An Introduction to Homological Algebra

Generalities concerning modules 1.1 Left modules and right modules 11.2 Submodules 31.3 Factor modules 31.4 A-homomorphisms 31.5 Some different types of A-homomorphisms 41.6 Induced mapp

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AN INTRODUCTION TO

HOMOLOGICAL

ALGEBRA

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Cambridge Books Online © Cambridge University Press, 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 15:15:00 BST 2012.

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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www Cambridge org Information on this title: www.cambridge.org/9780521058414

This publication is in copyright Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 1960 Reprinted 1962 This digitally printed version 2008

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-05841-4 hardback ISBN 978-0-521-09793-2 paperback

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Preface page ix

1 Generalities concerning modules

1.1 Left modules and right modules 11.2 Submodules 31.3 Factor modules 31.4 A-homomorphisms 31.5 Some different types of A-homomorphisms 41.6 Induced mappings 51.7 Images and kernels 61.8 Modules generated by subsets 71.9 Direct products and direct sums 91.10 Abbreviated notations 121.11 Sequences of A-homomorphisms 13

2 Tensor products and groups of homomorphisms

2.1 The definition of tensor products 162.2 Tensor products over commutative rings 172.3 Continuation of the general discussion 182.4 Tensor products of homomorphisms 192.5 The principal properties of HomA (B, G) 24

3 Categories and functors

3.1 Abstract mappings 303.2 Categories 313.3 Additive and A-categories 32

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Vi C O N T E N T S

3.4 Equivalences page 32 3.5 The categories &% and &% 33

3.6 Functors of a single variable 333.7 Functors of several variables 343.8 Natural transformations of functors 353.9 Functors of modules 363.10 Exact functors 383.11 Left exact and right exact functors 403.12 Properties of right exact functors 41

3.13 A ® A G and HomA {B, C) as functors 44

4 Homology functors

4.1 Diagrams over a ring 46

4.2 Translations of diagrams 474.3 Images and kernels as functors 484.4 Homology functors 524.5 The connecting homomorphism 544.6 Complexes 594.7 Homotopic translations 62

5 Projective and injective modules

5.1 Projective modules 635.2 Injective modules 675.3 An existence theorem for injective modules 715.4 Complexes over a module 755.5 Properties of resolutions of modules 7 75.6 Properties of resolutions of sequences 805.7 Further results on resolutions of sequences 84

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CONTENTS Vii

6 Derived functors

6.1 Functors of complexes page 90

6.2 Functors of two complexes 946.3 Right-derived functors 996.4 Left-derived functors 1096.5 Connected sequences of functors 113

7 Torsion and extension functors

7.1 Torsion functors 121

7.2 Basic properties of torsion functors 1237.3 Extension functors 1287.4 Basic properties of extension functors 1307.5 The homological dimension of a module 1347.6 Global dimension 1387.7 Noetherian rings 1447.8 Commutative Noetherian rings 1487.9 Global dimension of Noetherian rings 149

8 Some useful identities

8.1 Bimodules 1558.2 General principles 1568.3 The associative law for tensor products 1608.4 Tensor products over commutative rings 1618.5 Mixed identities 1648.6 Rings and modules of fractions 167

9 Commutative Noetherian rings of finite global dimension

9.1 Some special cases 1749.2 Reduction of the general problem 1849.3 Modules over local rings 189

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Viii CONTENTS

9.4 Some auxiliary results page 202

9.5 Homological codimension 2049.6 Modules of finite homological dimension 205

10 Homology and cohomology theories of groups and

monoids

10.1 General remarks concerning monoids and groups 21110.2 Modules with respect to monoids and groups 21410.3 Monoid-rings and group-rings 215

10.5 Axioms for the homology theory of monoids 21910.6 Axioms for the cohomology theory of monoids 221

10.7 Standard resolutions of Z 223

10.8 The first homology group 22910.9 The first cohomology group 23010.10 The second cohomology group 23810.11 Homology and cohomology in special cases 24410.12 Finite groups 24910.13 The norm of a homomorphism 25210.14 Properties of the complete derived sequence 256

10.15 Complete free resolutions of Z 259

Notes 266 References 278 Index 281

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Cambridge Books Onlinehttp://ebooks.cambridge.org

An Introduction to Homological Algebra

Northcott Book DOI: http://dx.doi.org/10.1017/CBO9780511565915

Online ISBN: 9780511565915 Hardback ISBN: 9780521058414

Paperback ISBN: 9780521097932

Chapter Preface pp ix-xii Chapter DOI: http://dx.doi.org/10.1017/CBO9780511565915.001

Cambridge University Press

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of pure mathematics has been generally recognized.

