE, xeM,

(7)

«(314) = $0”) = (MM.

so that for all s e E, the homothety h,: x »—-> sx of M is an endomorphism of the underlying A-module structure. Conversely, being given an E-module structure on M is equivalent to being given an A-module structure on M and an A-algebra homomorphimz s I—> h, of E into EndA(M) DEFINITION 3. Let E and F be two (unital) A-algebras and M a set with an E—module

structure and an F-moa'ule structure. M is called a (left) bimodule over the algebras E and F if:

(1) M is a bimodule over the underlying rings of E and F (II, § 1, no. 14-); (2) the two A-module structures underlying the E-module and F-module structures on M are identical. The latter condition says that if e and e’ are the unit elements of E and F respectively, then

(8)

(one)x = (ue')x for a. e A, x e M;

then out 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 I——> h; of E into EndA(M) and tn—-> hf of F into EndA(M)

such that hghé’ = hz'hf, for all s e E and teF. Consequently (no. 2, Proposition 5) an A-algebra homomorphism u s—> ha of E ®A F into EndA(M) is canonically derived such that hm; = hghfi’ = hfh; for s e E and te 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 (s®t).x=s(tx) = t(sx)

forseE,teFandxeM.

The given E-module and F-module structures on M can be derived from this (E ®A F) -module structure by restrictions of the ring ofscalars, corresponding to the two canonical homomorphisms E —> E (8,, F and F —> E 8)A F.

Conversely, if an (E l®A F)-module structure is given on M, by means of

the canonical homomorphisms E —> E ®AF and F —-> E (29,, 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 (8,, 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-

466

MODULES AND MULTIMODULES OVER TENSOR PRODUCTS or 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 andf: M —-> M’ is a bimodule homomorphism,fis also an (E (8,, 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 lgfl bimodule structure over E and F°. Examples. (1) Let B be an A-algebra; the ring B has canonically a (B, B)bimodule structure (II, § 1, no. 14, Example 1) and, if e is the unit element of B,

then (ow)x = x(ow) = ax for all x EB and all «6A; B can therefore be considered as a [gift 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 b’ in B,

(9)

(b ® I”) .x = bxb’

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 bimodule 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 ®A N, defined by x.(m ®n) = (x.m) ® 11,

y.(m ® 11) = m ® (11.11)

for x e E,y e F, m e M, n e N (II, § 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 ®A N, such that

(10)

(x ®y)-(m 8n) = (x-m) ® (y-n)

fore,yeF, meM, neN. In particular, taking M = E,, E, 8),, N has canonically an (E (8),, F)module structure; on the other hand, E ®A N is canonically identified with

E ®A (Fa ®F N) = (E ®A F) ®r N, where E 8),, F is considered as having its right F-module structure defined by the canonical homomorphism v: F-—>E ®AF; for x,x' in E, yEF, neN, x' ®n is thus identified with

(x’ ®e') ® n and (x ®y).(x’ 8) n') = (xx’) ® (gm) with ((xx’) ®y) Q; n. The (E ®A F)-module E, (8,, N is thus identified with the (E ®A F)-module derived from N by extending the scalars to E (8),, F by means of the homomorphism 1; (II, §5, no. 1). The canonical mapping n »—>e ® 11 of N into E, ®A N is identified with the canonical mapping 7: I—> (e ® e’) ® n of N into (E ®A F) ®F N; this is known to be an F-homomorphism.

467

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

With the same notation, let P be a right (E ®A F)-modulc; then there is a canonical Z-module isomorphism (ll)

P®E®AF (E, ®AN)—>P®,.N

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 ®s®Ar (E ®AF) and (E ®A F) ®FN with E (2;)A (F ®FN) and hence with E ®A N, which establishes the stated isomorphism (II, § 3, no. 8, Proposition 8 and II, §3, no. 4-, proposition 4). All the above extends to multimodule: (II, § 1, no. 14-). 4. TENSOR PRODUCT 0F ALGEBRAS OVER A FIELD

Let K be a commutative field and E, F two algebras over K whose respective unit elements e, e’ are non-zero. Then the homomorphism 1m: K—>E and 11F:K -—>- F (§ 1, no. 3) are injections which allow us to identify K with a subfield of E (resp. F). The canonical homomorphisms u: E —> E ®K F and 0: F——> E ®x F, defined by u(x) = x (8 e’ and v(y) = e ®y are injective (II, § 7, no. 9, Proposition 19) and allow us to identify E and F with subalgebras of E ‘81: F, both having as unit element the unit element e (8) e’ of E ®x F. In E ®x F, E n F = K (II, § 7, no. 9, Proposition 19). If E’ and F’ are subalgebras of E and F respectively, the canonical homomorphism E’ 8);: F’ —>E ®KF is injective and allows us to identify E’ ®K F’

with the subalgebra ofE ®x F generated by E’ U F’ (II, § 7, no. 7, Proposition 14). PROPOSITION 6. Let E, F be two non-zero algebra: over a commutative field, K, C

(resp. D) a subalgebra qf E (resp. F) and C’ (resp. D’) the eentralz'zer of C in E' (resp. D in F). Then the centralizer q ®K D in E 8),, F is C’ 8),; D’. It all reduces to verifying that an element 2 =. Z x, (8) y, of the centralizer ofC ®x D (x, e F, g, e F) belongs to 0' 8h: D’; we know that C, ®K D, = (C, ®x F) n (E ®x D’)

(II, §7, no. 7, Corollary to Proposition 14). The 3/, may be assumed to be linearly independent over K; for all x e C, of necessity (:4 ® e’)z = z(x ® e’), that is 2 (xx, — xix) ®y, = 0, whence xx, = x,x for all 1' (II, §3, no. 7, Corollary 1 to Proposition 7); hence of necessity x, e C’ for all i and therefore 2 e C' @x F; it can similarly be shown that z e E ®K D', whence the proposition. COROLLARY. IfZ and Z’ are the respective centre: QfE and F, the centre qfE 8’: F is Z ®x Z’. 468

TENSOR PRODUCT OF ALGEBRAS OVER A FIELD

§ 4.4

Let E and F be two subalgebras of an algebra G over a commutative field K; suppose that every element ofE commutes with every element of F. Then the canonical injections i: E ——> G, j: F —> G define a canonical homomorphism h = i (8) j: E ®x F -—> G (no. 2, Proposition 5) such that

(i ®j)(x®!l) = xy fore,yeF. DEFINITION 4-. Given an algebra G over a commutativefield K, two subalgebras E, F (J G are said to be linearly disjoint over K ifthey satisfi; thejbllowing conditions : (1) every elementq commutes with every element of F; (2) the canonical homomorphism of E ®x F into G is injective. PROPOSITION 7. Let G be an algebra over a commutative field K and E, F two subalgebras ofG such that every elementq commutes with every element ofF. For E and F to be linearly diy'oint over K, it is necessary and suflicient that there exist a basis of E over K which is afree subset ofGfor 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 81: F onto the subalgebra of G generated by E U F; (ii) E n F = K; (iii) everyfree subset of E (resp. F) over K is a free subset of G with its right or left F-module (resp. E-moa'ule) structure. The condition of the statement is obviously necessary, since every basis of E over K is a basis of E ‘81: F with its right F-module structure (II, §3, no. 7, Corollary 1 to Proposition 7). To see that the condition is sufficient, note that

the image H ofE ®x F under h is the set of sums Z x,y, = Zyix, in G, with x, e E and y, e F; if (a) is a basis of E over K, H is therefore also the submodule ofthe (right or left) F-module G, generated by (a). The condition ofthe statement therefore means that there exists a basis (ax) 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 (II, § 7, no. 1, Theorem 2).

COROLLARY 1. For the canonical homomorphism 1y“ E ®K F into G to be bijective, it is necessary and suflicient that there exist a basis of E over K which is a basis of the (right or left) F-module G.

Coiomy 2. Let E, F be two subalgebras (yo, offinite rank over K and such that every elementq commutes with every element ofF. For E and F to be linearly disjoint over K, it is necessary and suflicient that the subalgebra H of G generated by E U F be such that (12)

[I-I:K] = [E:K].[F:K].

This says that the rank over K of the suijective canonical homomorphism

469

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC. ALGEBRAS

h: E 691: F -——> H is equal to the rank ofE ®x F over K, which is equivalent to

saying that this homomorphism is bijective (II, § 7, No. 4-, Proposition 9). 5. TENSOR PRODUCT OF AN INFINITE FAMILY OF ALGEBRAS

Let A be a commutative ring and (Ewe; an arbitrary family of (unital) Aalgebras. For every finite subsetJ of I, let Ea denote the tensor product @I E, of the algebras E, of index 1' EJ; let 2, denote the unit element of E, and e, = £8; e, the unit element of E, ; letf“ denote the canonical homomorphism E, —> EJ for i eJ (no. 2, Proposition 5). IfJ, J' are two finite subsets of I such that J C J’, a homomorphism f”: EJ—>EJ, is canonically derived (no. 2, Proposition 5), by the condition f” of“ = f1“. for all i EJ. Moreover the uniqueness off“ implies that if J, J', J' are three finite subsets of I such that J C J’ C J”, then fr; = f”: of”. In other words, (Enfm) is a direct system of A—algebras whose indexing set is the right directed set 3(1) of finite subsets of I.

DEFINITION 5. The direct limit E of the direct system (EJ,f”) is called the tensor product 1y” thefamily QfA-algebras (E,),EI. If I is finite, E is identified with Q E,. By an abuse of notation, E is also

denoted by g E. even if I is infinite. For every finite subset J of I, let f, denote the canonical homomorphism ® E, —> g E, (writingf} instead offm) ; if e is the unit element of g E), then teJ fJ(eJ) = e for all J 6 3(1). It is immediate that if all the algebras E, are commutative, so is “I E).

PROPOSITION 8. (i) The homomorphismsfi: E, ——> E = 1% E, are such thatfor two indices 2', j such that 1' 9E j, fi(x,) andj}(x,) commute in E for all x, e E, and x, e E); further, E is generated by the union of the subalgebrasfi(E,). (ii) Let F be an A-algebra and, for all i e I, let u,: E, —>F be an A-algebra

homomorphism such that, for i aé j, u‘(x,) and u,(x,) commute in F fir all x, e E, and x, e E,. Then there exists one and only one A-algebra homomorphism u: E —> F such thatu, = uoflfiaralliel. (i) As, for every finite subset J of I, 1‘} =13 of“, the first assertion in (i) follows from no. 2, Proposition 5, taking J containing 1' and j; the second also follows from no. 2, Proposition 5, taking account of the fact that E is the union

of thef,(EJ) whenJ runs through 8(1). (ii) For every finite subset J of I, it follows from no. 2, Proposition 5 that there exists a unique homomorphism 14,: EJ —> F such that uJ of“ = u,

470

TENSOR PRODUCT or AN INFINITE FAMILY or ALGEBRAS

§4.5

for all iEJ; it immediately follows from this uniqueness property that, for J C J’, u, = uJ. of”; in other words, the u; form a direct system of homomorphisms. Let u = 13$q E——>F; then by definition uJ = u of, for every finite subsetJ of I and in particular at = u ofi for all i e I; the uniqueness of u follows from these relations and the fact that thefl(E¢) generate the algebra E. COROLLARY. Let (Emu, (Elite: be two families Qf A-algebras with the same

indexing set and,for all i e I, let at: El —> E; be an algebra hmnomorphism. Then there exists one and only one A-algebra homomorphism u: g E. —> g E: such that, for all i e I, the diagram

ii

if.

Eg—w-9E:

.183 E —.> 9682 E5 is eommutative,fi andf1 denoting the canonical homomorphism. It suflices to apply Proposition 8 to the homomorphismfi' 0 u.. The homomorphism a defined in the Corollary to Proposition 8 is denoted by g at. IfJ is any subset of I, Proposition 8 can be applied to the family (12),“ of canonical homomorphisms j}: Ei ——> g E, = E; a canonical homomorphism EJ —> E is derived which is also denoted byfJ and which, when] is finite, coincides with the homomorphism denoted thus above. Now let (x015; be an element of g E, such that the family (x, — ea,“ has I finite support H. It is immediate that, ifJ and J’ are two finite subsets of I containing H, then fJ((x¢)teJ) =jb"((xt)teJ')'

The common value of thej}((x,).gJ) for the finite subsets] D H of I is denoted by § x,. PROPOSITION 9. Let (Ems! be a family of A-algebras andfin“ each is I let 3‘ be a basis of E, such that the unit element e, belongs to B,. Let B be the set ofelements qfthe

form @ x,, where (in) runs through the set Qf elements of 1:! B, such that the family

(xi — e.) hasfinite support. Then B is a basis of the algebra g E, and this basis contains the unit element e.

For every finite subset J of I, let 13.J be the basis of EJ = 933 E1 the tensor product of the bases B, for i eJ (II, § 3, no. 9). It follows immediately from the

4-7l

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

definitions that B is the union of thefJ(BJ) when] runs through 8(1) and that fiW(BJ) C BJ. when J C J’; hence (BI) is a direct system of subsets of the E, and B = iii? BJ; the conclusion then follows from 11, §6, no. 2, Corollary to

Proposition 5.

The basis B is also called the tensor product'of the bases B, for ie I; when the conditions of Proposition 9 are fulfilled, the canonical homomorphisms fizEJ —> E = $9; E, are injective for every subset J of I, for if BJ is the basis of EJ the tensor product of the B, for {6], it is immediately verified that the restriction ofj} to B,, 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,)1‘,‘,, a finite sequence

ofelements ofE, 0‘0“,“ afinite sequence qfelements q andy an element q; suppose that (13)

my = tog/x, foil < i < n.

Then (14')

(”t-”a - - - x,,)y = (117‘: - - - A1:).1/(5‘1342 . - - x“).

The lemma being trivial for n = 1, we argue by induction on n 2 2. Now

(xixa- - may = (x1 - - .xn—1)(xny)

= (x1- --xn—1)(7snyxn) = M((x1- - 'xn—1)y)xm

which, by the induction hypothesis, is equal to

v

11101 ' - - M—1)y(x1---x.._1)xn = (7‘1 - - - ln—1M)y(x1 - - J’s-111.): whence the lemma.

Lemma 2. Let A be a commutative ring, E an A—algebra and (x,)1‘,‘,l and (31,) 1‘,“

twofinite sequences ofn elements ofE; suppose thatforl = ”10‘: - - - xn)y1(!/s-- -!/n)

=(H h1)(x1y1)(x2 xnxya . ya) [>1

and it then suffices-to apply the induction hypothesis to obtain (16). 472

COMMUTATION LEMMAS

§4.6

For every family A = (M) of elements of A, with l < j < i < n, and for every permutation a e 6,, we write (17)

e¢1()‘) = ‘>,_°-1(¢) a(j), yumyzm = xomxam = Ano).«(nxcun’iam = Numwynmynm

whence, by the induction hypothesis (using the fact that 1c(n) = n): yum/«(2) - - - yam) =(t F, be a graded algebra

homomorphism of type A“ Then, if h,:E, a‘Q E, and MGR—>553!) F, are the canonical homomorphism, there exists one and only one homomorphism of graded A-

algebras oftype A, gz‘Q E, —) Q F. such that g o h, = h; o g,fat all is I. Also, if each g, is bijeotioe, so is g.

It suffices to apply Corollary 1 to f, = h; o g,, noting that conditions (26) then follow from relations (25) applied to the hi. The homomorphism defined in Corollary 2 is also denoted by a® g, (if no confusion can arise); if, for each is I, G, is a third graded A-algebra of type A, and gs. —-> G, a graded algebra homomorphism, then

m)

(®w%®Q=®m

as follows immediately from (30). In the case of a skew tensor product of graded algebras of type Z, we write sgf, instead of B®f¢ for homomorphisms fizE, —-> F, of graded algebras of type Z; when I = {1, 2}, this homomorphism is also denoted by f1 ‘®fi; when I = [1, n] and all the E, (resp. F.) are equal and all thej; equal to the same homomorphismf, we write E70”.

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 478

TENSOR pnonuc'r RELATIVE TO COMMUTATION morons

§4.7

E,. If the ordering on I is changed, another multiplicative structure arises

on Q 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 ofthe algebra E1 ”(8A E2 onto the algebra E2 “(8,, E1 maps at, ® x2 to €2,1(a, B)x2 ® x1, where x1 is homogeneous of degree a and x2 homogeneous of degree (3. Let] be a subset of I and, for each i 6], consider the canonical homomorphism h,:E, -—> 6% E, = E. By virtue ofrelations (25) a canonical homomorphism

h:E’ = 8% E, —> E is derived canonically (by Proposition 10) from these homomorphism, such that, for all is], hf = h o h., h; being the canonical

homomorphism E,—->E’. Taking the total ordering on J induced by that chosen on I, we obtain

rigs) = 1:1 wt) = ® :4 where the middle term is the product of the ordered sequence (h.(x,)),eJ and where, in the right hand term, at; = x, for i eJ, x; = e, for i $J. PROPOSITION ll. (“associativity” of the tensor e-product). In the notation of Proposition 10, let (JA)”1, be a partition of I and write AA = H A, for all A e L. Let E; be a graded tensor e-product of type A, of the famcly (E,),:J,. (for some total ordering chosen on J,_). On the other hand, for A, pt in L and A aé y., we write, for “i. = (‘39)!e flit = (6!”e

(32)

slow, lift) =

,6; 16-! saw» [5!)-

Then (3;...) is a system of commutation factors over the A; with values in A and there exists one and only one homomorphism ofgraded algebras oftypeA 1): ® E, -—>- ® E,, such that

(33)

”(Q9 (559 ’61)) = ‘3’ Let.

6.1;,

for all (an) e 1:! E,, provided that the total ordering is taken on I which induces on each J,‘ the chosen total ordering, and which is such that, for )t < y. in L, i6], and j ei < j.

The fact that the 3;“, form a system of commutation factors is trivial. Let hugE, —>- E;, hizE; —> 3?; E; be the canonical homomorphism (for A E- L, ieJA) and write hf = hi 0 km; it will sufl'ice by virtue of the uniqueness of 4-79

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'TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

the solution of a universal mapping problem, to show that 3% El, and the hf satisfy the conditions of Proposition 10. Now, for all A e L, let fn; —> F be the unique algebra homomorphism such that f,_’o hm. = fl for all 1' SJ” We show that, for 7‘ 7e 9-: “it = (“diam an: ($01e fA (x7016u(xu) = Huh-M Pu)fu(xu)fh(xl)

(34')

for x,’., e E; (resp.:c’ 6 EL) homogeneous of degree a; (resp. fly); it sufiiees, by linearity, to verify it when xf,= (8‘15” xf,= I® x,, x, (resp. x,) being homogeneous of degree ac, (resp. [5,) in E, (resp. E,) for iEJM jeJu. But this follows from formula (30) which defines the f,1 and Lemma 3 of no. 6, taking account of hypothesis (26) and definition (32). There is therefore one and only one algebra homomorphism f:

® E; ——> F such that f 0 hi = f,_ for

all A e L; whencefo h’,’ =f, for all t e I and the uniqueness offis trivial. 8. TENSOR PRODUCT 0F GRADE!) ALGEBRAS OF THE SAME TYPES

Assuming the hypotheses of no. 7, Proposition 10 hold, suppose fiirther that all the A, are equal to the same commutative monoid A0; we can then consider on

the tensor s-product 8% E, the total graduation of type A0, associated with the graduation of type A = A}, on this algebra (II, §ll, no. 1); we shall call a@ E,, with this graduation, a graded tensor e-produot of type A0 of the family (Ema, of graded algebras of type A0.

Always preserving the notation of Proposition 10 of no. 7, suppose that F is also a graded A—algebra of type A0 and that the fi are homomorphism of graded algebras cf type A0. Then f: 28; E, -—> F is also a homomorphism qradod algebras oftype A0: for it follows from formula (30) (no. 7) that ifx, is homogeneous and of degree «.6130, @x, andg fi(x,) are both homogeneous of degree 2 a. 5 A0.

It can therefore be said that '® E“ with the total graduation of type A0, constitutes, together with the6 canonical mapping 4:, a solution of a

third universal mapping problem, where )3 is the species of graded Aalgebra of type A0, the morphisms are homomorphisms of graded algebras of type A0 and the or-mappings are the mappings nfi, where, in addition to conditions (26) (of no. 7), it is assumed that each f; is a homomorphism of graded algebras of type A0.

480

TENSOR PRODUCT or GRADED ALGEBRAS or THE SAME TYPES

§4.8

For every subsetJ of I, the canonical homomorphism 3@ E, —> 282 E, (no. 7) is, in fact, a homomorphism of graded algebras of type A0, as follows immediately from the above.

PROPOSITION l2. (“associativity” of the tensor s—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 A0; let 03);“, be a partition of I. With the notation of Proposition 11 of no. 7, suppose that the right hand side offormula (32)

(no. 7) depends only on the sums a; = 1%,. on, B: = I; (3;, for every ordered pair (A, p.) of distinct indioes, all ale A; and all (3,16 AL; let sided, :1) denote the right hand side of (32). Then (Si...) is a system of eommutationfaetors over the family (Aiken where A; = A0 for all A e L. [f E; is the graded tensor e—produet of type A0 of the family (Ethan: there exists one and only one isomorphism of graded algebras

oftype A0, wf’g‘ {928; 13., such that

(35)

w( (8) ( T’+°(M)

(§ 3, no. 1, Remark). For p > (II, §3, no. 9) and, when p phism of A ®A T‘1(M) onto of T’(M) ®AA onto T”(M)

0 and q > 0, mM is the associativity isomorphism = 0 (reap. q = 0), mo”, is the canonical isomorT"(M) (resp. m,_o is the canonical isomorphism (II, §3, no. 4-, Proposition 4). Then, for x, e M,

at e A, (x1 ® ‘ ' ' ®xp)°(xp+1 ® ' " ®xp+q)

(l) ot.(:v:1 ®

=x1®"'®xp®xp+1®"'®xp+l ®x,) = «(x1 ® ®,"r)°

It is immediate that the multiplication thus defined on T(M) is associative and admits as unit element the unit element 1 of A = T°(M). 4-84

rune-ram PROPERTIES OF THE 'I’ENSOR ALGEBRA

§ 5.2

DEFINITION I. For every module M over a commutative ring A, the tensor algebra of

M, denoted by T(M), or Tens(M), or TA(M), it the algebra u>0 GB T“(M) with the .

multiplication defined in (l). The canonical injection :T1(M) ->T(M) (II, §1, no. 12) (also denoted by it“) is called the canonical injection Qf M into T(M).

PROPOSITION 1. Let E be a (unital) A-algebra andf: M —> E on A-linear mapping. There exists one and only one A-algebra homomorphism g:T(M) —>E such that

f = 3 ° 4»

In other words, (T(M), d)) is a solution of the universal mapping problem (Set Theory, IV, §3, no. 1), where 2 is the species of A-algebra structure,

the oc-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).

For every finite family (x,)1 0, the mapping

uniqueness of g.

(#1, - - -: xn) Hf(x1)f(x2) -- -f(xn) of M“ into E is A-multilinear; hence there corresponds to it an A-linear mapping gn:T"(M) —> E such that

(2)

£041 @162 ® - -- ® x») =f(x1)f(12) - - -f(xn)-

Also we define the mapping gozT°(M) —>E as equal to 1),; (§ 1, no. 3), in other words go(a) = out for an EA. Let g be the unique A-linear mapping of T(M) into E whose restriction to T"(M) is g" (n 2 0); it is immediate that g o 4) = g1 = f and it remains to verify that g is an A-algebra homomorphism. By construction g(l) = e and it suflices by linearity to show that g(uv) = g(u)g(v) for ueT’(M) and veT" (M) (p > 0, q > 0); now it follows from formulae (1) and (2) that this relation is true when u = 3:, ® x2 (8) - - - ® 14,, and v ex,“ ® - - - ® x,” (where the x, belong to E). It is therefore true for u e T”(M) and v e T‘(M) by linearity. Remark. Suppose that E is a graded A-algebra of type Z, with graduation (En), and suppose also that .

(3)

f(M) C E1-

Then it follows from (2) that g(T”(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-module: and u: M ——> N

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

an A-linear mapping. There exists one and only one A-algebra homomorphism u’:T(M) —> T(N) such that the diagram M —u> N

ml

1%

T(M) 7 T(N) is commutative. Further, u’ is a graded algebra homomorphism. The existence and uniqueness of u’ follow from no. 1, Proposition 1, applied to the algebra T(N) and the linear mapping ¢N o u:M —> T(N); as u(M) T(M), T(P) —-> T(M) and T(N n P) —> T(M), canonical extensions of the canonical injections, are injective; if T(N), T(P) and T(N n P) are identified with subalgebras of T(M) by means of these homomorphism, 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’G—)(NnP),P = P'(—B(NnP);then

N+P=N'(—BP'®(NnP) and there exists by hypothesis a submodule M’ of M such that

M=M’®(N+P) =M'(-BN'€I—)P’(—B(NF\P) = M’QP’®N = M’®N’(—BP. In particular, N + P, N, P and N n P are direct factors in M, which implies, as has been seen above, that the canonical homomorphisms

T(N) —> T(M), T(N + P) —> T(M), T(N n P) —> T(M) T(P) —> T(M),

are injective. The three algebras T(N), T(P) and T(N n P) are thus identified with subalgebras of T(N + P) and the latter with a subalgebra of T(M);

4—87

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

III

writing Q = N n P, it remains to show that, if T(Q), T(N’ (43 Q) and T(P’ ® Q) are identified with subalgebras of T(N’ ® P’ 63 Q), then

(5)

T(N' (43 Q.) n T(P’ (B Q) = T(Q)-

Now, consider the commutative diagram

N’EDQ. —> N'GP'CJBQ Q

1" 63 Q

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' 63 P' 6) Q) (7)

.1

.

T(Q) ——;—> W ® Q) where r and s are surjective homomorphism (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’ GB Q.) n T(P’ 93 Q), then xeT(Q). Now the definition of the homomorphism 3 shows that its restriction to T(P’ 63 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’ {-B 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 a field (II, § 7, no. 3, Proposition 4-). Moreover, if N C P and N aé P, then T"(N) aé 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

T”(R) aé {0}COROLLARY. Let K be a commutative field and M a vector space over K. For every element 2 E T(M), there exists a smallest vector space N of M such that z e T(N) and N is offinite 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); 2

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 zeT(Q). 488

EXTENSION or THE RING OF SCALARS

§ 5.3

Let 8 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 ‘3 has a minimal element N (Set Theory, III, §6, no. 5). It remains to verify that every P e 3 contains N; now, 2 e T(P) n T(N) = T(P n N) (Proposition 4); in view of the definition of N, thisimplieaP =N,thatisP:> N. The subspace N of M is said to be associated with z. 3. EXTENSION OF THE RING OF SGALARS

Let A, A’ be two commutative rings and pzA —>A' a ring homomorphism. Let M be an A-module, M’ an A'-module and u : M —> M’ an A-homomorphism; as the canonical injection 4m. : M’ —> TA. (M’) is also an A-homomorphism

(by restriction of scalars), so is the composition M 31> M’ 1> TA,(M’). An A-algebra homomorphism TA(M) —-> 9*(TAI(M’)) is derived (no. 2), also denoted by T(u) : TA(M) —> TA.(M’), which is the unique A-homomorphism rendering commutative the diagram

M ——"——> M' (8)

(ml

lon-

TA(M) 760’) TA’(MI)

If czA’ —>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(o o u) = T(o) o T(u).

