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Joseph J. Rotman

An Introduction toHomological AlgebraSecond Edition

123

Joseph J. RotmanDepartment of MathematicsUniversity of Illinois at Urbana-ChampaignUrbana IL 61801USArotman@math.uiuc.edu

Editorial board:Sheldon Axler, San Francisco State UniversityVincenzo Capasso, Università degli Studi di MilanoCarles Casacuberta, Universitat de BarcelonaAngus MacIntyre, Queen Mary, University of LondonKenneth Ribet, University of California, BerkeleyClaude Sabbah, CNRS, École PolytechniqueEndre Süli, University of OxfordWojbor Woyczynski, Case Western Reserve University

ISBN: 978-0-387-24527-0 e-ISBN: 978-0-387-68324-9DOI 10.1007/978-0-387-68324-9

Library of Congress Control Number: 2008936123

Mathematics Subject Classification (2000): 18-01

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To the memory of my mother

Rose Wolf Rotman

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . x

How to Read This Book . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1 Introduction1.1 Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . 11.2 Categories and Functors . . . . . . . . . . . . . . . . . . . . . 71.3 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 2 Hom and Tensor2.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . 692.2.1 Adjoint Isomorphisms . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 3 Special Modules3.1 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . 983.2 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . 1153.3 Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.3.1 Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter 4 Specific Rings4.1 Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . 1544.2 von Neumann Regular Rings . . . . . . . . . . . . . . . . . . 1594.3 Hereditary and Dedekind Rings . . . . . . . . . . . . . . . . . 160

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4.4 Semihereditary and Prüfer Rings . . . . . . . . . . . . . . . . 1694.5 Quasi-Frobenius Rings . . . . . . . . . . . . . . . . . . . . . 1734.6 Semiperfect Rings . . . . . . . . . . . . . . . . . . . . . . . . 1794.7 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.8 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter 5 Setting the Stage5.1 Categorical Constructions . . . . . . . . . . . . . . . . . . . . 2135.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295.3 Adjoint Functor Theorem for Modules . . . . . . . . . . . . . 2565.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2735.4.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2885.4.2 Sheaf Constructions . . . . . . . . . . . . . . . . . . . . . . . 2945.5 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . 3035.5.1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Chapter 6 Homology6.1 Homology Functors . . . . . . . . . . . . . . . . . . . . . . . 3236.2 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . 3406.2.1 Left Derived Functors . . . . . . . . . . . . . . . . . . . . . . 3436.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.2.3 Covariant Right Derived Functors . . . . . . . . . . . . . . . 3646.2.4 Contravariant Right Derived Functors . . . . . . . . . . . . . 3696.3 Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . 3776.3.1 Čech Cohomology . . . . . . . . . . . . . . . . . . . . . . . 3846.3.2 Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . . . 392

Chapter 7 Tor and Ext7.1 Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4047.1.1 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4127.1.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157.2 Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4187.2.1 Baer Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4287.3 Cotorsion Groups . . . . . . . . . . . . . . . . . . . . . . . . 4387.4 Universal Coefficients . . . . . . . . . . . . . . . . . . . . . . 448

Chapter 8 Homology and Rings8.1 Dimensions of Rings . . . . . . . . . . . . . . . . . . . . . . 4538.2 Hilbert’s Syzygy Theorem . . . . . . . . . . . . . . . . . . . 4678.3 Stably Free Modules . . . . . . . . . . . . . . . . . . . . . . 4768.4 Commutative Noetherian Local Rings . . . . . . . . . . . . . 484

Contents ix

Chapter 9 Homology and Groups9.1 Group Extensions . . . . . . . . . . . . . . . . . . . . . . . . 4959.1.1 Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . 5009.1.2 General Extensions and Cohomology . . . . . . . . . . . . . . 5049.1.3 Stabilizing Automorphisms . . . . . . . . . . . . . . . . . . . 5149.2 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . 5199.3 Bar Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . 5259.4 Group Homology . . . . . . . . . . . . . . . . . . . . . . . . 5359.4.1 Schur Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . 5419.5 Change of Groups . . . . . . . . . . . . . . . . . . . . . . . . 5599.5.1 Restriction and Inflation . . . . . . . . . . . . . . . . . . . . . 5649.6 Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5729.7 Tate Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 5809.8 Outer Automorphisms of p-Groups . . . . . . . . . . . . . . . 5879.9 Cohomological Dimension . . . . . . . . . . . . . . . . . . . 5919.10 Division Rings and Brauer Groups . . . . . . . . . . . . . . . 595

Chapter 10 Spectral Sequences10.1 Bicomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 60910.2 Filtrations and Exact Couples . . . . . . . . . . . . . . . . . . 61610.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 62410.4 Homology of the Total Complex . . . . . . . . . . . . . . . . 62810.5 Cartan–Eilenberg Resolutions . . . . . . . . . . . . . . . . . 64710.6 Grothendieck Spectral Sequences . . . . . . . . . . . . . . . . 65610.7 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66010.8 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66610.9 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67510.10 Künneth Theorems . . . . . . . . . . . . . . . . . . . . . . . 678