The young mathematician, about to start on research, will beanxious to learn about homological ideas and methods, and one ofthe aims of this book is to help him to get started In trying to caterfor his needs, I have imagined such a reader as being familiar with thenotions of group, ring and field but still relatively inexperienced inmodern algebra For him, the account given here is self-containedsave in a small number of particulars which are mentioned below, andwhich need not discourage him

An introduction to homological algebra must, of necessity, be anintroduction to the book of Cartan and Eilenberg, for the student whowishes to go further will need to read their work; but much of greatinterest and value has been achieved even more recently, and some ofthis later work has been given a place in the following pages The list

of contents gives a fairly detailed picture of the main topics treated,but a few additional comments may be a help

Chapters 1-6 develop, in a leisurely manner, the results that areneeded to establish and illustrate the theory of derived functors, afterwhich follows an account of torsion and extension functors These arethe most important ones which are obtainable by the process ofderivation and, in a sense, the remainder of the book is concerned withtheir applications Such an application is the theory of global dimen-sion given at the end of Chapter 7, and here are included some im-portant results of M Auslander on Noetherian rings that havepreviously been available only in the original research paper

Chapter 9 deals with the structure of commutative Noetherian rings

t H Cartan and S Eilenberg, Homological Algebra (Princeton University Press,

1956).

http://dx.doi.org/10.1017/CBO9780511565915.001

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X PREFACE

of finite global dimension and represents one of the most satisfyingachievements of homological methods This, too, appears in a text-book for the first time Here, it must be admitted, the account is notcompletely self-contained, but considerable care has been taken inexplaining the results of Ideal Theory which are needed to supplementthe purely homological arguments This is the most ambitious chapter,and the author hopes that it will help to stimulate interest in com-mutative algebra The treatment given here was found successful in acourse of lectures in which the audience had no specialized knowledge

of classical Ideal Theory

Chapter 10 is an introduction to the homology and cohomologytheories of monoids and groups This, by itself, has a considerableliterature and was one of the earliest branches of our subject to bedeveloped The chapter can be read, if desired, before Chapter 9 anddoes not require any specialized knowledge of Group Theory, f Indeciding how far to go with this topic, I had in mind the student whomight wish to acquire some general background before proceeding tothe applications in some specialized field such as Class Field Theory.Nearly all the topics covered in the following pages were included

in a course of lectures given at Sheffield University When lecturing,

it is possible to digress at some length in order to explain the generalplan of development and the connexions with other branches ofmathematics Also one likes to mention important results connectedwith what one is discussing even if there is no time for a full treatment.Some of this supplementary material, which I hope will add to theenjoyment and interest of the main text, will be found in the Noteswhich follow Chapter 10

The final chapter has been much improved as the result of tions of J Tate with whom I had an opportunity of discussing it AtSheffield, I have been aided, at all stages, by my colleague H K.Farahat Of particular value has been his willingness to discuss points

sugges-of detail and to make helpful criticisms This work owes a great deal

to his continued interest I am also indebted to Sir William Hodge,who, when I first had the idea of writing an introduction to homo-logical methods, encouraged me to go ahead

f There is actually one reference to a result proved in Chapter 9, but there is no difficulty in taking this out of context.

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PREFACE Xi

Writing a book takes up much time and energy, and this one couldnever have been completed without the generous help of J J Kielywho typed the first draft from notes taken at lectures I am also greatlyindebted to my secretary, Mrs M Ludbrook, for the great care andpatience with which she cut innumerable exquisite stencils To both

of these I wish to express my thanks Their strenuous efforts made itunthinkable not to finish a work to which they had contributed somuch

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NOTE ON CROSS-REFERENCES

If the reader is referred to a result, say to Theorem 7,and no chapter or section is specified, then he is tounderstand that the reference is to Theorem 7 of thechapter in which he is reading When a reference ismade in one chapter to a result which occurs in another,the section in which it will be found is also given ThusLemma 4 of section (5.3) means the fourth lemma inChapter 5, and this will be found in the third sub-division of the chapter

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Paperback ISBN: 9780521097932

Chapter

1 - Generalities concerning modules pp 1-15

Chapter DOI: http://dx.doi.org/10.1017/CBO9780511565915.002

Cambridge University Press

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1 GENERALITIES CONCERNING

MODULES

Notation A denotes a ring, which is not necessarily commutative,

with an identity element 1

1.1 Left modules and right modules

The notion of a ring-module has, in recent years, come to be regarded

as one of the most important in modern algebra, and the theory ofring-modules is so extensive that it includes, for example, that ofvector spaces, ideals, algebras and group representations But, inspite of the fact that this concept covers so many widely differingstructures, there exists an elaborate and rich theory common to themall Of this theory, homological algebra forms an important part just

as, in topology, homology theory is a valuable system of results which

is valid for many different kinds of space In special situations onecan hope to extend, in certain particulars, such a universal body ofknowledge, and in this way arises the possibility of making usefulapplications On the present occasion, however, these are reservedfor the later sections of the book, and it is with very broadly basedideas that we shall be concerned for some time

For the reader's convenience, we begin with the idea of a module

on which the elements of a given ring A act as operators, it being posed that he is familiar with the concept of a ring and also of a group.Then we shall give an account of the more elementary notions whicharise out of the definition Probably the reader will already be familiarwith much that is said in the first chapter, but even so it will repayhim to glance through it because the opportunity is taken to pre-pare the ground for the introduction of new ideas in later chapters.Also, in section (1.10), we describe a slightly unusual kind ofnotation which, it is hoped, will make it easier to follow some of theproofs

sup-We come now to the first definition Let M be an additive abelian group, then M is called a left A-module if, for each element xoiM and

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2 GENERALITIES CONCERNING MODULES

each element A of A, there is defined a 'product' Xx which belongs

to M and satisfies the following axioms:

(i) \{x x + x 2 ) = Xx x + Xx 2 , (ii) (X ± + A2) x = X x x + A2 x, (iii) X 1 {X 2 x) = {X 1 X 2 )x, (iv) Ix = x.

