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1 W. M. L. Holcombe Algebraic automata theory2 K. Petersen Ergodic theory3 P. T. Johnstone Stone spaces4 W. H. Schikhof Ultrametric calculus5 J.-P. Kahane Some random series offunctions, 2nd edition6 H. Cohn Introduction to the construction ofclass fields7 J. Lambek and P. J. Scott Introduction to higher-order categorical

logic8 H. Matsumura Commutative ring theory9 C. B. Thomas Characteristic classes and the cohomology offinite

groups10 M. Aschbacher Finite group theory11 J. L. Alperin Local representation theory12 P. Koosis The logarithmic integral I13 A. Pietsch Eigenvalues and s-numbers14 S. J. Patterson An introduction to the theory of the Riemann

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bifurcations 'I36 L. Batten and A. Beutelspacher The theory offinite linear spaces37 Y. Meyer Wavelets and Operators38 C. Weibel An introduction to homological algebra39 W. Bruns and J. Herzog Cohen-Macaulay rings40 S. Martin Schur algebras and representation theory

AN INTRODUCTION TO

HOMOLOGICAL ALGEBRA

CHARLES A. WEIBELDepartment ofMathematics

Rutgers University

""~'"'' CAMBRIDGE;:: UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICAlE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge CB2 lRP, United Kingdom

CAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, United Kingdom

40 West 20th Street, New York, NY 10011-4211, USA10 Stamford Road, Oaldeigh, Melbourne 3166, Australia

© Cambridge University Press 1994

This book is in copyright. Subject to statutory exception andto the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe written pennission of Cambridge University Press.

First published 1994First paperback edition 1995

Reprinted 1997

Printed in the United States of America

Typeset in Times

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

Library ofCongress Cataloguing-ill-Publication Data is available

ISBN 0-521-43500-5 hardbackISBN 0-521-55987-1 paperback

To my wife, Laurel Van Leer, whose support is invaluable,and to my children, Chad and Aubrey, without whomthis book would have been completed much sooner.

Acknowledgments

I wish to express my appreciation to several people for their help in the for-mation of this book. My viewpoint on the subject comes from S. MacLane,S. Eilenberg, J. Moore, and R. Swan. The notes for the 1985 course weretaken by John Lowell, and many topics were suggested by W. Vasconcelos.L. Roberts and W. Vasconcelos used early versions in courses they taught; theyhelped improve the exposition. Useful suggestions were also made by L. Al-fonso, G. Cortinas, R. Fainnan, J.-L. Loday, J. P. May, R. McCarthy, S. Morey,R. Thomason, M. Vigue, R. Wilson, and the referees. Much of the typing wasdone by A. Boulle and L. Magretto.

vi

Contents

Introduction xi

Chain Complexes 1

1.1 Complexes of R-Modules 1

1.2 Operations on Chain Complexes 5

1.3 Long Exact Sequences 10

1.4 Chain Homotopies 15

1.5 Mapping Cones and Cylinders 18

1.6 More on Abelian Categories 25

2 Derived Functors 30

2.1 a-Functors 30

2.2 Projective Resolutions 33

2.3 Injective Resolutions 38

2.4 Left Derived Functors 43

2.5 Right Derived Functors 49

2.6 Adjoint Functors and Left/Right Exactness 51

2.7 Balancing Tor and Ext 58

3 Tor and Ext 66

3.1 Tor for Abelian Groups 66

3.2 Tor and Flatness 68

3.3 Ext for Nice Rings 73

3.4 Ext and Extensions 76('-

vii

viii Contents

3.5 Derived Functors of the Inverse Limit 80

3.6 Universal Coefficient Theorems 87

4 Homological Dimension 91

4.1 Dimensions 91

4.2 Rings of Small Dimension 95

4.3 Change of Rings Theorems 99

4.4 Local Rings 104

4.5 Koszul Complexes 111

4.6 Local Cohomology 115

5 Spectral Sequences 120

5.1 Introduction 120

5.2 Terminology 122

5.3 The Leray-Serre Spectral Sequence 127

5.4 Spectral Sequence of a Filtration 131

5.5 Convergence 135

5.6 Spectral Sequences of a Double Complex 141

5.7 Hyperhomology 145

5.8 Grothendieck Spectral Sequences 150

5.9 Exact Couples 153

6 Group Homology and Cohomology 160

6.1 Definitions and First Properties 160

6.2 Cyclic and Free Groups 167

6.3 Shapiro's Lemma 171

6.4 Crossed Homomorphisms and H I 174

6.5 The Bar Resolution 177

6.6 Factor Sets and H 2 182

6.7 Restriction, Corestriction, Inflation, and Transfer 189

6.8 The Spectral Sequence 195

6.9 Universal Central Extensions 198

6.10 Covering Spaces in Topology 203

6.11 Galois Cohomology and Profinite Groups 206

Contents ix

7 Lie Algebra Homology and Cohomology 216

7.1 Lie Algebras 216

7.2 g-Modules 219

7.3 Universal Enveloping Algebras 223

7.4 Hi and Hi 228

7.5 The Hochschild-Serre Spectral Sequence 232

7.6 H 2 and Extensions 234

7.7 The Chevalley-Eilenberg Complex 238

7.8 Semisimple Lie Algebras 242

7.9 Universal Central Extensions 248

8 Simplicial Methods in Homological Algebra 254

8.1 Simplicial Objects 254

8.2 Operations on Simplicial Objects 259

8.3 Simplicial Homotopy Groups 263

8.4 The Dold-Kan Correspondence 270

8.5 The Eilt~nberg-ZilberTheorem 275

8.6 Canonical Resolutions 278

8.7 Cotriple Homology 286

8.8 Andre-Quillen Homology and Cohomology 294

9 Hochschild and Cyclic Homology 300

9.1 Hochschild Homology and Cohomology of Algebras 300

9.2 Derivations, Differentials, and Separable Algebras 306

9.3 H2, Extensions, and Smooth Algebras 311

9.4 Hochschild Products 319

9.5 Morita Invariance 326

9.6 Cyclic Homology 330

9.7 Group Rings 338

9.8 Mixed Complexes 344

9.9 Graded Algebras 354

9.10 Lie Algebras of Matrices 362

x Contents

10 The Derived Category 369

10.1 The Category K(A) 369

10.2 Triangulated Categories 373

10.3 Localization and the Calculus of Fractions 379

10.4 The Derived Category 385

10.5 Derived Functors 390

10.6 The Total Tensor Product 394

10.7 Ext and RHom 398

10.8 Replacing Spectral Sequences 402

10.9 The Topological Derived Category 407

A Category Theory Language 417

Al Categories 417

A2 Functors 421

A3 Natural Transformations 423

A4 Abelian Categories 424

A5 Limits and Colimits 427

A6 Adjoint Functors 429

References 432

Index 435

Introduction

Homological algebra is a tool used to prove nonconstructive existence theo-rems in algebra (and in algebraic topology). It also provides obstructions tocarrying out various kinds of constructions; when the obstructions are zero,the construction is possible. Finally, it is detailed enough so that actual cal-culations may be perfonned in important cases. The following simple ques-tion (taken from Chapter 3) illustrates these points: Given a subgroup A of anabelian group B and an integer n, when is nA the intersection of A and nB?Since the cyclic group 7l../n is not flat, this is not always the case. The obstruc-tion is the group Tor(B / A, 7l../n), which explicitly is {x E B / A: nx = O}.

This book intends to paint a portrait of the landscape of homological alge-bra in broad brushstrokes. In addition to the "canons" of the subject (Ext, Tor,cohomology of groups, and spectral sequences), the reader will find introduc-tions to several other subjects: sheaves, lim I, local cohomology, hypercoho-mology, profinite groups, the classifying space of a group, Affine Lie alge-bras, the Dold-Kan correspondence with simplicial modules, triple cohomol-ogy, Hochschild and cyclic homology, and the derived category. The historicalconnections with topology, regular local rings, and semisimple Lie algebrasare also described.

After a lengthy gestation period (1890-1940), the birth of homological al-gebra might be said to have taken place at the beginning of World War IT withthe crystallization of the notions of homology and cohomology of a topolog-ical space. As people (primarily Eilenberg) realized that the same fonnalismcould be applied to algebraic systems, the subject exploded outward, touchingalmost every area of algebra. This phase of development reached maturity in1956 with the publication of Cartan and Eilenberg's book [CEl and with theemergence of the central notions of derived functors, projective modules, andinjective modules.

xi

xii Introduction

Until 1970, almost every mathematician learned the subject from Cartan-Eilenberg [CE]. The canonical list of subjects (Ext, Tor, etc.) came from thisbook. As the subject gained in popularity, other books gradually appeared onthe subject: MacLane's 1963 book [MacH], Hilton and Stammbach's 1971book [HSJ, Rotman's 1970 notes, later expanded into the book [Rot], andBourbaki's 1980 monograph [BX] come to mind. All these books covered thecanonical list of subjects, but each had its own special emphasis.

In the meantime, homological algebra continued to evolve. In the period1955-1975, the subject received another major impetus, borrowing topolog-ical ideas. The Dold-Kan correspondence allowed the introduction of simpli-cial methods, lim1 appeared in the cohomology of classifying spaces, spec-tral sequences assumed a central role in calculations, sheaf cohomology be-came part of the foundations of algebraic geometry, and the derived categoryemerged as the fonnal analogue of the topologists' homotopy category.

Largely due to the influence of Grothendieck, homological algebra becameincreasingly dependent on the central notions of abelian category and derivedfunctor. The cohomology of sheaves, the Grothendieck spectral sequence, lo-cal cohomology, and the derived category all owe their existence to these no-tions. Other topics, such as Galois cohomology, were profoundly influenced.

Unfortunately, many of these later developments are not easily found bystudents needing homological algebra as a tool. The effect is. a technologicalbarrier between casual users and experts at homological algebra. This book isan attempt to break down that barrier by providing an introduction to homo-logical algebra as it exists today.

This book is aimed at a second- or third-year graduate student. Based on thenotes from a course I taught at Rutgers University in 1985, parts of it wereused in 1990--92 in courses taught at Rutgers and Queens' University (thelatter by L. Roberts). After Chapter 2, the teacher may pick and choose topicsaccording to interest and time constraints (as was done in the above courses).

As prerequisites, I have assumed only an introductory graduate algebracourse, based on a text such as Jacobson's Basic Algebra I [BAI]. This meanssome familiarity with the basic notions of category theory (category, functor,natural transfonnation), a working knowledge of the category Ab of abeliangroups, and some familiarity with the category R-mod (resp. mod-R) of left(resp. right) modules over an associative ring R. The notions of abelian cat-egory (section 1.2), adjoint functor (section 2.3) and limits (section 2.6) areintroduced in the text as they arise, and all the category theory introduced inthis book is summarized in the Appendix. Several of the motivating exam-ples assume an introductory graduate course in algebraic topology but may

Introduction xiii

be skipped over by the reader willing to accept that such a motivation exists.An exception is the last section (section 10.9), which requires some familiaritywith point-set topology.