PROPOSITION 5. Let A, B be two commutative rings, 9 :A —> B a ring homomorphism and M an A-module. The canonical extension ¢zTB(B ®A M) ——>B ®A TA(M) ofthe B-linear mapping 1,, ® tbs ®A M—>B T(B ®A M) and the homomorphism h = T(i) :T(M) —> T(B 8),, M)

derived (cf. formula (8)) from the canonical A-linear mapping

i:M—>B ®AM. 489

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

As T°(B ®A M) is contained in the centre of T(B ®A M), Proposition 5 of §4, no. 2 can be applied and an A-algebra homomorphism 4/:B ®A T(M) —> T(B ®A M) is obtained such that, forfleB, x,eorl < i g n,

“(369061 ®x2®~~ ®xn)) =l3((1 (89:1) ®(1 6912) ®

®(1 ®xn)),

which shows immediately that 4" is also a B-algebra homomorphism. It suffices to prove that 4; o «(4' and 4/ o q; are the identity mappings on B (59‘, T(M)

and T(B ®A M) respectively. Now, these two algebras are generated by B ®A M and clearly q; o L)" and q/ o «.p coincide with the identity mapping on B ®A M, whence the conclusion. 4. DIRECT LIMIT OF TENSOR ALGEBRAS

Let (Au, (baa) be a directed direct system of commutative rings and (Mmfm) a direct system Oa-modules (II, § 6, no. 2); let A = lim A“ and M = lgn> M“,

which is an A-module. For a < B an Au-algebrahgmomorphism (no. 3, formula (8)) f5, = T(fm): TA,,,(M¢) —> TM(MB) is derived canonically from the Ant-homomorphism f5“: Ma——>MB and it follows fi'om (9) (no. 3) that (TM(M.,) ,f5“) is a direct system qu-algebms. On the other hand letf“: M“ ~> M be the canonical Act-homomorphism; an Aa-algebra homomorphism fé: TA,(Mu) —> TA(M) is derived (no. 3, formula (8)) and it also follows from (9) (no. 3) that thef; constitute a direct system of Act-homomorphism. PROPOSITION 6. The A-homomorphism f' = gfé:£n TA“(Mu) —> TA(M) is a graded algebra isomowlzism.

To simplify we write E = TA(M) and E’ = li_m)TAa(Ma) and let

go. = TA..(Ma) -> E' be the canonical Au-homomorphism. Clearly the composite Aa-linear map-

pings Ma 23) TA.(Ma) —g5-> E’ form a direct system and there is therefore one and only one A-linear mapping u = Egg“ 0 (but...) :M —> E’ such that u °.fiz = get ° 4,111“

for all a. This mapping itself factorizes uniquely (no. 1, Proposition 1) into M 31> E—-> E', where h is an A-algebra homomorphism. It will suffice to prove that h of’ =1Erandf’ o h = 1E-

To this end note that, for every index ac, (no. 3, formula (8))

h°fé°uu=h°i>u°fi=u°flz=gm°¢m. 4-90

TENSOR ALGEBRA OF A DmEcr SUM, A FREE MODULE AND A GRADED MODULE

§ 5.5

whence, by the uniqueness assertion of no. 1, Proposition 1, h °fl = get

for all at; it follows that (h of') ogm = gat for all at and hence h of’ =13, by definition of a direct limit. On the other hand, by virtue of no. 3, formula (8), f,°u°‘f; =floga°¢M¢ =fl°¢M¢ = ¢M°fou

whence again f’ o u = 4),, by definition of a direct limit; we conclude that f’ o h o 431»! = by, and the uniqueness property of no. 1, Proposition 1 gives f' o h = In. Proposition 6 can also be shown by observing that, for every integer n 2 1, there is a canonical A-module isomorphism lim HAM“) —» T:(M),

as follows by induction on n from II, § 6, no. 3, Proposition 7. It is immediately verified that these isomorphisms are the restrictions off’. 5. TENSOR ALGEBRA OF A DIRECT SUM. TENSOR ALGEBRA OF A FREE MODULE. TENSOR ALGEBRA OF A GRADE!) MODULE

Let A be a commutative ring and M = 5‘2 MA the direct sum of a family of A-modules. It follows from II, §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

I

MM®MM®---®MM->T“(M)=M®“

relative to all the sequences (1,) e L". Identifying MM ® MM ® . - - (8) M“ with this image, it is seen that T(M) is the direct sum of all the modules

Mh1®MM®"'®MM where It runs through N and, for each n, (M) runs through L”. We first deduce the following consequence: THEOREM 1. Let A be a commutative ring, M a free A-module and (amen a basis of M. Then the elements e, = e,“1 ®eM ® ®ew where s = (A1,...,7t,.)

runs through the set of all finite sequences of elements of L and e,a is used to denote the unit element of T(M), firm a basis of the A-module T(M). The elements of this basis are obviously homogeneous and the multiplica-

tion table is given by (10)

”set = est

where st denotes the sequence of elements of L obtained by juxtaposing the

sequences s and t (I, § 7, no. 2). 491

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

It is seen that the basis (e,) of T(M), with the multiplicative law (10), is canonically isomorphic to the flee monoid of the set L (I, §7, no. 2), the isomorphism being obtained by mapping each word s of this monoid to the element es. It follows (§ 2, no. 7) that the tensor algebra T(M) ofafree module M, with a basis of indexing set L, is canonically isomorphic to the free associative algebra of L over A. In particular (§ 2, no. 7, Proposition 7), for every mapping f: L —> E from L to an A-algebra E, there exists one and only one A-algebra

homomorphismf: T(M) -—> E such thatf(eh) = f(1). 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 II, § 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 (II, §2, no. 2, Proposition 4-) and hence T(M) is a direct factor of T(N) (no. 2); as T(N) is free (Theorem 1), this shows that T(M) is projective (II, §2, no. 2).

PROPOSITION 7. Let A be a commutative monoid, M a graded A-module of type A and (Mgdm its graduation. For every ordered pair (a, n) EA x N, let T“"‘(M) be the (direct) sum of the submodules Me,1 8) Ma: ® - -- ® M“. of T"(M) such it

that :21 a. = on; then (T°""(M))(u,,,,EAxN is the only graduation oftype A x N compatible with the algebra structure on T(M) and inducing an M = T1(M) the given gradation. It has been seen at the beginning of this no. that T(M) is the direct sum of the T“' "(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 T“'1(M) = T"“’1 for aeA, it follows immediately from the definitions that, for all n 2 l and all at EA, T“-"(M) C T’“-"; but since T(M) is also the direct sum of the T“"‘(M), this implies that T’“"‘ = T“-"(M) (II, § 1, no. 8, Remark). 6. TENSORS AND TENSOR NOTATION

Let A be a commutative ring, M an A—module, M" the dual of M (II, §2,

no. 3) and I and J two disjointfinite sets; the A-module {$1.1 E” where E1 = M if i 61, E, = M* if is], is denoted by T§(M); the elements of T§(M) are

called tensors of type (I,_]) over M. They are called contravariant if I = Q, covariant if I = g and mixed otherwise. Let I', I” be two subsets of I and J',J" two subsets of] such that I’ U I” = I,

492

TENSORS AND TENSOR NOTATION

§ 5.6

I’nI' = 2!, J’UJ" =J,J’nJ' = 25; then there is a canonical associativity isomorphism (II, § 3, no. 9)

(11)

T§(M) —>T§'I(M) ®AT KM)-

Considering the tensor algebra T(M 69 M*), it follows from no. 5 that T“(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, 11]. When I = [1,p] and J = [p + l,p + q] with integersp 2 0, q 2 0 (where we replace I (resp. J) by a when p = 0 (resp. q = 0)), the A-module §(M) is also denoted by T; (M); the A-modules T3(M) and T2(M) are therefore by definition the A—modules T"(M) and T"(M*) respectively. When I and J are arbitrary finite sets of cardinals p = Card(I) and q = Card(J), we give each of them a total ordering; then there exists an increasing bijection of I (resp. J) onto [1,11] (resp. [p + 1, p + q] and these bijections therefore define an isomorphism

T§(M) —> T5(M)When M is a finitely generated prty'ective A-module, it follows from II, §2, no. 7, Corollary 4- to Proposition 13 and II, §4, no. 4, Corollary 1 to Proposi-l tion 4- that there is a canonical isomorphism

(T3(M))* -> Ti'(M)Suppose now that M is a finitely generatedfree A-module and let (aha, be a basis of M (L therefore being a finite set). The basis of M“ dual to (9).) (II, §2, no. 6) is denoted by (taken The bases (em) and (e") of M and M“ respectively define (no. 5) a basis of T§(M), which we give explicitly as follows: given two mappings f: I —-> L and g:J —> L, let e? be the element 16%|] x‘ of T§(M) defined by

xi = em) ifieI,

x, = e9“) ifieJ.

When (f; 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 (e,_) of M. For z e T§(M), we can therefore write

(12)

z = (,2 «ten: .11)

where the a; are the coordinate forms relative to the basis (4); by an abuse of language, the 04(2) are called the coordinates of the tensor 2 with respect

to the basis (eA) of the module M. The on; constitute the dual basis of (3;), in

other words they are identified with the elements of the basis of T1 (M), associated with (ex). When I and J are complementary subsets of an interval

[1, n] of N, a; (or 04(2)) is denoted as follows: each element f(i) for i e I is 493

III

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 EJ ; similarly, g(i) for i eJ is written as a lower index in the i-th place with a dot in the i—th place for z e I. For example, for I = {1, 4}, J = {2, 3}, we

write a Go? iff(1) = 1 f(4) = 111%?) = v, g(3) = PLet (59)“, be another basis of M and P the matrix of passing from (’11) to (5,) (II, §10, no. 8). Then the matrix of passing from (e") to (5") (dual basis of (21)) is the contragredient ‘P‘1 of P (II, § 10, no. 8, Proposition 5). It follows (II, § 10, no. 10) that the matrix of passing from the basis (4') of T§(M) to the basis (5}) (where f (resp. g) runs through the set of mappings of I into L (resp. ofJ into L)) is the matrix

(13)

“(8) Q” whereQ,=PifieI,Q,=‘P‘1ifieJ.

The transpose of this matrix is therefore identified with

(14)

‘QJa, whereR,=‘P‘1ifieI,R,=PifieJ.

Suppose now that M is an arbitrary module. Let 1' 61, j G J and write I’ = I -- {i}, J’ = J — {j}; we shall define a canonical A-linear mapping

T§(M) —> THM), called contraction Qf the index i and the indexj. For this, note that the mapping of MI x (M*)", which associates with every family (10161“, where x; EM if

is I and x. e M* ifieJ, the element

(an, 1’) Ice a @414) x” of T}, (M), is A-multilinear; c, is the corresponding A-linear mapping. (15)

Suppose now that M is free and finitely generated and let (e,_)AeL be a basis

of M Given two mappings f: I ——> L, g: J —> L, letj; denote the restriction of —{j}; then by virtue of fto I’ = I — {i} and g, the restriction ofg toJ’ =J—

(12)

(16)

‘ i .;(.;)={0 ifmsgo)

4i iff(i) = £0)

The expression for the coordinates of c)(z) as a function of those of z is obtained; for every mappingf’ (resp. g’) of 1’ into L (resp. ofJ’ into L) and every A e L, let (1", A) (resp. (g’, A)) denote the mapping of I into L (resp. ofJ into L) whose restriction to I’ (resp. J') is f’ (resp. g’) and which takes the value A at the element 1 (resp. j). Then, if the coordinate forms relative to the basis

(e’ ) of TJ (M) are denoted by (3,..,

(17) 494

13. (4(2)) = Z «55': k; (z). AEL

TENSORS AND TENSOR NOTATION

§ 5.6

Example: qftemors. (1) Let M be afinitelygeneratedprojective A-module. We know (II, §4, no. 2, Corollary to Proposition 2), that there is a canonical A-module isomorphism 0,4: M* ®A M —> EndA(M)

such that 0M(x* (83 x) (for x e M, x* e M*) is the endomorphism

y H (y, x*>xHence, by means of 6M, TERM) (isomorphic to T: (M)) can be identified with the A-module EndA(M). Suppose that M is a free module and let (aha. be a basis of M; then the coordinates of a tensor 2 e M* ® M relative to the

basis (e“ 8) ex) of this module are denoted by 2;}. As 911(9" (8 ex) is the endomorphism y »—> (y, e">eA, the endomorphism u = 014(2) = 6,4,2“ Cfle” (8) ex) maps 1/ to 1.21:: ciky, 0'58»; writing 1/ = 32.: we obtain the relation

as)

M) = Z we“

in other words, the matrix ofthe linear mapping u = 6M(2) is that whose element in the row of index p. and column ofindex A 1': C73}. The definition of the trace of u (II, §4, no. 3) shows iImnediately that

now» = sic-z). Therefore the element 20 = Z 42" ® eh (whose coordinates {.15 are zero for A 96 p. and equal to l for A = pt), which is such that 0M(zo)= I", is the image of the element 1 E A= To0(M) under the mapping the transpose of the contraction

TERM) -> A. (2) Suppose always that M is a finitely generatedprry'ectioe A-module ; there is a canonical A-module isomorphism

u: M“ ®A M* -* (M ®A M)* (II, §4, no. 4, Corollary 1 to Proposition 2) and a canonical isomorphism 0: (M ®A M)* ®A M —> HomA(M ®A M, M) (II, §4, no. 2, Corollary to Proposition 2); also HomA(M ®A M, M) is canonically isomorphic to the A-module $2 (M, M; M) ofA-bilinear mappings ofM x M into M (II, § 3, no. 9). Composing these isomorphisms, a canonical isomorphism is obtained “:11, 2,(M) = M“ (8 M“ ® M—>$.,,(M,M;M) 4-95

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

such that, for x*,y* in M*, 2 e M, “(fl ®y* ® 2) is the bilinear mapping

(“a z’) ’—> (a, “X”: y*>zHence, by means of XM: TEEMM) (isomorphic to T;(M)) can be identified with the A-module $2(M, M; M). Suppose that M is a free A-module and let (eQAEL be a basis of M; then the coordinates of a tensor 2 e M* (8 M* ® M relative to the basis (e" (8) e“ (83 (3,) of this module are denoted by {if}. The bilinear mapping “(2) maps the ordered pair (eh, e“) to 2 CM» 0‘,

and therefore the m.“ are just the constants Qf structure of the (in general nonassociative) algebra defined on M by the bilinear mapping XM(z), with respect to the basis (9;) (§ 1, no. 7). Remark 2. Let (Mum (5,.)“1, 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 7t and the column of index y. is denoted by at“ and the element of the contragredient tP 1 appearing in the row of index A and the column of index p. is denoted by (3:. Then (in the notation introduced above) 5“ = 2 “can.

(19)

(20)

‘

i" = z [3269“

5:1 = (/Za) (1:! i‘:z))(1—Irs:2§?)e'

for all mappings f’: I —>L and g'. J—>L. The coordinates C’ of a tensor 2 e TJ(M) with respect to the basis (eh) can therefore be expressed 1n terms of the coordinates t; of z with respect to the basis (6,.) using the formulae

(21>

ra= (2.1.21 5%)(11 2259):;2.

The matrix P 1 of passage from the basis (6;) to the basis (eh) is the transpose of ’P' 1, so that [3: 15 the element which appears in the column of index A and the row of index p. of P 1.The calculation of 4 1n terms of the i, and

that of the C’ in terms of the Cf, are therefore made by replacing a“"by B: and K by at: in the above calculations and exchanging the roles offandf’ and those ofg and g’. Then

(22) 496

4 =20] az)e-;:

humor: or THE svmmo ALGEBRA or A MODULE

§ 6.1

r a) 9(1) (23)thz=uzn(;[;1 “a (1))(1—I {(0)23The 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 themselves because of this circumstance to suppress the summation signs.

§6. SYMMETRIC ALGEBRAS 1. DEFINITION OF THE SYMIIFJI‘RIG ALGEBRA OF A MODULE

DEFINITION 1. Let A be a commutative ring and M an A-module. The symmetric algebra of M denoted by S(M), or Sym(M), or SA(M), is the quotient algebra over A of the tensor algebra T(M) by the two-sided ideal 8’ (also denoted by 8M) generated by = x (81/- y ® x ofT(M), where x andy run through M. the elementsxy— yx— Since the ideal 3’ is generated by homogeneous elements of degree 2, it is a graded ideal (II, § 11, no. 3, Proposition 2); we write 5;, = 8' n T"(M); the algebra S(M) is then graded by the graduation (called comical) consisting of the S"(M) = T"(M)/8,',. Now 33 = 8; = {0} and hence S°(M) is canonically identified with A and 51(M) with T1(M) = M; we shall always make these identifications in what follows and denote by 4)’ or M, 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 ¢’(x), where .7: runs through M, generate S(M), the conclusion follows from § 1, no. 7. PROPOSITION 2. Let E be an A-algebra andf: M —> E an A-linear mapping such that

f(x)f(y) =f(y)f(x)f0r all any in M-

(1)

There exists one and only one A-algebra homomorphism g: S(M) —>E such that

f = g ° ’In other words, (S(M), 41') is a solution of the universal mapping problem (Set Theory, IV, § 3, no. 1), where 2 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 § 5, No. 1, Proposition 1

there CXiSts an A—algebra homomorphism g1: T(M) —> E such thatf = g1 o 43 497

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

all that needs to be proved is that g1 is zero on the ideal 8’, for then, ifp: T(M) —>- S(M) = T(M) [5’ is the canonical homomorphism, we can write g1 = g o [1, where g: S(M) —> E is an algebra homomorphism, and the conclusion will follow from the fact that p o (I) = 41’. Now the kernel ofg1 is a twosided ideal which, by virtue of (l) and the relation g1 o (I) = f; contains the elements x ® y —— 31 (8 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 (E,), and suppose also that the linear mapping f (assumed to satisiy (1)) is such that

(2)

f(M) C E1-

Then the relation g(x1x2. . .x,) = f(x1)f(x2) . . .f(x,) with the x‘ e M shows that g(SP (M)) C E, for allp 2 0 and henceg is a graded algebra homomorphism. (2) Every element of S(M) is a sum ofproducts ofthe form xlx2 . . .xn, where the x, 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!.l 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 fi'om the above remark and I,

§8, no. 2, Proposition 2. 2. FUNCTORIAL PROPERTIES OF THE SYMMETRIC ALGEBRA

PROPOSITION 3. Let A be a commutative ring, M and N two A-module: and u: M —> N an A-linear mapping. There exists one and only one A-algebra homomorphism u’: S(M) —-> S(N) such that the diagram

M ——"——> N “‘1

ltbh

S(M) 7+ S(N) is commutative. Moreover, u’ is a graded algebra homomorphism.

The existence and uniqueness of u’ follow from no. 1, Proposition 2 applied to the commutative algebra S(N) andf = (bfi o u: M —-> S(N); as

f(M) C $1(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 ——> P an A-linear mapping, then

(3) 4-98

5(1)”) = 5(0) ° 5(a)

FUNCTORIAL PROPERTIES or THE SYMMETRIC ALGEBRA

§ 6.2

for S(v) oS(u) is an algebra homomorphism rendering commutative the diagram M

tel

you

lee

S(M) W S(P) As S(M) contains M = 81(M), 5(a) is sometimes called the canonical extension of u to S(M). The restriction S"(u): S”(M) ——> S"(N) is such that

5"(u)(x1xa- - J») = “(x1)u(x2)- - .um) where the 1,6 M, since 5(a) is an algebra homomorphism and 51(u) = u; the restriction S°(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-. If u: M —>N is a sarjective A-linear mapping, the homomorphism S(u): S(M) —> S(N) is smjective and its kernel is the ideal of S(M) generated by the kernelP C M C: S(M) qfu.

We write 1) = T(u):T(M) —>T(N); we know (§ 5, no. 2, Proposition 3) that a is surjective and hence it follows from the definitions that v(3{u) = fife;

m is the kernel of I), then 31%) = a + at. As S(u) : T(M)/33, —> T(N)/3;, is derived from v by passing to the quotients, it is a suijective homomorphism

whose kernel is R’ = (Q + 35.1)]811,1. As R is generated by the kernel P of u (§ 5, no. 2), so is R’. If u: M -—> N is an infective 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 directfactor of S(N); the proof is the same as that for the analogous assertions for T(u) (§ 5, no. 2) replacing T by S.

PROPOSITION 5. Let N and P be two submoa'ules Qf an A-module M such that their sum N + P is a directfactor in M and their intersection N n P is a directfactor in N and in P. Then the homomorphism: S(N) —> S(M), S(P) —> S(M) and S(N n P) —-> S(M),

canonical extensions ofthe canonical injections, are injective; ifS (N), S(P) and S(N n P) are identified with subalgebras of S(M) by means of these homomorphism, then

(4)

S(N n P) = S(N) n S(P).

The proofreduces to that of § 5, no. 2, Proposition 4- replacing T by 5 throughout. The hypotheses of Proposition 5 always hold for arbitrary submodulcs

N, P ofM when A is a field.

499

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TENSOR AIGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG AIDEBRAS

COROLLARY. Let K be a commutative field and M a vector space over K. For every element 2 e S(M) there exists a smallest vector space N ofM such that z e S(N) and N isfinite-dimensional. The proof is derived from that of § 5, no. 2, Corollary to Proposition 4- replacing T by 5 throughout.

N is called the vector subspace of M associated with z. 3. n-th SYMNIETRIC POWER OF A MODULE AND SYRIMETRIG MULTILINEAR MAPPINGS

Let X, Y be two sets and 11 an integer 2 l. A symmetric mapping of X" into Y is any mapping f: X" —>Y such ‘that, for every permutation a e 6,, and every element (x,) e X“, ' (5)

f(xo(1), xa(2)s - - u 12:01)) =f(x1, x2, - - 's xn)'

As the transpositions which exchange two consecutive integers generate the group 6,. (I, §5, no. 7), it suffices that condition (5) hold when a is such a 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 Y” 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: S“(M) —> N (n 2 l) is associated the n-linear mapping

(6)

(x1, x2, . . ., x") Hgolxz. . .xn)‘

(where on the right-hand side the product is taken in the algebra S(M)), a biiective A-linear mapping is obtained of the A-module HomA(S”(M), N) onto the A—module of symmetric n-linear mappings of M" into N. Recall (II, §3, no. 9) that there is a canonical bijection of the A-module HomA(T"(M), N) onto the A-module 2,,(M, . . ., M; N) of all n-linear mappings of Mn into N obtained by associating with every A-linear mapping f: T“(M) —> N the n-linear mapping (7)

f:(x15x2’°-':xn) Hf(x1®x2®"'®xn)'

On the other hand, the A-linear mappings g: S"(M) —-> N are in one-to-one correspondence with the A-linear mappingsf: T"(M) —> N such that f is zero on 8;, by associating with g the mappingf = g o pn, where

Pu: T"(M) -> 5”(M) = “(M)/8; is the canonical homomorphism (II, §2, no. 1, Theorem 1). But as 8:, is a linear combination of elements of the form

(u1®ua®---®u,)®(x®y—y®x) ®(01®---®vn-p—2) 500

n-th svmnuc POWER ‘

§ 6.3

(x, y, at, v, in M), to say that the functionf is of the form g o p" means that the

corresponding n-linear functionf satisfies the relation

f—(ula ' - '1 up, My, 111, - ' °: vn-p-a) =f-(u1! ' ' " ”my: ‘7‘, ”1" ‘ '3 05-9-3);

in other words, by what has been seen above, this means that f— is symmetric; whence the proposition, taking account of the fact that p,,(x1 ®x2 ®' - - ®x,.) = xlxa. ..x,,

with the x,eM.

The A-module S“(M) is called the n-th symmetric power of M. For every Amodule homomorphism u: M —> N, the mapping S“(u) : S“(M) —> 5"(N) which coincides with S(u) on S"(M) is called the n-th symmetric power of u. Remark. Let 0' be a permutation in 6”; as the mapping (”1: x2: ' ' '2 xn) I—> xa'1(1)® x0_1(2) ® ' ‘ ‘ ® xd_1(fl)

of M“ into T"(M) is A—multilinear, it may be written uniquely as (x1: ‘ ' ', x11) l-—>u°(x1 ® x2 ® ' ' ' ®xn)s

where u‘, is an endomorphism of the A-module T"(M), also denoted by z I—-> a. 2. Clearly, if a is the identity element of 6”, no is the identity; on the other hand, writing3/, = rec—1m, we obtain, for every permutation r 6 6”, 3/51“) =xo'1('t'1(1))

and hence 1-. (6.2) = (1'6) .2; in other words, the A-module T"(M) is a left 6n-set under the operation (a, z) r—> a.z (I, § 5, no. 1). The elements of T"(M) such that a. z = z for all a e 6,, are called (contravariant) symmetric tensors of order 72; they form a sub-A-module S;(M) of T"(M).

For all z e T“(M), we write 3.2 = deg a. z and call 3.7. the symnwtn'zation of the tensor z; clearly s. z is a symmetricn tensor and therefore 2 I—> s. z is an endomorphism of T"(M) whose image S;(M) is contained in 5;,(M); in general, SZ(M) aé 5;,(M) (Exercise 5). If 2 is a symmetric tensor, then s. z = n !z; it follows that when n! is invertible in A, the endomorphism z r—> (n !) ‘ ls . z is a projector of T"(M) (II, § 1, no. 8), whose image is S;,(M) = S;(M); more-

over the kernel on this projector is just 8,2. For, obviously a(3,'.) C 8;. for all e e 6,, and 8;, is by definition generated by the tensors z -— p. z, where p is a transposition exchanging two consecutive numbers in [1, It]; also, if a, 1 are two permutations in 6,” then 2 -— (ar).z = z — 0.2 + e'.(z — r.z), whence it follows (since every permutation in 6,, is a product of transpositions ex-

changing two consecutive numbers) that z — 0'. z e 8; for all z e T"(M) and a- e 6”. Therefore (always supposing that n! is invertible in A), it is seen that

z — (n!)‘1s.z = ”2.5,. (n!)'1(z — 6.2) e 3;. for all zeT"(M), which proves our assertion. 501

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

When n! is invertible in A, the submodules S,’,(M) and 8;, of T"(M) are therefore supplementary and the restriction to S;(M) of the canonical homomorphism T"(M) —>- S"(M) = T"(M)/8,’l 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 identification 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 considered. 4. EXTENSION OF THE RING OF SCALARS

Let A, A’ be two commutative rings, 9: A ——> A’ a ring homomorphism, M an A-module, M' an A’-module andf: M —> M’ an A-homomorphz'sm (relative to p) of M into M’. The composite mapping M —f> M' 2-) SA[(M’) is an A-linear mapping of M into the commutative algebra 9*(SA(M’)) ; then there exists no. 1, Proposition 2) one and only one A-homomorphism of algebras f’: SA(M) —> SA. (M') rendering commutative the diagram

M—’——>M' 147w

Val

(8)

SAM) "’7’ SA’ (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’: SA: (M’) —-> SA-(M') the corresponding A’-homomorphism of algebras, then the composite A-homomorphism of algebras ,

f’ (9)

SA(M)

'—>

SA! (M)

,,

3' ‘——>

.

SA~(M)

corresponds to the composite A-homomorphism g of: M —>M" (relative to a o p).

PROPOSITION 7. Let A, B be two commutative rings, 91A —> B a ring homomorphism and M an A-module. The canonical extension

4“ 5303 ®A M) "> B ®A SA(M)

of the B—linear mapping 1,, ® 4);“: B ®A M —> B ®A SA(M) is a graded B-algebra isomorphism.

The proof is derived from that of § 5, no. 3, Proposition 5 replacing T by S and on by (bid.

502

DIRECT mm or SYMMETRIC ALGEBRAS

§ 6.5

5. DIRECT LIMIT OF SYMWTRIG ALGEBRAS

Let (Au, 4)”) be a directed direct system ofcommutative rings, (Maafaa) a direct system of Act-modules, A = ling Acl and M = HEM“. For at < (5, we derive canonically from the Aa-homomorphismfm: M, ——> MB an Act-algebra homomorphism (no. 4, formula (3)) flu: SA“(Ma) -—> SA,(MB) and it follows from (9) (no. 4-) that (SM(MG),fB’¢) is a direct system of Aa-algebras. On the other hand, 1;: M¢—>M be the canonical A-homomorphism; we derive (no. 4-, formula (8)) an Au-algebra homomorphism fl: SAC('M) —> SA(M)

and it also follows from (9) that thef; constitute a direct system of Aa-homomorphisms.

PROPOSITION 8. The A-homomorphism f’ = 11_m>fl: ligSAJMu) —> SA(M) is a graded algebra isomorphism.

