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xiiPreface to the Second Editionsingular 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 unfamiliar items therein. Some category theory appears throughout, but it makes amore brazen appearance in Chapter 5, where we discuss limits, adjoint functors, and sheaves. Although presheaves are introduced in Chapter 1, we do notintroduce sheaves until we can observe that they usually form an abelian category. 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, nowadays, 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 Z p 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 Robinson, 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 BookSome 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 Sections 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 important 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 homomorphismsdn 1dn Cn 1 Cn Cn 1 in which dn dn 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 thisxiii

xivHow to Read This Bookobservation only motivates the singular complex of a topological space. Amore perspicacious reason arises in Algebra. Every R-module M is a quotient 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 presentations 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 /K 1 for somefree module F1 (thus, K 1 can be thought of as relations among the relations;Hilbert called them syzygies). Now 0 K 1 F1 K 0 is exact;splicing it to the earlier exact sequence gives exactness ofd0 K 1 F1 F M 0(where d : F1 F is the composite F1 K F), for im d K ker(F M). Repeat: K 1 F2 /K 2 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 module by a resolution of it. Resolutions are exact sequences, and exact sequencesare complexes (if im dn 1 ker dn , then dn dn 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 homology 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 functors to resolutions. Even though resolutions are exact sequences, they becomemere complexes after applying Hom. Homological Algebra is a tool that extracts information from such sequences. As the reader now sees, the contextis interesting, and it puts flesh on abstract definitions.

1Introduction1.1 Simplicial HomologyHomological Algebra is an outgrowth of Algebraic Topology, and so we beginwith a historical discussion of the origins of homology in topology. Let X bean open set in the plane, and fix points a and b in X . Given a path1 β in XFig. 1.1 Two paths.from a to b, and given a pair P(x, y) and Q(x, y) of real-valued, continuouslydifferentiable functions on X , one wants to evaluate the line integral P d x Q dy.β1Let I [0, 1] be the closed unit interval. A path β in X from a to b is a continuousfunction β : I X with β(0) a and β(1) b; thus, a path is a parametrized curve. Apath β is closed at a if β(0) a β(1) or, what is the same thing, if β is a continuousmap of the unit circle S 1 into X with f : (1, 0) a.J.J. Rotman, An Introduction to Homological Algebra, Universitext,c Springer Science Business Media LLC 2009DOI 10.1007/978-0-387-68324-9 1, 1

2IntroductionCh. 1It is wise to regard β as a finite union of paths, for β may be only piecewise smooth; for example, it may be a polygonal path. For the rest of thisdiscussion, we ignore (necessary) differentiability hypotheses.A fundamentalquestion asks when the line integral is independent of thepath β: is β P d x Q dy β P d x Q dy if β is another path in X froma to b? If γ is the closed path γ β β [that is, γ goes from a to b via β,and then goes back via β from b to a, where β (t) β (1 t)], then theintegral is independent of the paths β and β if and only if γ P d x Q dy 0. Suppose there are “bad” points z 1 , . . . , z n deleted from X (for example, ifP or Q has a singularity at some z i ). The line integral along γ is affected byFig. 1.2 Green’s theorem.whether any of these bad points lies inside γ . In Fig. 1.2, each path γi is asimple closed path in X (that is, γi is a homeomorphism from the unit circleS 1 to im γi R2 ) containing z i inside, while all the other z j are outsideγi . If γ is oriented counterclockwise and each γi is oriented clockwise, thenGreen’s Theorem states that n P Q d xd y,P d x Q dy P d x Q dy x yγγiRi 1where R is the shaded two-dimensional region in Fig. 1.2. One is tempted tonwrite i 1 γi P d x Q dy more concisely, as γi P d x Q dy. Moreiover, instead of mentioning orientations explicitly, we may write sums anddifferences of paths, where a negative coefficient reverses direction. For simple paths, the notions of “inside” and “outside” make sense.2 A path maywind around some z i several times along γi , and so it makes sense to writeformal Z-linear combinations of paths; that is, we may allow integer coefficients other than 1. Recall that if Y is any set, then the free abelian group2 The Jordan curve theorem says that if γ is a simple closed path in the plane R2 ,then the complement R2 im γ has exactly two components, one of which is bounded.The bounded component is called the inside of γ , and the other (necessarily unbounded)component is called the outside.