Many of the modem applications of homological algebra are to algebraicgeometry. Inasmuch as I have not assumed any familiarity with schemes oralgebraic geometry, the reader will find a discussion of sheaves of abeliangroups, but no mention of sheaves of Ox-modules. To include it would havedestroyed the flow of the subject; the interested reader may find this materialin [Hart].

Chapter I introduces chain complexes and the basic operations one canmake on them. We follow the indexing and sign conventions of Bourbaki[BX], except that we introduce two total complexes for a double complex: thealgebraists' direct sum total complex and the topologists' product total com-plex. We also generalize complexes to abelian categories in order to facilitatethe presentation of Chapter 2, and also in order to accommodate chain com-plexes of sheaves.

Chapter 2 introduces derived functors via projective modules, injectivemodules, and a-functors, following [Tohoku]. In addition to Tor and Ext, thisallows us to define sheaf cohomology (section 2.5). Our use of the acyclicassembly lemma in section 2.7 to balance Tor and Ext is new.

Chapter 3 covers the canonical material on Tor and Ext. In addition, we dis-cuss the derived functor lim1 of the inverse limit of modules (section 3.5), theKiinneth Fonnulas (section 3.6), and their applications to algebraic topology.

Chapter 4 covers the basic homological developments in ring theory. Ourdiscussion of global dimension (leading to commutative regular local rings)follows [KapCR] and [Rot]. Our material on Koszul complexes follows [BX],and of course the material on local cohomology is distilled from [GLC].

Spectral sequences are introduced in Chapter 5, early enough to be able toutilize this fundamental tool in the rest of the book. (A common problem withlearning homological algebra from other textbooks is that spectral sequencesare often ignored until the last chapter and so are not used in the textbookitself.) Our basic construction follows [CE]. The motivational section 5.3 onthe Leray-Serre spectral sequence in topology follows [MacH] very closely.(I first learned about spectral sequences from discussions with MacLane andthis section of his book.) Our discussion of convergence covers several resultsnot in the standard literature but widely used by topologists, and is based onunpublished notes of M. Boardman.

In Chapter 6 we finally get around to the homology and cohomology ofgroups. The material in this chapter is taken from [Brown], [MacH], and [Rot].

xiv Introduction

We use the LyndonIHochschild-Serre spectral sequence to do calculations insection 6.8, and introduce the classifying space BG in section 6.10. The ma-terial on universal central extensions (section 6.9) is based on [Milnor] and[Suz]. The material on Galois cohomology (and the Brauer group) comes from[BAll], [Serre], and [Shatz].

Chapter 7 concerns the homology and cohomology of Lie algebras. AsLie algebras aren't part of our prerequisites, the first few sections review thesubject, following [lLA] and [Humph]. Most of our material comes from the1948 Chevalley-Eilenberg paper [ChE] and from [CE], although the emphasis,and our discussion of universal central extensions and Affine Lie algebras,comes from discussions with R. Wilson and [Will.

Chapter 8 introduces simplicial methods, which have long been a vital partof the homology toolkit of algebraic topologists. The key result is the Dold-Kan theorem, which identifies simplicial modules and positive chain com-plexes of modules. Applied to adjoint functors, simplicial methods give riseto a host of canonical resolutions (section 8.6), such as the bar resolution, theGodement resolution of a sheaf [Gode], and the triple cohomology resolutions[BB]. Our discussion in section 8.7 of relative Tor and Ext groups parallelsthat of [MacH], and our short foray into Andre-Quillen homology comes from[Q] and [Barr].

Chapter 9 discusses Hochschild and cyclic homology of k-algebras. Al-though part of the discussion is ancient and is taken from [MacH], most is new.The material on differentials and smooth algebras comes from [EGA, IV] and[Mat]. The development of cyclic homology is rather new, and textbooks on it([Loday],[HK)) are just now appearing. Much of this material is based on the

articles [LQ], [Connes], and [Gw].Chapter lOis devoted to the derived category of an abelian category. The

development here is based upon [Verd] and [HartRD]. The material on thetopologists' stable homotopy in section 10.9 is based on [A] and [LMS].

Paris, February 1993

1

Chain Complexes

1.1 Complexes of R-Modules

Homological algebra is a tool used in several branches of mathematics: alge-braic topology, group theory, commutative ring theory, and algebraic geometrycome to mind. It arose in the late l800s in the following manner. Let f and gbe matrices whose product is zero. If g . v = 0 for some column vector v, say,of length n, we cannot always write v = f . u. This failure is measured by thedefect

d = n - rank(f) - rank(g).

In modem language, f and g represent linear maps

f gU---+V---+W

with gf = 0, and d is the dimension of the homology module

H = ker(g)/f(U).

In the first part of this century, Poincare and other algebraic topologistsutilized these concepts in their attempts to describe "n-dimensional holes" insimplicial complexes. Gradually people noticed that "vector space" could bereplaced by "R~module" for any ring R.

This being said, we fix an associative ring R and begin again in the categorymod-R of right R-modules. Given an R-module homomorphism f: A ~ B,one is immediately led to study the kernel ker(f), cokernel coker(f), andimage im(f) of f. Given another map g: B ~ C, we can form the sequence

f gA ---+ B ---+ C.

2 Chain Complexes

We say that such a sequence is exact (at B) if ker(g) = im(f). This impliesin particular that the composite gf: A ---+ C is zero, and finally brings ourattention to sequences (*) such that gf = O.

Definition 1.1.1 A chain complex C. of R-modules is a family {Cn}nEl ofR-modules, together with R-module maps d = dn : Cn ---+ Cn-I such that eachcomposite dod: Cn ---+ Cn-2 is zero. The maps dn are called the differentialsof C.' The kernel of dn is the module of n-cycles of C., denoted Zn = Zn(C,).The image of dn+I: Cn+I ---+ Cn is the module of n-boundaries of C., denotedBn = Bn(C,). Because dod = 0, we have

for all n. The nth homology module of C. is the subquotient Hn(C,) = Zn/Bnof Cn. Because the dot in C. is annoying, we will often write C for C.'

Exercise 1.1.1 Set Cn = 7l../8 for n :::: 0 and Cn = 0 for n < 0; for n > 0let dn send x (mod 8) to 4x (mod 8). Show that C. is a chain complex of7l../8-modules and compute its homology modules.

There is a category Ch(mod-R) of chain complexes of (right) R-modules.The objects are, of course, chain complexes. A morphism u: C. ---+ D. is achain complex map, that is, a family of R-module homomorphisms Un: Cn ---+Dn commuting with d in the sense that un-Idn = dn-iUn. That is, such thatthe following diagram commutes

d d d d... ---+ Cn+I ---+ Cn ---+ Cn-I ---+

lu lu lud d d d

---+ Dn+I ---+ Dn ---+ Dn-I ---+

Exercise 1.1.2 Show that a morphism u: C. ---+ D. of chain complexes sendsboundaries to boundaries and cycles to cycles, hence maps Hn(CJ ---+ Hn(D,).Prove that each Hn is a functor from Ch(mod-R) to mod-R.

Exercise 1.1.3 (Split exact sequences of vector spaces) Choose vector spaces{Bn, Hn}nEl over a field, and set Cn = Bn EEl Hn EEl Bn-I. Show that theprojection-inclusions Cn ---+ Bn-I C Cn-I make {Cn} into a chain complex,and that every chain complex of vector spaces is isomorphic to a complex ofthis form.

1.1 Complexes of R-Modules 3

Exercise 1.1.4 Show that {HomR(A, Cn)} fonns a chain complex of abeliangroups for every R-module A and every R-module chain complex C. TakingA = Zn, show that if Hn(HomR(Zn, C)) = 0, then Hn(C) = O. Is the conversetrue?

Definition 1.1.2 A morphism C. ---+ D. of chain complexes is called a quasi-isomorphism (Bourbaki uses homologism) if the maps Hn(CJ ---+ Hn(D.) areall isomorphisms.

Exercise 1.1.5 Show that the following are equivalent for every C. :

1. C. is exact, that is, exact at every Cn.2. C. is acyclic, that is, Hn(C.) = 0 for all n.3. The map 0 ---+ C. is a quasi-isomorphism, where "0" is the complex of

zero modules and zero maps.

The following variant notation is obtained by reindexiIig with superscripts:Cn = C-n. A cochain complex G- of R-modules is a family {Cn} of R-modules, together with maps d n:Cn ---+ Cn+l such that dod = O. Zn(c") =ker(dn) is the module of n-cocycles, Bn(G-) = im(dn- l ) f; Cn is the mod-ule of n-coboundaries, and the subquotient Hn(C') = Zn IBn of Cn is the nthcohomology module of C'. Morphisms and quasi-isomorphisms of cochaincomplexes are defined exactly as for chain complexes.

A chain complex C. is called bounded if almost all the Cn are zero; ifCn =0 unless a :::: n :::: b, we say that the complex has amplitude in (a, b]. Acomplex C. is bounded above (resp. bounded below) if there is a bound b (resp.a) such that Cn =0 for all n > b (resp. n < a). The bounded (resp. boundedabove, resp. bounded below) chain complexes fonn full subcategories of Cb= Cb(R-mod) that are denoted Cbb, Cb_ and Cb+, respectively. The sub-category Cb~o of non-negative complexes C. (Cn = 0 for all n < 0) will beimportant in Chapter 8.

Similarly, a cochain complex G- is called bounded above if the chain com-plex C. (Cn = C-n ) is bounded below, that is, if Cn = 0 for all large n; G-is bounded below if C. is bounded above, and bounded if C. is bounded.The categories of bounded (resp. bounded above, resp. bounded below, resp.non-negative) cochain complexes are denoted Cbb, Cb-, Cb+, and Cb~o,respectively.

Exercise 1.1.6 (Homology of a graph) Let r be a finite graph with V vertices(VI, .. " vv) and E edges (el, ... ,eE). If we orient the edges, we can form theincidence matrix of the graph. This is a V x E matrix whose (ij) entry is +1

4 Chain Complexes

if the edge ej starts at Vi, -1 if ej ends at Vi, and 0 otherwise. Let Co be thefree R-module on the vertices, CI the free R-module on the edges, Cn =0if n =1= 0, 1, and d: CI ---+ Co be the incidence matrix. If r is connected (Le.,we can get from Vo to every other vertex by tracing a path with edges), showthat Ho(C) and HI (C) are free R -modules of dimensions 1 and V - E - 1respectively. (The number V - E - 1 is the number of circuits of the graph.)Hint: Choose basis {vo, VI - Vo, ... , Vv - vol for Co, and use a path from voto Vi to find an element of C I mapping to Vi - Vo.

Application 1.1.3 (Simplicial homology) Here is a topological applicationwe shall discuss more in Chapter 8. Let K be a geometric simplicial complex,such as a triangulated polyhedron, and let Kk (o::s k ::s n) denote the set ofk-dimensional simplices of K. Each k-simplex has k + 1 faces, which areordered if the set Ko of vertices is ordered (do so!), so we obtain k + 1 setmaps OJ: Kk ---+ Kk-I (0 ::s i ::s k). The simplicial chain complex of K withcoefficients in R is the chain complex C., formed as follows. Let Ck be the freeR-module on the set Kk; set Ck =0 unless 0 ::s k ::s n. The set maps OJ yieldk + 1 module maps Ck ---+ Ck-I, which we also call 0i; their alternating sumd = L(-I)i oi is the map Ck ---+ Ck-I in the chain complex C.. To see that C.is a chain complex, we need to prove the algebraic assertion that dod = O.This translates into the geometric fact that each (k - 2)-dimensional simplexcontained in a fixed k-simplex a of K lies on exactly two faces of a. Thehomology of the chain complex C. is called the simplicial homology of K withcoefficients in R. This simplicial approach to homology was used in the firstpart of this century, before the advent of singular homology.