The proof is the same as that of § 5, no. 5, Proposition 6 replacing throughout T by S and (b by ‘l” and taking account of the fact that a direct limit of commutative algebras is commutative. 6. SYMMETRIC ALGEBRA OF A DIRECT SUM. SYMluETRIC ALGEBRA OF A FREE MODULE. SYMMETRIC ALGEBRA OF A GRADED MODULE

Let A be a commutative ring, M = $911 MA the direct sum of a family of A-modules and jh: M,L —> M the canonical injection; we derive an A-homomorphism of algebras S(jk): S(MA) -—>S(M). Since S(M) is commutative, Proposition 8 of § 4, no. 5, can be applied to the homomorphism S(jh) and there therefore exists a unique algebra homomorphism

(10)

g: g; S(MA) —> S(M)

(also denoted by g“) such that S(jl) = g ofi for all A e L, where

A: W.) —> 5,8; S(M.) denotes the canonical homomorphism. PROPOSITION 9. The canonical homomorphism g (formula (10)) is a graded algebra isomorphism (cf. §4-, no. 8, Remark 1).

To prove that g is bijective, we define an algebra homomorphism

(11)

h: S(M) —> g; S(MA) 503

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

such that g o h and h o g are respectively the identity mappings on S(M) and goal 5((MA). For each A e L, let u,‘ be the composite linear mapping

M, i S((M,) ——> (8 S(M,_). There exists one and only one A-linear mapping a: M -—> § S(MA) such that u oj,v = uh for all A eL. 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(MA) such that h o (H, = 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 £9 S(ML) and hence h is a graded algebra homomorphism. For x; 6 MM

h(g(u;.(xi))) = h(g(fit(¢iti(xt)))) = h(5(jt)(ih(xt))) = h(ia(ji(xi))) = amt); as the u,(x,_) generate the algebra 1%, S(MA) (§ 4., no. 5, Proposition 8), h o g is certainly the identity mapping. Similarly,

g(h(it(i;.(xt)))) = g(ui(x;t)) = g(fi.(ih(xi))) = S(J'i)(in (xtl) é d>£i(j;.(xi)) and, as the elements ¢h01(x,_)) generate the algebra S(M), g o h is certainly the identity mapping.

Remark (1) Let N = g N, be the direct sum of another family of A-modules with L as indexing set and, for all A e L, let 0,“: M,,—>NA beanA-linear mapping, whence there is an A-linear mapping 1) = C13 0,: M -—> N (II, § 1, no. 6, Proposition 6). Then the diagram

(8) S(MA) —-“‘—> S(M)

LEI:

1% “ml

150:)

® S(N.) ——> S(N)

Ash

is commutative, as follows from the definitions (§ 4, no. 5, Corollary to Proposition 8).

The sub-A-module ofg; S(MA) with which S“(M) is identified by means of the isomorphism g can be described more precisely. For every finite subset] of L, we write E, = @1 S(MA), sO thug S(MA) = li__>mEJ relative to the

504

SYMMETRIC ALGEBRA or A DIRECT sum

§ 6.6

directed set 3(L) of finite subsets of L, by definition (§ 4-, no. 5). For every family v = (n1) EN‘” (thus having finite support) such that 2; n, = n and every finite subset J of L containing the support of the family v, we write

(12)

S""(M) = (8 WM»

so that the submodule E“, of elements of degree n in E5 is the direct sum of the S"”’(M) over all the families v of support contained in J and such than“; nA = n (§ 4, no. 7, Proposition 10 and §4, no. 8). As a convention we write 5'" “(M) = {0} for the families v whose support is not contained in J ; then EJ'n can also be called the direct sum of all the SJ'V(M), where v runs through the set H“ of all families v = (nL)REL such that 22:1. n, = n. Since 5" (MA) is identified with A, clearly also, for two finite subsets] C J’ of L and a family v of support contained in J, the canonical mapping SJ'V(M) —-> S"""(M) (restriction of the canonical mapping E, —-> E; to S’-"(M)) is bg'iectioe. If we write, for all v e H",

(13)

SV(M) = li_m> S’-"(M)

it is seen that, taking account of II, §6, no. 2, Proposition 5:

COROLLARY. The A-module S“(M) is the image under isomorphism (10) of the

submodule «y :8; S(M,) the direct sum of the submodules SV(M) for all the families

v = (n,) e N“) such that k; n, = n; ifJ is the support ofv, SV(M) is canonically isomorphic to @ S’h-(MA).

In general S"(M), A®J sn»(M,) and their image in s"(M) are identified. THEOREM 1. Let A be a commutative ring and M afree A-module with basis (9).)“ L. For every mapping a: L —> N offinite support, we write

(14)

e“ = Q ex“)

(product in the oomnmtatioe algebra S(M)). Then, when a run: through the set N“) of mappings of L into N, offinite support, the e“form a basis of the A-module S(M). As M is the direct sum of the M, = Ark, 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), thenx®y —y®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, no. 5, Theorem 1.

505

III

'I'ENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMME'I’RIC ALGEBRAS

The multiplication table of the basis (14) is given by

(15)

We” = e“”

where a + B is the mapping 7. I-—> «(2.) + (3(1) of L into N. In other words, the basis (e‘) of S(M), with the multiplicative law (15), is canonically isomorphic to the free commutative monoid N‘” derived from L; it follows (§ 2, no. 9) that the symmetric algebra S(M) of afiee module M with basis whose indexing set is L, is canonically isomorphic to the polynomial algebra A[(X;.);ter.], the canonical isomorphism being obtained by mapping e, to XL. In particular (§ 2, no. 7, Proposition 7), for every mappingf: L -—> E of L into a commutative A-algebra E, there exists one and only one A-algebra homomorphism f: S(M) —> E such

thatflei.) =f(1)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 II, § 1, no. 11, 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 §5, no. 5, Corollary to Theorem 1, replacing T by S.

PROPOSITION 10. Let A be a commutative monoid, M a graded A-module of type A and (Magda, its graduation. For every ordered pair (a, n) e A x N, let 5""(M) be the submodule of $"(M) the direct sum ofthe submodules 1% S"k(M¢,_), where (womb

runs through the m offamilies afintegers >0 such that A; n, = 71,] is its support and, jbr each (11,“), (“1.)heJ runs through the set offamilies q" such that 3‘, ark = or. Then (S°""(M))(‘,,_,De A w is the only graduation oftype A x N compatible with the algebra structure on S(M) and which induces an M = $1(M) the given graduation.

The fact that S(M) is the direct sum of the 5"- "(M) follows from the Corollary to Proposition 9; the rest of the proof is identical with the end of the proof of §5, no. 5, Proposition 7.

Suppose more particularly that A = Z and let S(M) be given the total graduation (of type Z) (II, §ll, no. 1) 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 5""‘(M) for q + m = n. Letf 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 o 4);, is a homomorphism of graded algebras of type Z, as follows from the formula

g(x1x2. . .33.) = f(x1)f(x,) . . .f(x,,) for homogeneous x, in M, from the hypothesis onf and from the definition of the graduation of type Z on S(M). 506

§7. EXTERIOR ALGEBRAS 1. DEFINITION OF THE EXTERIOR ALGEBRA OF A MODULE

DEFINITION 1. Let A be a commutative ring and M an A-rnodule. The exterior algebra

te, denoted by A (M) or Alt(M) 01' A,(M), is the algebra over A the quotient qfthe

tensor algebra T(M) by the two-sided ideal 3" (also denoted by 3%) generated by the elements 1: ® x, where x runs through M. Since the ideal 8" is generated by homogeneous elements of degree 2, it is a graded ideal (II, § 11, no. 3, Proposition 2); we write 3: = 3' n T”(M); the algebra /\(M) is therefore graded by the graduation (called canonical) con-

sisting of the A"(M) = T"(M)/5;’,. Then 33 = 3'; = {0} and hence /\°(M) is identified with A and A1(M) with T1(M) = M; in what follows we shall always make these identification and the canonical injection M —> /\ (M) will be denoted by (12" or (birPROPOSITION 1. Let E be an A-algebra andf: M —> E on A-linear mapping such that (f(x))2 = 0 for all x EM.

(1)

There exists one and only one A-algebra homomorphism g: A(M) —>E such that

f = e ° ”. In other words, (A (M), 4;”) is a solution of the universal mapping problem (Set Theory, IV, § 3, no. 1), where 2 is the species, of A-algebra structure, the a—mappings being the linear mappings from the A-module M to an A-algebra satisfying (1).

The uniqueness ofg follows from the fact that b”(M) = M generates A (M). To prove the existence of g, we note that by §5, no. 1, Proposition 1 there exists an A-algebra homomorphism g1: T(M) —> E such that f = g, o (b; we need to prove that g, is zero on the ideal 8", for then if

P: T(M) —> NM) = T(M)/s" is the canonical homomorphism, we can write g1 = g o p, where g: A(M) —> E is an algebra homomorphism and the conclusion will follow from the fact that p 0 ”. Now, the kernel of g1 is a two-sided ideal which, by virtue of (l) and the relation g, o 4) = j; contains the elements 1: ® 1: for x EM. This

completes the proof.

507

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

Remarks. (1) Suppose that E is a graded A-algebra oftype Z, ofgraduation (En) , and suppose also that the linear mappingf (assumed to satisfy (1)) is such that

f(M) c E,.

(2)

Then the relation g(x1x2'. . .x,,) = f(x1)f(x3) . . .f (x,) with the x, e M shows that g(/\’(M)) C E, for all p 2 O and hence g is a graded algebra homomorphism. (2) To avoid confiision, the product of two elements a, v of the exterior algebra A(M) is usually denoted by u A v and is called the exteriorproduct of u by v. The elements of A“(M) are therefore the sums of elements of the form x1 A x, A - - - A on with x! e M for l < i < n and are often called n-veotors. 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-h'near mapping. There exists one and only one A-algebra homomorphism

u': /\(M) —>/\(N) such that the diagram M—1—->N

Mai

16?:

NM) 7* MN) is commutative. Moreover, a” is a graded algebra homomorphism. The existence and uniqueness of I" follow from no. 1, Proposition 1 applied to the algebra A(N) and f = (I); o u: M —> A (N); for f(M) C N and hence f satisfies condition (1) by definition of 8;: as f(M) C A1(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 /\(u). If P is an A-module and v: N ——> P is an A-linear mapping, then

(3)

Now) = No) o/\(u)

for A (o) o A(u) is an algebra homomorphism which renders commutative the diagram

M

65.1

P

1%

NM) m NP) 508

FUNCTORIAL PROPERTIES OF THE EXTERIOR ALGEBRA

§ 7.2

Since A(M) contains M = A1(M), A(u) is sometimes called the canonical

extension ofu to A(M). The restriction Noe) : A"(M) --> A"(N) is such that

/\" T"(N) by passing to the quotients. PROPOSITION 3. If u: M -> N is a surjective A-linear mapping, the homomorphism

A(u): A(M) _> MN) is smjective and its kernel is the non-sided ideal (If/\(M) generated by thekemelP c: M c A(M) qfu. The proof is derived from that of § 6, no. 2, Proposition 4, replacing S by

/\ and 3’ by 3'. If u: M —> N is an injective linear mapping, it is not always true that A(u) is an injective mapping (§ 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 directfactor of N and then the image of A(u) (isomorphic to A(M)) is a directfactor of A(N); the proof is the same as that for the analogous assertions for T(u) (§ 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 directfactor in M and their intersection N n P is a directfactor in N

and in P. Then the homomorphism: A(N) —> A(M), A(P) —> NM) and A(N n P) -> A(M), canonical extensions of the canonical iry'ections, are injective; if

A(N), A(P) and A(N n P) are identified with subalgebras of/\(M) by means of these homomorphism, then

(5)

A(N n P) = A(N) n A(P).

The proof is derived from that of § 5, no. 2, Proposition 4 replacing T by A throughout. The hypotheses ofProposition 4 are always satisfied by arbitrary submodules N, P of M when A is afield. COROLLARY. Let K be a commutative field and M a vector space over K. For every element 2 e A(M), there exists a smallest vector subspace N ofM such that z e A(N) and N isfinite-dimensional. The proof is deduced from that of § 5, no. 2, Corollary to Proposition 4- replacing T by A throughout. N is called the vector subspace of M associated with the element 2 of A(M). 509

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRJC ALGEBRAS

3. ANTICOMMUTATIVITY OF THE EXTERIOR ALGEBRA

PROPOSITION 5. (i) Let (x,)1 «a, be afinite sequence ofelements ofthe module M;for every permutation o‘ in the symmetric group 6,, (6) ”car A ”am A A ”am = tu-xi A 742 A A x, where so denotes the signature of the permutation 0. (ii) If there exist two distinct indices 1', j such that x, = x,, the product x1 s A---Axn

is zero.

(i) First of all, since at A x = 0 for all x e M by definition of the ideal 3',

also, for x, y in M,

xAy+yAx=(x+y) A (x+y) -xAx—y/\y=0This establishes (6) in the case n = 2. The general case then follows from § 4,

no. 6, Lemma 3.

(ii) Under the hypothesis of (ii), there exists a permutation a e 6“ such that

0(1) = i and 6(2) = j; then the left hand side of (6) is zero for this permuta-

tion and hence so is the right hand side.

COROLLARY I. Let H, K be two complementary subsets of the interval [1, n] of N and let (1),)“,‘0, (jk)1 a(h') is

equal to v.

Conomv 2. The graded algebra /\(M) is alternating (§ 4, no. 9). It suffices to apply Proposition 13 of § 4, no. 9 to /\(M), taking as generating system the set M and using Proposition 5.

510

n-th EXTERIOR POWER OF A MODULE

§ 7.4-

PROposrrION 6. If M is a finitely generated A-module, /\(M) is a finitely generated A-module; also, ifM admits a generating system with n elements, then A’(M) = {0} for p > n. Let (x,)1‘y“ be a generating system of M. Every element of A’(M) is a linear combination of elements of the form x“ A x12 A "' A x1,

where the indices ik belong to (l, n]; by Proposition 5, these indices can be assumed to be distinct (otherwise the corresponding element is zero). If}; > n, there is no such sequence of indices and hence A”(M) = {0}. Ifp s n, these sequences are finite in number, which completes the proof. 4. n-th EXTERIOR POWER OF A MODULE AND ALTERNATING MULTILINEAR 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 g n "" 2s (8)

f(u1:--'2upaxa x: ”linuvn-p-n) = 0

forallx,theug(l gisp) andthev, (1 N the nlinear mapping f:(x1,'--,xn) Hf(x1®x2®m®xu)

(II, § 3, no. 9). On the other hand, the A-linear mappings g: A"(M) —> N are in one-to-one correspondence with the A-linear mappingsf: T“(M) —> N such thatf is zero on 8;, by associating with g the mappingf = g o p“, where

tn: T“(M) —> MM) = T"(M)/s: 511

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

is the canonical homomorphism (II, §2, no. 1, Theorem 1). But as 3; is a linear combination of elements of the form (“1 @112 ®"'®up) ®(x®x)®(”1®'“®”n—p—2)

(x, u,, v, in M), to say thatfis of the form g o [in means that the corresponding n-linear functionfsatisfies (8), in other words it is alternating. The A-module A" (M) is called the n-th exterior power of M. For every Amodule homomorphism u: M -—> N, the mapping

/\"(u) : A”(M) —-> A"(N) which coincides with /\(u) on /\"(M) is called the n-th exterior power qfu. COROLLARY I. For every alternating n-linear mapping g: M“ -—> N, for every permutation o' e 6", (10)

g(xa(1)s 360(2), ~ ~ - s ”00») = ea'g(""1: x29 ' - '2 xn)

jbr all x, e M; moreover if x, = xjfirr two distinct indices i, j, then g(x1, x2, . . ., x") = 0.

This is an obvious consequence of Proposition 7 and no. 3, Proposition 5. COROLLARY 2. Let (xi), «9, be a sequenee of n elements of M such that xlgA--0Ax,,=0;

then, for every alternating n-h'near mapping g: M" —> N, g(x1, . . . , x") = 0.

‘

COROLLARY 3. Let f: M"'1 -> A be an alternating (n - l)-linear form. If

(x,)1 A" is another ring homomorphism, M" an A”-module, gzM’ —> M" an A’-homomorphism (relative to a) and g”:/\A,(M’) —> AA»(M') the corresponding A”-homomorphism of algebras, then the composite A-homomorphism of algebras

(13)

f' 8" AM) —+ AMM), ——> /\u(M)n

corresponds to the composite A—homomorphism g of:M—> M” (relative to a o p).

PROPOSITION 8. Let A, B be two commutative rings, pzA —> B a ring hamomorphim and M an A—module. The canonical extension

aBas @A M) —> B @A AA(M)

q” the B-lz'near mapping 113 93 4:311:13 @A M—->B @A AA(M) is a graded Balgebra isomm'phism.

The proof is derived from that of § 5, no. 4, Proposition 5 replacing T by A and (bu by (lie6. DIRECT LIMITS OF EXTERIOR ALGEBRAS

Let (Au, (1),“) be a directed direct system of commutative rings, (wea) a direct system of Aa-modules, A = 1112A“ and M = 1i_1_r>1 M0,. For at < B, we

derive canonically from the Ant-homomorphism fnuzMa —> Me an Act-algebra

homomorphism (no. 5, formula (12)) fag: AA“(Ma) _> AAB(M,,) and it follows from (13) that (AAa(Mu)sl:x) is a direct system of graded Act-modules. On the other hand let A: Ma —> M be the canonical Au-homomorphism; we

derive (no. 5, formula (12)) a graded Act-algebra homomorphism f: :AAJMa) —> AA(M) and it also follows from (13) that the f; constitute a direct system of Aa-homomorphisms.

514

EXTERIOR ALGEBRA or A DIRECT SUM

§ 7.7

Pnoposmon 9. The A-homomorphism f” = gfizhg AA.(M,,) —> AA(M) is a graded algebra isomorphism. The proofis the same as that of§ 5, no. 4-, Proposition 6 replacing throughout T by A and (I) by (V and taking account of the fact that a direct limit of alternating algebras is alternating. 7. EXTERIOR ALGEBRA OF A DIRECT SUM. EXTERIOR ALGEBRA OF A GRADED MODULE

Let A be a commutative ring, M = “(Jab M,L the direct sum of a family of a family of A-modules and jkte- M the canonical injection; an A-homo-

morphism ofgraded algebras /\ (A) :/\ (MA) —) /\ (M) is derived. Since /\(M)

is anticommutative, Proposition 10 of §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 (jA) ; then there exists a unique algebra homomorphism

/\(M,‘) -> /\(M) g: “8§ A(M,H) such that u ojA = u,L for all AeL. The skew tensor product 3&1 /\(M,) is an alternating algebra: for, for L finite, this follows from § 4, no. 9, Proposition 14and, for L arbitrary, this follows from the definition of this product given in

§4, no. 8, Remark 1 and the fact that a direct limit of alternating graded alge515

III

'I'ENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

bras is alternating. As also u(M) is contained in the submodule of elements of degree 1 in the graded algebra I:Eg/VMQ, it follows from no. I , Proposition 1 and Remark 1 that there exists a unique graded algebra homomorphism.

h:/\(M>+‘® NM» ASL

such that h o (1);, = u. The verification of the fact that g o h and h o g are the identity mappings is then performed as in §6, no. 6, Proposition 9 replacing S by A and (If by 4)”.

Remark. Let N = $691. N,» be the direct sum of another family of A-modulcs with L as indexing set and, for all he L, let z),':M,~,-->N,L be an A-linear mapping, whence there is an A-linear mapping 0 = (‘P 12t —>N (II, § 1, no. 6, Proposition 7). Then the diagram

‘69 AM) 1‘-> NM) 3:. Audi

1A (a)

1%), /\(N,.) T /\(N) is commutative (cf. §4, no. 5, Corollary to Proposition 8).

The sub-A-module of “1%). /\ (MA) with which A”(M) is identified by means of the isomorphism g can be described more precisely. For every finite subset K

J of L, we write EJ = Lg /\(M,_), so that “1% /\(M,\) = li_m>EJ relative to the directed set E’y’(L) of finite subsets of L, by definition (§ 4, no. 8,

Remark 1). For every family v =‘(nA) eN‘L) (therefore with finite support)

such that A; n,“ = n and every finite subset J of L containing the support of the family v, we write

(16)

AWM) = 99 AWM.) eJ

so that the submodule EJ’n of elements of degree n in E, is the direct sum of the AJ’V(M) for all families v of support contained in J and such that Znh=n (§ 4-, no. 7, Proposition 10 and no. 8). By way of convention we write

AJ’V(M) = {0} for the families v whose support is not contained in J; then 516

EXTERIOR ALGEBRA OF A FREE MODULE

§ 7.8

E; n can also be called the direct sum of all the AJ'V(M), where v runs through

the set H, of all families it = (mm such that A; n, = n. Since /\°(M,) is identified with A, clearly also for two finite subsets J C J’ of L and a family

v of support contained in J, the canonical mapping AJ’V(M) —>/\J"V(M)

(restriction of the canonical mapping EJ—>EJ. to N'"(M)) is bijective. If we write, for all v e H,” V

0 such that A; n,L = h, J is its support and, for each (n,_), (1A.)AEJ runs through the set offamilies q" such that

A; e, = e. The; (/\“""(M))(.,,,,,eAxN is the only graduation oftype A x N compatible with the algebra structure on A(M) and which induces an M = A1(M) the given graduation. The fact that A (M) is the direct sum ofthe A“' "(M) follows from the Corollary to Proposition 10; the rest of the proof is identical with the end of the proof of § 5, no. 5, Proposition 7. 8. EXTERIOR ALGEBRA OF A FREE MODULE

THEOREM 1. Let M be an A-module with a basis (ether, Let L be given the structure of a totally ordered set (Set Theory, III, §2, no. 3, Theorem 1) andfor every finite subsetJ of L we write (18)

9J=3MA9MA”'A‘A.

517

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

where (Akh‘k‘n is the sequence of elements of J arranged in increasing order (Set Theory, III, §5, no. 3, Proposition 6) ; we write e2, = l, the unit element of A.

Then the eJ, where] runs through the set 3(L) offinite subsets of L, form a basis of the exterior algebra A (M). Since the eA generate the A-module M, every element of /\(M) is a linear combination of a finite number of products of the elements e,L and hence (taking account of no. 3, Proposition 5) is a linear combination of a finite number of elements eJ for J E 3(L). It reduces to proving that the eJ 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 eJ whose coeflicients in this relation are 560 is a finite subset K of L (since there is only a finite number of coeflicients

#0). Let N be the submodule of M generated by the e,“ such that )t GK; N is a direct factor in M, hence (no. 2) A(N) is identified with a subalgebra

of/\(M) and, if we show that the e, with J c K form a basis of /\(N), we shall obtain the desired contradiction. It therefore reduces to providing Theorem 1 when the basis of M is finite; we may therefore suppose that L = [1, m] C N. For each ieL, let M. be

the free submodule Ae, of M; M is the direct sum of the M, and /\(M,) is the direct sum of A°(M,) = A and A1(M,) = M, (no. 3, Proposition 6). Let /\(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, e,) of the /\(M,) (II, § 3, no. 7, Corollary 2 to Proposition 7); thus we obtain all the elements u1®u2® ... ®um

where either u, = 1 or u, = e,; if J is the set of indices 1' such that u, = e" ul ® uz ® - - - ® um is identical with eJ, which completes the proof.

Comment: 1. Suppose that L = (1, m]; then the basis (anew q(M) has 2m elements. For 1) > m, /\’(M) = {0}; /\’"(M) has a basis consisting ofa single element eL; for 0 < p < m the number of elements in the basis (e3) of A’(M) con;sisting of the e, such that Card (J) = p is (m) _

1’

m!

.

PW" “'10)!

This follows from Set Theory, III, §3, no. 5, PropOsition l2 and Set Theory,

III, §5, no. 8, Corollary 1 to Proposition 11.

We return to the case where the set L in Theorem 1 is arbitrary and give 518

0mm FOR LINEAR INDEPENDENCE

§ 7.9

explicitly the multiplication table (§ 1, no. 7) of the basis (4,). Given two finite subsets J, K of the totally ordered set L, we write

(19)

ifJ n K 9‘ ‘5 9.1.: = 0 9.1.: = ("Dv ifJ n K = fi

where v denotes the number of ordered pairs (1, p.) e] x K such that A > it. Then Corollary 1 to Proposition 4- of no. 2 immediately implies the relation (20) Note the formula (21)

‘J A 9x = PJ.x¢JuKPJ.xPx.J = (-1)"‘

when J nK = z, j = Card(J), k = Card(K) (no. 3, Corollary2 to Proposition 5.) COROLLARY 2. If M is a projective A-module, A(M) is a praiective A-module.

The proof is the same as that of §5, no. 5, Corollary to Theorem 1 re-

placing T by A. COROLLARY 3. Let M be a prigjective A-module and (1:4) 1‘,“ a finite sequence qf elements Qf M. For there to exist an M an alternating n-linearformfsuch that f(x1:x22---axn) 7e 0:

itisnecessaryandsuficientthatn A x... A

A xnaéO.

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 xlzA

AxnaéO.

Then A"(M) is projective (Corollary 2) and hence the canonical mapping

/\"(M) —> (A"(M))** is injective (II, §2, no. 7, Corollary 4- to Proposition 13); we conclude that there existsalinear form g:/\"(M) —>Asuch that g(x1 A x, A - - - A x") 95 0. Iff is the alternating n-linear form corresponding to g (no. 4-, Proposition 7), thenf(x1,. . .,x,,) 9e 0. 9. CRITERIA FOR LINEAR INDEPENDENCE

PROPOSITION 12. Let M be a pny'ective A-madule. For elements x1, x2, . . ., it» If M to be linearly dependent, it is necessary and suflicient that there exist A at: 0 in A such that (22)

MlaA‘°'Axn=0.

519

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

The condition is necessary (with no hypothesis on M), for if for example Ax, (with 7t aé 0) is a linear combination of :42, . . ., 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 7. 7’: 0. If MzAxaA---Axn=0, then the induction hypothesis implies that x2, . . ., x“ are linearly dependent and hence a fortiori x1, x2, . . ., x" are. If M2 A x3 A - - - A x“ 9E 0, it follows

from no. 8, Corollary 3 to Theorem 1 that there exists an alternating (n — 1)linear formf such thatf(M2 A x3 A - - - A x“) = p. as 0. Since x1A(7lx2)A ---Ax,,=0,

it follows from no. 8, Corollary 3 to Theorem 1 that 9.x, is a linwr combination of M2, x3, . . ., x”; hence x1, x2, . . ., x" are linearly dependent.

COROLLARY. Let M and N be two projective A-module: andf:M ——>‘ N an A-linear mapping. Iff is iry'ective, then A(f) : A(M) ——> A(N) is infective. We prove this first under the assumption that M is free; let (90“,, be a basis of M, so that (mama (formula (18)) is a basis of A(M). Suppose that the kernel of A(f) contains an element u = 2 age, 96 0. Let K be an element minimal among the finite subsets J such that a; aé 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 «J aé 0; for all J aé K such that «J aé 0, we have therefore by definition J n H =;é 0 and consequently (no. 8, formula (20)). u A an = +“K6KUH

belongs to the two-sided ideal of /\(M), the kernel of Nf). We write axon = 9M A em A A 57...; then “Kf(‘tl) Af(5M) A Af(‘7t..) = 05 by virtue of Proposition 12, the elementsf(eh) (I g i S n) of N are linearly dependent. But this contradicts the hypothesis thatf is injective (II, § 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 (II, § 2, no. 2, Proposition 4). Consider the linear mapping gzM @M’ —>N (43M C-BM' such thatg(x,y) = (f(x), 0,31), so that there is a commutative diagram

l l M—L—>N

PT>N69P

520

CRITERIA FOR LINEAR INDEPENDENCE

§ 7.9

where the vertical arrows are the canonical injections. Since g is injective and P is free, /\(g) is an injective homomorphism as has been seen above. Now, /\(j):/\(M) ->/\(P) is an injective homomorphism since M is a direct factor in P (no. 2). The composite homomorphism

A(M) 13’; /\(P) 1‘2) /\(N ear) is therefore injective and, as it is also equal to the composite homomorphism

/\(M) M A(N) fl> A(N @P) we conclude that /\(f) is injective. PROPOSITION 13. Let M be an A-module, N a direetfactar submodule 9," M which is free If dimension p and {u} a basis qf A’(N). For an element x e M to belong to N, it is necessary and sufiicient that u A x = 0. Let P be a submodule of M complementary to N and let 3/ e N, z e P be suchthatx=y + z.Thenu A x = u A z.As/\p(N) isfreeofdimension l,

the mapping 4):? —-> P g: /\"(N) such that w) = [2 ® a is bijective (II, §3, no. 4-, Proposition 4). On the other hand (no. 7, Proposition 10), the composition of the canonical homomorphisms

vbzP ® A”(N) —>/\(P) ® A(N) —>/\(M) is injective. The mapping t]; 0 M ofratio A e A determined uniquely (II, §2, no. 3, Proposition 5).