Of course, in the above, x, x l9 x 2 are arbitrary elements of M and

A, A1? A2 may vary freely in A while, in (iv), 1 denotes the identityelement of A

Right A-modules are defined similarly except that the product is written xX and the corresponding axioms are:

(i)' (x x + x 2 ) A = x x A + x 2 A,

(ii)' x(X ± + A2) = XX-L + xX 29

(iii)' (%X 1 )X 2 = x(X 1 X 2 ), (iv)' xl = x.

Suppose, for the moment, that A is a commutative ring and that M

is a left A-module, then we can turn M into a right A-module simply

by putting xX = Xx Conversely, every right A-module can be

re-garded as a left A-module Thus all modules over commutative ringsare virtually two-sided and the distinction between left and rightdisappears

In future, unless otherwise stated, we shall understand by a

A-module a left A-module However, our definitions and results will

also be applicable to right modules with the appropriate formalchanges A A-module which comprises only the zero element will bedenoted by 0

Let M be an additive abelian group, let x be an element of M, and let k be an integer Then hx has a well-defined meaning Also, with an

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Let N be a submodule of the A-module Jf, then, in particular, it is

a subgroup of M and therefore the cosets of N in M form the abelian group MjN Further, when x ± and x 2 belong to the same coset, then

x i ~~ X 2 *s a n element of N and so Xx ± — Xx 2 = X(x x — x 2 ) is also a member

of N If x is an element of M let us write x for the coset to which it

be-longs, then, by virtue of the above remark, we can define a 'product'

A#, where A e A, by writing Xx = Xx If this is done then MjN becomes

a A-module called the factor (or residue) module of Jf modulo N The mapping x->x, which carries each element into the coset to which it belongs, is called the natural mapping of Jf on to MJN, and when this

natural mapping occurs in a diagram it is sometimes convenient todraw attention to it by writing

nat

M • MjN.

1.4 A-homomorphisms

L e t / : M-*N be a mapping of the A-module Jf into the A-module N.

We say t h a t / i s A-linear or t h a t / i s a A-homomorphism if

where x v x 2 are arbitrary elements of Jf and A is any element of A

Remarks, (a) If N is a submodule of Jf then the 'inclusion map'

N->M, in which each element of N is mapped into itself, is a

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4 G E N E R A L I T I E S C O N C E R N I N G M O D U L E S

so that f(x) belongs to N Then / i s a mapping of M into N and it can

easily be verified t h a t / i s a A-homomorphism This particular

homo-morphism is written f x +/2 and, in using this notation, we have defined

'addition' for homomorphisms of M into N It is now a

straight-forward matter to verify that this set of homomorphisms, which isdenoted by HomA(Jf,iV), forms an abelian group This group willreceive a great deal of attention later

Now suppose, for the moment, that A is commutative and let /belong to HomA (M, N) For each element x of M write g(x) = A/(#), where A is some fixed element of A, then, because A is commutative,

g also belongs to HomA (M, N) The homomorphism g is denoted by

A/ and, with this definition of A/, HomA (M, N) becomes a A-module.

To summarize, we may say that in the general (non-commutative) case

Hom A (M, N) is an additive group, but, when A is commutative, we may,

if we wish, endow it with the structure of a A-module.

(e) Let /, f l9 f 2 be A-homomorphisms M ->N and let g, g l9 g 2 be

A-homomorphisms N-+L Then

(i) g{fi+h) = g

(iii) if A is a commutative ring, then (Xg)f = g(Xf) = \{gf),

where A is an arbitrary element of A

1.5 Some different types of A-homomorphisms

L e t / : M->N be a A-homomorphism.

Definition If f(x) =¥f(y) whenever x 4= y, then / is called a

mono-morphism.

Definition If/ maps M on to N, t h e n / i s called an epimorphism.

Definition If/ is both a monomorphism and an epimorphism, then

it is said to be an isomorphism and we write / : M&N In this case

the inverse mapping/-1 is an isomorphism N&M a n d / , / "1 are called

inverse isomorphisms.

It is well worth noting that the A-homomorphisms/ : M-+N and

g : N->M are inverse isomorphisms if and only if both gf andfg are

identity maps.f

f The identity map of a set maps the set on to itself and leaves each element fixed.

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INDUCED MAPPINGS 5

1.6 Induced mappings

By a pair (M f , M) will be meant a A-module M together with a module M', and by a homomorphismf: (M', M) -> (N f , N) of pairs will

sub-be meant a A-homomorphism/: M->N for which f(M') c; iV', where

/(If') denotes the image of Jf' Suppose that we have this situation

and that x l9 x 2 are elements of M which belong to the same coset of M'.

1 1 1 6 1 1 /(*l) -/(X,) = /(*! - *,) € f(M') £ tf',

and so/(x^ a,ndf(x 2 ) belong to the same coset of N' in N This

which is easily verified to be a A-homomorphism The m a p / * is called

the induced map, and it is characterized by the property that it makes the diagram *

We have just referred to the idea of a commutative diagram, but this

is a concept which requires some explanatory comment If

A >B

t

is a square diagram of modules and homomorphisms and we say that

it is commutative, then we mean that g<f> and af coincide Similarly,

the triangular diagram u

-rr- ^ T^

w\ / v

z

is commutative when vu = w Sometimes we have to deal with more

complicated figures such as

(h yh

A > B > G

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but they will always be composed, in a simple way, out of squares and triangles Such a diagram is said to be commutative when the

small component squares and triangles have this property, so that (for example) this will be the case in (1.6.1) when both erf '= g(J> and

rg = hi/r After this remark, the intention should be clear in all cases

which present themselves, though the reader should observe that we

do not give a definition of a commutative diagram that will apply to completely general situations.