Exercise 1.1.7 (Tetrahedron) The tetrahedron T is a surface with 4 ver-tices, 6 edges, and 4 2-dimensional faces. Thus its homology is the homol-ogy of a chain complex 0 ---+ R4 ---+ R6 ---+ R4 ---+ O. Write down the matricesin this complex and verify computationally that H2(T) ~ Bo(T) ~ R andHI(T) = O.

Application 1.1.4 (Singular homology) Let X be a topological space, andlet Sk = Sk(X) be the free R-module on the set of continuous maps fromthe standard k-simplex !:i.k to X. Restriction to the i tk face of !:i.k (0 ::s i ::s k)transforms a map !:i.k ---+ X into a map !:i.k-I ---+ X, and induces an R-modulehomomorphism OJ from Sk to Sk-I. The alternating sums d = L(-I)joi (fromSk to Sk-I) assemble to form a chain complex

d d d. .. ---+ S2 ---+ SI ---+ So ---+ 0,

1.2 Operations on Chain Complexes 5

called the singular chain complex of X. The nth homology module of S. (X) iscalled the nth singular homology of X (with coefficients in R) and is writtenHn(X; R). If X is a geometric simplicial complex, then the obvious inclusionC. (X) ---+ S. (X) is a quasi-isomorphism, so the simplicial and singular homol-ogy modules of X are isomorphic. The interested reader may find details inany standard book on algebraic topology.

1.2 Operations on Chain Complexes

The main point of this section will be that chain complexes fonn an abeliancategory. First we need to recall what an abelian category is. A reference forthese definitions is [MacCW].

A category A is called an Ab-category if every hom-set Hom.A(A, B) inA is given the structure of an abelian group in such a way that compositiondistributes over addition. In particular, given a diagram in A of the fonn

we have h(g +g')f = hgf + hg'fin Hom (A, D). The category Ch is an Ab-category because we can add chain maps degreewise; if Un} and {gn} are chainmaps from C. to D., their sum is the family of maps Un + gn}.

An additive functor F: B ---+ A between Ab-categories B and A is a functorsuch that each HomB(B', B) ---+ Hom.A(FB', F B) is a group homomorphism.

An additive category is an Ab-category A with a zero object (Le., an ob-ject that is initial and tenninal) and a product A x B for every pair A, B ofobjects in A. This structure is enough to make finite products the same as fi-nite coproducts. The zero object in Ch is the complex "0" of zero modulesand maps. Given a family {Aa} of complexes of R-modules, the product TIAaand coproduct (direct sum) EBAa exist in Ch and are defined degreewise: thedifferentials are the maps

IT da : IT Aa,n ---+ IT Aa,n-l and EB da : EBaAa,n ---+ EBaAa,n-l,a a

respectively. These suffice to make Ch into an additive category.

Exercise 1.2.1 Show that direct sum and direct product commute with ho-mology, that is, that EBHn(Aa) ~ Hn(EBAa) and TIHn(Aa) ~ Hn(TIAa) forall n.

6 Chain Complexes

Here are some important constructions on chain complexes. A chain com-plex B is called a subcomplex of C if each Bn is a submodule of Cn and thedifferential on B is the restriction of the differential on C, that is, when theinclusions in : Bn c; Cn constitute a chain map B ---+ C. In this case we canassemble the quotient modules CnlBninto a chain complex

d d d•.• ---+ Cn+llBn+l ---+ CnlBn ---+ Cn-llBn-l ---+

denoted CI B and called the quotient complex. If I: B ---+ C is a chain map, thekernels {ker(fn)} assemble to fonn a subcomplex of B denoted ker(f), andthe cokernels {coker(fn)} assemble to fonn a quotient complex of C denotedcoker(f).

Definition 1.2.1 In any additive category A, a kernel of a morphism I: B ---+C is defined to be a map i: A ---+ B such that Ii = 0 and that is universal withrespect to this property. Dually, a cokernel of I is a map e: C ---+ D, whichis universal with respect to having el = O. In A, a map i: A ---+ B is monicif ig = 0 implies g = 0 for every map g: A' ---+ A, and a map e: C ---+ D isan epi if he = 0 implies h = 0 for every map h: D ---+ D'. (The definition ofmonic and epi in a non-abelian category is slightly different; see A.I in theAppendix.) It is easy to see that every kernel is monic and that every cokernelis an epi (exercise!).

Exercise 1.2.2 In the additive category A = R-mod, show that:

1. The notions of kernels, monics, and monomorphisms are the same.2. The notions of cokernels, epis, and epimorphisms are also the same.

Exercise 1.2.3 Suppose that A = Ch and I is a chain map. Show that thecomplex ker(f) is a kernel of I and that coker(f) is a cokernel of I.

Definition 1.2.2 An abelian category is an additive category A such that

1. every map in A has a kernel and cokernel.2. every monic in A is the kernel of its cokernel.3. every epi in A is the cokernel of its kernel.

The prototype abelian category is the category mod-R of R-modules. Inany abelian category the image im(f) of a map I: B ---+ C is the subobjectker(coker f) of C; in the category of R-modules, im(f) = {feb) : b E B}.Every map I factors as

1.2 Operations on Chain Complexes

e mB ---+ im(f) ---+ C

with e an epimorphism and m a monomorphism. A sequence

f gA--+B---+C

7

of maps in A is called exact (at B) ifker(g) = im(f).A subcategory B of A is called an abelian subcategory if it is abelian, and

an exact sequence in B is also exact in A.If A is any abelian category, we can repeat the discussion of section 1.1

! to define chain complexes and chain maps in A-just replace mod-R by A!These form an additive category Ch(A), and homology becomes a functorfrom this category to A. In the sequel we will merely write Ch for Ch(A)when A is understood.

Theorem 1.2.3 The category Ch = Ch(A) ofchain complexes is an abeliancategory.

Proof Condition 1 was exercise 1.2.3 above. If f: B -+ C is a chain map, Iclaim that f is monic iff each Bn -+ Cn is monic, that is, B is isomorphic to asubcomplex of C. This follows from the fact that the composite ker(f) -+ Cis zero, so if f is monic, then ker(f) = O. So if f is monic, it is isomorphic tothe kernel of C -+ ClB. Similarly, f is an epi iff each Bn -+ Cn is an epi, thatis, C is isomorphic to the cokernel of the chain map ker(f) -+ B..

Exercise 1.2.4 Show that a sequence 0 -+ A. -+ B. -+ C. -+ 0 of chain com-plexes is exact in Ch just in case each sequence 0 -+ An -+ Bn -+ Cn -+ 0 isexact in A.

Clearly we can iterate this construction and talk about chain complexes ofchain complexes; these are usually called double complexes.

Example 1.2.4 A double complex (or bicomplex) in A is a family {Cp,q} ofobjects of A, together with maps

such that dh 0 dh = dV 0 dV= dVdh + dhd v = O. It is useful to picture thebicomplex C.. as a lattice

8 Chain Complexes

1 1 1d h d h.. . +-- Cp-l,q+l +-- Cp,q+l +-- Cp+l,p+l +-- ...

dVl dVl dVl

dh dh.. . +-- Cp-l,q +-- Cp,q +-- Cp+l,q +-- ...

dVl dVl dVl

d h d h... +-- Cp-l,q-l +-- Cp,q-l +-- Cp+l,q-l +-- . ..

1 1 1

in which the maps d h go horizontally, the maps d V go vertically, and eachsquare anticommutes. Each row C*q and each column Cp* is a chain complex.

We say that a double complex C is bounded if C has only finitely manynonzero terms along each diagonal line p + q = n, for example, if C is con-centrated in the first quadrant of the plane (a first quadrant double complex).

Sign Trick 1.2.5 Because of the anticommutivity, the maps dV are not mapsin Ch, but chain maps f*q from C*q to C*,q-l can be defined by introducing± signs:

fp,q = (-l)Pd~,q: Cp,q ---+ Cp,q-l.

Using this sign trick, we can identify the category of double complexes withthe category Ch(Ch) of chain complexes in the abelian category Ch.

TotEe(C)n = EB Cp,q.p+q=n

Totn(C)n = n Cp,q andp+q=n

Total Complexes 1.2.6 To see why the anticommutative condition dVdh +dhdv = 0 is useful, define the total complexes Tot(C) = Totn(C) and TotEe(C)by

The formula d = dh + dV defines maps (check this!)d: Totn(C)n ---+ Totn(C)n_l and d: TotEe(C)n ---+ TotEe(C)n_l

such that dod = 0, making Totn(C) and TotEe(C) into chain complexes. Notethat TotEe(C) = Totn (C) if C is bounded, and especially if C is a first quadrantdouble complex. The difference between Totn (C) and TotEe (C) will becomeapparent in Chapter 5 when we discuss spectral sequences.

1.2 Operations on Chain Complexes 9

Remark Totn(C) and Tot(f1(C) do not exist in all abelian categories; theydon't exist when A is the category of all finite abelian groups. We say thatan abelian category is complete if all infinite direct products exist (and soTotn exists) and that it is cocomplete if all infinite direct sums exist (and soTot(f1 exists). Both these axioms hold in R-mod and in the category of chaincomplexes of R-modules.

Exercise 1.2.5 Give an elementary proof that Tot(C) is acyclic whenever Cis a bounded double complex with exact rows (or exact columns). We will seelater that this result follows from the Acyclic Assembly Lemma 2.7.3. It alsofollows from a spectral sequence argument (see Definition 5.6.2 and exercise5.6.4).

Exercise 1.2.6 Give examples of (1) a second quadrant double complex Cwith exact columns such that Totn(C) is acyclic but Tot(f1(C) is not; (2) asecond quadrant double complex C with exact rows such that Tot(f1(C) isacyclic but Totn(C) is not; and (3) a double complex (in the entire plane) forwhich every row and every column is exact, yet neither Totn(C) nor Tot(f1(C)is acyclic.

Truncations 1.2.7 If C is a chain complex and n is an integer, we let Te:.nCdenote the subcomplex of C defined by

ifi < nifi =nif i > n.

Clearly Hi (Te:.nC) = 0 for i < nand Hi (Te:.nC) = Hi(C) for i 2: n. The com-plex Te:.nC is called the (good) truncation of C below n, and the quotientcomplex T

10 Chain Complexes

with differential (-I)Pd. We call C[p] the pth translate of C. The way toremember the shift is that the degree 0 part of C[p] is Cp. The sign conventionis designed to simplify notation later on. Note that translation shifts homology:

We make translation into a functor by shifting indices on chain maps. That is,if I: C ~ D is a chain map, then f[p] is the chain map given by the formula

I[P]n = In+p (resp, f[pt = r-p)·Exercise 1.2.7 If C is a complex, show that there are exact sequences ofcomplexes:

do -----+ z (C) -----+ C -----+ B(C) [-I] -----+ 0;

do -----+ H(C) -----+ CjB(C) -----+ Z(C)[~I] -----+ H(C)[-I] -----+ O.