DEFINITION l. The determinant of an endomorphism u of a free A-module M of finite dinwnrion n (II, §7, no. 2, Corollary to Proposition 3 and Remark 1), denoted by det u, is the scalar A such that A"(u) is the homothety of ratio A. By formula (4) of § 7, no. 2, det u is the unique scalar such that

(1)

“(9:1) A "(#2) A

A “(1%) = (detu)-x1 A x: A

A x»

for every sequence (x,)1 det(X(x1, . . ., 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, 524-

DETERMINANT or A SQUARE MATRIX

§ 8.3

the determinant of the new matrix is equal to that of the old multiplied by c,. 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 let (emu be a basis of M; for every automorphism u of M, if X is the matrix of u with respect to the basis (q), then

det(u) = det(X)

(7)

as follows immediately from the definitions.

When I = [1, n], the determinant of X is also denoted by det(gli)1 p arbitrary values and applying Cramer’s formulae (no. 7, formulae (37)) to calculate the

E: 0findexj < 1’We know (II, § 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 536

THE UNIMODULAR GROUP SL(n, A)

§ 8.9

condition of the statement, let a; (l s i < [2) denote the first p columns of L and y = (11.) the (n + l)-th column of M; since all the columns ofL 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 a” or also (Proposition 15) that a1 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 (4-1). The last assertion is then an immediate consequence of Proposition 13 of no. 7.

9. THE UNIMODULAR GROUP SL(n, A) Let Mn(A) denote the ring of square matrices of order 71 over A. Consider the mapping det:M,,(A) —> A. The group GL(n, A) of invertible elements of Mn(A) (isomorphic to the group of automorphisms of the A-module A" (II, § 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()t, 1,. . ., 1)) = )t

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 SLn(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 field. Recall that for l s i S n, l < j S n, E” denotes the square matrix of order n all of whose

elements are zero except the one in the row ofindex i and the column ofindex j, which is equal to 1; with In denoting the unit matrix of order n, we write

3410‘) = I,, + XE” for every ordered pair of distinct indices 1', j and all A 6A (II,§10, no. 13).

PROPOSITION 17. Let K be a commutative field. The unimodular group SL(n, K) is generated by the matrices 3,,(1) with i aé j and A e K. By II, § 10, no. 13, Proposition 14-, we know that every matrix in GL(n, K)

is a product of matrices of the form B,,().) and a matrix of the form diag(l, l, . . ., l, at) with aeK*. Now it is immediate that det(B,,()\)) = l and det (diag(l, . . ., l, at)) = at (no. 6, Example 2); whence the proposition. 537

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

COROLLARY. The group SL(n, K) is the group of commutator: of GL(n, K), except in the case where n = 2 and where K is afield with '2 elements. As SL(n, K) is the kernel ofthe homomorphism det ofGL(7:, K) to a commutative 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*, B,,(A) belongs to P. Now, BUOQ is a conjugate of B,,(l) in GL(n, K) since 3:10) = Q.B,,(1) .Q‘l, where Q denotes the matrix with respect to the canonical basis (e,) of the automorphism o of K" such that o(e,) = M, o(e,,) = ck for I: ye 1'. On the other hand, let u“ (for i 96 j) be the automorphism of K" such that uu(e,) = —e,, u,,(e,) = e” u,,(e,,) = ck for k ${i, j}, which belongs to SL(n, K); then 3,,(1) = U,,B,,(-A)U,;1, where U“ is the matrix ofu“ with respect to the canonical basis. Similarly, if 1 < i < j, then B11(>\) = U1;Bu(l) U131 and finally, for 2 p.x of A[X] x M into M is A-bilinear, it defines canonically an A-linear mapping (1» A[X] @A M -—> M such that

4>(1_'® 1) = [M = 90000-

(44)

On the other hand, A[X] ®A M has canonically an A[X]-module structure (II, §5, no. I); we shall denote this A[X]-module by M[X] ; the mapping 4): M[X] —-> Mu is A[X]-linear since, for p, q in A[X] and x e M,

(WU! ® x)) = MW) 8”) = (910-16 = 40000000) = WWI ® 1)Moreover, u is an A[X]-endomorphism of M", for

"(P-x) = “(141000) = (uP(x))(x) =P-u(x)Finally, an A[X]endomorphism 17 ofM[X] is defined by writing (II, § 5, no. 1)

(45)

17(1) ® ’6) = 11 ® “(xl

Moreover it follows from formulae (44) and (45) that that A[X]-linear mappings u, 11 and 0. On the other hand, for all x e M and

all!) EA[XL

’(§(p ® 16)) = MP (8 306)) = P(u')(g(x)) = g(P(u)(x)) = g(¢(p 8) x)) and

'7'(g(!’ ® 56)) == 17(1) ®g(x)) = 1’ ® u'(g(x)) = 11 ® g(u(x)) = 4315(1) ® x)) which proves the commutativity of diagram (48). ll. CHARACTERISTIC POLYNOMIAL OF AN ENDOMORPHISM

Let M be a free A-module of dimension n and u an endomorphism of M. Consider the polynomial ring in two indeterminates A[X, Y] and the A[X, Y]module M[X, Y] = A[X, Y] (8A M; let I? be the endomorphism of the A[X,Y]-module M[X,Y] canonically derived from a (II, §5, no. 1). It follows from no. 5, Proposition 11 that

(49)

det(X — Y1?) = 2; (—1)rrr(/\’(u))xn-IYI

for if Uis the matrix ofu with respect to a basis (e,)1‘.‘,.ofM, Uisthe matrix ofl'i with respect to the basis (1 ® e‘)1« K’ [X] derived from h. In particular,

(21)

for K' = K[X], we obtain, writing A[X] = A ®x K[X],

(22) PcA,K(a; X) = NAIXJIKIX](X " “)More generally, ifK’ is a commutative K-algebra and A' = A (8,; K’, then, for all x e A', PCAIK(a; x)

= NA'IKr(x —' 0).

Examples. (l)_ Let A be a quadratic algebra over K of type (a, [3) and (e1, e2) a basis oftype (a: B) (§ 2: no. 3)- For x = $91 + 7192, Til/1:0”) = 2E + an and NA,x(x) = £3 + [351; — om”; these functions are therefore identical with the Cayley trace and norm of x (§ 2, no. 24). (2) Let A be a quaternion algebra over K. A direct calculation allows us to 54-4

PROPERTIES OF NORMS AND TRACES IN AN ALGEBRA

§9.4-

verify that TrA,K(x) = 2T(x) and NA,K(x) = (N(x))2, where T and N are the Cayley trace and norm (§ 2, no. 4). (3) Let A = Mn(K) and let the canonical basis (EH) ofA (II, § 10, no. 3) be

arranged in lexicographic order. Then it is immediately seen that for every matrix X = Z] guEu the matrix (oforder n2) of the endomorphism Y» XY is of the form

0 O

X

0

0

0

X

whence TrA,K(X) = 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 afinite basis. For an element a e A to be invertible, it is necessary andsuflieient that its NMK (a) be invertible in K. If a admits an inverse a’ in A, then NAIK(a)NA/x(a') = NAIK(M’) = NAIKU) = l

by formula (3) of no. 1. Conversely, if NA,K(a) is invertible, the endomorphism h: x -—> ax is bijective (§ 8, no. 2, Theorem 1). Then there exists a’ e A such that aa' = 1; then h(a’a — l) = aa’a — a = (aa' - l)a = 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 aeA, PcA,3(a; a) = 0. This follows immediately from the Gayley-Hamilton theorem (§ 8, no. 1], Proposition 20). PROPOSITION 5. Let A be a K-algebra and m a two-sided ideal qf A. Suppose that A0 = A/m is afree K-module qffinite dimension n, that there exists an integer r > 0 such that m' = {0} and that m‘ ‘ 1/m‘ is afree Ao-module offinite dimension syfiir l < i g r. Lets = :1 + . - - + s, andflralla eAletao denotethealassqfamod. m. ThenAisa flee K-madule qf dimension ns and, for all a e A,

(23)

TrA(a) = S-Trso(ao),

NA“) = (Nio(ao))‘

PcA(a; X) = (PcAo(ao; X))'.

By virtue of II, § 1, no. 13, Proposition 25, m"1/m‘ is a free K-module Of dimension ns,. Hence Proposition 1 of no. 2 can be applied with P, = m“ 1/m‘; 545

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

this shows in the first place that A is a free K-module of dimension n(s1 + - - - + 5,.) = ns. Moreover, the hypothesis implies that the A-module P. is isomorphic to a direct sum ofst submodules isomorphic to the A-module A0; by Proposition 1 ofno. 2, therefore NP‘(a) = NM(a)’t; finally therefore

NAM = NAM. In this formula NAo(a) is defined by considering AD as a left A-module and it is equal to the determinant of the K-linear mapping x v—> ax ofA0 into itself; but, as ax = aox for xe, NAo(a) = NAo(ao), which completes the proof of formula (23) for the norm. The two others are shown analogously. PROPOSITION 6. Let A be a commutative K-algebra admitting afinite basis over K and V an A-module admitting afinite basis over A. Then V admits afinite basis over K andfor every A-endomorphism u qf V, if ax is the mapping a considered as a K-endomorphism qfV,

(24)

Tran) = Trimaran,

det(ua = NA/x(d°t(“))

Pc("x3 X) = NAIXJIKISXJ(PC("; X»-

Let (a.)1‘.‘m be a basis ofA over K and (e,)1‘“,, a basis ofV over A; then

(ate,) is a basis ofV over K (11, § 1, no. 13, Proposition 25). On the other hand the third offormulae (24-) can be deduced from the second applied to the endomorphism X — a of the A[X]-module A[X] ®A V (§ 8, no. 10). It will therefore suffice to show the first two formulae in (24). We shall first establish the following lemma: Lemma l.LetX,, (1 sign, 1 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’) = (014) -a' + to, des(a)a' (d'a')-

It obviously suflices 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 8 (a, a’, a”) 1—» (da, d ’a’, d 'a”) 551

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

of A G) A’ 6-) A” into B {-9 B’ {-9 B”). Relation (4) can then be written more simply (5) d(a.a’) = (do) .a’ + ehdegmfl. (da’).

The e-derivations of (A, A', A”) into (B, B’, B”) of given degree form a sub— K-module of the K-module of graded linear mappings Homgrx(A {—B A’ (43 A”, B (43 B’ @ B”). When e(ac, fl) = l for all a, (5 in A, we say simply derivation (or K-derioation) instead of ederivation. Derivations form a sub-K-module of

HomK(A (+3 A’ e A', B a; B’ ea B”). When A = Z and s(p. g) = (—1)", every s-derivation of even degree is a derivation; every e-den'vation of odd degree is often called an antidm'vation (or K-antiderivatian); an antiderivation d therefore satisfies

(6)

d(a.a') = (do) .a’ + (— l)“°“°)a. (da’)

for a homogeneous element a e A. Remarks. (1) The notion ofderivation can be defined for non-graded modules by agreeing to give these modules the trivial graduation. (2) If only e-derivations of given degree 8 are considered, the commutation factor 5 may be disposed of as follows: the bilinear mapping 1;: A x B' ->- B" is modified by replacing it by the bilinear mapping mg: A x B’ —> B” such that, for every homogeneous a in A and all If e B’, 1““: I”) = so, des(n))‘2(a: ”>Then d is a derivation relative to the bilinear mappings u, 11, A5.

The general definition of e-derivations given above is especially used in two cases: Case (I): A = B, A’ = B’, A” = B” and the three bilinear mappings u, 11, 7&2 are equal to the same mapping. Case (II): A = A’ = A”, B = B’ = B”, so that (for y.) A is a graded algebra and the two K-bilinear mappings. (7)

AlsA—>B,

A,:AxB—>B

are such that the corresponding K-linear mappings B (8),; A —-> B, A ®x B —> B are graded of degree 0. An a-derivation of degree 8 ofA into B is then a graded

K-linear mapping (1: A —> B of degree 8, such that for every homogeneous x in A and every y e A, we have the relation

(3) 552

11W) = (My + sa.ae¢(a)x(dy)-

EXAMPLES or DERIVATIONS

§ 10.3

Consider in particular in case (II) the case where A is a unital associative Kalgebra and Al and 12 are the external laws of an (A, A)-bimodule (§ 4-, no. 3, Definition 3). This holds notably when A and B are two unital associative Kalgebras, 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 )‘1: (b: a) H bp(a),

)‘2: (a: b) H 9(a)b

for a e A, b e B.

Cases (I) and (II) have the following casein common: consider a graded Kalgebra A, take B = A, mappings (7) both being multiplication on A. We then speak of an e-derivation (or (K, a) -derivation) q" the graded algebra A: it is a graded K-linear mapping of A into itself, of degree 8, satisfying (8) for every homogeneous x in A and all y e A. In particular ifA is a graded ring, considered as an (associative) Z—algebra, we speak of the e-derivation of the 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 xeA,yeA

holds for such a derivation d: A —-> B. 3. EXAIKPLES 0F DERIVATIONS

Example 1. *Let A be an R-algebra ofdifferentiable mappings ofR into R and let x0 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 fn—> Df(x0) is a derivation, since (Functions of a Real Variable, I, § 1, no. 3)

(D(fg))(xo) = (Df(xo))g(xo) +f(xa)(Dg(xo))-* Example 2. *Let X be a differentiable manifold of class C” and let A be the graded R-algebra of difierenfial forms on X. The mapping which associates with every difi'erential form a) on X its exterior differential dm is an antiderivation of degree +1 (Difl’erentiable and Analytic Manifolds, R, §8).* Example 3. Let A be an associative K-algebra. For all a GA, the mapping x n——> ax — xa 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 (§ 7, no. 1). *It will be seen in § 11, no. 9 that, for all x e M, the right interior product i (x) is an antiderivation of A of degree — l.* 553

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TENSOR ALGEBRAS, EXTERIOR ADGEBRAS, SYMMETRIC ALGEBRAS

Example 5. Returning to the general situation of Definition 1 of no. 2, let K be another commutative ring and p: K—> K a ring homomorphism; let

A, A’, A”, B, B’, B" denote the graded K-modules obtained respectively from A, A’, A", B, B’, B” by extending the ring of scalars to K (II, § 11, no. 5); we derive from (1., Al and A2 K—bilinear mappings fi:AxA’—>A”,

ilsA’-—>B',

i,:XxB'—>B'

by considering the tensor products by 11-; of the corresponding K-linear mappings to u, A, and 12 (II, § 5, no. 1). Then, ifd is an e-derivation of degree §of (A, A', A”) into (B, B', B”), the mapping :2 = d ® 1;: ofA (-B A’ (-B K” into

B G) B’ (49 B” is an s—derivation of degree 8 of (A, A', X”) into (B, B’, B”). Example 6. Let A be a graded K—algebra oftype Z; a graded linear K-mapping of degree 0, d: A—>A, is defined by taking, for x" 6A,,(n e Z), d (xn) = nxn. This mapping is a derivation of A, since, for x, e A,, xq e A“, d(xpxa) = (P + q)x9xa = d(xr)xq + xrd(xa)-

4. COWGSITION 0F 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 (1.: A x A’ —>A' corresponding to a graded K-linear mapping of degree 0, A (8),; A’ —> A". The graded endomorphismsfofA ® A’ (-B 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 Endn(A (-B A' 6—) A”) (§ 3, no. 1, Example 2). In particular two e-derivations of (A, A’, A”) can be composed, but it should not be thought that the composition of two s-derivations is an s-derivation. On every graded algebra B of type A is defined the s-bracket (or simply bracket when 5 = l) of two homogeneous elements u, v, by the formula (10) [14, v], = no — adesuIdegavu (denoted simply by [u, v] if e = 1). By extending this mapping by linearity, a K—bilinear mapping (u, o) n—> [u, o], of B x B into B is obtained. Then, for homogeneous u and v in B [”1 u]: = _‘de¢u.de¢u[us v]:-

Applying this definition to the graded algebra Endn(A (-BA' (-BA'), the e-bracket of two graded endomorphism is thus defined. PROPOSITION 1. Let d1, (12 be two e-derioations of (A, A’, A”) of respective degree:

81, 8,. Then the s—bracket [‘11: d2]: = “’1 ° ‘12 — 551.5242 ° ‘11

is an s—derivation ofdegree 81 + 82. Moreover, ifd is an s-derioation of (A, A’, A”) of degree 8 and ifs“, = —1, then (12 = do d is a derivation. 554-

COMPOSITION OF DERIVATIONS

§ 10.4-

Suppose I: e A is homogeneous of degree 5; for ally e A', d1(da("!l)) = ((4142) (1))!1 + ‘61.6a+:(d2x) ((11.11)

(ll)

+ 362 :(dlxxday) + 361+62.:x((41¢12)(!/)) talnng account of formulae (1) of no. 1. Exchanging the roles of (1; and d2, we obtain, after simplifications again using (1) (no.1), (d1d2) (xy) — $61.62(d2d1)(xy) = ((d1d2)("))y — 351.63((d2d1) (5'))! + 56; +62. z"((d1d2) (11)) — 351. 6355; +52. :x((d2d1) (11))

that is, writing d = [db algLz and 8 = 81 + 83,

d(xy) = (My + so,:x(dy) which proves that d is an e-derivation. ' On the other hand, if, in (11), we let d1 = d2 = d, 81 = 8, = 8 and a“ = —l, we obtain, since then 35.“: = —ea,¢ by (1),

WW) = “My + $25.:x(d’zl) and as $25,; = 1 it is seen that d2 is a derivation.

COROLLARY. Suppose thatA = 2. Then: (i) The square (J an antiderivation is a derivation. (ii) The bracket of two derivations is a derivation. (iii) The bracket of an antiderivation and a derivation 9" even degree is an antiderivation. , (iv) If (11 and d2 are antidm'vations, dldz + dzdl is a derivation.

Under the hypotheses of the beginning of this no. consider now a finite sequence D= 0101‘..." of pairwise permutable derivations of (A, A’, A'). For every POIYnomial P (X1:

9 Xn) m the algebra K[X1s“: Xu]; the element

P(d1,..,a',,) ofEndn(A 6-) A’€-) G—)'A') is then defined (§ 2, no. 9); its abbreviated notation is P(D).

PROPOSITION 2. With the above hypotheses and notation, consider 2n indeterminate: T1, . . ., Tn, T1,. . ., T,’, and for every polynomial F EK[X1, . . ., Xn] write F(T) = F(T19 ‘ ' ', Tn), F(T’) = F(Tia - - -: T9,!) and

_ Suppose that

F(T+T’)=F(T1+T1,...,T,,+T;,).

M + T') = Z Q.(T)R. E be an e-den'vation of degree 8 and a a homogeneous element oa ofdegree on. Then the mappingx I—> a(dx) is an e—den'uation ofdegree 8 + (X.

We write d’(x) = a(dx); for x homogeneous of degree E, in A and y EA, by virtue of (23) and (1) (no. 1),

WW) = 4((dx)y) + su.aa(x(a’y)) = (4(dx))y + eo+a.z("a)(dy) = (d'x)y + ¢o+u.:x(d'y)Proposition 7 says that the K-module of e-derivations of A into E is a graded I Zg-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 x I“) [a, x]8

(no. 4, formula (10)) which is graded of degree deg a. PROPOSITION 8. Let A be a graded K-algebra. (i) For every e-derioation d:A—>A andevery homogeneouselement a q, (24)

[de ade(a)]e = ade(da)‘

(ii) Ifthe algebra A i: associative, ad,3 (a) is an a-derioation ofA of degree deg(a). (i) Suppose that d is of degree 8, let a = deg a and write f = [d, ad,,(a)],. For every homogeneous element x e A of degree E, we have, by (1) (no. 1),

f(x) = 4W - 8(06, 5)“) - so. «(4011) - Bu.s+:(dx)a) = (da)x + e5_aa(dx) — e¢'=(dx)a — s°+¢.:x(da)

- 65.41%) + saddx)“

= (da)x — sumgxwa) = [da,x],. (ii) For all x homogeneous of degree 5 and all y homogeneous of degree

n in A, ada(a)(xy) = «(1a) - sa.:+n(xy)a = (ax _ contra”! + annexe!!! _ sunny“)

= ada(a)(x) -y + 8a,:x-ads(a)(y) taking account of (l) and the associativity of A. 559

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

When A is associative, ad,(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 ofA,

(25)

lads“), ads(b)]s = ads([a, 5].).

It suflices to replace d by ad,(a) and ad,(a) by ad,(b) in (24-). If deg a = 0:, deg b = B, formula (25) is equivalent to the following relation for every homogeneous element c e A of degree 7

(25)

Eula, [5, 6L]; + 58m”: [6, 01313 + 61. alt, [4, 51:]: = 0

called the Jacobi identity. 7. FUNCTORIAL PROPERTIES

In this no., all algebras are assumed to be associative and unital and every algebra homomorphism 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 0:E—>F a graded A-homomorphism of A-bimodules (relative to 9), (3f degree 0. Then: (i) For every s—derivation d':B——>F, d’ o p:A—> p*(F) is an s-derivation ofthe same degree. (ii) For every e-derivation d :A—> E, 0 o d :A—> p*(F) is an e-derivation of the the same degree. The two assertions follow immediately from the relations

d’(9(xy)) = d'(9(x)9(v)) = d'(9(x))9(y) + €5’.:P(x)d'(P(!I)) 0(d(xy)) = “(My + 65,9601”) = WMPU) + eo.:9(x)9(dy) for x e A homogeneous of degree E and y e A, 8 and 8' denoting the respective degrees of d and d’.

COROLLARY. Let S be a generating system ofthe algebra A. In order that d’ o p = 6 o d, it is necessary and suficient that d’(p(x)) = 6(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 (II, § 1, no. 14, Example 1). PROPOSITION 10. Under the conditions ofProposition 9, fin an s—derivation d’ :B -—> F

to be A-linear fir the left (resp. right) A-module structures on B and 9*(F), it is necessary and suficient that d’ be zero on the subalgebra p(A) of B. 560

FUNCTORIAL PROPERTIES

§ 10.7

We perform the proof for left A-module structures. For a e A, b e B,

womb) = we)» + p(a)d'b and hence if d’ o p = 0, d’ is linear for the left A—module structures on B and 9*(F). Conversely, if this is so, in particular

d' B is the homomorphism defining the K-algebra structure on B). Now let A, B, G be three graded K—algebras, pzA -—> B, azB -—~> 0 two graded algebra homomorphisms and G a graded (C, C)-bimodule; if DA(B, G), D3(C, G) and DA(C, G) denote the respective K-modules DA.,(B, 0* (G)), DE ‘,(C, G) and DAWN, (C, G), DB(C, G) is clearly a sub-K-module ofDA(C, G)

since c(p(A)) C 603). PROPOSITION 11. Under the above conditions, there is an exact sequence of K-homomorphisms

(27)

o —> D,(G, G) _"_> DA(C.', G) —’> 13,03, G)

where u is the canonical injection and o the homomorphism d »—> d o 0' (Proposition 9). The kernel of v is the set of derivations d:C —> G such that d (a(b)) = O for all b e B, which is precisely the image of u. 8. RELATIONS BETWEEN DERIVATIONS AND ALGEBRA HOMOMORPHISMS

We suppose again in this no. that Case (II) of no. 2 holds and the graded Kalgebra A is not assumed to be associative. Given an element 8 E A, consider the graded K-module E(8) (II, § 11, no. 2) such that (E(8))u = Eu+a

for all p. EA. We define on the graded K-module A Q 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(8)

(28)

(a. w: x') = (aa',

+ W)

the verification of the fact that this multiplication defines a graded ring struc-

ture is immediate.

561

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

The projection11: (a, x) -—> a is called the augmentation ofthe algebra A Q E(8) and is a graded algebra homomorphism. The graded K-linear mappings g:A —> A (43 E(8) of degree 0 such that the composition

A —‘—> A ea 13(3) —"> A is the identity 1A are the mappings of the form x .-—> (x,f(x)), wheref: A —> E is a graded K-linear mapping of degree 8. PROPOSITION 12. For a graded K-h'near mapping f:A—>E of degree 8 to be an e—derivatian, it is necessary and safl'ieient that the mapping xI-> (x,f(x)) qf A into A ® E(8) be a graded K-algebra homonwtphism. Using the fact that for x homogeneous in A and y e A

(W,f(xy)) = (n(1‘)) ~ (y,f(y)), we obtain, taking account of (23), the equivalent relation

f(xy) =f(x) -y + ea.m(x)xf(y), whence the proposition. PROPOSITION 13. For the algebra A {-9 E(8) to be associative and unital, it is necessary and .mflz'eient that A be associative and unital, and that the mapping: (a, x) l~> a.x and (a, x) I—-> x.a define an E an (A, A) -bim0dule structure; the unit element qf A ® E(8) is then (1, 0). If an element (a, m) e A ® E(8) is Wn'tten as unit element of this algebra, it is immediately found that u must be the unit element of A; writing (0, x), we obtain u.x = x.u = x for e (a, m) (0, x)= (0, x). (u, m) 0).(u, m) = (11,0), we obtain m = O. The (u, 0) (u, m) and, writing (u, fact that A'Is associative when A EB E(8) is follows from the fact that the aug-

mentation is a sqective 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) (0, J4)) = ((a’ 0)(a', 0))(0, It); finally the condition a. (x.a') = (a.x) .a' is equivalent to

(a, 0)((0, x) (a', 0)) = ((0, 0)(0, x))(a’, 0)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)-bimadule. Let do:A —-—> E be a 562

EXTENSION or DERIVATIONS

§ 10.9

derivation cf the ring A into the A-moa'ule E and dl—>E an additive group homomorphism such that,for allaeAandallxeM,

(29) -

d1(ax) = ad1(x) + do(a).x,

andfirther, when B = S(M),

(30)

x.d1(y) + d1(x).y = y.d1(x) + d1(y).x

for all x, y in M, and, when B = /\(M), (31)

x.d1(x) + d1(x).x = 0

for all x e M. Then there exists am and only one derivation d ofB (considered as a Z-algobm) {m the (B, B)-bimodule E such that d [A = do and dlM = d1. We take on the Z-module B {-9 E the associative Z-algebra. structure defined

by

(b, t)(b’, t’) = (w, bt’ + tb’) which has (1, 0) as unit element (no. 8, Proposition 13). Under the canonical injection t +—> (0, t), E is identified with a two-sided ideal of B Q E such that E9 = {0}. On the other hand, the mapping ho: B ® E defined by ho(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 h1(x) = (x, d1(x)), it follows from the definition of ho and (29) that hl (ax) = ho(a)h1(x) ; in other words hl is an A-linoar mapping ofM into B (B E. Then there exists one and only one A-algebm homomorphism, h:B—>B ® E such that h|M = 1:1 (and necessarily h|A = ho): for, ifB = T(M), this follows from § 5, no. 1, Proposition 1; if B = S(M), condition (30) shows that h(x)h(y) = h(y)h(x) for all x, g in M and the conclusion follows from§ 6, no. 1, Proposition 2; finally if B = A(M), condition (31) shows that (h(x))’ = 0 for all x e M, since at A x = 0 and the conclusion follows from §-7, no. 1, Proposition 1. The homomorphism h is such that the composition p o h: B —> B with the augmentation p: B ® E —> B is the identity 13, forp o h and 13 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) = (b, (1(6)) for all 6 EB and the mapping b v—>d(b) ofB 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 of type A; the K-algebras T(M), S(M) and /\(M) are given the corresponding graduation: of type A' = A x Z (§ 5, no. 5, Proposition 7, § 6, no. 6, Proposition 10 and ‘§ 7, no. 7, Proposition 11). On the otherhandMicgiventhegraduationoftg/pe A’ mhmtm,1 = Maforall ateA and, = {O}forac e A andn #1. Lot e' be a oommutationfaotor over A’. 563

m

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

(i) Let E be a graded (left and right) T(M)-bimodule of type A’; for all 8 EA and every integer n e Z, every graded K—linear mappingf:M —> E cfdegree 81’ = (8, n) can be extended uniquely to an e’-derivation d :T(M) —> E of degree 8’. (ii) Let E be a graded S(M)-module of type A'; for a graded K—linear mapping f:M —> E of degree 8’ to be extendable to an e’-derivation d:S(M) —> E of degree 8’, it is necessary and sufieient that, for every ordered pair (1:, y) of homogeneous elements of M, (33)

x -f(1/) + 56'. A of the algebra A,for each i, a K-linear mapping d,:M, ——> M, and a K-linear mapping D:P —> P, so that, fbr l < i < n, (do, (1,, ah) is a K-den'vation q" (A, M‘, M,) into itself and (do, D, D)

is a K-dm'vatz'on of (A, P, P) into itself. Then there exists a K-linear mapping D’ :H -—> H such that (do, D', D’) is a K-den'vation of (A, H, H) into itself and ' (37)

D(f(x1, . . ., 19.))

n

= (Difxxla ' ' 'a x») + zlflxb - - ': xl—l’ dlxb xi+1s - - -a xn)

forallx,eM,fbrl g i g nandfeH.