Let us return to the consideration of induced mappings Let

/ : M ->N be a homomorphism which maps a submodule M' of M into the zero element of N so that (M', M) -> (0, N) is a map of pairs.

In this case the induced map/* maps MjM' into N and is

character-ized by the fact that the diagram

1.7 Images and kernels

Let/ : M-+N be a A-homomorphism and let us write

Im (f)=f(M),

Ker(/)=/-i(0),

so that Im (/), which is called the image off, consists of all elements of the form/(#), where xeM, while Ker(/), the so-called kernel off, is

made up of all elements that are mapped into zero In addition, we

define the coimage off and the cokernel off by means of the formulae

Coim(/) = ilf/Ker(/), Coker(/) = 2V7Im(/).

Let us observe that / is a monomorphism if and only if Ker (/) = 0, while for / to be an epimorphism we require that Coker (/) = 0.

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IMAGES AND KERNELS 7

Accordingly, / is an isomorphism when and only when Ker (/) andCoker (/) are both null modules

Theorem 1 Let f: M-+N be an epimorphism Then the induced

map f* : Jf/Ker (f)->N is an isomorphism.

Proof Since/is an epimorphism so is / * ; hence it remains to be

shown t h a t / * is a monomorphism, i.e that Ker (/*) = 0 Let x be an element of M and let x be the coset of x modulo Ker (/) Then x is an entirely general element of ikf/Ker (/) If now x belongs to Ker (/*) then 0 =f*(x) —f(x) so that x is an element of Ker (/) and therefore

x — 0 Accordingly, Ker (/*) = 0 and the theorem follows.

Since iHf -> Im (/) is an epimorphism with kernel Ker (/), the theoremshows that there is an isomorphism

For this reason Coim (/) does not often appear However, in certainspecial situations the two concepts play genuinely different roles

1.8 Modules generated by subsets

Let M be a A-module and \u^\ ieI a family of elements of M, the system / of parameters being arbitrary The subset of M, consisting of all

elements which can be written in the form

i

where each A^ is an element of A and A^ = 0 for almost all i (that is, A^ = 0 for all i with at most a finite number of exceptions), forms a submodule of M This submodule is called the submodule of M generated by [%li€j If this submodule happens to coincide with M itself then [u i ] ieI is called a system of generators of M.

Let [u i ] i€l be a given system of generators of M If now, for each element x of M, the A^ for which

x —

are uniquely determined, then [%]^€j is called a base of M A module

which admits a base is called /reef

Let F be a free A-module with base [u i ] ieI , let N be a A-module, and let [v i ] i£l be a family of elements of N indexed with the same

system / of parameters Then there always exists a unique

A-homo-morphism / : F-+N such that

t A module, which consists only of a zero element, is to be regarded as a free

module with an empty base.

2-2

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for all i of / Indeed, / is defined by

Let [WiJtejr be a family of symbols Consider the set of all formal sums 2 ^i^i, where each A^ is an element of A and A^ is zero for almost all i For such formal sums we define addition and multiplication (by

elements of A) in the obvious manner If this is done the result is a

A-module Let us identify Wj with the formal sum J^S^w^ where

Sy = 0 if i =£j and 8^—1 Each element of the module has then a unique

representation in the form 2 A ^ , hence the module is free and has

i

l w i1iei a s a base This module is called the free module generated by the symbols [w 4 ] i€l

Theorem 2 Given any A-module M there exists a free module F with

an epimorphism F->M If M can be generated by m elements (0 ^ m < oo) then F can be chosen with a base of m elements.

Proof Let [%k€Z be a system of generators of M,f let [ ^ ]l € / be a

similarly indexed system of distinct symbols, let F be the free module

generated by [vjf€j, and l e t / : F-^M be the A-homomorphism for which f(v 4 ) = u t Then/has the properties required by the theorem.

Theorem 3 Let F be a free A-module, p : F-^N a A-homomorphism,

and q : M->N an epimorphism of A-modules Then it is possible to find a A-homomorphism <j> : F->M such that the diagram

f Note that we can certainly find a system of generators of M Indeed, the set of

all the elements of M forms such a system.

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DIRECT PRODUCTS AND DIRECT SUMS 91.9 Direct products and direct sums

The notions of a direct sum of modules and a direct product of modules,which we discuss in the present section, are of fundamental importance

for our theory Let [M^^j be a family of A-modules, the set / of

parameters being quite arbitrary We consider the families [ m ji € /

where, for each i, m i is an element of M t For such families we define

addition and multiplication (by elements of A) by means of the sponding operations on individual components This produces aA-module, which is written n -M* a n (i called the direct product of the

Here pj maps an element mj into that element of JJ M i whose jth

component is m;- and whose remaining components are zero On the

other hand, q$ maps an element of n - ^ into its jth component These mappings have the property that q i p i = identity and q t pj = 0 if i 4= j

For the direct sum we have similar canonical mappings

Mj-l^M^Mj (jel), which not only satisfy g i f i = identity and g^f i = 0 for i+j 9 but forwhich we have in addition

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into 2/i(^)« If the homomorphism ^ A M i ->M is an isomorphism

such that g i f i = identity and g j f i = O when i + j (1.9.3)

In fact if [m i ] ieI belongs to 2 J^, then the mapping g i will carry

for all elements m of M.