Exercise 1.2.8 (Mapping cone) Let I: B ~ C be a morphism of chain com-plexes. Form a double chain complex D out of I by thinking of I as a chaincomplex in Ch and using the sign trick, putting B [-I] in the row q = I andC in the row q = O. Thinking of C and B[-I] as double complexes in theobvious way, show that there is a short exact sequence of double complexes

8o -----+ C -----+ D -----+ B [- I] -----+ O.

The total complex of D is cone(f'), the mapping cone (see section 1.5) ofa map I', which differs from I only by some ± signs and is isomorphicto I.

1.3 Long Exact, Sequences

It is time to unveil the feature that makes chain complexes so special from acomputational viewpoint: the existence of long exact sequences.

Theorem 1.3.1 Let 0~ A,~ B,~ C, ~ 0 be a short exact sequence 01chain complexes. Then there are natural maps a: Hn(C) ~ Hn-l(A), calledconnecting homomorphisms, such that

is an exact sequence.

1.3 Long Exact Sequences 11

Similarly, if 0 -+ A'~ B·~ C· -+ 0 is a short exact sequence ofcochain complexes, there are natural maps a: Hn(c) -+ H n+1(A) and a longexact sequence

...~W-1(C)~ W(A) --L. W(B)~ Hn(C)~ H n+1(A) --L. ....

Exercise 1.3.1 Let 0 -+ A -+ B -+ C -+ 0 be a short exact sequence of com-plexes. Show that if two of the three complexes A, B, C are exact, then so isthe third.

Exercise 1.3.2 (3 x 3 lemma) Suppose given a commutative diagram

0 0 0

1 1 1o -----+ A' -----+ B' -----+ C' -----+ 0

1 1 1o -----+ A -----+ B -----+ C -----+ 0

1 1 1o -----+ A" -----+ B" -----+ C" -----+ 0

1 1 10 0 0

in an abelian category, such that every column is exact. Show the following:

1. If the bottom two rows are exact, so is the top row.2. If the top two rows are exact, so is the bottom row.3. If the top and bottom rows are exact, and the composite A -+ C is zero,

the middle row is also exact.

Hint: Show the remaining row is a complex, and apply exercise 1.3.1.

The key tool in constructing the connecting homomorphism a is our nextresult, the Snake Lemma. We will not print the proof in these notes, becauseit is best done visually. In fact, a clear proof is given by Jill Clayburgh at thebeginning of the movie It's My Turn (Rastar-Martin Elfand Studios, 1980). Asan exercise in "diagram chasing" of elements, the student should find a proof(but privately-keep the proof to yourself!).

Snake Lemma 1.3.2 Consider a commutative diagram of R-modules of theform

12 Chain Complexes

A' -----7 B' -----7 C' -----7 0

/1 d h1i

0 -----7 A -----7 B -----7 C.

If the rows are exact, there is an exact sequence

aker(f) ---+ ker(g) ---+ kerCh) -+ coker(f) ---+ coker(g) ---+ coker(h)

with adefined by the formula

Moreover, if A' ---+ B' is monic, then so is ker(f) ---+ ker(g), and if B ---+ C isonto, then so is coker(f) ---+ coker(g).

Etymology The tenn snake comes from the following visual mnemonic:

ker(f) ----+ ker(g) ----+ kerCh)

1 1 1

\

JI

II

• ) • ) ~,; II

I

/1_---------1-------- 1,I

• ) • ) •

I

1 1 1I

I

I\ ,-----~ coker(f) ----+ coker(g) ----+ coker(h).

Remark The Snake Lemma also holds in an arbitrary abelian category C. Tosee this, let A be the smallest abelian subcategory of C containing the ob-jects and morphisms of the diagram. Since A has a set of objects, the Freyd-Mitchell Embedding Theorem (see 1.6.1) gives an exact, fully faithful embed-ding of A into R-mod for some ring R. Since aexists in R-mod, it exists inA and hence in C. Similarly, exactness in R-mod implies exactness in A andhence in C.

1.3 Long Exact Sequences

Exercise 1.3.3 (5-Lemma) In any commutative diagram

A' -----+ B' -----+ C' -----+ D' -----+ E'

A-----+B-----+C-----+D-----+E

13

with exact rows in any abelian category, show that if a, b, d, and e are isomor-phisms, then c is also an isomorphism. More precisely, show that if b and dare monic and a is an epi, then c is monic. Dually, show that if b and d areepis and e is monic, then c is an epi.

We now proceed to the construction of the connecting homomorphism aofTheorem 1.3.1 associated to a short exact sequence

O--+A--+B--+C--+O

of chain complexes. From the Snake Lemma and the diagram

0 0 0

1 1 1o -----+ Zn A -----+ Zn B -----+ Zn C

1 1 1o -----+ An -----+ Bn -----+ Cn -----+ 0

dl dl dlo -----+ An-l -----+ Bn-l -----+ Cn-l -----+ 0

1 1 1An-l Bn-l Cn-l

-----+ 0-----+ -- -----+dAn dBn dCn

1 1 10 0 0

we see that the rows are exact in the commutative diagram

An Bn Cn-----+ 0-----+ -- -----+ --

dAn+l dBn+l dCn+l

dl d dlZn-l(A)

fZn-l(b)

go -----+ -----+ -----+ Zn-l (C).

14 Chain Complexes

The kernel of the left vertical is Hn(A), and its cokernel is Hn-l (A). Thereforethe Snake Lemma yields an exact sequence

The long exact sequence 1.3.1 is obtained by pasting these sequences together.

Addendum 1.3.3 When one computes with modules, it is useful to be able topush elements around. By decoding the above proof, we obtain the followingformula for the connecting homomorphism: Let z E Hn (C), and represent it by

a cycle c E Cn. Lift the cycle to bE Bn and apply d. The element db of Bn-lactually belongs to the submodule Zn-l (A) and represents a(z) E Hn-l (A).

We shall now explain what we mean by the naturality of a. There is acategory S whose objects are short exact sequences of chain complexes (say,in an abelian category C). Commutative diagrams

o -----+ A -----+ B -----+ C -----+ 0

o -----+ A' -----+ B' -----+ C' -----+ 0give the morphisms in S (from the top row to the bottom row). Similarly, thereis a category .c of long exact sequences in C.

Proposition 1.3.4 The long exact sequence is a functor from S to.c. That is,for every short exact sequence there is a long exact sequence, and for everymap (*) ofshort exact sequences there is a commutative ladder diagram

a a... -----+ Hn(A) -----+ Hn(B) -----+ Hn(C) -----+ Hn-l (A)-----+

a-----+ Hn(C')

a-----+

Proof All we have to do is establish the ladder diagram. Since each Hn is afunctor, the left two squares commute. Using the Embedding Theorem 1.6.1,we may assume C = mod-R in order to prove that the right square commutes.Given z E Hn(C), represented by c E Cn' its image z' E Hn(C') is representedby the image of c. If bE Bn lifts c, its image in B~ lifts c'. Therefore by 1.3.3a(z') E Hn-l (A') is represented by the image of db, that is, by the image of arepresentative of a(z), so a(z') is the image of a(z).

1.4 Chain Homotopies 15

Remark 1.3.5 The data of the long exact sequence is sometimes organizedinto the mnemonic shape

This is called an exact triangle for obvious reasons. This mnemonic shapeis responsible for the term "triangulated category," which we will discuss inChapter 10. The category K of chain equivalence classes of complexes and

.. maps (see exercise 1.4.5 in the next section) is an example of a triangulatedcategory.

Exercise 1.3.4 Consider the boundaries-cycles exact sequence 0 -+ Z -+C -+ B(-1) -+ 0 associated to a chain complex C (exercise 1.2.7). Show thatthe corresponding long exact sequence of homology breaks up into short exactsequences.

Exercise 1.3.5 Let f be a morphism of chain complexes. Show that if ker(f)and coker(f) are acyclic, then f is a quasi-isomorphism. Is the converse true?

Exercise 1.3.6 Let 0 -+ A -+ B -+ C -+ 0 be a short exact sequence of dou-ble complexes of modules. Show that there is a short exact sequence of totalcomplexes, and conclude that if Tot(C) is acyclic, then Tot(A) -+ Tot(B) is aquasi-isomorphism.

1.4 Chain Homotopies

The ideas in this section and the next are motivated by homotopy theory intopology. We begin with a discussion of a special case of historical impor-tance. If C is any chain complex of vector spaces over a field, we can alwayschoose vector space decompositions:

Cn=ZnE9B~,

Zn = Bn E9 H~,

B~ ~ Cn/Zn = d(Cn) = Bn-I;

H~ ~ Zn/Bn = Hn(C).

Therefore we can form the compositions

16 Chain Complexes

to get splitting maps sn: Cn -+ Cn+l, such that d = dsd. The compositions dsand sd are projections from Cn onto Bn and B~, respectively, so the sum ds +sd is an endomorphism of Cn whose kernel H~ is isomorphic to the homologyHn(C). The kernel (and cokernel!) of ds + sd is the trivial homology complexHAC). Evidently both chain maps H*(C) -+ C and C -+ H*(C) are quasi-isomorphisms. Moreover, C is an eXact sequence if and only if ds + sd is theidentity map.

Over an arbitrary ring R, it is not always possible to split chain complexeslike this, so we give a name to this notion.

Definition 1.4.1 A complex C is called split if there are maps Sn: Cn -+ Cn+lsuch that d = dsd. The maps Sn are called the splitting maps. Ifin addition Cis acyclic (exact as a sequence), we say that C is split exact

Example 1.4.2 Let R = 71. or 71./4, and let C be the complex

2 2 2 2... -+ 71./4 -+ 71./4 -+ 71./4 -+ ....

This complex is acyclic but not split exact. There is no map s such that ds + sdis the identity map, nor is there any direct sum decomposition Cn ~ Zn E9 B~.

Exercise 1.4.1 The previous example shows that even an acyclic chain com-plex of free R-modules need not be split exact.

1. Show that acyclic bounded below chain complexes of free R-modulesare always split exact.

2. Show that an acyclic chain complex of finitely generated free abeliangroups is always split exact, even when it is not bounded below.

Exercise 1.4.2 Let C be a chain complex, with boundaries Bn and cycles Znin Cn. Show that C is split if and only if there are R-module decompositionsCn ~ Zn E9 B~ and Zn = Bn E9 H~. Show that C is split exact iff H~ = O.

Now suppose that we are given two chain complexes C and D, togetherwith randomly chosen maps Sn: Cn -+ Dn+l' Let In be the map from Cn to Dndefined by the fonnula In = dn+1sn + Sn-ldn .

d dCn+l ----+ Cn ----+ Cn-l

s/ /1 s/Dn+1 ----+ Dn ----+ Dn-l

d d

1.4 Chain Homotopies

Dropping the subscripts for clarity, we compute

dl = d(ds + sd) =dsd = (ds + sd)d = Id.

Thus I =ds + sd is a chain map from C to D.