We show that forfe H, the mapping f of M1 x M2 x - - - x M, into P defined by (37) is A-multilinear. For a E A, (f)(ax1, x2, . . ., x") = D(qf(x1, . . ., x")) —f(d1(ax1), x2, . . ., x”)

—a Zafoc" . . ., x,_1, dm, l, . . ., x“) and by hypothesis

D(t£f(xn - - -, x2!» = (doa)f(x1, - - -, x») + aD(f(161, ~ - '3 79.)) and d1(ax1) = (doa)x1 + a.d1x1, which gives

(f)(ax1,x2,...,x,,) = a.(f)(x1,...,x,,) and linearity in each of the x, is proved similarly. On the other hand, .

(D'(Qf))("12- .., x») = D(4{(x1, - - .,x,.))

’

‘21 af(x1, - - ~a J‘s-1, dixia “H1: ~ . '9 ’51-) = (doa)f(x13 - - -: xn» + aD(f(x19 ' ' -: xn»

—' ‘21 #(xb - - -: xi-l: dtxb xi+13 - ' u xn) = (doa)f(x1: - - -: xii) + “(D7) (x1: ' ' -: xii)

in other words

D’(af) = (doa)f + “(D7) 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, a', d) of (A, M, M) is derived a K—derivation (do, (1*, d*) of (A, M“, M*) such that

do = (dm, m*> + (m, d*m*) for meM and m“ EM“. The K-linear mapping of M @M“ into itself (38)

566

PROBLEM FOR DERIVATIONS: NON-COMMUTATVE CASE

§ 10.10

which is equal to d on M and to d* on M* then satisfies condition (29) and there is therefore a K—dm'vation D of the A-algebra T(M (-B 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 Tf,(M) of T(M (-9 M*) (§ 5, no. 6) is a K-endomorphism of 1?,(M) such that (do, d}, d}) is a K—derioatian of (A, TKM), T5(M)). Moreover,

for ie I, j EJ, ifWe write I’ = I — {i}, J' = J — {j}, itisimmediatelyverified that for the contraction 0} (§ 5, no. 6)

use» = dsxcxz» for all z emM). (4) Let M, (l < i < 3) be three A-modules and, for each i, suppose that (do, (1,, d,) is a derivation of (A, M,, M,); applying Proposition 16 again for n = l, a derivation (do, d,” d”) of (A, H“, H”), where H" = HomA(M,, M,), is derived for each ordered pair (1',j) . With this notation, for u e HornA(M1, M2) and v e HomA(M2, M3),

(39)

- anew) = (423v) ou + 0°(d121‘)

as is immediately verified from the definitions. 10. UNIVERSAL PROBLEM FOR DERIVATIONS; NON-COMMUTATIVE CASE

Throughout the rest of § 10 all the algebras are assumed to be associative and unital ’ and all the algebra hommnorphisms are assumed to be unital. Let A be a K—algebra; the tensor product A ®KA has canonically an (A, A)-bimodule structure under which

(40)

96-0! ®v)-y = (W) 83 (v1!)

for all x, y, u, v in A (§ 4-, no. 3, Example 2). The K-linear mapping m:A ®x A ~> A corresponding to multiplication in A (and hence such that m(x (8 y) = xy) is an (A, A)-bimodule homomorphism; its kernel I is therefore a sub-bimodule of A ®x A.

Lemma 1. The mapping Syn—>1: ®l — l ®xis a K-derivatz'on ofAintoI and I is generated, as a left A—module, by the image of 8A. The first assertion follows from the fact that

(xy)®1-l®(xy)=(x®1-1®x)-y+x-(y®l—l®y) by (40). On the other hand, if the element 2 x, (8 y, (for x", y, in A) belongs to I, by definition 2 x,y, = 0 and hence

2,: (7‘1 @111) = 2&0 ®yi —y¢ @1) by (40), which completes the proof of the lemma. 567

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TENSOR ALGEBRAS, EXTERIOR AIGEBRAS, SYMMETRIC ALGEBRAS

PROPOSITION 17. The derivation 8,, 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 = f0 8A. Note first that, for every (A, A)-bimodule homomorphism f:I -—> E, f0 8A is a derivation (no. 7, Proposition 9). Conversely, let d: A -—> E be a Kderivation; then we prove first that if there exists an (A, A)-bimodule homomorphismf: I —-> E such that d = fo 8A,fis uniquely determined by this condition for the definition of 8,, gives

f(x®l —l®x) =dx

and our assertion follows from the fact that the image of 8,, already generates I as a left A-Inodule (Lemma 1): hence of necessity f(Z xi ®yt) = Zia-fa ®y1 *1/18”) = ‘s-dyt-

Conversely, as the mapping (x, y) r—> —x.dy of A x A into E is K-bilinear, there exists one and only one K-linear mapping n ®KA—>E such that g(x ® y) = -x.dy; it suffices to verify that the restriction f of g to I is Alinmr 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 2 x, (8) y, e I

and x e A, then

2 max) = Z Max + 2 (mm but since 2‘: x43], = 0 by definition of I, this completes the proof.

We have thus defined a canonical K-module isomorphismfu—>f a 8A

Horn“. “(1, E) —> DK(A, E) the left hand side being the K—module of (A, A) -bimodu1e homomorphisms of A into E. 11. UNIVERSAL PROBLEM FOR DERIVATIONS; COMMUTATIVE 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 (8,; A is identical with its (A ®K A)-module structure arising fi'om the commutative ring structure on A ®x A, since in this case, for x, y, u, v in A,

x-(u ®v)-y = (W) 69 (W) = (’6'!) ® (W) = (x ®y)(u ‘80)The kernel 8 of m is therefore in this case an ideal of the ring A (8): A and,

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PROBLEM FOR DERIVATIONS; COMMUTATIVE CASE

§ 10.11

as m:A ®KA—>A is surjective, (A (8),: A)/8 is isomorphic to A; ifalso E is considered as an (A (8),; A)-module by means of m (in other words the (A ®x A)-module m*(E)), the (A, A)-bimodule homomorphism 5 —>- E are

identified with the (A ®x A)-module homomorphisms 5 -> E (§ 4, no. 3), in other words there is a. canonical K—module isomorphism.

H°m(A.A)(5: E) "> HomA 8; AW: E). On the other hand, SE = {0}, for the elements 1 (8) x — 1: ® I generate 8 as an (A ®x A)-module (no. 10, Lemma 1) and, for all z e E, (l®x-x®l)z=0 by virtue of the definition of the (A ®x A)-module structure on E. Since 5 is contained in the annihilator ofthe (A ®x A)-module E and the ((A ®x A) [8)module structure on E is by definitionjust the initial A—module structure given on E, there is, taking account of the canonical isomorphism of

5 ®x ((A SH: ANS) onto 95/52 (§ 4, no. 1, Corollary 1 to Proposition 1), a canonical K-module isomorphism HomA 3‘ A(8, E) —> HomA(5/82, E). Taking account of Proposition 17 of no. 10, it is seen that we have proved the following proposition: PRoposmON 18. Let A be a commutative K-algebra and 5 the ideal the kernel of the surjective canonical homomorphism m:A ®KA—>A, so that A is isomorphic to (A ®K A)/8 and 8/32 has canonically an A—module structure. Let dAmzA——>I/I’ be the K—linear mapping which associates with every x e A the class ofx ® 1 — 1 ® or modulo 3*. The mapping dug 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: 5/82 —> E such that D = g o (1,“. The A-module 8/32 is called the A-moa'ule q-diflerentials ofA and is denoted by QK(A); for all x e A, dA,K(x) (also denoted by dx) is called the diflerential of x; 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-module QK(A). Proposition 18 shows that the mapping g +—> g o dMK is a canonical A-module isomorphism ‘l’A:HomA('QK(A): E) _’ DK(A: E)

(the A-module structure on DK(A, E) being defined by Proposition 7 of no. 5). The ordered pair (QK(A), dMK) is therefore the solution of the universal mapping problem where Z is the species of A-module structure and the

cat-mappings the K-derivations from A to an A-module (Set Theory, IV,

§ 3, no. 1).

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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 +—> D | 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, Homx(M, E) is canonically isomorphic to

H0msh(x ® 1), where h e Homsm,(M ®x S(M), E) (II, §5, no. I). Let D, be the K-derivation S(M) ->M ®x S(M) whose restriction to M is the canonical homomorphism x I—-> x ® 1; every K-derivation D: S(M) —> E can therefore be written uniquely as h 0 Do with h e Homsm(M ®K S(M), E).

By the uniqueness of a solution of a universal mapping problem, it is seen that there exists a unique S(M)-module isomorphism' s ®K S(M) —> QK(S(M)) such that D, o o) = d504,“; in other words, for all x e M, m(x ® 1) = dx. In particular, if M is afree K-madule with basis (e1) “L, QK(S(M)) is a free S(M)-module with basis the set (J diflerentials deh. Consider in particular the case where L = [1, n], so that S(M) is identified with the polynomial algebra K[X1,. . ., Xn] (§ 6, no. 6); for every polynomial P EK[X1, . . .,X,,], we

can write uniquely

dP = n‘PJlX, with DP 6 K[X1, . . ., Xn] and, by virtue ofthe above, the mappings P I—> DiP are the K-dm'vations Of K[Xl, . . ., X1] corresponding to the coordinate forms 8P on QK(S(M)) for the basis (dX,); we also write .5): instead of DP and this is called the partial derivative of P with respect to X,. 12. FUNCTORIAL PROPERTIES OF K-DIFFERENTIALS

PROPOSITION 19. Let '

vi

if

A ——u—> A’

be a commutative diagram of commutative ring homomorphism, 1) (reap. 11') making 570

FUNGTORIAL PROPERTIES or K-DIFFERENTIALS

§ 10.12

A (rcsp. A’) into a K-algebra (resp. K’-algebm). There exists (me and only one A-linear mapping 0: QK(A) —> Qx.(A’) rendering commutative the diagram A —-—u—> A’ 11A]:

'JA'Ix'

WA) —,—> 934A? (Imp o u is a K-derivation of A with values in the A-module QK.(A’); the existence and uniqueness of u then follow from Proposition 18 of no. 11. The mapping 0 of Proposition 19 will be denoted by 9(a); if there IS a commutative diagram of commutative ring homomorphisms

K —°> K’ —i+ K' 111 ‘

n’l

111'

A —u-) A’ 7—) A” it follows immediately from the uniqueness property of Proposition 19 that 9(u’ o u)= 9(u') o (2(a). Since 93 (A’) is an A’-module, from 9(a) we derive canonically an A’linear mapping

(41)

nontoxic) or A'—>sl(A')

such that (2(a) is the composition of 00 (u) and the canonical homomorphism iA: QK(A) —> QK(A) (8A A’. For every A’-module E’, there is a commutative diagram

HomA.(Qx,(A’), E’) w HomA.(ox(A) @A A’, E’) 0A ° 7A

‘bA’

Dx.(A’, E')

C(u)

DK(A, E)

where C(u) 18 the mapping D r—>D o u (no. 7, Proposition 9) and 1A is the canonical isomorphism Hom(iA, 1w) :HomA,(QK(A) ®A A’, E’) —>- HomA(QK(A), E’);

this follows immediately from Proposition 19 and the definition of the isomorphisms A and im‘57!

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

PROPOSITION 20. Suppose that A' = A ®1< K’, with 13’2K' ——>A' and u:A->A’ the canonical homomorphirms. Then the A'-lz'near mapping S2001) :QK(A) ®A A’ —> 9K,(A’) is an isomorphism. By virtue of the fact that in diagram (4-2) the vertical arrows are bijectivc, it reduces to proving that, for every A'-modu1e E', the homomorphism C(u) :D v—> D o u in diagram (42) is bijectivc (II, § 2, no. 1, Theorem 1). Now Hom(u, IE): HomK (A ®x K', E') —> Homx(A, E’) is an isomorphism (II, § 5 no. 1, Proposition 1) and C(11) 13 its restriction to D]; (A’, E’) and hence is injectivc; moreover, iff A’ —> E’ is a K’-linear mapping such that ,

fo u :A ——> E’ is a K-dcrivation, it follows immediately from the fact thatf 1s K’-linear and

the fact thatf((x ®1)(y ® 1)) = (y ®1)f(x ® 1) + (x ®1)f(y ® 1) for

x, y in A, thatf18 a K’-dm'vation, the elements an- ® 1 for x e A forming a generating system of the K’-module A’; this completes the proof that 0(a) is bijective.

From now on we confine our attention to the case where p:K —> K’ is the identity mapping of K; every K-algebra homomorphism u:A —> B is therefore mapped to a B-linwr mapping (43)

(20(14) :QK(A) ®K B —> QK(B).

On the other hand, we can consider the B-module of A-dg'fi'erential: QA(B) since B is an A-algebra by means ofu ; the canonical derivation d3,A:B ——> QA(B) being afortion' a K-derivation, it factorizes uniquely into 9,,

B——> ——>nx “" -—> 9.03) where 9., is a B—linear mapping (no. 1 1, Proposition 18) . For every B-module E, there is a commutative diagram

Hom..(91 DK(B, E)

where ju is the canonical injection (no. 7); this follows immediately from Proposition 18 of no. 11. PROPOSITION 21. The sequence zy‘ B-linear mappings

(45) is exact.

572

QK(A) 69, B fl 93(3) —+ 12,03) —> o

mmc'rORIAL PROPERTIES or K-DIFFERENTIALS

§ 10.12

It reduces to verifying that, for every B-module E, the sequence

0 —> Hom,(n,(B), E) m Hom3(QK(B), E)

“————+°"“ °‘" "’ Homt(nx(A) oils, E) is exact (II, §2, no. 1, Theorem 1); but by virtue of the fact that in the commutative diagrams (4-2) and (44) the vertical arrows are isomorphisms, it suffices to show that the sequence

0 ——> DA(B, E) L DK(B, E) ffl» DK(A, E) is exact, which is just Proposition 11 of no. 7.

We consider now the case where the K~algebra homomorphism uzA —-> B is suijeetioe; if 3 is its kernel, B is then isomorphic to A/8. We consider the

restriction d|8:3—->QK(A) of the canonical derivation d = dun and the composite A-linear mapping

d’:3 £1) nx(A) —"‘—> 9,,(A) @A B. Then d'(8’) = 0, since, for x, y in 8,

d’(xy) = d(xy) 691 = (My +yidx) ®l = dy 69 u(#) + dx 69 M) = 0 since u(x) = u(y) = 0. Hence we derive from d', by passing to the quotient, an A-linmr mapping

a: 3/32 —-> QK(A) (g), B and as 8 annihilates the A-module 3/33, a is a B-linear mapping. PROPOSITION 22. Let 3 be an ideal of the commutative K-algebra A, B = A/8 and uzA -> B the canonical homomorphism. The sequence of B-linear mappings

(46)

5152 1—» EMA) sin 3‘1 (has) —>0

is then exact.

Note that QK(A) ®AB is identified with QK(A)/8QK(A) and that the image of a is the image of (1(3) in this quotient module; the quotient of 9AA) ®AB by Im(z-i) is therefore identified with the quotient QK(A)/N, where N is the sub-A-module generated by 89K(A) and (1(3). Moreover, the composite mapping

A ‘—*"+ aim) ——> nx(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 :3 ——> QK(A) /N.

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG 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 Kderivation D: B —-> E, there exists a unique B-linear mapping g: QK(A) /N —> E such that D = g o Do. But, the composite mapping D o u: A —> E is a Kderivation (no. 7, Proposition 9) and hence there exists one and only one Alinear mappingf: QK(A) —-> E such thatfo (INK = D o a. This relation shows already thatfis zero on d(8); as also SE = {0} since E is a B-module,fis zero on 893(A); hencefis zero on N and defines, when passing to the quotient, a B-linear mapping g: (2,;(A) /N —> E such that g a D0 = D; the uniqueness ofg follows from the uniqueness off It must not be thought that, even if u :A —> B is an injective homomorphism,

90(u) : QK(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 itsfield offiaotions and u:A —> B the canonical injection. Then (20 (u) :QK (A) ®A B —> (2,; (B) is an isomorphism. Using the fact that in diagram (4-2) the vertical arrows are bijective, it reduces to proving that, for every vector space E over B, the mapping C(u) :DK(B, E) 9DK(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).

§ll. 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*" will denote the graded A-module of type N, whose homogeneous elements 9" degrees n are the A-linearforms which are zero on Mkfor all k as n. 1. COGEBRAS

DEFINITION l. A eogebra over A (or A-cogebra, or simply cogebra if no eonfiu‘ion can arise) is a set E with a structure defined by giving thefollowing: (1) an A-moa'ule structure on E; (2) an A-linear mapping ezE ——> E ®A E called the coproduet of E.

DEFINITION 2. Given two eogebras E, E', whose coproduets are denoted respectively by e and e', a morphism of E into E’ is an A-linear mapping u;E —> E' such that (l)

574

V

(u®u)oc;c'ou,

§ 11.1

COGEBRAS

in other words, it renders commutative the diagram q-linear mappings

E ———”———> E’ c

c’

E 69,15 ”.73? E’ ®AE’ 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 ®A A (II, §3, no. 5) defines an A—cogebra structure on A. (2) Let E be a cogebra, 0 its coproduct and o- the canonical automorphism of the A-module E ®AE such that a(x ®y) = 3] ® I: for e, yeE; the A-linear mapping a o 6 defines a new cogebra structure on E. With this structure E is called the opposite cogebra to the given cogebra E. (3) Let B be an A-algebra and let m:B (8),, B -—> B be the A-linear mapping defining multiplication on B (§ 1, no. 3). The transpose ‘m is then an Alinear mapping of the dual B* of the A-module B into the dual (B ®A B)* of the A-module B (8.4 B. If also B is a finitely generated projective A-module,

the canonical mapping [1. :B* (8,, B* —-> (B ®A B)* is an A-module isomorphism

(II, §4-, no. 4); the mapping 0 = 971 o ‘m is then a coproduct defining a cogebra structure on the dual B* of the A-module B. (4) Let X be a set, A“) the A-module of formal linear combinations of elements of X with coeflicients in A (II, § 1, no. 11) and (ex)xex the canonical

basis of Am. An A-linear mapping c:A‘x’ —>A‘x’ ®A A“) is defined by the

condition e(e,,) = 0:: ® 5x and a canonical cogebra structure is thus obtained on A“). (5) Let M be an A-module and T(M) the tensor algebra of M (§ 5, no. 1); by II, § 3, no. 9 there exists one and only one A-linear mapping 6 of the A-module T(M) into the A-module T(M) ®A T(M) such that, for all n 2 0,

(3)

0(x1x2 . . . x") = 0‘?“ (37l . . .xp) ® (xpfl . . .x,,)

for all x, e M (x11:2 . . . .7:" 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 n—-> (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) (§ 6, no. 2, Proposition 3). On the other hand, in §6, no. 6 we defined a canonical graded algebra isomor-

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phism h: S(M x M) -—>- S(M) 8)A S(M); by composition we therefore obtain an A-algebra homomorphism

0 = I”°5(A) “S(M) 9 S(M) ®A S(M), thus defining on S(M) a eogebra structure. For all x e M, by definition S(A) (x): (f, x) and the definition of 1: given in § 6, no. 6 shows that h((x,x)) = x®l + l ®x. It follows that e is the unique algebra homomorphism such that, for all x e M, (4-) c(x)=x®l+l®x. As e is an algebra homomorphism, it follows that, for every sequence (2:01“ a of n elements of M,

(5) C(xix2---xn)=,ll(x¢®l+18%) = Z (at,1 . . .xlp) ® (x,1 . . --"1..-,) the summation on the right hand side of (5) being taken over all ordered pairs of strictly increasing sequences (in some cases empty) i1 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. COASSOGIATIVITY, COCOMMUTATIVITY, COUNIT

Let E be a cogebra, 6 its coproduct, N, N', N' three A-modules and m a bi-

linear mapping of N x N’ into N'. Let mzN ®A N’ —> N' denote the A-linear mapping corresponding to m. If q -—> N, s —> N’ are two A-linear mappings, we derive an A-linear mapping u ® v:E 8),, E —>N ®A N’ and a composite A-linear mapping of E into N":

(11)

m(u,v):E —‘> E ®AE “—8—; N ®AN’ 1» N'.

Clearly we have thus defined an A-bilinear mapping (u, v) I—-> m(u, v) of HomA(E, N) x HomA(E, N’) into HomA(E, N”). When E is a graded cogebra, N, N’, N" graded A-modules of the same type and m a graded homomorphism of degree I: of N (8,, N’ into N", then, if u (resp. v) is a graded homomorphism of degree p (resp. q), m(u, v) is a graded homomorphism of degree p + q + 1:.

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 M” into N (resp. N', N"). Given a multilinear mapping u: M” —> N and a multilinear mapping 1): M“ —->N’, the above method allows us to deduce a multilinear mapping m(u, o) : MP “1 -—> N” called the product (relative to m) of u and v. Formulae (3) (no. 1) and (l I) show that, for x1, . . ., ref-q in M, (”‘(us 0))(x1: ' ' -: xp+q) = m(u(x1: ' - -2 xv): ”(xrnls - - -: xp+q))'

(2) Take E to be the graded cogebra S(M) (no. 1), 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 M” into N (§ 6, no. 3). Then we derive from a symmetric multilinear mapping u: M” ->N and a symmetric multilinear mapping 0: M" —> N’ a symmetric multilinear mapping m(u, 1)): MP” ——> N”, also denoted (to avoid confusion) by u.mv (or even u. v)

and called the symmetric product (relative to m) of u and v. Formulae (6) (no. 1) and (11) show that, for x1,. . ., x,+¢ in M,

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GOASSOCIATIVITY, cocomwmnvrrv, COUNIT

§ 11.2

m("(xo(1)a ' - '3 x60»): ”(xow+1), - - u ”can” ("an”) (*1: ' ‘ ': 7694.4) = Z 6

the summation being taken over permutations o- e 6,” which are increasing on each ofthe intervals [1,12] and [p + l,p + q]. (3) Take E to be the graded cogebra A(M) (no. 1). Then we deduce similarly from an alternating multilinear mapping a: M’ —-> N and an alternating multilinear mapping 1): M" ——> N’ an alternating multilinear mapping m(u, 0): MP” —> N”, also denoted by u A m v or u A v and called the alternating product (relative to m) of u and o. Formulae (9) (no. 1) and (11) show in this case that, for 1:1,. . ., x“, in M, (u A m 1’) (x1, - - -a x9”) = Z som(u(xa(1)’ - - -: ”cool: ”(”own): - - -: xctp+a)))

the summation again being taken over permutations 666,,” which are increasing on each ofthe intervals [1,p] and [p + 1,9 + 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 multiplication in B, so that fitzB (8),, B —-> B is a graded A-linear mapping of degree 0. Thus a graded A-algebra structure is obtained on the graded A-module HomgrA(E, B) = C. In particular, suppose that B = A (with the trivial graduation), so that HomgrA(E, A) is the graded dual 'E*"‘, which thus has a graded A-algebra structure.

Let F be another graded cogebra o’ its coproduct and F a graded cogebra morphism (no. 1); then the canonical graded morphism

HomgrAE, B) is a graded algebra homomorphism. For a, o in HomgrA(F, B) and x e E,

($(uv))(x) = (W)((x)) = "1((u ® v)(c’(¢l>(x))))-

But by hypothesis ewe» = (h o 4») oh», hence (14 ® v)(6'((x))) = ($00 ® tl>(v))(0(x)) and therefore $(uo) = $(u)(o), which proves our assertion. In particular, the graded transpose '¢:F*"—>E*" is a graded algebra , homomorphism.

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Remark. Suppose that the E, are finitely generated prtzjeotive A-modules, so

that the graded A-modules (E ®A E)*" and E*“‘ ®A E*“' can be canonically identified (II, §4, no. 4, Corollary 1 to Proposition 4-). If also the A-modules A ®A A and A are then canonically identified (II, § 3, no. 4-),

the linear mapping E*" ®A E*"‘ —> E*“‘ which defines multiplication in E*“' can be called the graded transpose of the coproduet e.

PROPOSITION I. Let E be a eogebra over A. In order that, jbr every associative Aalgebra B, the A-algebra HomA(E, B) be associative, it is necessary and sufieient that the eapraduet czE -—> E 8)“ E be such that the diagram

E——c——>E®AE

(12)

'

l

1138‘

E ®AE 78—1: E ®AE ®AE

be commutative. Let B be an associative A-algebra and u, v, to three elements of

C = HomA(E, B). Let m3 denote the A-linear mapping B ®AB ®AB-—>B which maps b (8 b’ ® b" to bb'b”. By definition of the product on the algebra C, (uv)w is the composite mapping

EéE®Efl>E®E®EmB®B®B—E—>B whilst u(vw) is the composite mapping

E—‘——>E®E—“‘—®1>E®E®E'L®IflB®B®B—£—>B. It follows that if diagram (12) is commutative, the algebra HomA(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, v, w of E into B such that the mapping ma 0 (u (3) v (8 w) of E ® E (8 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 o (u ® v ® w) is then the canonical mapping E ® E ® E = T"(E) —-> T(E), which is injective. When the cogebra E satisfies the condition of Proposition 1, it is said to be eoassoeiative. Examples. (4) It is immediately verified that the cogebra A (no. 1, Example (1) the cogebra A“) (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 (§ 1, no. 3). Conversely, the

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GOAssoc're, COCOMMUTATIVITY, COUNIT

§ 11.2

same argument and the canonical identification of the A-module B with its bidual (II, § 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 commutativity of the diagram M ———i-—> M x M A

lnxA

M x M ——> M x M x M In AX

the functorial properties ofS(M) (§ 6, no. 2) and /\(M) (§ 7, no. 2), which give the corresponding commutative diagrams

r

S(M)

5“”

S(M x M)

S(A)

lsmuxl)

S(MxM)WS(MxMxM)

(14)

NM)

/\(M x M)

A“)

Ami

(Amen)

~/\(MxM)m/\(MXMXM) and the existence and functoriality of canonical isomorphisms for symmetric and exterior algebras of a direct sum (§ 6, no. 6 and § 7, no. 7). PROPOSITION 2. Let E be a oogebra over A. In order that, for every commutative A-algebra B, the A-algebra HomA(E, B) be commutative, it is necessary and sufioient that the coproduet czE —> E (8‘ E be such that the diagram

/\ E

E 8),, E —°> E ®A E

(where o- is the symmetry homomorphism such that 5(x ®y) = 3/ ® 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, a two elements ofC = HomA(E, B).

581

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

By definition of the product in C, uv and Du are respectively equal to the composite mappings

E—s‘

man—Hm B®B——>’"

B

and

E—L>E®E"—®~">B®B—m—>B. It follows that if diagram (15) is commutative the algebra HomA(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 (n ® a) :E (3) E —>B is injective. Take B to be the algebra S(E {-9 E) and u (resp. o) the composition of the canonical mapping E (43 E —> S(E (B E) and the mapping x x—> (x, 0) (mp. xn—-> (0,x)) of E into E (43E. If h:S(E) ® S(E) —>S(E (8) E) is the canonical isomorphism (§ 6, no. 6, Proposition 9) and A: E—>S(E) is the canonical mapping, then h‘1 o m o (n ® a) = A (8) A. Now A ® A is injective, for ME) is a direct factor of S(E) (II, § 3, no. 7, Corollary 5 to Proposition 7).