Next let [iV^]i€ j be a family of A-modules and suppose that we have

in which an element n of N is mapped into [qi(n)] i€l If now the

A-homomorphism i V ^ n ^ is an isomorphism, then we say that

i

(1.9.6) is a protective representation of N as a direct product In such a

situation there exist uniquely determined A-homomorphisms

Pi-.N^N (iel) (1.9.7)

with the properties that

q i p i = identity, q i p i = 0 when i=t=J (1.9.8)

Pi Qi

We say then that J^->JV->^ (iel) (1.9.9)

is a complete representation of N as a direct product of A-modules Again let Y be a A-module and let [Y i ] ieI be a family of submodules

If each element y of Y has a unique representation in the form

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DIRECT PRODUCTS AND DIRECT SUMS 11

where y i is an element of Y i and y i = 0 for almost all i, then we say that

Y is the internal direct sum of the submodules [j^]i€/ There then exists

a canonical isomorphism 2 Y i « Y, between Y and the external direct

Theorem 4 Let M i XM Q XM i (iel) (1.9.11)

be a system of A-modules and A-homomorphisms which satisfy

9ih = identity and gjfi = O if i+j.

Then (1.9.11) is a complete representation of M as a direct sum if and only if we have

Proof Suppose that (1.9.12) holds and consider the homomorphism

in which [m i ] ieI is mapped into 2 / i (mi ) - We shall show

representa-Now suppose that / is finite Let [m i ] ieI be an arbitrary element of

II M t , then [ m ji € l is the image of S/i(^i) m the mapping M -> H M i

i i i

determined by the g t Thus, in any event, M -> n M i is an epimorphism

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Next assume that if-> Yl M i is an isomorphism, that is to say, that

i

(1.9.11) is a direct product representation Then, since

we see that m and 2 / * 9i(^) &re elements of M with the same image in

i

n i ^ Accordingly m = 2 / i 9i( m )> which means that (1.9.12) holds.

i i

Finally, assume that (1.9.12) holds To complete the proof, we need

only show that M -> ]J M i is a monomorphism, that is to say we must

i

show that if g^m) = 0 for all i, then m = 0 But this is obvious

because

Definition Let 5 be a submodule of a A-module if If M is the

internal direct sum of two submodules, one of which is B, then we say that B is a direct summand of Jf.

Definition A monomorphism L -> Jf is called eKratf if Im ( i -> ilf) is

a direct summand of Jf An epimorphism i f ->N is called direct if its kernel is a direct summand of M In this case we say that N is a direct factor of M.

1.10 Abbreviated notations

Suppose that we have a A-homomorphism / : M->N If there is no other homomorphism of i f into N under consideration it is sometimes convenient to denote the homomorphism itself by MN We call this the abbreviated notation for homomorphisms In this notation the composition of two homomorphisms, say L->M and M-+N, is written as (LM) (MN) or LMN For example, the statement that the

Proposition 1 A monomorphism L->M is direct if and only if there

is a A-homomorphism M->L such that LML = identity An phism M->N is direct if and only if there is a A-homomorphism N ->M such that NMN = identity.

epimor-Cambridge Books Online © epimor-Cambridge University Press, 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 15:31:46 BST 2012.

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ABBREVIATED NOTATIONS 13

Proof We shall prove the first part of the proposition, the proof

of the second part being similar Assume first that L -> M is direct Put A — Im (LM), then there exists a submodule B of M such that

M = A+B (direct sum) The monomorphism L^M determines an isomorphism of L on to A or, in other words, LMA is an isomorphism, where M ->A is the canonical mapping corresponding to the decomposition M = A+B Let AL be the inverse isomorphism and put

ML = MAL Then

LML = (LM) (ML) = (LM) (MAL) = (LMA) (AL) = identity Conversely, suppose that we have a A-homomorphism ML such that LML = identity Put A = Im (LM) and B = Ker (ML) We shall show that M = A+B (direct sum) Let m be an element of M then mMLM = a, where a is an element of A Also

(m-a)ML = mML-mMLML = mML-mML = 0.

Thus m — a = b, where b is an element of B This shows that every element of M can be written in the form a + b, where a is an element of

A and b is an element of B Now suppose that a' + V = 0, where a' is

an element of A and b' is an element of B We shall show that a' = V = 0, thereby proving that M = A+B (direct sum) To do this

we observe that 0 = (a! + b')ML = a'ML But a' = ILM for a suitable element I of L, consequently 0 = a'ML = ILML = Z, and therefore

a 1 = ILM = 0 Since this implies that b' — 0 the proof is complete.

Corollary Any epimorphism M->F, where F is free, is direct.

Proof Let FF denote the identity map of F By Theorem 3 it is

possible to find a A-homomorphism F->M such that the diagram

is commutative Then FMF = FF = identity and now the corollary

follows from the theorem

1.11 Sequences of A-homomorphisms

Let L-+M-+N (1.11.1)

be a three-term sequence of A-modules and A-homomorphisms We

shall say that (1.11.1) is a Q-sequence if LMN = 0, that is, if

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Again, we shall say that (1.11.1) is exact if Im (LM) = Ker (MN) As important examples let us note that L -> M is a monomorphism if and only if 0 -> L -> ilf is exact, while M-+N is an epimorphism if and only

if M->N-+0 is exact.