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

Special Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

Preface to the Second Edition

Homological Algebra has grown in the nearly three decades since the first edi-tion of this book appeared in 1979. Two books discussing more recent resultsare Weibel, An Introduction to Homological Algebra, 1994, and Gelfand–Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfandand Manin divide the history of Homological Algebra into three periods: thefirst period ended in the early 1960s, culminating in applications of Homo-logical Algebra to regular local rings. The second period, greatly influencedby the work of A. Grothendieck and J.-P. Serre, continued through the 1980s;it involves abelian categories and sheaf cohomology. The third period, in-volving derived categories and triangulated categories, is still ongoing. Bothof these newer books discuss all three periods (see also Kashiwara–Schapira,Categories and Sheaves). The original version of this book discussed the firstperiod only; this new edition remains at the same introductory level, but itnow introduces the second period as well. This change makes sense peda-gogically, for there has been a change in the mathematics population since1979; today, virtually all mathematics graduate students have learned some-thing about functors and categories, and so I can now take the categoricalviewpoint more seriously.

When I was a graduate student, Homological Algebra was an unpopularsubject. The general attitude was that it was a grotesque formalism, boringto learn, and not very useful once one had learned it. Perhaps an algebraictopologist was forced to know this stuff, but surely no one else should wastetime on it. The few true believers were viewed as workers at the fringe ofmathematics who kept tinkering with their elaborate machine, smoothing outrough patches here and there.

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Preface to the Second Edition xi

This attitude changed dramatically when J.-P. Serre characterized regularlocal rings using Homological Algebra (they are the commutative noetherianlocal rings of “finite global dimension”), for this enabled him to prove thatany localization of a regular local ring is itself regular (until then, only spe-cial cases of this were known). At the same time, M. Auslander and D. A.Buchsbaum also characterized regular local rings, and they went on to com-plete work of M. Nagata by using global dimension to prove that every regularlocal ring is a unique factorization domain. As Grothendieck and Serre revolu-tionized Algebraic Geometry by introducing schemes and sheaves, resistanceto Homological Algebra waned. Today, it is just another standard tool in amathematician’s kit. For more details, we recommend C. A. Weibel’s chapter,“History of Homological Algebra,” in the book of James, History of Topology.

Homological Algebra presents a great pedagogical challenge for authorsand for readers. At first glance, its flood of elementary definitions (whichoften originate in other disciplines) and its space-filling diagrams appear for-bidding. To counter this first impression, S. Lang set the following exerciseon page 105 of his book, Algebra:

Take any book on homological algebra and prove all the theoremswithout looking at the proofs given in that book.

Taken literally, the statement of the exercise is absurd. But its spirit is ab-solutely accurate; the subject only appears difficult. However, having rec-ognized the elementary character of much of the early material, one is oftentempted to “wave one’s hands”: to pretend that minutiae always behave well.It should come as no surprise that danger lurks in this attitude. For this rea-son, I include many details in the beginning, at the risk of boring some readersby so doing (of course, such readers are free to turn the page). My intent istwofold: to allow readers to see that complete proofs can, in fact, be writtencompactly; to give readers the confidence to believe that they, too, can writesuch proofs when, later, the lazy author asks them to. However, we must cau-tion the reader; some “obvious” statements are not only false, they may noteven make sense. For example, if R is a ring and A and B are left R-modules,then HomR(A, B) may not be an R-module at all; and, if it is a module, it issometimes a left module and sometimes a right module. Is an alleged functionwith domain a tensor product well-defined? Is an isomorphism really natural?Does a diagram really commute? After reading the first three chapters, thereader should be able to deal with such matters efficiently.

This book is my attempt to make Homological Algebra lovable, and Ibelieve that this requires the subject be presented in the context of other math-ematics. For example, Chapters 2, 3, and 4 form a short course in module the-ory, investigating the relation between a ring and its projective, injective, andflat modules. Making the subject lovable is my reason for delaying the formalintroduction of homology functors until Chapter 6 (although simplicial and

xii Preface to the Second Edition

singular homology do appear in Chapter 1). Many readers wanting to learnHomological Algebra are familiar with the first properties of Hom and tensor;even so, they should glance at the first chapters, for there may be some unfa-miliar items therein. Some category theory appears throughout, but it makes amore brazen appearance in Chapter 5, where we discuss limits, adjoint func-tors, and sheaves. Although presheaves are introduced in Chapter 1, we do notintroduce sheaves until we can observe that they usually form an abelian cat-egory. Chapter 6 constructs homology functors, giving the usual fundamentalresults about long exact sequences, natural connecting homomorphisms, andindependence of choices of projective, injective, and flat resolutions used toconstruct them. Applications of sheaves are most dramatic in the context ofSeveral Complex Variables and in Algebraic Geometry; alas, I say only a fewwords pointing the reader to appropriate texts, but there is a brief discussionof the Riemann–Roch Theorem over compact Riemann surfaces. Chapters 7,8, and 9 consider the derived functors of Hom and tensor, with applications toring theory (via global dimension), cohomology of groups, and division rings.