17

Definition 1.4.3 We say that a chain map I: C ---+ D is null homotopic ifthere are maps Sn: Cn ---+ Dn+1 such that 1= ds + sd. The maps {sn} arecalled a chain contraction of I.

Exercise 1.4.3 Show that C is a split exact chain complex if and only if theidentity map on C is null homotopic.

The chain contraction construction gives us an easy way to proliferate chain

maps: if g: C ---+ D is any chain map, so is g + (sd + ds) for any choice ofmaps Sn' However, g + (sd + ds) is not very different from g, in a sense thatwe shall now explain.

Definition 1.4.4 We say that two chain maps I and g from C to D are chainhomotopic if their difference I - g is null homotopic, that is, if

1- g=sd +ds.

The maps {snl are called a chain homotopy from I to g. Finally, we say thatI: C ---+ D is a chain homotopy equivalence (Bourbaki uses homotopism) ifthere is a map g: D ---+ C such that gl and I g are chain homotopic to therespective identity maps of C and D.

Remark This terminology comes from topology via the following observa-tion. A map I between two topological spaces X and Y induces a map1*: SeX) ---+ S(n between the corresponding singular chain complexes. Itturns out that if I is topologically null homotopic (resp. a homotopy equiv-alence), then the chain map 1* is null homotopic (resp. a chain homotopyequivalence), and if two maps I and g are topologically homotopic, then 1*and g* are chain homotopic.

Lemma 1.4.5 If I: C ---+ D is null homotopic, then every map 1*: Hn(C) ---+Hn(D) is zero. If I and g are chain homotopic, then they induce the samemaps Hn(C) ---+ Hn(D).

Prool It is enough to prove the first assertion, so suppose that I = ds + sd.Every element of Hn(C) is represented by an n-cycle x. But then I(x) =d(sx). That is, I(x) is an n-boundary in D. As such, I(x) represents 0 inHn(D). 0

18 Chain Complexes

Exercise 1.4.4 Consider the homology H*(C) of C as a chain complex withzero differentials. Show that if the complex C is split, then there is a chainhomotopy equivalence between C and H*(C). Give an example in which theconverse fails.

Exercise 1.4.5 In this exercise we shall show that the chain homotopy classesof maps form a quotient category K of the category Ch of all chain complexes.The homology functors Hn on Ch will factor through the quotient functorCh ---+ K.

1. Show that chain homotopy equivalence is an equivalence relation onthe set of all chain maps from C to D. Let HomK(C, D) denote theequivalence classes of such maps. Show that HomK(C, D) is an abeliangroup.

2. Let f and g be chain homotopic maps from C to D. If u: B ---+ C andv: D ---+ E are chain maps, show that vfu and vgu are chain homotopic.Deduce that there is a category K whose objects are chain complexes andwhose morphisms are given in (1).

3. Let fo, II, go, and gi be chain maps from C to D such that fi is chainhomotopic to gi (i = 1,2). Show that fo + II is chain homotopic togO + gl· Deduce that K is an additive category, and that Ch ---+ K is anadditive functor.

4. Is K an abelian category? Explain.

1.5 Mapping Cones and Cylinders

1.5.1 Let f: B. ---+ C. be a map of chain complexes. The mapping cone off is the chain complex cone(f) whose degree n part is Bn- I E9 Cn. In orderto match other sign conventions, the differential in cone(f) is given by theformula

deb, c) = (-d(b), d(c) - f(b», (b E Bn-I, c E Cn)'

That is, the differential is given by the matrix

[-dB-f

Bn-I -----+ Bn-2

o ] E9 ~- E9 .+dc :Cn -----+ Cn-I

+

1.5 Mapping Cones and Cylinders 19

Here is the dual notion for a map I: B· -+ C- of cochain complexes. Themapping cone, cone(f), is a cochain complex whose degree n part is Bn+1 E9Cn . The differential is given by the same formula as above with the same signs.

Exercise 1.5.1 Let cone(C) denote the mapping cone of the identity map ideof C; it has Cn-l E9 Cn in degree n. Show that cone(C) is split exact, withs(b, c) = (-c, 0) defining the splitting map.

Exercise 1.5.2 Let f: C -+ D be a map of complexes. Show that I is nullhomotopic if and only if I extends to a map (-s, f): cone(C) -+ D.

1.5.2 Any map f*: H*(B) -+ H*(C) can be fit into a long exact sequenceof homology groups by use of the following device. There is a short exactsequence

80-+ C -+ cone(f) --')- B[-l] -+ 0

of chain complexes, where the left map sends c to (0, c), and the right mapsends (b, c) to -b. Recalling (1.2.8) that Hn+! (B[ -1]) ~ Hn(B), the homol-ogy long exact sequence (with connecting homomorphism a) becomes

... --+ Hn+I(cone(J))~ Hn(B)~ Hn(C) --+ Hn(cone(f))~ Hn-l(B)~ ....

The following lemma shows that a = 1*, fitting 1* into a long exact sequence.

Lemma 1.5.3 The map a in the above sequence is 1*.

Prool If bE Bn is a cycle, the element (-b, 0) in the cone complex lifts b via8. Applying the differential we get (db, fb) = (0, fb). This shows that

arb] = [fb] = I*[b].

Corollary 1.5.4 A map I: B -+ C is a quasi-isomorphism if and only if themapping cone complex cone(f) is exact. This device reduces questions aboutquasi-isomorphisms to the study 01split complexes.

Topological Remark Let K be a simplicial complex (or more generally a cellcomplex). The topological cone CK of K is obtained by adding a new vertexs to K and "coning off" the simplices (cells) to get a new (n + I)-simplexfor every old n-simplex of K. (See Figure 1.1.) The simplicial (cellular) chaincomplex C. (s) of the one-point space {s} is R in degree 0 and zero elsewhere.C.(s) is a subcomplex of the simplicial (cellular) chain complex C.(CK) of

cone Cf

20 Chain Complexes

Figure 1.1. The topological cone CK and mapping cone Cf.

L

the topological cone CK. The quotient C. (CK) j Co (s) is the chain complexcone(C.K) of the identity map of C.(K). The algebraic fact that cone(CoK) issplit exact (null homotopic) reflects the fact that the topological cone CK iscontractible.

More generally, if f: K ---+ L is a simplicial map (or a cellular map), thetopological mapping cone Cf of f is obtained by glueing CK and L together,identifying the subcomplex K of CK with its image in L (Figure 1.1). This isa cellular complex, which is simplicial if f is an inclusion of simplicial com-plexes. Write C. (Cf) for the cellular chain complex of the topological map-ping cone Cf. The quotient chain complex C.(Cf)jC.(s) may be identifiedwith cone(f*), the mapping cone of the chain map f*: C.(K) ---+ C.(L).

1.5.5 A related construction is that of the mapping cylinder cyl(f) of a chaincomplex map f: B. ---+ Co' The degree n part of cyl(f) is Bn E9 Bn-l E9 Cn, andthe differential is

deb, b', c) = (d(b) + b', -deb'), d(c) - f(b'».

That is, the differential is given by the matrix

Bn+ > Bn-1

dB idB 0 E9 ;( E90 -dB 0 Bn-1 ) Bn-2

0 -f de E9 ~ffi+Cn > Cn-1

1.5 Mapping Cones and Cylinders

The cylinder is a chain complex because

21

dB -dB

d2BfdB -def

Exercise 1.5.3 Let cyl(C) denote the mapping cylinder of the identity mapide of C; it has Cn EEl Cn-! EEl Cn in degree n. Show that two chain mapsf, g: C -4 D are chain homotopic if and only if they extend to a map (f, s, g):cyl(C) -4 D.

Lemma 1.5.6 The subcomplex of elements (0,0, c) is isomorphic to C, andthe corresponding inclusion a: C -4 cyl(f) is a quasi-isomorphism.

Proof The quotient cyl(f)ja(C) is the mapping cone of -idB, so it is nUll-homotopic (exercise 1.5.1). The lemma now follows from the long exact ho-mology sequence for

ao -----+ C -----+ cyl(f) -----+ cone(-idB) -----+ O.

Exercise 1.5.4 Show that f3(b, b', c) = feb) + c defines a chain map fromcyl(f) to C such that f3a = ide. Then show that the formula s(b, b', c) =(0, b, 0) defines a chain homotopy from the identity of cy1(f) to af3. Concludethat a is in fact a chain homotopy equivalence between C and cyl(f).

Topological Remark Let X be a cellular complex and let I denote the interval[0,1]. The space I x X is the topological cylinder of X. It is also a cell com-plex; every n-cell en in X gives rise to three cells in I x X: the two n-cells,o x en and I x en, and the (n + I)-cell (0,1) x en. If C.(X) is the cellularchain complex of X, then the cellular chain complex C. (l x X) of I x X maybe identified with cyl(ide.x), the mapping cylinder chain complex of the iden-tity map on C.(X).

More generally, if f: X -4 Y is a cellular map, then the topological map-ping cylinder cyl(f) is obtained by glueing I x X and Y together, identifyingo x X with the image of X under f (see Figure 1.2). This is also a cellularcomplex, whose cellular chain complex C.(cyl(f» may be identified with themapping cylinder of the chain map C. (X) -4 C. (Y).

The constructions in this section are the algebraic analogues of the usualtopological constructions I x X ~ X, cyl(f) ~ Y, and so forth which wereused by Dold and Puppe to get long exact sequences for any generalized ho-mology theory on topological spaces.

22

[xX

----------

Chain Complexes

lxX

y

Cyl(f)

Figure 1.2. The topological cylinder of X and mapping cylinder cyl(f).

Here is how to use mapping cylinders to fit f* into a long exact sequenceof homology groups. The subcomplex of elements (b, 0, 0) in cyl(f) is iso-morphic to B, and the quotient cyl(f) / B is the mapping cone of f. Thecomposite B ---+ cyl(f)~ C is the map f, where fJ is the equivalence ofexercise 1.5.4, so on homology f*: H(B) ---+ H(C) factors through H(B) ---+H(cyl(f». Therefore We mayconstruct a commutative diagram of chain com-plexes with exact rows:

C

f/ i.eo ----+ B ----+ cyl(f) ----+ cone(f) ----+0

i a8

o ----+ C ----+ cone(f) ----+ B[-I] ----+ O.The homology long exact sequences fit into the following diagram:

-a -a... -----+ HnCB) -+ HnCcyl(f)) -+ HnCcone(f)) -----+ Hn-lCB) -+ ...

III t'\.. III III0... -+ Hn+1CB[-1]) ---+ HnCC) -+ HnCcone(f) ) ---+ HnCB[-I]) ---+a a

Lemma 1.5.7 This diagram is commutative, with exact rows.

Proof It suffices to show that the right square (with -a and 8) commutes.

1.5 Mapping Cones and Cylinders 23

Let (b, c) be an n-cycle in cone(f), so deb) = 0 and feb) = d(c). Lift it to(0, b, e) in cyl(f) and apply the differential:

d(O, b, e) = (0 + b, -db, de - fb) = (b, 0, 0).

Therefore amaps the class of (b, e) to the class of b = -8(b, e) in Hn-l (B).