More generally, a sequence

which may be finite, infinite, or semi-infinite, will be called a O-sequence

if every triplet L r ->L r+1 ->L r+2 is a O-sequence, and it will be called

exact if every triplet is exact Let us observe that if L is a submodule

of M then the canonical sequence

is exact Particularly important are exact sequences of the slightly

Definition An exact sequence (1.11.2) is said to be direct or to split

if Im (LM) = Ker (MN) is a direct summand of M.

For example, if A +B is the direct sum of A and B, then the

canon-ical exact sequence 0 ^ A

splits Or again, any exact sequence of the type

where F is free, splits This follows from the corollary to Proposition 1.

Proposition 2 Let 0->L-*M->N->0 (1.11.3)

be a given sequence of A-modules and A-homomorphisms Then in order that (1.11.3) should be a split exact sequence it is necessary and sufficient that there should exist A-homomorphisms M->L and N->M such that LML = identity, NMN = identity, MLM + MNM = identity,]

LMN = 0, NML = 0 j

(1.11.4)

Remark I t should be observed that, by Theorem 4, (1.11.4) is

equivalent to the statement that

L-*M^L, N->M->N

is a complete representation of M as a direct sum.

Proof Assume that (1.11.3) is a split exact sequence Then

Im (LM) = Ker (MN) = A (say)

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SEQUENCES OF A-HOMOMORPHISMS 15

and M = A + B (direct sum) for a suitable submodule B of M Let

MA and BM be the canonical homomorphisms associated with this direct decomposition, then LMA and BMN are isomorphisms Denote

by AL and NB the inverse isomorphisms and put ML — MAL,

Im (LM) c Ker (MN) Next let m be an element of Ker (MN), then

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TENSOR PRODUCTS AND GROUPS

OF HOMOMORPHISMS

Notation A denotes a ring, which is not necessarily commutative

but which possesses an identity element, and Z denotes the ring of

integers

2.1 The definition of tensor products

In section (1.4) we had occasion to observe that if B and C are left

A-modules then HomA (B, C) has the structure of an abelian group.

This is an important example of a method of obtaining a new modulefrom two given modules, and we shall have quite a lot more to sayabout HomA (B, C) in section (2.5) For the present, however, we shall

study a construction which yields an abelian group when we are givenboth a right A-module and a left A-module In this construction,which is one of the most important of modern algebra, the resultinggroup is known as the ' tensor product' of the two A-modules

Coming now to the details, suppose that we have aright A-module B and a left A-module C We shall use Z(B, C) to designate the free JZ-module generated by the set of symbols (6, c), where b belongs to B and c belongs to C Denote by Y(B, C) the smallest submodule of Z(B, C) which contains all elements of each of the forms

(i) (b 1 + b 2 ,c)-(b v c)-(b 2 ,c) 9

(ii) (6, c x + c2) - (6, cx) - (6, c2),(iii) (6A,c)-(6,Ac),

where b, b l9 b 2 belong to B, c, c l9 c 2 belong to C and A belongs to A P u t

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DEFINITION OF TENSOR PRODUCTS 17

Remarks, (a) If c is kept fixed, the mapping b -> b ® c is a

Z-homo-morphism of B into B® A C, and if 6 is kept fixed c -> 6 ® c is a

Z-homo-morphism of C into 2? ® A C Accordingly, if m is an integer then

(mb) ® c = m(6 ® c) = b ® (me).

In particular 0 ® c = 0 = & ® 0

(6) Since every element of Z(B, C) is a finite sum 2 m ^ , c^), we see

i

that every element of B ® A C is a finite sum Sm^frj ® c^) But

hence e^en/ element ofB® A Cis a finite sum of elements of the form b ® c.

2.2 Tensor products over commutative rings

Suppose, for the moment, that A is a commutative ring and let A

belong to A There is a Z-homomorphism Z(B, C)-*B ® A C in which

(6, c) maps into (6A) ® c In this homomorphism elements of the form

(b 1 + b 2 ,c)-(b v c)-(b 2y c)

and (6, cx + c2) - (6, cx) - (6, c2)

are mapped into zero Also the image of any element of the form

is zero, because A is commutative Thus our homomorphism vanishes

on Y(B, C) and so there is induced a Z-homomorphism

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18 TENSOR PRODUCTS

In other words, we assert that our definition of Xx makes B ® A C into

a A-module But in order to prove (2.2.3), (2.2.4) and (2.2.5), we need

only consider the case in which x has the form b ® c (This follows from (2.2.1) and the fact that every element of B ®A0 is expressible

as a finit6 sum of terms of the form b ® c.) However, by virtue of

(2.2.2), all of (2.2.3), (2.2.4) and (2.2.5) are trivial in this case

Summary In the general (non-commutative) case B® A C is a

^-module, but when A is commutative it can be given the structure

of a A-module by writing

A(6 ® c) = (6A) ® c = b ® (Ac).

2.3 Continuation of the general discussion

We now abandon the assumption that A is commutative and suppose,

as before, that B is a right A-module and C a left A-module A itself

can, of course, be regarded both as a right A-module and as a leftA-module

Theorem 1 There is a canonical Z-isomorphism

f: A® A C&C

in which f (A ® c) = Ac If A is commutative then f is a A-isomorphism.