Learning Homological Algebra is a two-stage affair. First, one must learnthe language of Ext and Tor and what it describes. Second, one must beable to compute these things and, often, this involves yet another language,that of spectral sequences. Chapter 10 develops spectral sequences via exactcouples, always taking care that bicomplexes and their multiple indices arevisible because almost all applications occur in this milieu.

A word about notation. I am usually against spelling reform; if everyoneis comfortable with a symbol or an abbreviation, who am I to say otherwise?However, I do use a new symbol to denote the integers mod m because, nowa-days, two different symbols are used: Z/mZ and Zm . My quarrel with thefirst symbol is that it is too complicated to write many times in an argument;my quarrel with the simpler second symbol is that it is ambiguous: when p isa prime, the symbol Zp often denotes the p-adic integers and not the integersmod p. Since capital I reminds us of integers and since blackboard font is incommon use, as in Z,Q,R,C, and Fq , I denote the integers mod m by Im .

It is a pleasure to thank again those who helped with the first edition.I also thank the mathematicians who helped with this revision: MatthewAndo, Michael Barr, Steven Bradlow, Kenneth S. Brown, Daniel Grayson,Phillip Griffith, William Haboush, Aimo Hinkkanen, Ilya Kapovich, RandyMcCarthy, Igor Mineyev, Thomas A. Nevins, Keith Ramsay, Derek Robin-son, and Lou van den Dries. I give special thanks to Mirroslav Yotov whonot only made many valuable suggestions improving the entire text but who,having seen my original flawed subsection on the Riemann–Roch Theorem,patiently guided my rewriting of it.

Joseph J. RotmanMay 2008Urbana IL

How to Read This Book

Some exercises are starred; this means that they will be cited somewherein the book, perhaps in a proof.

One may read this book by starting on page 1, then continuing, page bypage, to the end, but a mathematics book cannot be read as one reads a novel.Certainly, this book is not a novel! A reader knowing very little homology (ornone at all) should begin on page 1 and then read only the portion of Chapter 1that is unfamiliar. Homological Algebra developed from Algebraic Topology,and it is best understood if one knows its origins, which are described in Sec-tions 1.1 and 1.3. Section 1.2 introduces categories and functors; at the outset,the reader may view this material as a convenient language, but it is very im-portant for the rest of the text.

After Chapter 1, one could go directly to Chapter 6, Homology, but I don’tadvise it. It is not necessary to digest all the definitions and constructions inthe first five chapters before studying homology, but one should read enoughto become familiar with the point of view being developed, returning to reador reread items in earlier chapters when necessary.

I believe that it is wisest to learn homology in a familiar context in which itcan be applied. To illustrate, one of the basic constructs in defining homologyis that of a complex: a sequence of homomorphisms

→ Cn+1 dn+1−→ Cn dn−→ Cn−1 →in which dndn+1 = 0 for all n ∈ Z. There is no problem digesting sucha simple definition, but one might wonder where it comes from and why itis significant. The reader who has seen some Algebraic Topology (as in ourChapter 1) recognizes a geometric reason for considering complexes. But this

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observation only motivates the singular complex of a topological space. Amore perspicacious reason arises in Algebra. Every R-module M is a quo-tient of a free module; thus, M ∼= F/K , where F is free and K ⊆ F is thesubmodule of relations; that is, 0 → K → F → M → 0 is exact. If Xis a basis of F , then (X | K ) is called a presentation of G. Theoretically,(X | K ) is a complete description of M (to isomorphism) but, in practice, it isdifficult to extract information about M from a presentation of it. However, ifR is a principal ideal domain, then every submodule of a free module is free,and so K has a basis, say, Y [we also say that (X | Y ) is a presentation]. Forexample, the canonical forms for matrices over a field k arise from presenta-tions of certain k[x]-modules. For a general ring R, we can iterate the ideaof presentations. If M ∼= F/K , where F is free, then K ∼= F1/K1 for somefree module F1 (thus, K1 can be thought of as relations among the relations;Hilbert called them syzygies). Now 0 → K1 → F1 → K → 0 is exact;splicing it to the earlier exact sequence gives exactness of

0 → K1 → F1 d−→ F → M → 0(where d : F1 → F is the composite F1 → K ⊆ F), for im d = K =ker(F → M). Repeat: K1 ∼= F2/K2 for some free F2, and continuing theconstruction above gives an infinitely long exact sequence of free modulesand homomorphisms, called a resolution of M , which serves as a generalizedpresentation. A standard theme of Homological Algebra is to replace a mod-ule by a resolution of it. Resolutions are exact sequences, and exact sequencesare complexes (if im dn+1 = ker dn , then dndn+1 = 0). Why do we need theextra generality present in the definition of complex? One answer can be seenby returning to Algebraic Topology. We are interested not only in the ho-mology groups of a space, but also in its cohomology groups, and these ariseby applying contravariant Hom functors to the singular complex. In Algebra,the problem of classifying group extensions also leads to applying Hom func-tors to resolutions. Even though resolutions are exact sequences, they becomemere complexes after applying Hom. Homological Algebra is a tool that ex-tracts information from such sequences. As the reader now sees, the contextis interesting, and it puts flesh on abstract definitions.

Contents(Preface to the Second Edition!!)(How to Read This Book!!)

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