When the cogebra E satisfies the condition of Proposition 2, it is said to be oocommutative. Examples. (5) It is immediate that the cogebra A (no. 1, Example I) and the cogebra' A“) (II, § 11, no. 1, Example 4-) are cocommutative. It follows fiom 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 (II, §2, no. 7)), this follows

from the fact that the commutativity ofdiagram (14-) is equivalent by transposition to that of the diagram which expressed the commutativity of B (§ 1, no. 3). PROPOSITION 3. Let E be a cogebm over A. In order that,fir every unital A-algebra B, the A-algebra HomA(E, B) be unital, it is necessary and rufioient that there exist a linearform Y on E rendering commutative the diagrams

E—‘->E®AE (16)

\

1mm

A ®A E

E—‘—>E®AE x

112®Y E ®AA

where ozE —>E ®AE is the eoproduet and h’ and h" the canonical isomorphism

582

COASSOCIATIVITY, COCOMMUTA'I‘IVITY, COUNIT

§ 11.2

(II, § 3, no. 4, Proposition 4). The unit qf HomA(E, B) is then the linear mapping x +—> y(x)l (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, nzA —> B the canonical mapping and v = 'q o y the element of the A-algebra C = HomA(E, B). For every

element u e C, u!) is the composite mapping

(17)’

E—‘—>E®E—‘“—9—1E®A—"31>B®BL>B.

Then uv = m o (14 ® 7;) o h” = u. It is similarly proved that vu = u and hence v is unit element of C. Conversely, let the A-module A EB E be given a unital algebra structure such that (a, x) (a', x') = (aa’, ax’ + a’x) for a, a' in A and x, x' in E. Let B denote the A-algebra thus defined and let G be the A-algebra HomA(E, B). Suppose that C is unital and let em »—> (7(x), A(x)) be its unit element (where 7(x) e A and A(x) e E). On the other hand letfbe the element x »—> (0, x) of C. An immediate calculation shows thatfe is the element

x '-> (0, (h')'1((1n ® Y)(6(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 commutativity 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 HomA(E, A). A cogebra with a counit is called eounital.

Examples. (6) The identity mapping is the counit of the cogebra A; on the cogebra A“) (no. 1, Example 4-) the linear form V such that flex) = l for all x e X is the counit. On the cogebra T(M) (resp. S(M), A(M)) the linear form 7 such that y(1) = l and 7(2) = O for z in the T"(M) (resp. S”(M), /\"(M)) for n 2 l 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 7: x* v—> (e, x") is the counit, for this form is just the transpose of the A-Iinear mapping 1),: E n——> lie of A into B and by transposition the commutativity of diagrams (l6) follows from that of the diagrams which express (using 1),) the fact that e is unit element of B (§ 1, no. 3); the same argument moreover shows that conversely, if the cogebra B* admits a counit 7, the transpose of 7 defines a unit element e = ”7(1) of B. PROPOSITION 4. Let E be a cogebra admitting a counit y and suppose that there exirt: in E an element e such that 7(a) = l ; then E is the direct sum qf the sub-A—module: Ae and E, = Kcr(y) and 18 { 0(e) E e ®e(mod.E., ®E.,) ( ) e(x)sx®e+e®x(mod.E.,®E.,) foralley. 583

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

The first assertion is immediate, for 7(x — y(x)e) = 0 and the relation flue) = 0 implies a = 0. Let e(e) = Z 5, (X) t,, so that 9 = Z 700’: = Z YM)‘:

by (16) and l = 7(e) = 270070,). Therefore

2 (s; — me o (a — was) = Z s. o t. - 2 e o my. "‘ Z WAS; ® 0

+ Z Me «8 woe which, by the above relation, is just 6(a) — e (8 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 ® 6) C9 ((A6) 83 Ev) 69 (E1 ® (Ad) 93 (E1 (8 Ev) allowsustowrite, forey, e(x) = Me (8e) + (e ®y) + z ®e) + uwhere u = Z 0, (8 w,, y, z and the o, and w, belong to Ev. The definition of the counit ‘Y then gives x = M + y = Ae + z and, as fix) = O, necessarily A = 0, x =-- y = z, whence the second relation of (18). Remark. Let C be a counital coassociative A—coalgcbra, B a unital associative A-algebra and M a left B-module. The A-bilinear mapping (b, m) s—> bm of B x M into M defines an A-bilinear mapping HomA(C, B) x HomA(C, M) —> HomA(C, M) by the general procedure described at the beginning of this no. It is immediately verified that this mapping defines on HomA(C, M) a left module structure over the ring HomA(C, B). 3. PROPERTIES OF GRADE!) GOGEBRAS OF TYPE N

PROPOSITION 5. (i) Let E be a graded eogebra admitting a counit y; then 7 is a homogeneous linearform of degree 0. (ii) Suppose fierther that there exist: an element eeE such that E, = Ae and

7(e) = 1. Then the kernel E, qf'y isequal to E+ = “ZIE,” e(e) = e ® e and (19)

for alle+. 584

vc(x)sx®e+e®x(mod.E+ ®E+)

BIGEBRAS AND SKEW-BIGEBRAS

§ 11.4

(i) It suffices to verify that fix) = 0 for at SE“, for all n 2 1. Since a is a graded homomorphism of degree 0,

(20)

co) = Z (Z!!u®zt.n~1)

o S(M)“r

‘/\(u) :/\(N)*“ _> /\(M)*=r are graded algebra homomorphism. We now note that the dual M"I of M is identified with the submodule of

elements of degree 1 in T(M)*"' (resp. S(M) *3“, A(M)*“). It therefore follows from the universal property of the tensor algebra (§ 5, no. 1) and the universal property of the symmetric algebra (§ 6, no. 1) that there exists one and only one graded algebra homomorphism 61:T(M*) —> T(M)*" which extends the canonical injection M* —> T(M) “r, and one and only one graded algebra homomorphism 05;S(M*) —> S(M) *3”

which extends the canonical injection M* —> S(M)"! On the other hand, the canonical injection of M* in the opposite algebra to A(M) *" is such that the square of every element of M“ is zero; hence (§ 7, no. 1, Proposition 1) there exists one and only one graded algebra homomorphism

6A :/\v= E (x?! xl>

(in other words 01- restricted to T2(M*) is just the canonical homomorphism of II, §4, no. 4) 13

(29 bis)

7|

«sates. . . act), x1e. . .x.> = 2(1—I)

(30 bis) T(M)"‘gr and 0A:/\(M*) —>(/\(M)*¢')° are bijective. Also the graded dual /\(M)*“ is then equal to the dual /\(M)* ofthe Amodule /\ (M). Suppose first that M has afim'te basis (egmmm and let (ef)1 M whose composition is the identity 1M. We deduce a commutative diagram

T""" T(L*) ——>T"”’ TM) ml 0.1 01l TIM)": W T(L) *3: W T(M)*sr and an analogous commutative diagram where T is replaced by A. The proposition then follows from the following lemma: Lemma 2. Let

X__"_>Y__'L_)X

’1

”l

’1

X’TY’TX’

be a commutative diagram qf sets and mapping: such that v o u and v’ o u’ are the identity mappings qf X and X' respectively. Then, if g is injective (resp. smjective,

resp. biieetive), so isf. u is injective since I) 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 a’ o u’ is; hence, if g is suijective, fo v = v’ o g is surjective and thereforef is smjective. The last assertion of Proposition 7 follows from the fact that A(M) is then a finitely generated A-module (§ 7, no. 3, Proposition 6 and II, § 11, no. 6, Remark). We now examine what can be said concerning the homomorphism 03 when

M is projective and finitely generated. Suppose first that M admits a finite basis (e,)1 A defines the multiplication on A and a is the coproduct of S(M). In other words, a“, is just the coefficient of e“ 8) efl when c(e") is 592

THE GRADED DUALS T(M)*", S(M)*“‘ AND /\ (M)*“'

§ 11.5

written in terms of the basis of S(M) ® S(M) consisting of the e‘ (8) e", where E and 1; run through N’". But since 0 is an algebra homomorphism,

cw) = U (com = I} (e. 8: 1 + 1 ® eavby formula (4-) of no. I; this gives

(31)

cw) = W, (a, 1M s a"

where we write

(5* T "3' (cf. § 10, no. 4, formula (18)). (32) «a. 11)) = H—— Hence we obtain the multiplication table (33)

“club = «as 3))ua-i-B'

0n the other hand, if (efh‘Km is the basis of M*, dual to (a), it follows from formula (29 bis) that, for all at eN’",

(34)

85 (e*°‘) = a la,

in the notation of § 6, no. 6. Hence the homomorphism 05 is bijective if and only ifthe atlum form a basis of S(M) *3”, or also if the elements ad] are invertible. PRoposrrION 8. Suppose that the ring A is an algebra over the field Q of rational numbers; then, for everyfinitely generated projective A—module M, the homomorphism 65:5(M*) ——> S(M)*" is biieetiae. 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-modulc and p :A —> B a commutative ring homomorphism. Then therc is a commutative diagram of graded B-algebra homomorphisms

T((M*) (T(M)*")(n> T(vn)

0mm

T((M(B))*) 712‘" T(M(B))*3“ where the first row is a homomorphism composed of the homomorphism 01- ® lB:T(M*) ®A B —> T(M)*" and the canonical isomorphism

T((M*)m)) -> T(Ml‘) ®A B 593

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

(§ 5, no. 3, Proposition 5). It is immediately verified, using formula (28) and the definition of the homomorphism DE (II, §5, no. 4-), that this diagram is

commutative. When M is afinitely generated projective A-module, M0,) is a finitely generated projective B-module (II, §5, no. 1, Corollary to Proposition 4) and all the homomorphisms of the above diagram are biiective (Proposition 7 and II, § 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 5 also consists of bijective homomorphisms (Proposition 8). 6. INNER PRODUCTS: CASE OF ALGEBRAS

Let E = $30 E, be a graded A-algebra of type N and P a graded A-module of type Z; for every homogeneous element x e E,” lefl multiplication by x is an A-linear mapping e(x) of E into itself which is graded of degree ,0. For every element u eHomgrA(E, P), the right inner product of u by x, denoted by u L x, is the element u o e(x) of HomgrA(E, P). We also write (i(x))(u) = u L x and we see that i(x) is a graded endomorphism of degree [2 of the graded A-module

HomgrA(E, P). If now x = ”a x, 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 11), we write {(x) = 1201' (x,), which is therefore an endomorphism of the A-module HomgrA(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 associatioity of the algebra E goes over to the relation e(xy) = e(x) o e(y) for x, y homogeneous; whence, by definition of i(x),

(35)

1' (xy) = i (1/) ° 1'0!)

first for x, y homogeneous and then, by linearity, for arbitrary x, y in E 3 this may also be written

(36)

(uLx)Ly=uL(xy)

for x, y in E and u eHomgrA(E, P); as on the other hand clearly i(l) is the identity mapping (since this follows from e(l) = In) and x v—> i (x) is A-linear, it is seen that the external law of composition (x, u) r——> u L x (x e E,

ueHomgrA(E, P)) defines, with addition, a right E-module structure on HomgrA(E, P).

594

INNER PRODUCTS: CASE OF ALGEBRAS

§ 11.6

In particular we consider the case P = A, HomgrA(E, P) being in this case the graded dual E*"‘ of E; i(x) is then the graded transpose of the A-linear mapping e(x) (II, § 11, no. 6), in other words, for all x, y in E, u e E*“,

(u L MD = (u, Icy)-

(37)

With the convention at the beginning of the paragraph, note that, if x 6 Ep, 2' (x) is an endomorphism of E"“‘r qf degree —p.

For every homogeneous element x e E,” right multiplication by x is similarly denoted by e’(x) and the element u o e’(x) of HomgrA(E, P), called the lefl inner product ofu by x, by x _| u; we write i’(x) = x .J u and £’(x) is therefore a graded endomorphism of HomgrA(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 my) = e'(y)e'(x>, (38)

i’(xy) = 1"(#6) ° i’(y)

which may also be written (39)

x_I(y_|u)=(xy)_1u

and shows that the external law of composition (x, u) n-—> x J u defines, with addition, a lqfl E-moa'ule structure on HomgrA(E, P). The associativity of E implies on the other hand that e(x) o e’(y) = e'(y) c e(x) for x, y homogeneous in E, whence the relation

(40)

(yJu)Lx=y.J(uLx)

so that the two external laws of composition on HomgrA(E, P) define on this set an (E, E)-bimoa'ule structure (II, § 1, no. 14-). When we take P = A, i’(x) is the graded transpose of e’ (x); in other words, for all x, y in E, u e E*“‘,

(41)

nandthat,forp E ®A E is the coproduct and m: A ®A A —> A defines the multiplication. In other words, if, for x e E,

c(x) = 2 y, (8 2,, we can write (canonically identifying A (8),, E and E)

zi=x by virtue of the definition 'of counit. As the mapping u r—> i (u) is linear, it is seen that the external law of composition (a, x) -—-> x L u defines a right EN“module structure on E.

Similarly we define, for all u e E’m‘, the endomorphism of E

(52)

1"(1‘) = (In 8“!) °0

and, for all x e E, we write

(53)

(i' = u 4 x

and this element of E is called the lgft inner product of x by u. As above it is seen 598

INNER PRODUCTS: CASE OF COGEBRAS

§ 11.7

that the external law (u, v) -—> u .I It defines a lefl E*“—module structure on E. Moreover, these two structures are compatible, in other words,

(54-)

(q)|_v=u.J(xI_v)

for u, v in E*" (II, § 1, no. 14). With the same notation as above, the left hand side of (54) is ‘2; u(z‘)v(y,’j)311', and the right hand side is 12" v(y,)u(z{’k)z{k; their equality follows from the fact that they are the respective images ofthe left and right hand sides of (51) under the linear mapping g of E ® E (8 E into E such

that g(x ® y ® 2) = ”(x)u(Z)y-

It is therefore seen that the two external laws of composition on E defines on this set an (E*", E*") -bimodule structure.

When the cogebra E is eoeommutative, then u _l x = x L u for all x e E and u e E*"; when it is antieoeommatative (§ 4-, no. 9) and u e E: and x 6 Ed, we canwrite c(x) = by hypothesis

Z (2%, ® lac-1) withyu and zuinE, for allj and then

0‘j

whence

(55)

.

MIC) '- “ = (x '— Wu»;

and similarly

(55)

a J M") = MW") -' x).

In other words, (i) is an F*"—lwmomorphism of the (E*"‘, E*")-bimodule E into the (F*", F*"')-bimodule F, relative to the ring homomorphism t4): F*Kl‘ 9 E*KP.

599

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMME'I'RIC ALGEBRAS

Examples. In particular the above can be applied when E is one of the graded cogebras T(M), S(M) or/\ (M) for an A-module M (no. 1, Examples 5, 6 and 7). To find explicitly the bimodule structures thus obtained, we again identify a homogeneous elementfofdegree n in T(M)*" (resp. S(M) *3”, reap. /\(M)*") with an n-linearfirm (resp. symmetric n-lz'nearfom, reap. altemating n-linearform, also called an nfonn) on M". It suffices to express the products

(1:; ‘8) 7:, ®' ' '® x,) Lf(resp. (x1162. - 4‘12) Lf; rap. (:41 A x, A - ~ - A x,) Lf) for every finite sequence (x,) 1‘“, 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: (x1 ®x2®"'®xp) Lf=f-' (x1 ®x2®'”®xp) = 0

(xlxz...x,) Lf=f_l (xlxz...x,,) = 0 for p < n.

(57)

(x; AxaAmA xp)Lf=f-I(x1 A xaAmA x.) =0 For [I 2 n, we have respectively (58)

(11x2. . .xp) Lf = Z f(xo(1), - . ., xa(n))x°(n+1). . NV”)

(59)

(60)

'

n+1 ®-.-®xp (x1 ®x2 ®H'®xp) Lf=f(x1,...,xn)x

(“'1 A 7‘2 A ’ ' ' A xv) Lf= z e¢:IJ‘”(°"'a(1)a ' ' -: xa(n))xa(n+1) A ' ' ' A ”0(9)

(where, in (59) and (60), the summations are taken over the permutations

066, which are increasing on each afthe intervals [1, n] and [n + 1,p]ofN); and similarly (61) f-' (x1 ® 9‘2 ®' ‘ ' ® xp) =f(xp-n+1s - ' -: xp)x1 ® 7‘2 ®' ' ‘ ® xp-n

(62)

f4 (xl- - 4’22) = ;f(xa(p-n+1» - - -, x0(p))xa(1)"'xa(p—a)

(63) fJ(x1Ax2A-~Axp) 2 e)xa(n+1)~ - “”000

i(fx§A-~-Ax,f).(x1s/xn-Ax,,) fl-

= (‘DMPIW Z 5::(111: (3": x0(j)>)xo(n+1) A ' ' ' A x661)

where, the formulae (67) and (68), 6 runs through the set of permutations. a e 6,, which are increasing on the intervals [1, n] and [n + l, 1)]. We can also write, in the inner product notation,

(t L u*, 0*) = (t, 01-(u*v*))

for te T(M), u*, 0*

in T(M“)

(t L 14*, 0*) = (t, 05(u*o*))

for

in

te S(M), u“, 0*

S(M*)

M** defines an algebra homomorphism T(cM) :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. lqfl A(M)-module) structure. The explicit formulae giving the external laws of these modules are derived immediately 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. /\(M*)). PROPOSITION 11. The canonical homomorphism 01:T(M*) —> T(M) *9

(resp.

05:5(M) —>S(M)*", resp. 6A:/\(M*) —>/\(M)*sr) is a right T(M)-module (resp. right S(M)-module, resp. lqfl A(M)~madule) homomorphism.

604

INNER PRODUCTS IN FINITELY GENERATED FREE MODULE

§ 11.10

We show first that, for 2* e T(M*) and te T(M),

(69)

07(2“ L t) = 07(z*) L t.

Since M is a generating system of the algebra T(M), we need only prove (69) when t = x e M; moreover we can restrict our attention to the case where

z* = x? ®c‘ ®- - - ® xfi, where the xf eM*, and then, by (66) with the roles of M and M* interchanged, 2* L x = (x, xf)x’; ®- - - ® x3. Therefore, forallya,...,y,inM, P

«W: ._ xix/2 on e- - - m,» = 5. e“‘LuB

As for the right inner product ofan 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

6‘" I. e*”= 0

(75)

if 0% ? B

e“ L e*”= ——e°"‘ B if

—T—)! (a

°‘ > ‘3.

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 0201‘,“ allows us to identify the algebra S(M) with the polynomial algebra A[X1,. . ., Xn] (§ 6, no. 6); formula (75) shows that the inner product by 5*“ is just the differential operator D“ = Din’. . .D“", where D, = 6/6X, for l < i< n (§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

(§ 7, no. 8, Theorem I); similarly /\(M*) has as basis the elements e3. It follows from formula (68) of no. 9 that (76)

if K¢J {9:Je:=0 e* .1: —(—- I)?" 1””pKJ_KeJ_K if KC] and p= Card(K),

where pK_J_K is the number defined by formula. (19) of § 7, no. 8. There are analogous formulae with the roles of M and M* interchanged. 606

ISOMORPHISMS

§ 1 1.1 l

u. ISOMORPHISMS BETWEEN A’(M) AND A""(M*) FOR AN n-DIMENSIGNAL FREE MODULE M

Pnososmon 12. Let M be a free A-module qfdimension n; let e e A”(M) be an element forming a basis ref/WM) and bet e* be the element q"(M*) sueh that {(—1)» a(x, y, z) is alternating (§ 7, no. 3). (c) For every ordered pair of elements x, y ofF, xzy = x(xy) and yet2 = (yx)x. Clearly (a) implies (c). We show that (0) implies (b): by definition (§ 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). To prove that (b) implies (a), we use the following 4- lemmas: Lemma 2. Let E be an A-algebra such that the trilinear mapping (1‘, y, z) v—> a(x, y, z) is alternating, S a generating system of E and U a sub-A-module of E containing S and suchthatsU C UandUsC UforallseS. Tt = E. The set U’ of x e E such that xU C U and Ux C U is obviously a. sub-Amodule ofE, which contains S by hypothesis. 0n the other hand, for x, y in U’ and u e U, by hypothesis

(mu = mu) + (my, a) = x(vu) - a(x, M) = x(vu) — (My + x(uy) EU; 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 afortiori

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, v, w belong to H.

Lemma 3. Suppose that the mapping a is alternating. IfH is a strongly associative subset ofF, the subalgebra ofF generated by H is strongly associative.

As the set of strongly associative subsets of F is inductive, it suffices to prove that ifH is a maximal strongly associative subset of F, H is then a subalgebra ofF. As H is obviously a sub-A-module of F, it suffices to verify that for any two elements u, v of H, H U {uv} is also strongly associative, for by virtue of the

definition OfH, this will imply uv eH. Now, for all 2 SH and all t e F, by (2)

a(uv, t, z) — a(u, vt, z) + a(u, v, tz) = 0 612

APPENDIX

since H is strongly associative; as u, v, z are in H, also a(u, vt, z) = a(u, v, tz) = O, 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 a, v, w 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 ofF. Then the subalgebra ofE generated by X U Y is associative. LetZ bethesetofzeEsuch thata(u, v, z) = OforallueXandveY, 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 v eY, uZ C Z,vZ C Z,Za C ZandZv C Z.Now,foru,u’inX,veYandzeZ,by

(2)

a(u’u, z, o) — a(u', uz, v) + a(u’, u, zv) = a(u’, u, z)v + u’a(u, 2, v) = 0 by virtue of the fact that X is strongly associative and the definition of Z. But as X is strongly associative, a(a’, a, zv) = 0 and since u’u e X, a(u’u, z, o) = 0 by definition ofZ. Hence a(u’, uz, v) = 0, which shows that 142 C Z. Applying (2) now with (p, q, r, s) = (v, z, u, u’), we obtain similarly Zu C Z. Interchanging the roles ofX and Y and using the fact that a is alternating, we obtain vZ C Z and Zv C Z; whence the lemma.

It now suffices, to prove that (b) implies (a) in Proposition 1, to take X= {x}andY = {y},usingLemma4.

DEFINITION 1. An algebra F is called alternative if it satisfies the equivalent conditions ofProposition 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 8A A’ obtained from F by extending the scalars (§ 1, no. 5) is an alternative A’-algebra, as

follows from condition (b) of Proposition 1.

V

2. ALTERNATIVE CAYLEY ALGEBRAS

PROPOSITION 2. Let A be a ring, F a Cagley A-algebra, e its unit element, szx |-> 7: its conjugation and NzF —>- A its Cayley norm (§ 3, no. 4). (i) For F to be alternative, it is necessary and suficient that,fir every ordered pair of elements at, y ofF, xzy = x(xy). 613

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

(ii) If F 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 suflioient that N(x) be invertible in Ae; the inverse of x is then unique and equal to N(x)‘17r; denoting this by x",

x“(xy) = x(x' y) = y or allyeF. The condition xzy = x(xy) is obviously necessary for F to be alternative (no. 1, Proposition 1). Conversely, if it holds for every ordered pair of elements of F, applying it to E and y, it gives say = 2&9); applying the conjugation s to this relation, we obtain ya:2 = (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, EEG and similarly

37 e G. Then N(xy) = (xy)(:7g) = xygj = N(y)x:‘c = N(y)N(x), using the fact that N(y) e Ae. This proves (ii).

Finally we prove (iii). IfN(x) is invertible in Ae and we write x’ = N(x) ' 13, then xx' = x’x = e, for N(x) = x2 = Tex. 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:'c 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 it” = x”(xx') = (x”x)x’ = x’, whence the uniqueness assertion. The formulae x‘1(xy) = x(x‘1y) = y follow from the fact that x'l, 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, 7 an element of A and F the Cayley extension of E defined by y and the conjugation of E (§ 2, no. 5, Proposition 5). For F to be alternative, it is necessary and suflieient that E be associative. Let u = (x, y), o = (x’, y') be two elements of F (where x, y, x’, y’ are in E). Then (§ 2, no. 5, formula (27))

(3)

“’0 = ((x’ + YWW + 771w? + yx), (ye? + ex)? + y'(x2 + 7%)) “(140) = (x(xx' + fly) + fix? + 53%“??? + Yiy') + (w? + 3190*)-

Using the fact that 373/ and 7x: + x are in Ae, examining these formulae shows that the associativity of E implies uav = u(uv) and hence the fact that F is alternative (Proposition 2). Conversely, if F is alternative, the equation ‘ 14% = u(uv) applied when 3/ = 0 gives

(c + ex)? = ya?) + (WMNow the left hand side is equal to (yT(x)):'c’ = y(:‘c’T(x)) = yfi’x + 5’5); 614-

APPENDIX

comparing with the right hand side, we obtain (yi')x = y(fi’x), which proves the associativity of E, since at, y and J?’ are arbitrary elements of E. 3. OCTONIONS

Let E be a quatemion algebra of type (a, (3, 7) over A (§ 2, no. 5, Example 2) and let 8 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 (cc, 3, y, 8). By Proposition 3 of no. 2, F is an alternative algebra. It hasva basis (e,)o .5, $7 of 8 elements, .

defined by 90 = (e: O):

‘1 = (i: 0):

92 = (j: 0):

33 = (k, 0)

e; = (0,9),

as = (0,5),

as = (0,1'),

67 = (03k)

where (e, i, j, k) is the basis of E defined loc. cit; clearly e0 (also denoted by e) is '7

the unit element of F. If u = :20 Jim is an element of F (with the 5, EA), formulae (23), (24) and (31) of§ 2, no. 5, give for the conjugate, trace and norm of the octonion u

a = (a. + sane. — ,2 ta. (4)

TA“) = 250 + BE: NF(“) = 53 + {55051 " “5i - 7(83 + 55253 - «8%)

— 8(52 + 35355 — “33) + Y3(€§ + [55057 — dig)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 § 2, no. 5 give

TF((W')“') = TW'x') + WWW”) + WWW) + 8T(F'y'x)

Tp(ux' is an automorphism of the ring A and (x‘)‘ = x“ for all x e A and s and tin 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 (b,), by the relation

(Zxabs)(2ysbs) = Zagwwz)». 8

where the a“ belong to A. Show that this law and addition on Am define a ring structure on this set, provided the am satisfy the conditions as.tast,u = 414%,";

for all s, t, u in S. The element 12, is unit element of this ring if also “3.3 = “3,2 = l

for all s e S; in that case A can be identified with a subring of A‘s) and if C denotes the subring of A consisting of the elements invariant under all the automorphisms xI—>x', AG" is a C-algebra, called the crossed product of the ring A and the monoid S, relative to thefactor system (a t) . If the factor system (am) is replaced by (:16; a”), where, for all s e S, c, is an invertible element at

of A and c, = l, the new crossed product obtained is isomorphic to the crossed product defined by the factor system (a,_,). 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 x' = a? 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(I) can be identified with the sequences (1101:,“ (11 an arbitrary integer

21) 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. 621

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

(ii) For all Ic < n — I, the number of symbols P which appear in the subsequencc (at) 1 ‘g‘ k is 2 k/2.

(Use Set Theory, I, Appendix to associate with each sequence (at) an element

of M(I); for example, to the sequence PPnsy corresponds the word

«(macaw)»

1[ 13. Let A be a commutative ring and E an A-module. An A-module E,I is defined inductively by writing

E1 = E,

En = 69 (E, ®AE¢) forn 2 2. y+q=n

We write ME = “@1 E, and an A-algebra structure is defined on ME by setting, for x, e E, and xq 5 Eq, xpxq = x, ® ace 6 EH“. (a) 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 extendingf. ME is called thefree algebra if the A-module E. (b) An integer a(n) is defined by induction on n by the formulae

no), we = gunman) ifn 2 2. Show that En is isomorphic to the direct sum of a(n) modules isomorphic to E8".

(0) Letf(T) be the formal power series ,2; a(n)T". Show that

f(T) = T + (f(T))’*Deduce the formulae

f(T) =

1— v21 —4T and “(1‘) = 2"_11.3.5..,.l'(2n — 1) 'II

(d) If I is any set and E = A“), show that LibA(I) is identified with ME. There are canonical homomorphisms M(I) 3> N“) —l> N whose composition is length in M(I) ; 0(m) is called the multidegree of m eM(I) ; the mapping co defines on LibA(I) a graduation of type N“). Show that the set (LibA(I))¢ of homogeneous elements of multidegree a: EN“) under this graduation is a free A-module of rank equal to

«(co = «02) 3—: = 2-fi5__g.:3 where ac! = g (ot(i))! and n = gafi) = 1(a) is the length ofac.