1.5.8 The cone and cylinder constructions provide a natural way to fit thehomology of every chain map f: B ---+ C into some long exact sequence (see1.5.2 and 1.5.7). To show that the long exact sequence is well defined, we needto show that the usual long exact homology sequence attached to any shortexact sequence of complexes

O---+B~C~D---+O

agrees both with the long exact sequence attached to f and with the long exactsequence attached to g.

We first consider the map f. There is a chain map q;: cone(f) ---+ D definedby the formula q;(b, e) = gee). It fits into a commutative diagram with exactrows:

8o ---+ C ---+ cone(f) ---+ B[-l] ---+ 0

l ao ---+ B ---+ cyl(f) ---+ cone(f) ---+ 0

1,8 19"f go ---+ B ---+ C ---+ D ---+ o.

Since f3 is a quasi-isomorphism, it follows from the 5-lemma and 1.3.4 that q;is a quasi-isomorphism as well. The following exercise shows that q; need notbe a chain homotopy equivalence.

Exercise 1.5.5 Suppose that the B and C of 1.5.8 are modules, consideredas chain complexes concentrated in degree zero. Then cone(f) is the complex

o---+ B .::.4 C ---+ O. Show that q; is a chain homotopy equivalence iff f : B CC is a split injection.

To continue, the naturality of the connecting homomorphism aprovides uswith a natural isomorphism of long exact sequences:

24 Chain Complexes

a a~ HnCB) ~ HnCcyl(f)) ~ HnCcone(f)) ~ Hn-lCB) ~ ...

1= 1= 111a a~ HnCB) ~ Hn(C) ~ HnCD) ~ Hn-lCB) ~

Exercise 1.5.6 Show that the composite

-8Hn(D) ~ Hn(cone(f» ---+ Hn(B[-I]) ~ Hn-I(B)

is the connecting homomorphism ain the homology long exact sequence for

o-+ B -+ C -+ D -+ O.

Exercise 1.5.7 Show that there is a quasi-isomorphism B[-I] -+ cone(g)dual to cpo Then dualize the preceding exercise, by showing that the com-posite

a ~Hn(D) -+ Hn-I(B) -=+ Hn(cone(g»

is the usual map induced by the inclusion of Din cone(g).

Exercise 1.5.8 Given a map f: B -+ C of complexes, let v denote the in-clusion of C into cone(f). Show that there is a chain homotopy equivalencecone(v) -+ B[-I]. This equivalence is the algebraic analogue of the topolog-ical fact that for any map f: K -+ L of (topological) cell complexes the coneof the inclusion L C Cf is homotopy equivalent to the suspension of K.

Exercise 1.5.9 Let f: B -+ C be a morphism of chain complexes. Show that

the natural maps ker(f)[-l]~ cone(f)~ coker(f) give rise to a longexact sequence:

a a f3 a...~ Hn-l(ker(f))~ Hn(cone(f)) ---* Hn(coker(f))~ Hn-2(ker(f)) .. ·.

Exercise 1.5.10 Let C and C' be split complexes, with splitting maps s, s',If f: C -+ C' is a morphism, show that cr(c, c') = (-s(c), s'(c') - s' fs(c»defines a splitting of cone(f) if and only if the map f*: H*(C) -+ HAC') iszero.

1.6 More on Abelian Categories

1.6 More on Abelian Categories

25

We have already seen that R-mod is an abelian category for every associativering R. In this section we expand our repertoire of abelian categories to includefunctor categories and sheaves. We also introduce the notions of left exact andright exact functors, which will fonn the heart of the next chapter. We give theYoneda embedding of an additive category, which is exact and fully faithful,and use it to sketch a proof of the following result, which has already beenused. Recall that a category is called small if its class of objects is in fact a set.

Freyd-Mitchell Embedding Theorem 1.6.1 (1964) If A is a small abeliancategory, then there is a ring R and an exact, fully faithful functor fromA into R-mod, which embeds A as a full subcategory in the sense thatHomA(M, N) ~ HomR(M, N).

We begin to prepare for this result by introducing some examples of abeliancategories. The following criterion, whose proof we leave to the reader, isfrequently useful:

Lemma 1.6.2 Let C c A be a full subcategory ofan abelian category A1. C is additive 0 E C, and C is closed under EB.2. C is abelian and C c A is exact C is additive, and C is closed under

ker and coker.

Examples 1.6.3

1. Inside R-mod, the finitely generated R-modules form an additive cate-gory, which is abelian if and only if R is noetherian.

2. Inside Ab, the torsionfree groups form an additive category, while thep-groups form an abelian category. (A is a p-group if (Va E A) somepna = 0.) Finite p-groups also form an abelian category. The category(11../p)-mod of vector spaces over the field 7L/p is also a full subcategoryofAb.

Functor Categories 1.6.4 Let C be any category, A an abelian category.The functor category A C is the abelian category whose objects are functorsF: C -+ A. The maps in AC are natural transformations. Here are some rele-vant examples:

1. If C is the discrete category of integers, AbC contains the abelian cate-gory of graded abelian groups as a full subcategory.

26 Chain Complexes

2. If C is the poset category of integers (... -+ n -+ (n + 1) -+ ...) thenthe abelian category Ch(A) of cochain complexes is a full subcategoryofAc.

3. If R is a ring considered as a one-object category, then R-mod is the fullsubcategory of all additive functors in AbR •

4. Let X be a topological space, and U the poset of open subsets of X. Acontravariant functor F from U to A such that F(0) = {OJ is called apresheaf on X with values in A, and the presheaves are the objects ofthe abelian category AUoP = Presheaves(X).

A typical example of a presheaf with values in ~-mod is given by CO(U) ={continuous functions f: U -+ ~}. If U C V the maps C°(V) -+ C°(U) aregiven by restricting the domain of a function from V to U. In fact, CO is asheaf:

Definition 1.6.5 (Sheaves) A sheaf on X (with values in A) 'is a presheaf Fsatisfying the

SheafAxiom. Let {Ud be an open covering of an open subset U of X.If {fi E F(Ui)} are such that each fi and Ii agree in F(Ui n Uj), thenthere is a unique f E F(U) that maps to every fi under F(U) -+ F(Ui)'

Note that the uniqueness of f is equivalent to the assertion that if f E F(U)vanishes in every F(U;), then f = O. In fancy (element-free) language, thesheaf axiom states that for every covering {Ui} of every open U the followingsequence is exact:

diff0-+ F(U) -+ nF(Ui) -+ nF(Ui n Uj).

i

1.6 More on Abelian Categories 27

O(U) (resp. O*(U» is the group of continuous maps from U into ([ (resp.([*). Then there is a short exact sequence of sheaves:

0-+ 7l.~ 0 ~ 0* -+ o.

When X is the space ([*, this sequence is not exact in Presheaves(X) becausethe exponential map from ([ = O(X) to O*(X) is not onto; the cokemel is7l. = Hl(X, 7l.), generated by the global unit liz. In effect, there is no globallogarithm function on X, and the contour integral 2;;;} fez) dz gives theimage of fez) in the cokemel.

Definition 1.6.6 Let F: A -+ B be an additive functor between abelian cat-egories. F is called left exact (resp. right exact) if for every short exact se-quence 0 -+ A -+ B -+ C -+ 0 in A, the sequence 0 -+ F(A) -+ F(B) -+F(C) (resp. F(A) -+ F(B) -+ F(C) -+ 0) is exact in B. F is called exact ifit is both left and right exact, that is, if it preserves exact sequences. A con-travariant functor F is called left exact (resp. right exact, resp. exact) if thecorresponding covariant functor F': Aop -+ B is left exact (resp.... ).

Example 1.6.7 The inclusion of Sheaves(X) into Presheaves(X) is a leftexact functor. There is also an exact functor Presheaves(X) -+ Sheaves(X),called "sheafification." (See 2.6.5; the sheafification functor is left adjoint tothe inclusion.)

Exercise 1.6.3 Show that the above definitions are equivalent to the follow-ing, which are often given as the definitions. (See [Rot], for example.) A (co-variant) functor F is left exact (resp. right exact) if exactness of the sequence

o-+ A -+ B -+ C (resp. A -+ B -+ C -+ 0)

implies exactness of the sequence

o-+ FA -+ FB -+ F C (resp. FA -+ FB -+ F C -+ 0).

Proposition 1.6.8 Let A be an abelian category. Then HomA(M, -) is a leftexact functor from A to Ab for every M in A That is, given an exact sequence

o-+ A~ B~ C -+ 0 in A, the following sequence of abelian groups isalso exact:

O H f. g.-+ om(M, A) ---+ Hom(M, B) ---+ Hom(M, C).

28 Chain Complexes

Prool If a E Hom(M, A) then I*a = loa; if this is zero, then a must bezero since I is monic. Hence f* is monic. Since g 0 f = 0, we have g*f*(a) =go f 0 a = 0, so g*f* = O. It remains to show that if f3 E Hom(M, B) is suchthat g*f3 = g 0 fJ is zero, then fJ = loa for some a. But if go f3 = 0, thenf3(M) s;; f(A), so fJ factors through A.

Corollary 1.6.9 HomA (-, M) is a left exact contravariantfunctor.

Proof HomA(A, M) = HomAop(M, A).

Yoneda Embedding 1.6.10 Every additive category A can be embedded inthe abelian category Ab

Aopby the functor h sending A to hA = HomA(-, A).

Since each HomA(M, -) is left exact, h is a left exact functor. Since thefunctors hA are left exact, the Yoneda embedding actually lands in the abeliansubcategory I:- of all left exact contravariant functors from A to Ab wheneverA is an abelian category.

Yoneda Lemma 1.6.11 The Yoneda embedding h reflects exactness. That is,

a sequence A~ B L C in A is exact, provided that for every M in A thefollowing sequence is exact:

Proof Taking M = A, we see that f3a = f3*a*(idA) = O. Taking M = ker(f3),we see that the inclusion t: ker(fJ) -,)- B satisfies f3*(t) = f3t = O. Hence thereis a a E Hom(M, A) with t = a*(a) = aa, so that ker(f3) = im(t) f: im(a).

We now sketch a proof of the Freyd-Mitchell Embedding Theorem 1.6.1;details may be found in [Freyd] or [Swan, pp. 14-22]. Consider the failure of

the Yoneda embedding h: A -,)- AbAop to be exact: if 0 -,)- A -,)- B -,)- C -,)- 0is exact in A and MEA, then define the abelian group W(M) by exactness of

0-,)- HomA(M, A) -,)- HomA(M, B) -,)- HomA(M, C) -,)- W(M) -,)- O.

In general W (M) =1= 0, and there is a short exact sequence of functors:

W is an example of a weakly effaceable functor, that is, a functor such thatfor all MEA and x E W(M) there is a surjection P -,)- M in A so that the

1.6 More on Abelian Categories 29

map W(M) ---,)- W(P) sends x to zero. (To see this, take P to be the pullbackM Xc B, where M ---,)- C represents x, and note that P ---,)- C factors throughB.) Next (see loco cit.), one proves:

Proposition 1.6.12 If A is small, the subcategory W of weakly effaceablefunctors is a localizing subcategory ofAbAo

pwhose quotient category is £.