Remark There is, of course, a similar canonical isomorphism

B ® A A « B for right A-modules.

Proof The Z-homomorphism of Z(A, C) into C in which (A, c) is

mapped into Ac is easily seen to vanish on Y(A, C) and so it induces

a Z-homomorphism «

in which/(A ® c) = Ac Since/(I ®c) = c for all c in C, we see t h a t / i s

an epimorphism Suppose now that x belongs to A ® A C Then, with

an obvious notation,

i i i

or x = 1 ® c for a suitable element c of C If therefore

then c = 0 and hence x = 0 Thus / is not only an epimorphism but

also a monomorphism, that is to say, it is an isomorphism

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CONTINUATION OF THE GENERAL DISCUSSION 19

Assume next that A is commutative, that A belongs to A and that

x belongs to A ® A C Writing x in the form x — 1 ® c we have

and therefore/is a A-isomorphism

2.4 Tensor products of homomorphisms

As before let B and C denote right and left A-modules respectively and let / : B->B' and g : C->C be A-homomorphisms Then the

2-homomorphism Z {B,C)->B> ® A C>,

in which (6, c) is mapped into/(6) ® g(c), is seen to vanish on Y(B, C).

It therefore induces a Z-homomorphism

(f®9) :B® A C->B'® A C for which (/® g) (b ® c) =/(&) ® g(c) The mapping/® g is called the tensor product of/ and g When A is commutative/® g is not only a

Z-homomorphism but also a A-homomorphism

It should be noted that i f / , / i , /2 are A-homomorphisms B->B' and

g, g v g 2 are A-homomorphisms C->C" then

and / ® 0 = 0,and 0 ® ? = 0,

as may be seen by applying both sides of each equation to a general

element of the form b ® c Further, if A is a commutative ring and A

belongs to A, then

(iii) (A/)®g = Mf®g)=f® (Xg).

Theorem 2, Ifi:B-+B andj : C->C are identity maps then

is an identity map Iff : B->B',f : B'->B", g :C->C',g f : <7->C"

are A-homomorphisms, then

The proof of this theorem is completely trivial

Corollary / / / : B&B',g : C&C are A-isomorphisms, then

f®g : B® A C->B'®A<7

is an isomorphism.

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Proof Let (/> : B' &B and ft : C &C be the inverse isomorphisms.

Then, by the theorem, (/® g) {$ ® ^) and (§J ®i/r)(f® ^) are both

identity maps, and this implies, as we saw in section (1.5), that

f®g and (j)®^r are inverse isomorphisms.

Theorem 3, Let the A-homomorphisms

Remarks For the notion of a complete representation see section

(1.9) It should be noted that, in Theorem 3, / and J need not be finite.

Also, when A is commutative, (2.4.3) is a complete representation

of B ® A C as a direct sum of A-modules.

Proof Let i, V belong to / and j , f to J, then

which is an identity map if (i',j f ) = (i,j) and is null otherwise.

Accordingly, by Theorem 4 of section (1.9), we need only show that

for each element x of B® A C, and this will follow if we establish it first in the case when x has the form b ® c Assume therefore that

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TENSOR PRODUCTS OF HOMOMORPHISMS 21

Hence 2 (/, ® ^ ) ( ^ ® ^ ) a; = S (/< ® 0;) (h ® ty)

This completes the proof of the theorem

Corollary Let C be a free left A-module with base [yji €j and let B

be an arbitrary right A-module Then each element of B® A C has a unique representation in the form

h 7i

i

where b i belongs to B and b i = 0 for almost all i.

Remark There is, of course, a similar result when B is free and C is

where b i belongs to B and 6^ = 0 for almost all i Thus b ® c has a

representation of the required form and therefore the same is true for

every element of B ® A C.

Suppose now that 2 (bi ® yd = 0, where b i belongs to B and b i = 0

i

for almost all i We wish to show that b i = 0 for all i Since [yi] ieI is

a base of C, the inclusion maps Ay^ -> (7 form an injective tion of C as a direct sum Consequently the maps B ® A Ay^ ->B ® A C,

representa-to which they give rise, constitute a representation of B® A C as a

direct sum of Z-modules and therefore determine an isomorphism

as an element of B® A Ay if is zero Now there is a A-isomorphism

A « Ay^ in which an element A of A is mapped into Ayt- and this yields

an isomorphism B® A A « 2?®AAyi But, by Theorem 1, we also

have a canonical isomorphism B & B ® A A and so, on combining, we obtain an isomorphism B & B® A Ay i In this isomorphism the element b i of B maps into b i ® y t , which is zero Consequently b i = 0

and this completes the proof

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Remarks H e r e / * is the tensor product of/ and the identity map

of C while g*, <f>*, ijr* are defined similarly Note that no assertions

are made concerning the kernels of/* and ^*

Proof We shall only prove (2.4.4) since the proof of (2.4.5) is

similar By Lemma 1, g* is an epimorphism Further, since gf = 0,

we have g*f* = 0 which shows that Im (/*) <= Ker (</*) Suppose now that b" belongs to B" and c to (7 Choose an element b in B so that g(b) = b" We assert that the image q(b ® c) of b ® c in the natural

depends only on b", c and not on the choice of b For suppose that we have g{b^) = b" then g{b — b^) = 0 and therefore b — b ± =f(b') for some b' in B' Thus

b®c-b 1 ®c= /(&') ®c= /*(&' ® c) e Im (/*),

and so g(& ® c) = g(6x ® c) as stated

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TENSOR PRODUCTS OF HOMOMORPHISMS 23

