An Introduction to Homological Algebra

AN INTRODUCTION TO HOMOLOGICAL ALGEBRA Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:15:00 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:15:00 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 AN INTRODUCTION TO HOMOLOGICAL ALGEBRA BY D G NORTHCOTT PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF SHEFFIELD w CAMBRIDGE AT THE UNIVERSITY PRESS 1962 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:15:00 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 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 © Cambridge University Press 1960 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 recordfor this publication is available from the British Library ISBN 978-0-521-05841-4 hardback ISBN 978-0-521-09793-2 paperback Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:15:00 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 CONTENTS Preface page ix Generalities concerning modules 1.1 Left modules and right modules 1.2 Submodules 1.3 Factor modules 1.4 A-homomorphisms 1.5 Some different types of A-homomorphisms 1.6 Induced mappings 1.7 Images and kernels 1.8 Modules generated by subsets 1.9 Direct products and direct sums 1.10 Abbreviated notations 12 1.11 Sequences of A-homomorphisms 13 Tensor products and groups of homomorphisms 2.1 The definition of tensor products 16 2.2 Tensor products over commutative rings 17 2.3 Continuation of the general discussion 18 2.4 Tensor products of homomorphisms 19 2.5 The principal properties of HomA (B, G) 24 Categories and functors 3.1 Abstract mappings 30 3.2 Categories 31 3.3 Additive and A-categories 32 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:16:32 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 Vi CONTENTS 3.4 Equivalences page 32 3.5 The categories &% and &% 3.6 Functors of a single variable 33 3.7 Functors of several variables 34 3.8 Natural transformations of functors 35 3.9 Functors of modules 36 33 3.10 Exact functors 38 3.11 Left exact and right exact functors 40 3.12 Properties of right exact functors 41 3.13 A ®A G and HomA {B, C) as functors 44 Homology functors 4.1 Diagrams over a ring 46 4.2 Translations of diagrams 47 4.3 Images and kernels as functors 48 4.4 Homology functors 52 4.5 The connecting homomorphism 54 4.6 Complexes 59 4.7 Homotopic translations 62 Projective and injective modules 5.1 Projective modules 63 5.2 Injective modules 67 5.3 An existence theorem for injective modules 71 5.4 Complexes over a module 75 5.5 Properties of resolutions of modules 77 5.6 Properties of resolutions of sequences 80 5.7 Further results on resolutions of sequences 84 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:16:32 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 CONTENTS Vii Derived functors 6.1 Functors of complexes page 90 6.2 Functors of two complexes 94 6.3 Right-derived functors 99 6.4 Left-derived functors 109 6.5 Connected sequences of functors 113 Torsion and extension functors 7.1 Torsion functors 121 7.2 Basic properties of torsion functors 123 7.3 Extension functors 128 7.4 Basic properties of extension functors 130 7.5 The homological dimension of a module 134 7.6 Global dimension 138 7.7 Noetherian rings 144 7.8 Commutative Noetherian rings 148 7.9 Global dimension of Noetherian rings 149 Some useful identities 8.1 Bimodules 155 8.2 General principles 156 8.3 The associative law for tensor products 160 8.4 Tensor products over commutative rings 161 8.5 Mixed identities 164 8.6 Rings and modules of fractions 167 Commutative Noetherian rings of finite global dimension 9.1 Some special cases 174 9.2 Reduction of the general problem 184 9.3 Modules over local rings 189 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:16:32 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 Viii CONTENTS 9.4 Some auxiliary results page 202 9.5 Homological codimension 204 9.6 Modules of finite homological dimension 205 10 Homology and cohomology theories of groups and monoids 10.1 General remarks concerning monoids and groups 211 10.2 Modules with respect to monoids and groups 214 10.3 Monoid-rings and group-rings 215 10.4 The functors A° and Ao 217 10.5 Axioms for the homology theory of monoids 219 10.6 Axioms for the cohomology theory of monoids 221 10.7 Standard resolutions of Z 223 10.8 The first homology group 229 10.9 The first cohomology group 230 10.10 The second cohomology group 238 10.11 Homology and cohomology in special cases 244 10.12 Finite groups 249 10.13 The norm of a homomorphism 252 10.14 Properties of the complete derived sequence 256 10.15 Complete free resolutions of Z 259 Notes 266 References 278 Index 281 Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:16:32 BST 2012 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511565915 Cambridge Books Online © Cambridge University Press, 2012 Cambridge Books Online