That is, there is an exact "reflection" functor R from AbAop

to £- such thatR(L) = Lfor every left exact Land R(W) ~°iffW is weakly effaceable.Remark Cokernels in £- are different from cokernels in AbAo

p, so the inclu-

sion £- C AbAop

is not exact, merely left exact. To see this, apply the reflectionR to (*). Since R(hA) = hA and R(W) ~ 0, we see that

is an exact sequence in £-, but not in AbAoP •

Corollary 1.6.13 The Yoneda embedding h: A ---,)- £- is exact and fully faith-ful.

Finally, one observes that the category £- has arbitrary coproducts and hasa faithfully projective object P. By a result of Gabriel and Mitchell [Freyd,p. 106], £- is equivalent to the category R-mod of modules over the ringR = Hom.c(P, P). This finishes the proof of the Embedding Theorem.

Example 1.6.14 The abelian category of graded R-modules may be thoughtof as the full subcategory of (OiE£' R)-modules of the form fBiE£,Mi. Theabelian category of chain complexes of R-modules may be embedded inS-mod, where

S = (IT R)[d]/(d2 = 0, {dr = rd}rER, {dei = ei-ldliE&')'iE£'

Here ei: 0 R ---,)- R ---,)- 0 R is the i lh coordinate projection.

2

Derived Functors

2.1 a-Functors

The right context in which to view derived functors, according to Groth-endieck [Tohoku], is that of a-functors between two abelian categories AandB.

Definition 2.1.1 A (covariant) homological (resp. cohomological) a-functorbetween A and B is a collection of additive functors Tn: A ---,)- B (resp.Tn: A ---,)- B) for n 2: 0, together with morphisms

defined for each short exact sequence 0 ---,)- A ---,)- B ---,)- C ---,)- 0 in A. Here wemake the convention that Tn = Tn = 0 for n < O. These two conditions areimposed:

1. For each short exact sequence as above, there is a long exact sequence

8 8Tn+I(C) ---* Tn(A) ---,)- Tn(B) ---,)- Tn(C) ---* Tn-I(A) ...

(resp.

In particular, To is right exact, and TO is left exact.

30

2.1 a-Functors 31

2. For each morphism of short exact sequences from 0 ~ A' ~ B' ~

C' ~°to°~ A ~ B ~ C ~ 0, the a's give a commutative diagram8

Tn-l (A') Tn(C/)8

rn+I(A' )Tn (C') ~ ~

1 1 resp. 1 18 8

rn+I(A).Tn (C) ~ Tn-I(A) Tn(C) ~

Example 2.1.2 Homology gives a homological a-functor H* from Ch2:o(A)to A; cohomology gives a cohomological a-functor H* from Ch2:o(A) toA.

Exercise 2.1.1 Let S be the category of short exact sequences

O~A~B~C~O

in A. Show that ai is a natural transformation from the functor sending (*) toTi(C) to the functor sending (*) to 1i-I(A).

Example 2.1.3 (p-torsion) If p is an integer, the functors To(A) = AIpA and

TI (A) = pA == {a E A : pa = O}

fit together to form a homological a-functor, or a cohomological a-functor(with TO = TI and T 1 = To) from Ab to Ab. To see this, apply the SnakeLemma to

O~A~B~C~O

O~A~B~C~O

to get the exact sequence

Generalization The same proof shows that if r is any element in a ring R,then To(M) = M IrM and TI (M) = rM fit together to form a homological a-functor (or cohomologicala-functor, if that is one's taste) from R-mod to Ab.

32 Derived Functors

Vista We will see in 2.6.3 that Tn(M) = Tor:(Rjr, M) is also a homolog-ical a-functor with To(M) = MjrM. If r is a left nonzerodivisor (meaningthat rR = {s E R : rs = O} is zero), then in fact Torf(Rjr, M) = rM andTor: (Rjr, M) = 0 for n 2: 2; see 3.1.7. However, in general rR i= 0, whileTorf(Rjr, R) = 0, so they aren't the same; Torf (M, Rjr) is the quotient ofrM by the submodule (rR)M generated by {sm : rs = 0, s E R, m EM}. TheTorn will be universal a-functors in a sense that we shall now make precise.

Definition 2.1.4 A morphism S -,)- T of a-functors is a system of naturaltransformations Sn -,)- Tn (resp. sn -,)- Tn) that commute with a. This is fancylanguage for the assertion that there is a commutative ladder diagram con-necting the long exact sequences for S and T associated to any short exactsequence in A.

A homological a-functor T is universal if, given any other a-functor S and anatural transformation fo: So -,)- To, there exists a unique morphism Un: Sn -,)-Tn} of a-functors that extends fo.

Acohomological a-functor T is universal if, given S and f O: TO -,)- So,there exists a unique morphism T -,)- S of a-functors extending fO.

Example 2.1.5 We will see in section 2.4 that homology H*: Ch::,:o(A) -,)- Aand cohomology H*: Ch::,:o(A) -,)- A are universal a-functors.

Exercise 2.1.2 If F: A -,)- B is an exact functor, show that To = F and Tn = °for n i= 0 defines a universal a-functor (of both homological and cohomologi-cal type).

Remark If F: A -,)- B is an additive functor, then we can ask if there is any a-functor T (universal or not) such that To = F (resp. TO = F). One obviousobstruction is that To must be right exact (resp. TO must be left exact). Bydefinition, however, we see that there is at most one (up to isomorphism)universal a-functor T with To = F (resp. TO = F). If a universal T exists, theTn are sometimes called the left satellite functors of F (resp. the Tn are calledthe right satellite functors of F). This terminology is due to the pervasiveinfluence of the book [eEl.

We will see that derived functors, when they exist, are indeed universal a-functors. For this we need the concept of projective and injective resolutions.

2.2 Projective Resolutions

2.2 Projective Resolutions

33

An object P in an abelian category A is projective if it satisfies the followinguniversal lifting property: Given a sUljection g: B ---,)- C and a map y: P ---,)- C,there is at least one map f3: P ---,)- B such that y = g 0 f3.

B--+C--+O

We shall be mostly concerned with the special case of projective modules(A being the category mod-R). The notion of projective module first appearedin the book [CE]. It is easy to see that free R-modules are projective (lift abasis). Clearly, direct summands offree modules are also projective modules.

Proposition 2.2.1 An R-module is projective iff it is a direct summand of afree R-module.

Proof Letting F(A) be the free R-module on the set underlying an R-moduleA, we see that for every R-module A there is a surjection :rr: F(A) ---,)- A. IfA is a projective R-module, the universal lifting property yields a map i: A ---,)-F(A) so that:rri = lA, that is, A is a direct summand of the free module F(A).

Example 2.2.2 Over many nice rings (;E, fields, division rings, ...) everyprojective module is in fact a free module. Here are two examples to showthat this is not always the case:

1. If R = Rl X Rz, then P = Rl X 0 and 0 x Rz are projective because theirsum is R. P is not free because (0, I)P = O. This is true, for example,when R is the ring ;E/6 = ;E/2 x ;E/3.

2. Consider the ring R = M n (F) of n x n matrices over a field F, actingon the left on the column vector space V = F n . As a left R-module, Ris the direct sum of its columns, each of which is the left R-module V.Hence R s:: V EfJ ••• EfJ V, and V is a projective R-module. Since any freeR-module would have dimension dnz over F for some cardinal numberd, and dimF(V) =n, V cannot possibly be free over R.

Remark The category A of finite abelian groups is an example of an abeliancategory that has no projective objects. We say that A has enough projectivesif for every object A of A there is a surjection P ---,)- A with P projective.

34 Derived Functors

Here is another characterization of projective objects in A:

Lemma 2.2.3 M is projective ijf HomA(M, -) is an exact functor. That is,ijfthe sequence ojgroups

0---,)- Hom(M, A) ---,)- Hom(M, B)~ Hom(M, C) ---,)- 0

is exact for every exact sequence 0 ---,)- A ---,)- B ---,)- C ---,)- 0 in A

Proof Suppose that Hom(M, -) is exact and that we are given a surjec-tion g: B ---,)- C and a map y: M ---,)- C. We can lift y E Hom(M, C) to {3 EHom(M, B) such that y = g*{3 = g 0 f3 because g* is onto. Thus M has theuniversal lifting property, that is, it is projective. Conversely, suppose M isprojective. In order to show that Hom(M, -) is exact, it suffices to show thatg* is onto for every short exact sequence as above. Given y E Hom(M, C),the universal lifting property of M gives f3 E Hom(M, B) so that y = g 0 f3 =g*({3), that is, g* is onto.

A chain complex P in which each Pn is projective in A is called a chaincomplex ofprojectives. It need not be a projective object in Ch.

Exercise 2.2.1 Show that a chain complex P is a projective object in Chif and only if it is a split exact complex of projectives. Hint: To see that Pmust be split exact, consider the surjection from cone(idp) to P[-l]. To seethat split exact complexes are projective objects, consider the special case0---,)- PI ~ Po ---,)- O.

Exercise 2.2.2 Use the previous exercise 2.2.1 to show that if A has enoughprojectives, then so does the category Ch(A) of chain complexes over A.

Definition 2.2.4 Let M be an object of A. A left resolution of M is a com-plex p. with Pi = 0 for i < 0, together with a map E: Po ---,)- M so that theaugmented complex

d d d E···~P2~PI~Po~M---,)-0

is exact. It is a projective resolution if each Pi is projective.

Lemma 2.2.5 Every R -module M has a projective resolution. More gener-ally, ifan abelian category A has enough projectives, then every object M inA has a projective resolution.

2.2 Projective Resolutions 35

0 0 0 0

\. / \. /M3 M I

/ \. d /d\. d... ~ P3 ) Pz ) PI ) Po ~M---+O\. / \. /

Mz Mo

/ \. / \.0 0 0 0

Figure 2.1. Forming a resolution by splicing.

Proof Choose a projective Po and a surjection EO: Po --,)- M, and set Mo =ker(Eo). Inductively, given a module Mn-[, we choose a projective Pn anda surjection En: Pn --,)- Mn-I. Set Mn = ker(En), and let dn be the compositePn --,)- Mn-I --,)- Pn-I. Since dn(Pn) = Mn-I = ker(dn-I), the chain complexp. is a resolution of M. (See Figure 2.1.)

Exercise 2.2.3 Show that if p. is a complex of projectives with Pi = 0 fori < 0, then a map E: Po --,)- M giving a resolution for M is the same thing asa chain map E: p. --,)- M, where M is considered as a complex concentrated indegree zero.

Comparison Theorem 2.2.6 Let p.~ M be a projective resolution of M

and f': M --,)- N a map in A Then for every resolution Q.~ N of N thereis a chain map f: p. --,)- Q. lifting f' in the sense that 1] 0 fo = f' 0 E. Thechain map f is unique up to chain homotopy equivalence.

E

. .. --+ P2 --+ PI --+ Po --+ M --+ 0

1/'. .. --+ Q2 --+ QI --+

T/Qo --+ N --+ 0

Porism 2.2.7 The proof will make it clear that the hypothesis that P --,)- M bea projective resolution is too strong. It suffices to be given a chain complex

with the Pi projective. Then for every resolution Q --,)- N of N, every mapM --,)- N lifts to a map P --,)- Q, which is unique up to chain homotopy. This

36 Derived Functors

stronger version of the Comparison Theorem will be used in section 2.7 toconstruct the external product for Tor.