Consider the Z-homomorphism

Z(B'',C)->(B®AC)IIm(f*)

in which (5",c) is mapped into q(b ® c) This homomorphism is seen

at once to vanish on Y(B", C), and so it induces a Z-homomorphism

u : J3''®A<7^(£®A<7)/Im(/*),

in which u(b" ®c) = q(b ® c), where b is any element of B such that g(b) = b" Again, since Im(/*) c Ker(^*), we have 0*(lm(/*)) = 0, consequently g* induces a Z-homomorphism

Let b belong to B and c to C and write q(b ® c) = x Then, since every element of (B ®AO)/Im(/*) is a finite sum of elements having the

same form as x, it will suffice to show that uv(x) = x But

v(x) = g*(b ® c) = g(b) ® c and so uv(x) = u(g(b) ® c) = q(b ® c) = x.

This completes the proof of the theorem

/ 9 Theorem 5 Let 0 ->B'->B-> B" ^ 0

be an exact sequence of right A-modules and let F be a free left A-module Then the sequence

f* 9*

0->£r ® A F^B ® A F->B" ® A F^0,

to which this situation gives rise, is exact Furthermore, there is a sponding result in which the roles of left and right modules are inter- changed.

corre-Proof By Theorem 4, we need only show that / * is a

monomor-phism Suppose then that x belongs to B' ® A F and that/*(#) = 0.

We have to show that x = 0.

Let [yi] i€l be a base of F then, by the corollary to Theorem 3,

# = S (^i ® 7i)> where b\ belongs to B' and b\ = 0 for almost all i Thus

and therefore, by the same corollary, /(6^) = 0 for all i But / is a monomorphism; accordingly b\ = 0 for all i and consequently x = 0.

3-2

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2.5 The principal properties of HomA(2?, C)

We now abandon the assumption that B and G are right and left

modules respectively and suppose instead that both are left modules Then, as we saw in section (1.4), HomA (B, G) is an abelian

A-group which, when A is commutative, can be regarded as a A-module

It will appear, from the results to be established shortly, that the

tensor product B (g) A 0 of the preceding sections, and HomA (B, G) of

the present one are to some extent complementary concepts

Theorem 6 The mapping f : HomA (A, B)-^B defined byf(<f>) =

is a Z-isomorphism When A is commutative f is a A-isomorphism.

Proof It is clear, with an obvious notation, that

hence/is a Z-homomorphism Further, if/(0) = 0, then 0(1) = 0 and

SO

#(A) 0(A1) A0(1) = 0

for all A in A Accordingly <f> — 0, and this shows that / is a morphism Next suppose that b belongs to B, then the mapping

mono-A->A6 is a A-homomorphism ^ : A-^J5 For this homomorphism

f(cj)) = 0(1) = b, and now the proof that / is a Z-isomorphism is

complete

Finally, if A is commutative and A belongs to A, then

and s o / i s not only a Z-isomorphism but also a A-isomorphism Thisestablishes the theorem

Consider the way in which HomA (B, C) is transformed when the modules B and G, from which it is constructed, are subjected to A-homomorphisms To this end assume t h a t / : B' ->B and g : C-+C

are A-homomorphisms and let ^ be a variable element of HomA (B, C)

so that ^ is a A-linear mapping of B into (7 Then gcfif is a morphism of B' into C" and therefore it belongs to ~&om A {B',C).

A-homo-The mapping 0-><7^/is now seen to be a Z-homomorphism, usually

denoted by Horn (/, g), of HomA (B, C) into HomA (B f , C") Thus we have

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PRINCIPAL PROPERTIES OF Hom A (B, C) 25

There are certain elementary properties of Horn (/, g) which need

to be noted L e t / , ^ , ^ be A-homomorphisms B'-^B and let g, g v g 2

be A-homomorphisms C->C Then, by the definition,

(i) Horn (A +/2, g) = Horn (f l9 g) + Horn (/2, g)

and Hom(O,0) = O;

(ii) Horn (/, g x + g 2 ) = Horn (/, g x ) + Horn (/, g 2 )

and Horn (/, 0) = 0.Also, when A is commutative and A belongs to A,

(iii) Horn (A/, g) = A Horn (/, g) = Horn (/, Kg).

Theorem 7 / / i : B-+B and j : C->C are identity maps then

Kom(i,j)istheidentitymapofH.om A (B,C) Iff: B'->BJ f : B"->B',

g : C->C, g' : C ->C" are all A-homomorphisms, then

Horn (//', g'g) = Horn (/', g') Horn (/, g).

Proof The first assertion is trivial Suppose now that ^ belongs to

To see this we need only observe that if cj) : B&B' and i/r : C" « C

are the inverse isomorphisms then, by the theorem,

Horn (/, g) Horn ($, i/r) and Horn (0, ^) Horn (/, </)

are identity maps

fi 9i

Theorem 8 Let B i -^B-^B i (iel) (2.5.1)

be a complete representation of B as a direct sum and let

Cj (jeJ) (2.5.2)

be a complete representation of C as a direct product of A-modules Then

Horn {g^Pi) Horn (f i9 q^

HomA(B i9 Cj) • HomA(B, C) > HomA(B i9 C,) (ielje J)

(2.5.3)

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