http://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 PREFACE The past ten years or so have seen the emergence of a new mathematical subject which now bears the name Homological Algebra To begin with, it was the concern of a few enthusiasts in certain specialized fields but, since the publication of Cartan and Eilenberg's now famous book,f its importance for several of the main branches of pure mathematics has been generally recognized The young mathematician, about to start on research, will be anxious to learn about homological ideas and methods, and one of the aims of this book is to help him to get started In trying to cater for his needs, I have imagined such a reader as being familiar with the notions of group, ring and field but still relatively inexperienced in modern algebra For him, the account given here is self-contained save in a small number of particulars which are mentioned below, and which need not discourage him An introduction to homological algebra must, of necessity, be an introduction to the book of Cartan and Eilenberg, for the student who wishes to go further will need to read their work; but much of great interest and value has been achieved even more recently, and some of this 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 are needed to establish and illustrate the theory of derived functors, after which follows an account of torsion and extension functors These are the most important ones which are obtainable by the process of derivation and, in a sense, the remainder of the book is concerned with their applications Such an application is the theory of global dimension given at the end of Chapter 7, and here are included some important results of M Auslander on Noetherian rings that have previously been available only in the original research paper Chapter deals with the structure of commutative Noetherian rings t H Cartan and S Eilenberg, Homological Algebra (Princeton University Press, 1956) Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:18:36 BST 2012 http://dx.doi.org/10.1017/CBO9780511565915.001 Cambridge Books Online © Cambridge University Press, 2012 NOTES 269 Consider now a second continuous mapping g : E->E' and let us suppose t h a t / and g are homotopic This means, in crude intuitive terms, t h a t / can be changed into g by continuous variation The usual topological analysis of this situation then shows that the translations induced by / and g are homotopic in the purely algebraic sense defined in section (4.7) The connecting homomorphism considered in section (4.5) is encountered naturally in the development of what is called the relative homology theory.^ Let F be a subspace of E then, since a p-simplex on F can be regarded as a ^-simplex on E9 GV(F) is a submodule of GV(E); further, the boundary operator dj, on GV(E) becomes, on restriction, the corresponding operator on GV(F) Put GV(E9F) = G9(E)/G9(F) then d^ induces a homomorphism CP(E9F)-+CV_1(E9F) and in this way we obtain a complex > Cv+l(E, F) -» C9(E9 F) -+ CV_X(E9 F) -> Denote its pth homology group by HV(E9F) This is what is called the relative pth homology group of E modulo F though, in topology, it is more usual to define it in terms of subgroups of CV(E) The obvious mappings -> C9(F) -> C9(E) -> C9(E9 F)^0 make an exact sequence out of our complexes consequently there arises, by Theorem of section (4.6), an exact sequence • • -> H9(F) -> H9(E) -> H9(E9 F) -> HV_X(F) -> , and this is known as the homology sequence of the pair (E9 F) For a discussion from the topological point of view, together with illustrations of some of the ways in which it can be applied, the reader is referred to Wallace (32) CHAPTER Injective modules were introduced by Baer (5) (under another name, and with a different definition from the one now customarily given) who also established the fundamental result that every module is a submodule of an injective module We make some observations on Z-injective modules Let be a Z-module then is said to be divisible if, given any # € and any integer n={=0, there always exists 0'€0 such that nd' = One has then the following result: a Z-module is injective if and only if it is divisible Indeed a divisible module can be proved injective in essentially the same way as we established the injective property of the group Cl of the rational numbers modulo the integers To see the converse, suppose that is Z-injective and let 6e One obtains a homomorphism/ : Z -> by putting/(m) = m6 Since 0is injective,/can be extended to a homomorphism / : R -> 0, where R is the field of rational numbers here considered as an additive group Clearly, if m == j is an integer, then and this shows that is divisible The above characterization of Z-injective modules makes possible a simple proof that an arbitrary Z-module D is a submodule of a Z-injective module t See, for example, Wallace (32), ch v, § 19 NHA Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:59:24 BST 2012 http://dx.doi.org/10.1017/CBO9780511565915.012 Cambridge Books Online © Cambridge University Press, 2012 270 NOTES For we can suppose that D = F/H, where F is a free Z-module and H is one of its submodules Let {Ui}i€i be a base for F and let Ff be the free JR-module with the same base F', regarded as a Z-module, has F as one of its submodules and now D = F/H is a submodule of F'/H But the latter is clearly divisible and therefore, by the previous remarks, injective Eckmann and Schopf (15) have used these ideas to give a particularly neat proof of the corresponding result in the general case We sketch their argument very briefly Let A be a A-module (where A is an arbitrary ring with an identity element) then, as Z-module, A can be regarded as a submodule of a Z-injective module Consider Homz(A,i) and Homz(A, 0) By regarding A as a right A-module these acquire the structure of left A-modules and then Homz(A,^l) is a submodule (with respect to A) ofHom^ (A, 0) Now the latter is A-injective by the same reasoning as was used to prove Lemma of section (5.3) Finally, if a,€ A and we denote by fa the element of Homz(A,A) which maps A of A into Xa, then a-*/o is a A-monomorphism A ->Homz(A, A) Combining these observations, we see that A can be considered as a submodule of the A-injective module Hom^ (A, 0) The treatment in the text combines arguments to be found in (15) and (33) There is one further result of Eckmann and Schopf which will be described Suppose that A is a A-module and X is a A-injective module containing A, X is called a minimal injective extension of A if, whenever X' is any other injective extension, there exists a A-monomorphism X -> X' leaving each element of A fixed By a most elegant argument, it is shown that every A-module has a minimal injective extension and this is unique to within a A-isomorphism over A, Our discussion of injective modules makes use of Zorn's lemma Since this and certain other related results are (in the opinion of the author) not as widely known among young mathematicians as is desirable, a little will be said here about the forms of transfinite induction which will be needed These are easily explained and, if not already familiar, accounts of them can be read at leisure afterwards, for example, in chapter of Birkhoff (7) The basic notion is that of a, partially ordered set Let E be a set of objects and let us use x, y, z, etc., to denote elements of E Suppose now that we have a relation which holds between some pairs of elements of E and let us write x ^ y if it holds between x and y This relation is said to define a partial ordering if the following conditions are all satisfied: (i) x^xfor each x in E; (ii) x < y and y^x together imply that x = y; (iii) whenever x^y and y^z then # < z For example, if E consists of all the subsets of a set X and we write x < y if the subset x is contained in the subset y9 then this gives a partial ordering on E We make one general observation, namely, that each subset of a partially ordered set is itself partially ordered by the original ordering relation This is called the induced ordering Let < be a partial ordering on a set E, then E is said to be totally ordered or simply ordered if, whenever x,yeE, then either x ^ y or y ^ x For instance, the real numbers are totally ordered with respect to the relation 'less than or equal to \ Definition A totally ordered set E is said to be well ordered if every nonempty subset of E has &firstelement Thus to say that E is well ordered is to assert that, given any non-empty subset Y, there exists rj € Y such that rj < y for all yeY The well-ordering principle, which will be used on a small number of occasions, can now be stated Cambridge Books Online © Cambridge University 2010 Downloaded from Cambridge Books Online by IP 160.94.45.156 on Mon Apr 02 Press, 15:59:24 BST 2012 http://dx.doi.org/10.1017/CBO9780511565915.012 Cambridge Books Online © Cambridge University Press, 2012 NOTES 271 It asserts that any given set can be well ordered by means of a suitable ordering relation Before stating Zorn's lemma, it will be convenient to define one further concept Definition A partially ordered set E is called an inductive system if, for every subset Y which is totally ordered with respect to the induced ordering relation, there exists an element 0) (belonging to E and depending on Y) such that y < o) for all yeY It is customary to express this by saying that E is an inductive system when every totally ordered subset is bounded above in E The result known as Zorn's lemma asserts that a non-empty inductive system possesses at least one maximal element In other words (and it is important to make the significance of maximal quite clear) if E is a non-empty inductive system, then there exists £eE such that §