Proof We will construct the In and show their uniqueness by induction on n,thinking of 1-1 as f'. Inductively, suppose Ii has been constructed for i :S nso that li-Id = dk In order to construct In+l we consider the n-cycles ofP and Q. If n = -1, we set Z-I(P) = M and Z-I(Q) = N; if n 2: 0, thefact that In-Id = din means that In induces a map I~ from Zn(P) to Zn(Q).Therefore we have two diagrams with exact rows

d d... ---* Pn+l ---* ZnCP) ---*0 0---* ZnCP) ---* Pn ---* Pn-I

31 llh and llh lin lln-Id

... ---* Qn+1 ---* ZnCQ) ---* 0 0---* ZnCQ) ---* Qn ---* Qn-l

The universal lifting property of the projective Pn+l yields a map In+l from

Pn+l to Qn+l, so that dln+l = I~d = Ind. This finishes the inductive step andproves that the chain map I: P ---7 Q exists.

To see uniqueness of I up to chain homotopy, suppose that g: P ---7 Q isanother lift of f' and set h = I - g; we will construct a chain contraction{sn: Pn ---7 Qn+d of h by induction on n. If n < 0, then Pn = 0, so we setSn = 0. If n = 0, note that since 1]hO = E(f' - I') = 0, the map ho sends Po toZo(Q) = d(QI). We use the lifting property of Po to get a map so: Po ---7 QIso that ho = dso = dso + s-ld. Inductively, we suppose given maps si(i < n)so that dSn-1 = hn-l - Sn-2d and consider the map hn - Sn-Id from Pn toQn' We compute that

d(hn - Sn-Id) = dhn - (hn-l - Sn-2d)d = (dh - hd) + sn-2dd = 0.

Therefore hn - Sn-Id lands in Zn(Q), a quotient of Qn+l. The lifting propertyof Pn yields the desired map Sn: Pn ---7 Qn+l such that dSn = hn - Sn-Id.

d dPn -+ Pn-l -+ Pn-2

and

d

Qn+l -+ Zn(Q) -+ °Here is another way to construct projective resolutions. It is called the Horse-shoe Lemma because we are required to fill in the horseshoe-shaped diagram.

2.2 Projective Resolutions

Horseshoe Lemma 2.2.8 Suppose given a commutative diagram

0

1

Pz p{ p.'E'

A'... ---+ ---+ 0 ---+ ---+ 0

liAA

lJrA

p!j p lI P~'E"

A"... ---+ 1 ---+ ---+ ---+ 0

10

37

where the column is exact and the rows are projective resolutions. Set Pn =P~ EfJ P~'. Then the Pn assemble to form a projective resolution P of A, andthe right-hand column lifts to an exact sequence ofcomplexes

0----+ P'~ P~ P"----+O,

where in: P~ ----+ Pn and :Trn: Pn ----+ p~' are the natural inclusion and projection,respectively.

Proof Lift E" to a map p~' ----+ A; the direct sum of this with the mapiAE': P~ ----+ A gives a map E: Po ----+ A. The diagram (*) below commutes.

0 0 0

1 1 1

ker(E') P~E'

A'o ---+ ---+ ---+ ---+ 0

1 1 1(*)

Eo ---+ ker(E) ---+ Po ---+ A ---+ 0

1 1 1

ker(E") p'"E"

A"o ---+ ---+ 0 ---+ ---+ 0

1 1 10 0 0

38 Derived Functors

The right two columns of (*) are short exact sequences. The Snake Lemma1.3.2 shows that the left column is exact and that coker(E) = 0, so that Po mapsonto A. This finishes the initial step and brings us to the situation

0

1pi d' ker(E/)... ---+ 1 ---+ ---+ 0

1ker(E)

1pll d" ker(E")... ---+ 1 ---+ ---+ 0

1o.

The filling in of the "horseshoe" now proceeds by induction.

Exercise 2.2.4 Show that there are maps An: P~' ---,)- P~-l so that

[d

l A]d = 0 d" '

I [pi] = [dl(pl) + A(pll)]i.e., d p" d" (p") .

2.3 Injective Resolutions

An object I in an abelian category A is injective if it satisfies the followinguniversal lifting property: Given an injection f: A ---,)- B and a map a: A ---,)- I,there exists at least one map f3: B ---,)- I such that a = f3 0 f.

We say that A has enough injectives if for every object A in A there is aninjection A ---,)- I with I injective. Note that if {Ia} is a family of injectives,then the product IT I a is also injective. The notion of injective module wasinvented by R. Baer in 1940, long before projective modules were thought of.

2.3 Injective Resolutions 39

Baer's Criterion 2.3.1 A right R-module E is injective if and only if forevery right ideal J ofR, every map J ---,)- E can be extended to a map R ---,)- E.

Proof The "only if" direction is a special case of the definition of injective.Conversely, suppose given an R-module B, a submodule A and a map a: A---,)-E. Let E be the poset of all extensions a ' : A' ---,)- E of a to an intermediatesubmodule A S; A' S; B; the partial order is that a ' :s a" if a" extends a ' .By Zorn's lemma there is a maximal extension a' : A' ---+ E in E; we have toshow that A' = B. Suppose there is some b E B not in A'. The set J = {r E

R: br E A'} is a right ideal of R. By assumption, the map J~ A'~ Eextendt'to a map f: R ---+ E. Let A" be the submodule A' + bR of Banddefine a": A" ---,)- E by

a"(a + br) = a'(a) + f(r), a E A' and r E R.

This is well defined because a'(br) = f(r) for br in A' n bR, and a" extendsai, contradicting the existence of b. Hence A' = B.

Exercise 2.3.1 Let R = 71../m. Use Baer's criterion to show that R is an in-jective R-module. Then show that 71../d is not an injective R-module whendim and some prime p divides both d and mid. (The hypothesis ensures that

71../m i= 71../d EfJ 71../e.)

Corollary 2.3.2 Suppose that R = 71.., or more generally that R is a principalideal domain. An R-module A is injective iff it is divisible, that is, for everyr i= 0 in R and every a E A, a = br for some b E A.

Example 2.3.3 The divisible abelian groups Q and 71.. p oo = 71..[ iJ/71.. are in-

jective (71..[i J is the group of rational numbers of the form a/pn, n 2: 1). Everyinjective abelian group is a direct sum of these [KapIAB,section 5]. In partic-ular, the injective abelian group Q/71.. is isomorphic to EfJ71..p oo •

We will now show that Ab has enough injectives. If A is an abelian group,let I (A) be the product of copies of the injective group Q/71.., indexed by theset HomAb(A, Q/71..). Then I (A) is injective, being a product ofinjectives, andthere is a canonical map eA: A ---,)- I (A). This is our desired injection of A intoan injective by the following exercise.

Exercise 2.3.2 Show that eA is an injection. Hint: If a E A, find a mapf: a 71.. ---+ Q/71.. with f(a) i= 0 and extend f to a map f': A ---,)- Q/71...

40 Derived Functors

Exercise 2.3.3 Show that an abelian group A is zero iff HomAb(A, 0../71..) =O.

Now it is a fact, easily verified, that if A is an abelian category, then theopposite category AOP is also abelian. The definition of injective is dual tothat of projective, so we immediately can deduce the following results (2.3.4-2.3.7) by arguing in AOP.

Lemma 2.3.4 The following are equivalent for an object I in an abeliancategory A:

]. ] is injective in A2. ] is projective in AOP.3. The contravariant functor HomA(-, 1) is exact, that is, it takes short

exact sequences in A to short exact sequences in Ab.

Definition 2.3.5 Let M be an object of A A right resolution of M is acochain complex ]. with ]i = 0 for i < 0 and a map M ---,)- ]0 such that theaugmented complex

is exact. This is the same as a cochain map M ---,)- ]', where M is considered asa complex concentrated in degree O. It is called an injective resolution if eachIi is injective.

Lemma 2.3.6 If the abelian category A has enough injectives, then everyobject in A has an injective resolution.

Comparison Theorem 2.3.7 Let N ---,)- ]. be an injective resolution ofNand1': M --* N a map in A Thenfor every resolution M ---,)- E' there is a cochainmap F: E' ---,)- I' lifting 1'. The map f is unique up to cochain homotopyequivalence.

O-+N-+ -+ ...

Exercise 2.3.4 Show that ] is an injective object in the category of chaincomplexes iff ] is a split exact complex of injectives. Then show that if Ahas enough injectives, so does the category Ch(A) of chain complexes overA. Hint: Ch(A)OP ~ Ch(AOP).

2.3 Injective Resolutions 41

We now show that there are enough injective R-modules for every ringR. Recall that if A is an abelian group and B is a left R-module, thenHomAb(B, A) is a right R-module via the rule fr: b 1--+ f(rb).

Lemma 2.3.8 For every right R -module M, the natural map

is an isomorphism, where (rf)(m) is the map r 1--+ f(mr) .

.~

Proof We define a map J1, backwards as follows: If g: M ---+ Hom(R, A) isan R-module map, J1,g is the abelian group map sending m to g(m)(l). Sincer(J1,g) = g and J1,r(f) = f (check this!), r is an isomorphism.

Definition 2.3.9 A pair of functors L: A ---+ B and R: B ---+ A are adjoint ifthere is a natural bijection for all A in A and B in B:

r = rAB : Homs(L(A), B)~ HomA(A, R(B».

Here "natural" means that for all f: A ---+ A' in A and g: B ---+ B' in B thefollowing diagram commutes:

Homs(L(A'), B)

1·HomA(A', R(B»

Lf*--+

f*----+

Homs(L(A), B)

1·HomA(A, R(B»

Homs(L(A), B')

HomA(A, R(B'».

We call L the left adjoint and R the right adjoint of this pair. The above lemmastates that the forgetful functor from mod-R to Ab has HomAb(R, -) as itsright adjoint.

Proposition 2.3.10 If an additive functor R: B ---+ A is right adjoint to anexact functor L: A ---+ B and I is an injective object of B, then R(l) is aninjective object of A (We say that R preserves injectives.)

Dually, if an additive functor L: A ---+ B is left adjoint to an exact functorR: B ---+ A and P is a projective object of A, then L(P) is a projective objectof B. (We say that L preserves projectives.)

Proof We must show that HomA(-, R(l» is exact. Given an injectionf: A ---+ A' in A the diagram

42 Derived Functors

Homs(L(A'), l)Lf*

Homs(L(A), l)~

1= 1=HomA(A', R(l»

f*------+ HomA(A, R(I»

commutes by naturality of r. Since L is exact and I is injective, the topmap Lf* is onto. Hence the bottom map f* is onto, proving that R(l) is aninjective object in A.

Corollary 2.3.11 If I is an injective abelian group, then HomAb(R, l) is aninjective R-module.

Exercise 2.3.5 If M is an R-module, let I (M) be the product of copies of10 = HomAb(R, (Q/::E), indexed by the set HomR(M, 10)' There is a canonicalmap eM: M ---,)- I (M); show that eM is an injection. Being a product of injec-tives, I (M) is i