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 .7: S y such that the relations x < y and x’ S y’, or x < 3/ and x’ < 3/, imply xT x’ < y Ty’. 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[(X)iel] is an integral domain-

(6) If A is an integral domain and f and g two polynomials in A[(X),el] such that fg is homogeneous and #0, show that f and g are homogeneous. In particular, the invertible elements of the ring A[(X,),eI] are the invertible elements of the ring A. 15. Let A be a commutative ring and 71 an integer >2 such that for all a e A the equation nx = a has a solution x e A. Let m be an integer >1 and u a polynomial in A[X] ofthe form X’"” + f(X) wherefis ofdegree < mn — l ; show that there exists a polynomial v eA[X] of the form X’" + w, where w is ofdegree u(v) is a law of composition which is associative and left distributive with respect to addition and multiplication in A[X]. If A is an integral domain, show that the relation u(v) = 0 implies that u = 0 or that v is constant and that, if u aé O and the degree of z: is >0, the degree of 11(0) is equal to the product of the degrees of u and o. 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 5(X) = aX + b, with a 9e 0 in K 623

TENSOR ALGEBMS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

and b E K (cf. Exercise 17). Conversely, show that being given an automorphism o- of K and two elements a, b of K such that a 9e 0 defines one and only one automorphism s of K[X] such that SIK = o- and 5(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 KM], 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, y.) (N, p.’) = (M’, Np. + pl).

19. Prove the polynomial identity in 8 indeterminates

(XE+X§+X§+X2)(Y§+Y§+Y§+YE)

,

= (X1Y1 " X2 2 " XaYa " X4Y4)2

+ (X1Y2 + X2Y1 + X3Y4 " Xiyala + (XiYa + XaYi + X4Y2 _ X2Y4)2 + (X1Y4 + X4Y1 + XzYa " X3Y2)2(Apply formula (32) of no. 5 to a suitable quaternion algebra.)

20. Show that there exists no polynomial identity in 6 indeterminates

(X? + X: +X§)(Y§ + Y: + Y3) = P2 + Q2 + R2 where P, Q, R are three polynomials with coefficients in Z with respect to the indeterminates X, and Y,. (Observe that 15 = 3 .5 cannot be written in the form [)2 + q2 + r”, where p, q, r are integers.)

21. Let S be a multiplicative monoid with identity element 2, 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 :58, the number n of terms in a finite sequence (t,)1 a“ of elements distinctfrom e and such that ill; . . . tn = s is less than a finite number v(s) depending only on 3. Show that, for an element x = 8; «,9 of the total algebra of S over a ring A to be invertible, it is necessary and sufficient that at, be invertible in A (reduce it to the case where at, = l and use the identity e—z"+1=(e —-z)(e+ z+

+ 2"».

22. If K is a commutative field, show that in the ring of formal power series K[[X1, X2, . . . , Xp]] there is only a single maximal ideal, equal to the

set of non-invertible elements. On the other hand, show that in the polynomial ring K[Xl, . . .,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 > O,

(X + Y)u(X, Y) = xmvm. 24-. Let K be a conmmtative field and let k be an integer such that k 9!: 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 I) e K[[X]] such that o" = u. 25. Let A be a ring, commutative or otherwise, and let a be an endomorphism of A. On the additive product group E = AN an internal law of composition

is defined by writing (was) = (n), where 7,. = ,2, «perms (with the convention

0°(E) = E). (a) Show that this law defines, with addition on E, a ring structure on E, admitting as unit element a the sequence (09,) where do = l and an = O for n 2 l. The mapping which associates with every EEA the element (an) e E, such that do = 5, an = O for n 2 l, is an isomorphism of A onto a

subring of E, with which it is identified. Then we write 1'; «90 instead of

(an) and X’fl = c’(fl)Xp for all fleA; if u = 2:0 anX" aé 0, the smallest integer n such that a” 75 0 is called the order of u and denoted by w(u). (b) If A is a ring with no divisor of 0 and a is an isomorphism of A onto a subring of A, show that E is a ring with no divisor of 0 and that o)(uv) = m(u) + (9(0) foru 96 OandvaéO. c For a = Z anX" to be invertible, it is necessa ry and sufficient that a 0 i=0

be invertible in A. (d) Suppose that A is a field and a an automorphism of A. Show that E admits a field of left fractions (I, §9, Exercise 15) and that in this field F every element 960 can be written uniquely in the form uX'”, where u is an element of E of order 0.

TI *26. (a) Let A and B be two well-ordered subsets of R (which are necessarily countable; cf. General Topology, IV, §2, Exercise 1). Show that A + B is well-ordered and that, for all 0 EA + B, there exists only a finite number

of ordered pairs (a, b) such that a e A, b e B and a + b = o (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 KR, consider the

vector subspace E consisting of the elements (ax) such that the set (depending 625

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

on (0%)) of x e R such that ax aé 0 is well-ordered. For any two elements (ax), Z «,9, (a sum which is (fix) of E, we write (¢x)([3x) = (yx), where Yx = y+z=x composition defines, with of law this that Show (a)). of virtue by meaningful addition, afield structure on E; the elements of E are also denoted by “ER a,Xt and called formal power series with well-ordered real exponents, with coefficients in K.* §3 1. Let A be a commutative monoid written additively with the identity element denoted by 0, all of whose elements are oanoellable. Let E be a graded

A-algebra of type A and suppose that E admits a unit element e = “2A e, where e“ e E“. Show that necessarily e = on 6 E0.

§4 {I 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 EndE(M), v e EndF(N), there exists one and only one G—endomorphism w of M ®AN such that w(x ®y) = u(x) ® v(y) for x e M, y e N and we can write w = (Mu ® 0), where 4) is an A-linear mapping (called canonical) of EndE(M) ®A EndF(N) into EndG(M (8,, N). Suppose in what follows that A is afield. (a) Show that the canonical mapping (1) is injective (consider an element z = Z a, ® 0,, where the 0,6 EndF(N) are linearly independent over A and

write (d>(z))(xa ® 3/) = 0 for every element x“ of a basis of M over A and all y e N).

(b) Suppose that M is afinitelygeneratedfree E—module. Showthat the mapping (1: 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. (0) Suppose that M is a free E-module and that there exist a y e N and an infinite sequence (on) ofendomorphisms ofthe F-module N such that the vector subspace (over A) of N generated by the 0,,(y) is of infinite rank over A. Show that, if the mapping :1) is bijective, every basis of M over E is necessarily finite. (d) Suppose still that M is a flee E-module. Show that, for the Gmodule M ®AN to be faithful, it is necessary and sufficient that the Fmodule N be faithful (consider a basis of E over A). 11 2. Let K be a commutative field, E, F two fields whose centre contains

626

EXERCISES

K, M a left vector space over E and N a left vector space over F; let G = E ‘81: F. (a) Let x aé 0 be an element of M and y at: 0 an element of N. Show that

x ®y is free in the G-module M ®KN (use Exercise 1(d) and the transitivity of EndE(M) (resp. EndF(N)) in the set of elements of M (resp. N) distinct from 0). (b) Show that M ®K N is a free G-module. (If (mu) is a basis of M over E and (713) a basis of N over F, show that the sum of the G-modules G(m‘,I ® n9) is direct, using a method analogous to that of (a) )

§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] be two finite sets, I0 = I U {at},Jo = J U {B}, where at and B are two elements not belonging to I and J respectively, and suppose that Io and Jo are disjoint. Let E be a vector space of finite rank over a commutative field K and let 2 be an element of T§(E). For all i e I, let W; be the subspace of E* consisting of the x* e E* such that c§(z ® x*) = 0 (at, being the contraction of index i e I and index (Bejo in T§°(E)) and let V, be the subspace of E orthogonal to Wi. For all j 6], let W, be the subspace of E consisting of the x e E such that 07 (z 8) x) = 0 (of being the contraction of index a e lo and index j e] in T§°(E)) and let V; be the subspace of E* orthogonal to W,. Show that 2 belongs to the tensor product (1:8; V,) (8 (£83 V;) canonically identified with a subspace of T§(E). Moreover, if (Ui)ieI is a family of subspaces of E, (U§),eJ a family of subspaces of E* such that 2 belongs to the tensor product (@ U‘) ® (99% U;), identified with a subspace of THE), then V, C U‘ and V; C U; for all i e I andj 6] (cf. II, § 7, no. 8, Proposition 17). 3. Let M be a finitely generated projective A-module. For every endomorphism u of M, let 17 denote the tensor 61—41(11) e M“ (8;; M corresponding to it. (a) For every element xeM, show that u(x) = c§(x 63) ii), where .7: ® ii belongs to M ® M" (8 M = T§;;s’(M). (1)) Let u, v be two endomorphisms of M and w = u o I); show that

17) = 63(0” ® 11), where 17 ® :2 belongs to TgfiXM).

(c) For every automorphism s of M and every tensor 2 e T§(M), a tensor

3.2 e T§(M) is defined by the condition that if z = kg?” zk where zk e M

forkeI and zkeM" forkej,thens.z=k§nwith 2;, =s(zk) ifkeI, 627

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

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z,’, = '3'1(z,,) if k6]. Show that the group GL(M) operates by the law (3, z) »—-> s. z on the A-module TJ(M), the mappings z v—> s. z being automorphisms of this A-module. Moreover, for every ordered pair of indices i 61,

J 6], c§(s.z) = s.c§(z). If M is projective and finitely generated, show, in the notation of Exercise 2, that 5.17 = (s o u 03—1)". 4. Let M be a finitely generated projective A-module; for everyabilinear mapping u of M x M into M, let u denote the tensor 0M‘1(u) in'l"‘1‘”2,(M)‘. Show that for the algebra structure on M3 defined by u to be associative, it is necessary and suflicient that the tensors 043(u (8) u) and 026(u ® u) correspond under the canonical associativity isomorphism of Th. 2, a,(M) onto T“? 3, 4, (M)

11 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-commutative K-algebra.

(5) Let u, v be two elements of T(E). Show that, if there exist two elements a, b ofT(E) such that ua = vb 9E 0, one ofthe elements u, v is a right multiple of 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 = e1141 + - - . + enun, v = e101 + - ~ - + can”, where (eh) is a basis of E. Conclude by arguing by

induction on the smallest degree of the homogeneous components of M.) Deduce that, if dim(E) > 1, the ring T(E) admits no field of left fractions (I, §9, Exercise 15).

(a) Show that for two non-zero elements u, v of T(E) to be permutable, it is necessary and sufficient that there exist a vector x 7‘: 0 in E such that u

and I) belong to the subalgebra of T(E) generated by x (use (b)). Deduce that, if dim(E) > 1, the centre of T(E) is equal to K. 11 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 (43 F), where E @ F is considered as a vector space over K;

E (resp. F) is identified with a vector subspace of T(E {-9 F) under the canonical injection. Let 3 be the two-sided ideal of T(E ® F) generated by the elements x1 ® x2 — (“1372) and 3/1 (8%: "‘ (311%): where x, E E: 11:5 F: i = 1: 2;

let E at F denote the quotient algebra T(E 6—) F) [3 ; it is called the free product of the K-algebras E and F; let (I: denote the canonical homomorphism of T(E 6-) F) onto E a: F and ac and [3 the restrictions of G, 628

EXERCISES

v:F —> G into a K—algebra G, there exists one and only one K-homomorphism w:E*F—->Gsuchthatu = wouandv = woB (whichjustifiesthetermino-

logy). (b) Let ZRfl be the vector subspace of T(E {-9 F) the sum of the T"(E {-9 F)

for 0 S k < n and write 3,, = 8 n R". Show that, if we consider a basis of E

consisting of l and a family (fiber, and a basis of F consisting of 1 and a family (Anew then a supplementary subspace of 3,, + R“; is obtained in R,‘ by considering the subspace of R1' with basis the elements of the form 21 ® 23 ® - -- ® 2,, where, either 22,“ isequal to one ofthe exfor 2j + l S n

and 2,, is equal to one of thefl, for 27' < n, or 22,” is equal to one of thej,‘l for 2j + l < n and 22, is equal to one of the e,_ for 2j < n (argue by induction on 72). (5)1431: Pa = “R10 C E*F§ then P» C Pan: P0 = K: Pt c Ph+k and E a: F is the union of the P“. Show that there is a canonical vector space isomorphism, for n 2 l, Pn/Pn-1—>(G1®G2®

®Gn)®(G2®G3®

®Gu+1)

where we write G, = E/K for 1: odd, G, = F/K for I: even. In particular the

homomorphism ac and (3 are injective, which allows us to identify E and F

with sub-K-algebras of E at F. P; (resp. Pg) is used to denote the inverse images in P ofGl ®G2 ® ®Gfl (resp. G2®G3 ® ®Gn+1); the

height of an element 2 e E as F, denoted by h(z), is the smallest integer such that z e P"; if h(z) = n and z e P; (resp. 26 Pg), 2 is called pure odd (resp.

pure even). Then P" = P; + P; and P; n P: = P".1 for n > 1.

(d) Show that, if u, v are two elements of E s: F such that h(u) = r, h(v) = s, then h(uv) S h(u) + [2(0) and that h(uv) = [1(a) + [1(0) unless u and v are

pure and, either 1 is even and u, v are not qf the same parity, or r is odd and u, v

are 9“ the same parity. (If for example r is even, uePfi, vePg, show that an e P,+,_1 is possible only ifu e P,_1 or v e P _1.)

(e) Suppose that E has no divisors of zero and, in the notation of (a'), suppose that r is even, it pure even and 11 pure odd. Show that h(zw) = r + s — 1,

unless there exist two invertible elements x, y of E such that uy e P,_1 and

xb eP,_1. (Let ((1,) (resp. (b,)) be the basis of the supplementary subspace of 3, + R,_.1 in R1 (resp. of 8, + R,_1 in Rs) considered in (1)); write u = Z (a,)x, and v = Z y,(b,), where x, and y, are in E, and show that, if an e P,+._2, then necessarily xiy, e K.) Treat similarly the other cases

where h(uv) < h(u) + h(v) when E and F are assumed to have no divisors of zero. (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

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TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

1[ 7. Let E be a vector space of finite dimension 7: over a field K. For every integer r 2 l, the symmetric group 6, operates linearly on the tensor product EQ’ by the action (a, z) 1—) 0.2 such that a.(x1 ® x2 ‘3) ' ' ' ® xr) = xa“(1)® xa'1(2)® ' ' ' ® xa'1(r)

for every sequence of r vectors (x,)1‘;‘, of E. Let u, (z) = a. 2. Show that for every sequence (v,)1 L _> o. T"(M’)——> For every subsetJ of the set {1, 2,. . ., n}, we write

PJ = N1 ®N2 ®

®Nm

whereN,=M'1fzeJ,N‘Mifi¢J

(PE, is therefore equal to T“ (M)) and QJ= N1 ®N2 ®

®Nm

whereN,=M'1fzeJ,N‘M'ifi¢J

(Qg'IS therefore equal to T"(M') .) For every integer i such that 0
T"(E) is the canonical mapping, the n—linear mapping ag o (1) is also called the skewsymmetrization of the n-Iinear mapping g o 4;. Deduce from Exercise 8(a) that if E is free every alternating n-linear mapping of E" into F is the skewsymmetrization of an n-linear mapping of E" into F.

{I 10. Let E be an A-module with a generating system of n elements. Show that every alternating n-linear mapping of E" into an A-module F is the skewsymmetrization of an n—linear mapping of E“ into F. (Let a, (l S j S n) be the generators of E and duA" —> E the homomorphism such that (M6,) = a, for all j, where (e,) is the canonical basis of A". Iff:E" -—> F is an alternating n-linear mapping, so is fi:(x1, . . ., x“) n—>f(¢(x1), . . ., d>(xn)); show that if go is an n-linear mapping of (A")" into F such that f0 = ago, go can also be written as (x1, . . ., x”) n—> g(¢[>(x1), . . ., t1>(x,,)), where g is an n-linear mapping of E“ into F.) 11 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 + AZ”. Show that there exists an alternating bilinear mapping of E2 into F, which is the skewsyrnmelrization of no bilinear mapping of E2 into F, but that in T2(E) the module 8; is equal to the kernel of the endomorphism 21—) az (consider T’(E) as a quotient 635

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMNEETRIC. ALGEBRAS

module of T2(A3) and show that the kernel of z I—-> az is the canonical image of the module of symmetric tensors S§(A3)). 1[ 12. In the notation of Exercise 11, let B be the quotient ring

A/(AX2 + AY2 + AZ“) and let G be the B-module E ®A B. Show that in T3(G), the module 8; is distinct from the kernel of the endomorphism z v—> 02 (method analogous to that of Exercise 11). 13. (a) Let M be a graded A-module of type N. Show that there exists on the algebra S(M) (resp. /\(M)) one and only one graduation of type N under which S(M) (resp. /\(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 add), the graded algebra A(M) thus obtained is alternating (§ 4, no. 9, Definition 7). (b) Given a graded A-module of type N, consider the following universal mapping problem: 2 is the species of anticommutative graded algebra structure (§ 4, no. 9, Definition 7), the morphisms are A-homomorphisms of graded algebras (§ 3, no. 1) and the a—mappings are the graded homomorphisrns of degree 0 of M into an anticommutative graded A-algebra. Let M‘ (resp. M+) denote the graded submodule of M generated by the homogeneous elements of odd (resp. even) degree. Show that the graded algebra ' G(M) = A(M‘) ®A S(M”) is anticommutative; a canonical injection j of M into this algebra is defined by writingj(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) = /\(M-) a. SM) 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 anticommutative 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 x1, . . ., x" be linearly independent elements of M; for every subset H = {in . . ., in} of m g n elements of

the interval [1, n], where £1 < 1'2 < - - - < im, we write xH=x,1 A an, A

Axlm.

Show that when H runs through the set of subsets of [1, n] with m elements,

the xn are linearly independent in /\(M). 636

EXERCISES

15. Let M be an n-dimensional free A-module. Show that every generating system ofM with n elements is a basis ofM.

§8 1. In a matrix U of order n, if, for each index 2', the column of index i is

replaced by the sum ofthe columns ofindex #1", the determinant of the matrix obtained is equal to (— l)”'1(n — 1) det U. Ifin U, for each i, the sum of the columns of index aéi is subtracted from the column of index i, the determinant

obtained is equal to —(n -— 2)2"'1 det U. 2. Let A = det(og,) be the determinant of a matrix of order n; for l < i < n — l andl ux of Ker(f) into the dual (M,,,(K))* is

injective; complete the proof of (*) by noting that the image under X n—> u, of Ker(f) is contained in the orthogonal of Im(f)

1[ 22. Let F be a mapping of Mn(K) into a set E, where K is a commutative field, such that, for every triple of matrices X, Y, Z in Mn(K), F(XYZ) = F(XZY).

We except the case M2(F2); show then that there exists a mapping (I) of K into E such that F(X) = (F(det(X)). (Using the Corollary to Proposition 17 (no. 9), show first that, for U eSLn(K), F(X) = F(XU); deduce that, in GLn(K), F(X) depends only on det(X). On the other hand, if, for r g n — l , we write I,

O

X' ' (0 0) show that there exists a matrix Y of rank 1 such that X,_1 = 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 x1, . . . , x" are elements of E, we write [x1x2' - in] =

Z oauy’caar - -"o(n)-

0'55"

Show that if E admits a finite generating system over A, there exists an integer N such that [x1x2. . .xN] = O for all elements at, (l < j < N) of E. 24. Let X, (l s i < m) be square matrices of the same order n over the commutative ring A. Show that if m is even then (Exercise 23)

Tr[X,X2. . .Xm] = 0 64-3

III

TEN'SOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

and ifm is odd Tr[X1X2. . .Xm] = m-Tr(Xm[X1X2. o .Xm_1]).

(If A is the cyclic permutation (l, 2, . . ., m), C the cyclic subgroup of 6,, generated by A and H the subgroup of 6", consisting ofthe permutations leaving m invariant, note that every permutation in 6,, can be written uniquely as or with a e H and 1 e C; then use the fact that the trace of a product of matrices is invariant under cyclic permutation of the factors.)

11 25. Let A be a commutative ring containing the field Qof rational numbers and let X be a square matrix of even order 271 2 2 over A. Let L be the subset {1, 2,. .., n + l} of {1, 2, . . ., 2n} and L’ its complement. Show the

following formula (“Redei’s identity”) :

(*) dams) mam) = Z (—1) 5—1—— deturm) damn?) it

(" ‘ l) r

where H runs through the set of subsets ofn elements of {1, 2, . . ., 2n} containing 1 and K the set of subsets of n elements of L and r = Card(H n L’). (Evaluate the coefficient of a term xc(1).1xa(2).2~ . .xmnm“ in the right-hand side of (*) for an arbitrary permutation o- e 62", noting that this coefficient can only be 750 ifH = 6(K); show that it can only be 960 ifo-(L) = L.) 26. (a) In the notation of Proposition 19 (no. 10), suppose that there exist two A[X]-linear mappings g1, g2 ofM[X] into M’ [X] such that 4/ o g2 = gl 0 4:; show that there then exists an A[X]-Iinear mapping g of Mu into M;. such that

(b’ o g1 = d)’ o g. Further, if g1 and g2 are A[X]-isomorphisms of M[X] onto M’[X_], g is an A[X]-isomorphism of Mu onto M,’,. (b) In the notation of no. 10, the endomorphisms u, u’ are called equivalent if there exist two isomorphisms f1, f3 of M onto M such that u' ofz = f1 0 u; they are called similar ifthere exists an isomorphismfofM onto M’ such that u’ of = f o u. Deduce from (a) that, for u and u' to be similar, it is necessary and sufficient that the endomorphisms X — a and X — 12’ (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 e1, e2 such

that a? = e1, e192 = 22, age, = e2 = 0; let A0 be the opposite algebra to A.

Show that there exists in A elements at such that TrA,K(x) 9e TrA0,x(x) and NAIKO‘) 95 NA°IX(x)' 64-4

EXERCISES

§10 1. IfA and B are K-algebras and a: and [3 two K-homomorphisms of A into B, an (at, B)-derivation of A into B is a K-linear mapping (1 of A into B satisfying the relation

(My) = (dx)°t(y) + BMW) for Jay 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”, gizA x A—>A is multiplication, A1: B x A —> B is the K-bilinear mapping (2, x) I——> za(x) and An: A x B —> B the K-bilinear mapping (x, z) l-> [5(x) 2. Suppose henceforth that A and B are commutativefields and that (1 ye [3. Show that there then exists an element 11 EB such that d is the (at, B)-derivation x n—> ba(x) — [3(x)b. (It can be assumed that d aé 0. Show first that the kernel NofdisthesetofxeAsuchthatoc(x) = [3(x) andthenshowthatforxgéNthe element 17 = d(x) /(1(x) - B(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 :1 of A into B satisfying the relation

d(xy) = (dx)a(y) + «(x)(dy) for w in A, in other words, a derivation on the sense of Definition 1 with A = A’ = A', B = B’ = B”, d = d' = d", y. multiplication in A, A, the mapping (z,x) r—rza(x) of B x A into B and A2 the mapping (x, z) I—nx(x)z of A x B into B. Show that if B is a commutative field, an extension of K, and d 96 0, there existsb aé OinBsuchthat

a(xy) =a(x)cc(y) + b(dx)(dy)

for x,y in A.

If b = c2 with c e B, 0 9E 0, there exist two homomorphism j; g of A into B such that oc(x) = (f(x) + g(x))/2, dx = (f(x) —- g(x))/2c. Otherwise, there exists a quadratic algebra (§ 2, no. 3) B’ over B, a p. e B’ such that p? = c,

and a homomorphism f of A into B’ such that or.(x) = (f(x) +f—(x—))[2,

dx = (N) —f(x»/2.

11 3. Let K be a field, commutative or otherwise, E the left vector space

K9“) and let (en),,eN be the canonical basis of this space. 0n the other hand let a be an endomorphism of K and d a (a, Ila-derivation (Exercise 1) of K into itself, in other words an additive mapping of K into itself such that

(“811) = (46001) + 50112)645

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIG ALGEBRAS

III

The elements ofE are linear combinations ofproducts due" and a multiplication is defined on E (Z-bilinear mapping of E x E into E) by the conditions

(«60036,») = (den—1)(0(fl)em+1 + (mm) for n 2 1 (Wo)(fiem) = (“WmThus an (in general non-commutative) ring structure is defined on E with an as unit element (identified with the element 1 of K). Ifwe write X = e1, then on = X" for n 2 1, so that the elements of E can be written as

a0 + a1X+---+ ac", with the multiplication rule X0: = a(oc)X + (dot)

for

or EK.

This ring is denoted by K[X; a, d] and is called the ring Qf nan-commutative polynomials in X with coqfiicient: in K, relative to a and d. For a non-zero element 2 = Z at,X’ of E, the degree dcg(z) is the greatest integerj such that cc, aé O. (a) Show that if u, v are two elements #0 of E, then:

m) 95 0 and deg(uv) = deg(u) + deg(v); deg(u + v) < sup(deg(u), deg(v)) if u + v 96 0. Moreover, if deg(u) 2 deg(v), there exist two elements w, r of E such that u = wv + r and, either 1' = 0, or deg(r) < deg(v) (“Euclidean division” in E). (12) Conversely, let R be a ring such that there exists a mapping u I—> deg(u) of the set R’ of elements 95 0 of R into N, satisfying the conditions of (a). Show that the set K consisting of 0 and the non-zero elements u ER such that deg(u) = 0 is a (not necessarily commutative) field. If K 9e 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 atjx’ with a, e K. (To see that every element of R is of this form, argue by reductio ad absurdum by considering an element a e R which is not of this form and for which deg(u) is the smallest possible.) Prove that there exist an endomorphism a of K and a (a, Ila-derivation d of K such that xac = c(ac)x + do: for all (X e K. Deduce that R is isomorphic to a field or a ring K[X; a, d].

4. Let M be a graded K-module of type N; show that every graded endomorphism of degree r can be extended uniquely to a derivation of degree r of the universal anticommutative algebra G(M) (§ 7, Exercise 13).

* 5. Let K be a field of characteristic 1; > 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 QK(A) ®A B ——> QK (B) is not injective... 646

EXERCISES

§u 1. IfV is a finitely generated projective A-modulc, so is End (V), canonically identified with V ®A V* (II, §4, no. 2, Corollary to Proposition 2). If every element u e End(V) is associated with the A—linear fonn 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 Aalgebra structure on End(V) defines by duality (no. 1, Example 3) an Acogebra 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 Mn(A) for which the coproduct is given by G(Eu) = 2E,“ ®E,k. This cogebra structure and the algebra structure on M,,(A) do not define a bigebra structure.

2. Show that in Definition 3 (no. 4), condition (3) (relative to graded bigebras) 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 (8) E —)- E,

c the coproduct E —> E (8) 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

(a, v) e H0mA-aig.(E: B) x Homi..1..(E, B) the A-algebra homomorphism

EJ+E®EEKB®BILB (where m3 is the product in B) defines a monoid structure on Horn“.15 (E, B). (b) In order that, for every commutative A-algebra B, Homkm. (E, B) be a

group, it is necessary and sufficient that there exist an A-algebra homomorphism i: E ——> Esuch thati(e) = eand the composite mappings m o (1r: (8) i) o c and m o (2' ® IE) 0 c are both equal to the mapping x I—> y(x)e. i is then called an inversion in E; it is an isomorphism of E onto the opposite bigebra E°. 5. Let E1, E2 be two bigebras over A. Show that on the tensor product

E1 ®A E2 the algebra and cogebra structures the tensor product of those on E1 and E2 define on E1 ®A E2 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) (§ 7, Exercise

647

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

13); deduce a canonical graded algebra homomorphism G (M*) —-> G(M) *" 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

(dam-1 = Ng) °‘1°/\(‘g)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 1|

0 of A(M) into an A-module G, we can write a = 32;) 0,, where v, is the 9

7|

restriction of v to /\(M); we then set '40 = go (—l)’vp, so that 7721) = I). 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 4) is the isomor-

phism of /\(M) onto /\(M*) defined starting with e = e1 A e, A . - . A en, show the formula

WM = Mu) o -1 o/\(u). X.9 The adjoint of a square matrix X of order n over A is the matrix =(det(X")) of minors ofX oforder n — 1. Show that ifX 15 invertible then dit(X)= (det X)" 1 and that every minor det (XE x) of X of order p is given by the formula

(I)

det (52m) = (det Irv-«X313»

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 (13—: 0 between minors of an arbitrary invertible square matrix X of order 12 over A another identity (I) = 0 can be derived called the complement of (I) = 0, by applying the identity (I) = 0 to the minors of

the adjoint X of X and then replacing the minors of X by fimctions of the minors of X using identity (1) of Exercise 9. In this way show the following identity:

det(X"‘) det(X1") — det(X"‘) det(X”‘) = (det X) det(X“"‘") for i < j where X"'1 M“ 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.

648

EXERCISES

{I 11. Let (D = 0 be an identity between minors of an invertible square matrix X of order n over a commutative ring A, (I) =2 0 the complementary identity (Exercise 10) and I: an integer >0. Let Y be an invertible square matrix of order n + k and let Yo be the submatrix of the adjoint matrix Y

formed by suppressing in Y 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 (A,,) of order n, show the identity det C = (det A) (det B)"-1 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 71). State the identities obtained by extending the Laplace expansion (§ 8, no. 6) and the identity of Exercise 2 of § 8. {I 12. Let X= (5“) be a square matrix of order n over a commutative field

K; let H be a subset of [1, n] with ,0 elements and H’ the complement of H m [1, n], suppose that, for every ordered pair of indices h e H, k e H’, 2 Ethan: = 0-

Show that, for all subsets L, M of [1, n] with [1 elements, PM. M’ det(X1..n) det(XM',n') " PL.L' det(XM,E) det(XL.E’) = 0(‘Consider the columns x" of index I: e H in X as vectors in E = K" and the columns x: of index k e H' as vectors in E* and show that the (n — p)-form xii. is proportional to ¢,(xn).)

13. Let 1" and A be two determinants of invertible matrices of order n over a commutative ring and F” the determinant obtained by replacing in I‘ the column of index i by the column ofA of indexj. Show that det(Pu) = Pit-1A.

(Expand F” by the i-th column and use Exercise 9.) 649

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

III

14-. Let M be a free A-module of dimension n > 1. Show that every isoP

"-P

73—?

p

morphism q; of /\(M) onto A (M*) such that A (a) o q) = q; 0 Nu) for every

automorphism u of M, of determinant equal to l, is one ofthe isomorphisms (b, defined in Proposition 12 of no. 11 (proceed as in § 7, Exercise 4).

15. Let E be a free A-module of dimension 1:. Let 2 be a p-vector over E and 2* a (p + q)-form on E; z can be canonically identified (§ 7, Exercise 8) with the skewsymmetn'zation 021 of a contravariant tensor 21 oforderp and z* with the skewsymmetrization of a covariant tensor of order p + g. If, in the mixed tensor 21 . 2*, the k—th contravariant index and the (p + k) -th covariant index are contracted for l < k S 1), show that the covariant tensor of order q thus obtained is skewsymmetrized and is canonically identified with the 9form 2* I. z, up to sign.

16. Let E be a free A-module of dimension n. The regressive product of

x 6 A03) and y e /\(E) with respect to a basis {e} of/\(E), denoted by x v y, -1

is the element ((x) A My», the isomorphism 4) being relative to 2. This product is only defined to within an invertible factor, depending on the basis {e} chosen. Show that ifx is a p-vector and y a q-vector, x V y = 0 ifp + q < n andthat,ifp + q 9 12,1: v yisa (p + q — n)-vectorsuchthat y V x = (_l)(n—a—q)x V y.

The regressive product is associative and distibutive with respect to addition and defines on /\(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 2 be a p—vector #0 over E and let Va be the vector subspace of E consisting of the vectors x such that z A x = 0. Then dim(Vz) < 1); if (1:1); ‘ I“ is a free system of vectors of Va, there exists a (p — q)-vector 1) such

—1 that z = v A x1 A - - - A xq. If E is finite-dimensional, then Vz = {>(M2), where M: is the orthogonal in E* ofthe subspace Mz associated with 2. For 2 to be a pure p—vector, it is necessary and suflicient that dim(Vz = p and then a —

a.

((1) Let 0 be a pure [J-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 2 p and that there exist a (q — p)-vector u such that w = u A 0. In order that V n W = {0}, it is necessary and sufficient that 650

EXERCISES

v A w aé 0; the subspace V + W is then associated with the pure (,0 + q)vector v A w.

(6) Let u and v be two pure p—vectors and U and V the subspaces associated respectively with u and v. For u + v to be pure, it is necessary and sufiicient thatdim(UnV) 2]) — l. 18. Let z = ; «Hen be a non-zero pure p-vector of an n-dimensional vector

space E, expressed using its components with respect to an arbitrary basis (a)1 «Q, of E. Let G be a subset of [1, n] with [1 elements such that MG 9’: 0 ; let (ih)1 u ..I x, the left annihilator of this C*-module is a two-sided ideal 8’ of the algebra 0*. Show that if E is finite-dimensional over K, C*/8’ is finite-dimensional. (b) Deduce from (a) that for every element xe C, the subcogebra of 0 generated by x is finite-dimensional. (Note that the C*-module E generated by

x is finite-dimensional and that, if 8’ is its left annihilator, xe 8'°, the ortho-

gonal of 8' in C.) (27) Let B be an associative unital algebra over a commutative field K and let m: B (3)]; B —-> B be the K-linear mapping defining the multiplication on B. The tensor product 3* ®K B* (where B* is the dual vector. space of the vector space B) is canonically identified with a vector subspace of (B ®K B)* (II, § 7, no. 7, Proposition 16).

(a) Show that for an element w e B* the following conditions are equivalent: (l) ‘m(w) eB“ ® B*; (2) the set ofx J w, where x runs through B, is a finite-dimensional subspace ofB*; (3) the set of w L x, where :4 runs through B, is a finite-dimensional subspace of B*; (4) the set of x .1 w L y, where x and y

run through B, is contained in a finite-dimensional subspace of B*.

653

III

TENSOR ALGEBRAS, EXTERIOR ALGEBRAS, SYMMETRIC ALGEBRAS

(b) Let B’ C B* be the set of w e B* satisfym'

' g the equivalent conditions in

(a). Show that 'm(B’) C B’ 8’: B'. (For wEB', write ‘m(w) = 2: u, @114,

where for example the u, are linearly independent in 3*; show that then 0‘ E B’ for all i using the associativity of B and arguing by reductio ad absurdum.) B' is therefore the largest cogebm contained in B*, for which ‘11: is the coproduct. B’ is called the dual cogebm of the algebra B. When B is finite-dimensional, B’ = B*. (a) Show that if B is a commutative field of infinite dimension over K, B’ = {0}. (Note that the orthogonal of the set of x .I w, for same w e B*, at

running through B, is an ideal of B.) (d) Let Bl, B2 be two K-algebras and f: B1 —> B2 a K-homomorphism of algebras. Show that ff(B§) C B; and that |‘f, restricted to 3;, is a cogebra homomorphism of B; into B1.

(e) If C is a coassociative counital cogebra over K, the canonical injection C —> C“ of the vector space G into its bidual maps 0 onto a subspace of (C*)’, the dual cogebra of the algebra 0* dual to C, and is a cogebra homomorphism of G into (C*)’. APPENDIX

1. Show that in alternative Cayley A-algebra (notation of §2, no. 4-) T((xy)z) = T(x(yz)). (Reduce it to proving that a(x, y, z) = a(2,_17, 5).) 2. Let F be an alternative A-algebra, in which the relation 2:: = 0 implies x = 0. Show that, for all x, y, z in F, x(yz)x = (xy) (zx) (Moufang’s identity). (In the identity

(402 +y(zx) = “M + (MK successively replace 3/ by xy and 2 by zx.) 3. Let F be an alternative Cayley A-algebra, in which the relation 2:: = 0 implies x = 0. Show that if x e F is invertible then, for all y, z, tin F,

(N(x) + N(y))(N(7-) + N0» = N(x2 + y?) + N0“ - (“Mr W (use Exercises 1 and 2).

654

HISTORICAL NOTE

(Chapters II and III)

(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 = 17: 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 conceming equations of the first degree had long been relegated to elementary teaching, 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 threeT and the rule offalse 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 ofA—ryabhata, the Arabs, Leonard of TCf. J. TROPFKE, Geschichte der Ekmentar-Mathematik, 1. Band, 2te Ausgabe,

Berlin-Leipzig (W. de Gruyter), 1921, pp. 150—155.

655

HISTORICAL NOTE ON CHAPTERS II AND III

Pisa and the countless “calculation books” of the Middle Ages and the Renaissance; but they never constituted more than a small part, for the use ofpractical 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. Historical 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 ([1], 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 algebraic 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 ox = b, 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 ofan 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 mathematicians 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 enunciates [4-(b)] the classification ofproblems into problems with a single solution, problems which reduce to an equation in two unknowns, an equation in three

656

HISTORICAL NOTE ON CHAPTERS II AND III

unknowns, etc. ; and he adds: the first consist ofdetermining 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 ofa surface appropriate to the question; here surfaces as loci have their genesis and similarlyfor the rest”, loo. 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 ofalgebra 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 transformation of coordinates in the plane and in space, which Fer-mat cannot have failed already to have perceived, is put in relief for example by Euler ([8 (11)], 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 (loo. oit., 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 geometrically 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 [10] and Bezout [l l] ; 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 considered 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 ofthe non-homogeneous equation is the sum of a particular solution and the general solution of the corresponding homogeneous 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 Ecole Polytechnique throws some light ([14],

657

HISTORICAL NOTE ON CHAPTERS H AND III

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) = O of a linear differential equation L( y) = 0, an example typical of duality by virtue of the relation

s(y) dx = fL*(z)ydx, valid for y and 2 zero at the extremities of the interval of integration; more precisely, and 30 years before Gauss defined explicitly the transpose ofa 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 I yz a'x). At the same time and again with Lagrange [9 (0)], 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, 3/), when x and y are given all inten values, does not change when a linear substitution with integral coefficients, of determinant l, 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 diflicult for us to appreciate, isolates the notion ofequivalence and that ofclass 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 ([l2(a)], p. 304). From this

moment onwards, the arithmetic study and the algebraic study of quadratic forms, in 2, 3 and later 21 variables, that of bilinear forms which are closely

related to them and more recently the generalization of these notions to an infinity ofvariables 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 Disquiritiones (cf. Historical Note to I), of the theory offinite conunutative groups, which occur there in four different ways, in the additive group of integers modulo m (for m an integer), in the multiplicative group of integers prime to m 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 already noted, it is clearly as commutative groups, or rather as modules over Z, that Gauss treats all these groups and studies their structure, their relations of isomorphism, etc. In the module of “complex integers” a + bi, he is later seen studying an infinite module over Z, whose isomorphism he no doubt perceived with the module of periods (discovered by him 1n the complex

domain) of elliptic fimctions; in any case this idea already appears neatly in Jacobi’5 work, for example in his famous proofofthe impossibility of a function with 3 periods and his views on the problem of inversion of Abelian integrals 658

HISTORICAL NOTE ON CHAPTERS H AND III

[l6(b)], to result soon in the theorems of Kronecker (cf. Historical Note to General Topology, 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 successors and soon under the name of the principle ofduality, 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 ofthe century), then at least Chasles did not fail [17] to perceive its relation with the notion of reciprocal spherical triangles, introduced into spherical trigonometry by Viete ([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 afl'ine 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 mathematicians of the first order; and it is diflicult 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 II AND III

equipollences” and taking its definitive form with Grassmann, M6bius and Hamilton; at the same time, under the name of “barycentric calculus”,

Mobius gives a version of it suitable for the needs ofprojective geometry [18]. 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 geometrically 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 prevented the Greeks from extending the notion of number to ratios of incom-

mensurable magnitudes. Hence the language and ideas relating to n-dimensional 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. 321), “without recourse to any metaphysical nation”);

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” 2' = V —l and perhaps thus open up vaster domains just as fertile as that ofthe complex numbers. Gauss himself was convinced ([l2(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’s'l' 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 1‘ Of. the interesting preface of his Lectures on quaternions [20] where he retraces the whole history of his discovery.

660

HISTORICAL NOTE ON CHAPTERS H AND III

field ofreal 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 oflater) that a matrix is only an abridged notation for a linear substitution, just as Gauss denoted the form (xx2 + 2bXY + c 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 everything 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 stiH 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 ofits 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 axiomatized) 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

dimV+dimW=dim(V+W) +dim(VnW) (lac. 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 ofall those already envisaged by Gauss which are generated by roots ofunity, then algebraic number fields and modules ofalgebraic 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 ofthe characteristic properties 661

HISTORICAL NOTE ON CHAPTERS II 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 discriminant) play an essential role at many points; and with Dedekind at any rate the methods are destined to become typically “hypercomplex”; Dedekind himselfmoreover, without setting himselfthe 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 ofideal, 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 ofmodules”) an equivalent notion, gives the first examples ofmodules over more general rings than Z; and in the work of the same authors, and then ‘ Hilbert, in particular cases the notion ofgroup 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 ofthe general principles on the solution of systems of linear equations, principles which due to the lack of the notion of rank, had escaped Jacobi/f 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 framework 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) coeflicients, which are also elucidated, in an obscure manual, with the minute care characteristic of him, by the famous author of Alice in Wonderland; as for Kronecker, he disdains to publish these results and leaves them to his

colleagues and disciples; the word “rank” itselfis 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 detenninants (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 1‘ Concerning the classification of systems of 1: equations in n unknowns when the

determinant is zero, he says ([16(a)], p. 370): “paullo prolixum In'detur negotium” (it could not be elucidated briefly).

662

HISTORICAL NOTE ON CHAPTERS II AND 111

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” ofwhich 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 “cogredien ” or “contragredient” variables, which is equivalent to the introduction of tensors; the latter becomes explicit and is popularized when, under the inspiration of the theory of invariants, Ricci and Levi-Civita, in 1900, introduce 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 Pfafl‘ 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 Poincaré [30] of the calculus of exterior differential forms. Poincaré introduces the latter, in order to form his integral

invariants, as the expressions which appear in multiple integrals, whilst Car-tan, 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 skewsymmetric 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 ofrepresentations ofthe linear group; and thus, for example, the substantial identity between the definition ofdeterminants 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 II AND III

generality theorems which were known only in special cases. Peano, one of the

creators of the axiomatic method and also one ofthe first mathematicians fully to appreciate the work ofGrassmann, 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, oflinear mappings 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 transposition 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 introducing (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 ofa 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 arithmetical 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 1' 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 ofthese 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 ofvector 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 noncommutative, 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. O. NEUGEBAUER, Vorlesungen fiber Geschichte der anti/cm Mathematilc, Vol. I:

Vorgriechische Mathematik, Berlin (Springer), 1934. 2. Euclidis Elementa, 5 vols., ed. J. L. Heiberg, Lipsiae (Tcubner), 1883—88. 2 his. T. L. HEATH, The thirteen book: Qf Euclid ’3 Elements. . ., 3 vols.,

Cambridge, 1908. 3. FRANCISCI VmTAE, Opera mathematica . . ., Lugduni Batavorum (Elzevir),

1646. 4. P. FERMAT, Oeuvres, vol. 1, Paris (Gauthier-Villars), 1891: (a) Ad locos

planos et solidos Isagoge (pp. 91—110; French transl. ibid., vol. III, p. 85); (b) Appendix ad methodum. . . (pp. 184—188; French transl., ibid.,_vol. III, p. 159). 5. R. DESGARTES, La geometric, Leyde (Jan Maire), 1637 (=0euvres, 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 éuénemens des rencontres d’un cone auec un plan (pp. 103—230). 7. G. W. LEIBNIZ, Mathematische Schnflen, ed. C. I. Gerhardt, vol. I, Bea

(Asher), 1849. 8. L. EULER: (a) Introductio in Analysin Infinitorum, vol. 2, Lausannae, 1748

(=Opera Omnia (1), vol. IX, Zfirich-Ieipzig—Berh'n (O. Fussli and B. G. Tcubner), 1945); (b) Imtitutianum Caleuli Integralis, vol. 2, Petropoli,

1769 (=0pera Omnia (1), vol. XII, Leipzig-Bea (B. G. Tcubner), 1914-). 9. J.-L. LAGRANGE, Oeuvres, Paris (Gauthier-Villas), 1867—1892: (a) Solu-

tions analytiques de quelques problémes sur les pyramides triangulaires, 665

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vol. III, pp. 661—692; ([1) Solution de difl‘érents problemes de calcul integral, vol. I, p. 471; (c) Recherches d’arithmétique, vol. III, pp. 695—795. . 10. G. CRAMER, Introduction d l’analyse des lignes caurbes, Geneva (Cramer and Philibert), 1750. 11. E. BEZOUT, The’orie ge’ne’rale des équatians algébriques, Paris, 1779.

12. C. F. GAUSS, Werke, Géttingen, 1870—1927: (a) Disquisitiones arithmeticae, vol. I; (b) Selbstanzeige zur Theoria residuorum biquadraticorum, Commentatio secunda, vol. II, pp. 169—178. 13. A.-L. CAUCHY, Mémoire sur les fonctions qui ne peuvent obtenir que dcux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu’elles renfennent, J. Ec. Polytech, cahier

17 (volume X), 1815, pp. 29—112 (=0euvres completes (2), vol. I, Paris (Gauthier-Villars), 1905, pp. 91—169). 14-. A.-L. CAUCHY, in Lepons de oaleul dzjférentiel et de calcul intégral, re’dige’es principalement d’aprés les me'thades de M. A.-L. Cauchy by Abbé Moigno, vol. II, Paris, 1844. 15. P. G. LEJEUNE-DIRICHLET, Werke, vol. 1, Berlin (G. Reimer), 1889,

pp. 619—644. 16. G. G. J. JACOBI, Gesammelte Werke, Berlin (G. Reimer), 1881—1891: (a) De formatione et proprietatibus determinantium, vol. III, pp. 355-392; (b) De functionibus duarum variabilium. . ., vol. II, pp. 25—50. 17. M. CHASLES, Aperpu historique sur l’origine et le développement a'es me'thodes en géamétrie . . ., Bruxelles, 1837. 18. A. F. Monms, Der bmyzmtrische Calcul . . ., Leipzig, 1827 (=Gesam1rwlte Werke, vol. I, Leipzig (Hirzel), 1885). 19. H. (3mm: (0) Die lineale Ausa'ehnungslelzre, ein neuer Zweig der Mathematik, dargestellt and dureh Anwendungen aaf die fibrigen Zweige a'er Mathematik, wie aueh aaf die Statik, Mechanik, die Lehre vom Magnetirmus and die

Kristallonomie erliz‘utert, Leipzig (Wigand), 1844- (=Gesammelte Werke, vol. I, lst Part, Leipzig (Teubner), 1894); (b) Die Ausdehnungslehre, vallstd’na’ig and in stronger Farm bearbez'tet, Berlin, 1862 (=Gesammelte Werke, vol. I, 2nd Part, Leipzig (Teubner), 1896). 20. W. R. HAMILTON, Lectures on quatemions, Dublin, 1353.

21. J. J. SYLVES'I’ER, Collected Mathematical Papers, vol. I, Cambridge, 1904: No. 25, Addition to the articles . . ., pp. 145—151 (=Phil. Mag., 1850). 22. A. CAYLEY, Collected Mathematical Papers, Cambridge, 1889—1898: (a) Sur quelques theoremes de la géométrie de position, vol. I, pp. 317—328 (=Crelle’s J., vol. XXXI (1846), pp. 213—227) ; (b) A memoir on the theory ofmatrices, vol. II, pp. 475—496 (=Phil. Trans., 1858). 23. K. WEIERSTRASS, Mathematische Werke, vol. II, Berlin, (Mayer und Miller),

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1895: Zur Theorie des aus n Haupteinheiten gebildeten complexen Grossen, pp. 311—332. 24. R. DEDEKIND, Gesammelte mathematische 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., 1861). 26. L. KRONEGKER, Vorlesungen iiber die Marie der Determinanten . . ., Leipzig (Teubner), 1903. 27. G. PEANO, Calcalo geometrica secondo l ’Ausdelmungstheorie di Grassmann preceduta dalle operazioni della logica deduttiva, Torino, 1888. 28. G. RICCI and T. LEVI-CIVITA, Méthodes dc calcul difl‘érentiel absolu ct leurs applications, Math. Ann., vol. LIV (1901), p. 125. 29. E. CARTAN, Sur certaines expressions difi'érentielles et le probléme dc Pfafi', Ann. E.N.S. (3), vol. XVI (1899), pp. 239—332 (= Oeuvres cample‘tes, vol. III, Paris (Gauthier-Villars), 1953, pp. 303—396). 30. H. POINcARfi, Les méthades nauvelle: de la me’canique oéleste, vol. III, Paris

(Gautlfier-Villars), 1899, Chapter XXII. 31. O. TOEPLITZ, Ueber die Auflosung unendlichvieler linearer Glcichungen mit unendlichvielen Unbekannten, Rend. Circ. Mat. Pal., vol. XXVIII (1909), pp. 88—96.

667

INDEX OF NOTATION

at +31, x.y,xy,xTy,x_Ly: I,§l, no. 1. X T Y, X + Y, XY (X, Y subsets): I, § 1, no.1.

X T a, a T X (X a subset, a an element): 1, § 1, no. 1. “IA-’5‘, Ix“, T35” “41‘ x, 4—5:“; J—xm u; xxx, 2x” 2"” Ex” 1;]:a

Hx,: I, §1, no. 2. “T‘qx x,,‘T an: I, §l, no. 2. x,T x,” T- --T x4: I, §1, no. 3. I". x, I. x", 71:: (neN):I, §I, no. 3. T x“, T x“: I, § 1, no. 5.

0 B a ring homomorphism): II, §5, no. 1. 9* (u), um, (p: A ——> B a ring homomorphism, u an A-module homomorphism): II,§5, no.1. dimK E, dim E, [E: K] (E a vector K-space): II, § 7, no. 2.

diInA E, dim E (E an A-module any two bases of which are equipotent): II, § 7, no. 2. codimEF, codim F (F a vector subspace of a vector space E): II, § 7, no. 3. rg(u) (u a linear mapping of vector spaces): II, § 7, no. 4.

rg(u) (u an element of a tensor product of vector spaces): II, § 7, no. 8. dimKE, dim E (E an affine space over a field K): II, §9, no. 1. a + t, t + a (a a point, t a translation ofan affine space): II, §9, no.1. 672

INDEX OF NOTATION

b — a (a, b points of an affine space): II, § 9, no. 1. 02:1,)“, ((x,)“,1 a family ofpoints of an afline space, (A), e, a family of scalars,

offinite support, such that Z A, = l or Z A, = 0): II, §9, no. 1. P(V), A(V) (V a vector space): II, §9, no. 5. Pn(K), An(K) (K a field): II, §9, no. 5. dimK P(V), dim P(V) (V a vector K-space): II, § 9, no. 5. K, 00 (K afield): II, §9, no. 9. PGL(V), PGLn(K), PGL(n, K) (K a field, V a vector space) : II, § 9, no. 10.

'M (Ma matrix): II, § 10, no. 1. M’ + M ' (M ’, M ' matrices over a commutative group): II, § 10, no. 2. f(M', M”), M'M' (M’, M' matrices): II, § 10, no. 2.

Eu (matrix units): 11, § 10, no. 3. 6(M), M" (M a matrix, 0- a ring homomorphism): II, § 10, no. 3. M (x), x (x an element of a finitely generated free module): II, § 10, no. 4. M (u) (u a homomorphism of a free module into a free module): II, § 10, no. 4. M (x), M (u) (matrices relative to decompositions as direct sums): II, § 10, no. 5.

M (u) (u a semi-linear mapping): II, § 10, no. 6. Mn(A), In 1,, (A a ring): II, § 10, no. 7. GLn(A), GL(n, A) (A a ring): II, § 10, no. 7. ‘X ‘1 (X an invertible square matrix): II, § 10, no. 7. diag(al)lela diag(a1! (13, - ' '9 an): 112 § 109 no. 7-

X1 ® X2 (X1, X2 matrices over a commutative ring): II, § 10, no. 10.

Tr(X) (X a square matrix over a commutative ring): II, § 10, no. 11. rg(X) (X a matrix over a field): II, § 10, no. 12. deg(x) (x an element of a graded group): II, § 11, no. 1. M00) (M a graded module, A0 an element of the monoid of degrees): II, § 11, no. 2. HomgrA(M, N) (M, N graded modules over a graded ring A): II, § 11, no. 6. EngrA(M), M*" (M a graded module): II, § 11, no. 6. M:N (M. N modules): II, § 1, Exercise 24. [Z 1;] (a, b, c, d points on a projective line): II, § 9, Exercise 11.

SL(E) (E a vector space): II, § 10, Exercise 12. PSL(E) (E a vector space): II, § 10, Exercise 14. x.y, xy (multiplication in an algebra): III, § 1, no. 1. E0 (E an algebra): III, § 1, no. 1. Homkm, (E, F) (E, F A-algebras): III, § 1, no. 1. El!) (b a two-sided ideal of an algebra E): III, § 1, no. 2. E (E an algebra): III, § 1, no. 2. 1),, Vin: 71 (E an algebra, c a unit): III, § 1, no. 3.

673

INDEX OF NOTATION

T(u), N(u): III, §2, no. 3. 12: III, §2, no. 4-. H: III, §2, no. 5.

LibA(I), LibasA(I), LibascA(I): III, §2, no. 7. U((x,),fl): III, §2, no. 8.

A[(xc)1exJ. Alida“: 111. §2, nO- 9-

A[(X,),e,]: III, §2, no. 9. A[X,, . . ., Xn]: III, §2, no. 9. A[X], A[X, Y], A[X, Y, Z]: III, § 2, no. 9. X" (v a multiindex): III, §2, no. 9. Z «5,19,, 2: 2,.5 (3 elements of a monoid): III, § 2, no. 10. A[[X,]],.1: III, §2, no. 11.

2 am: III, § 2, no. 11. w(u), 0301) (u a formal power series): III, § 2, no. 11.

@3313. E. o. E2 09- ~69. E... E. ® E. ®- ~® E. (E. A-algebras): 111, §4, no. 1. E0” (E an algebra): III, §4, no. 1. g E, (I an infinite set, E, algebras): III, §4, no. 5. ® an g x. (u, algebra homomorphism, x, elements, I infinite): III, § 4, no. 5.

tel

28; E‘, 28; fl, “G9“ (E,, G graded algebras, f, graded algebra homomorphisms, e a system of commutation factors): III, §4, no. 7.

‘§ 11., E' (3., F, scan, 2% fi,f. =®f,, “f”: 111, §4, no. 7. ®" M, T“(M), Tens"(M), T’A(M), T(M), Tens(M),TA (M) (M anA-module): III, §5, no. 1. T(u), T”(u) (u a linear mapping): III, §5, no. 2.

T§(M), T:(M) (M a module): III, §5, no. 6. ' «5(2), 4: III, § 5, no. 6. «5;? III, § 5, no. 6. of: III, §5, no. 6. S(M), SA(M), Sym(M): III, §6, no 1. 3’, 3h, 5,2: III, §6, no. 1. S"(M), S"(u), S(u) (M a module, u a linear mapping): III, § 6, nos. 1 and 2. 8.2: III, §6, no. 3. a“ (at a multiindex): III, §6, no. 6.

/\(M), /\..(M), Alt(M): 111, § 7, no. 1. 3”, 837‘, 8;: III, § 7, no. 1.

674

INDEX OF NOTATION

A"(M): III, § 7, no. 1. u A 152:, A x; A-uA x,:III,§7,no. l.

A(u), A"(u) (u a linear mapping): III, § 7, no. 2. xx (H a subset of[l, n]): III, § 7, no. 3. u (x1, . . ., :2), . . .,x,,): III, § 7, no. 4.

0.2: III, § 7, no. 4. A1,(M), A;(M): III, §7, no. 4. 4,: III, §7, no. 8. aux: III, § 7, no. 8.

det(u) (u an endomorphism): III, §8, no. 1. det(x1, x2, . . ., x") (x, vectors in an n-dimensional free A-module): III, §8, no. 1.

det(X) (X a matrix): III, §8, no. 3. det(gu)1