1.1 Simplicial Homology3with basis Y is an abelian group G[Y ] in which each element g G[Y ] hasa unique expression of the form g y Y m y y, where m y Z and onlyfinitely many m y 0 (see Proposition 2.33). In particular, Green’s Theoreminvolves the free abelian group G[Y ] with basis Y being the (huge) set of allpaths σ : I X . Intuitively, elements of G[Y ] are unions of finitely many(not necessarily closed) paths σ in X .Consider those ordered pairs (P, Q) of functions X R satisfying Q/ x P/ y. The double integral in Green’s Theorem vanishes for suchfunction pairs: mγ m i γi P d x Q dy 0. An equivalence relation oni m i σi G[Y ], call β andG[Y ] suggests itself. If β m i σi and β β equivalent if, for all (P, Q) with Q/ x P/ y, the values of their lineintegrals agree: P d x Q dy P d x Q dy.ββ The equivalence class of β is called its homology class, from the Latin wordhomologia meaning agreement. If β β m σ σ , where m σ 0 im σ is the boundary of a two-dimensional region in X , then β β P d x Q dy 0; that is, β P da Q dy β P da Q dy. In short, integration is independentof paths lying in the same homology class.Homology can be defined without using integration of function pairs.Poincaré recognized that whether a topological space X has different kinds ofholes is a kind of connectivity. To illustrate, suppose that X is a finite simplicial complex; that is, X can be “triangulated” into finitely many n-simplexesfor n 0, where 0-simplexes are points, say, v1 , . . . , vq , 1-simplexes arecertain edges [vi , v j ] (with endpoints vi and v j ), 2-simplexes are certain triangles [vi , v j , vk ] (with vertices vi , v j , vk ), 3-simplexes are certain tetrahedra [vi , v j , vk , v ], and there are higher-dimensional analogs [vi0 , . . . , vin ]for larger n. The question to ask is whether a union of n-simplexes in Xthat “ought” to be the boundary of some (n 1)-simplex actually is such aboundary. For example, when n 0, two points a and b in X ought to bethe boundary (endpoints) of a path in X ; if, for each pair of points a, b X ,there is a path in X from a to b, then X is called path connected; if thereis no such path, then X has a 0-dimensional hole. For an example of a onedimensional hole, let X be the punctured plane; that is, X is the plane withthe origin deleted. The perimeter of a triangle ought to be the boundary ofa 2-simplex, but it is not if contains the origin in its interior; thus, X has aone-dimensional hole. If the interior of X were missing a line segment containing the origin, or even a small disk containing the origin, this hole wouldstill be one-dimensional; we are not considering the size of the hole, but thesize of the possible boundary. We must keep our eye on the doughnut and notupon the hole!

4IntroductionCh. 1The triangle [a, b, c] in Fig. 1.3 has vertices a, b, c and edges [b, c], [a, c],[a, b]; its boundary [a, b, c] should be [b, c] [c, a] [a, c]. But edges areoriented; think of [a, c] as a path from a to c and [c, a] [a, c] as thereverse path from c to a. Thus, the boundary is [a, b, c] [b, c] [a, c] [a, b].Assume now that paths can be added and subtracted; that is, view paths aslying in the free abelian group C1 (X ) with basis all 1-simplexes. Then [a, b, c] [b, c] [a, c] [a, b].Similarly, define the boundary of [a, b] to be [a, b] b a C0 (X ), thefree abelian group with basis all 0-simplexes, and define the boundary of apoint to be 0. Note that ( [a, b, c]) ([b, c] [a, c] [a, b]) (c b) (c a) (b a) 0.a b cdFig. 1.3.The rectangle .The rectangle with vertices a, b, c, d is the union of two triangles,namely, [a, b, c] [a, c, d]; let us check its boundary. If we assume that is a homomorphism, then ( ) [a, b, c] [a, c, d] [a, b, c] [a, c, d] [b, c] [a, c] [a, b] [c, d] [a, d] [a, c] [a, b] [b, c] [c, d] [a, d] [a, b] [b, c] [c, d] [d, a].Note that the diagonal [a, c] occurred twice, with different signs, and so itcanceled, as it should. We have seen that the formalism suggests the use ofsigns to describe boundaries as certain alternating sums of edges or points.Such ideas lead to the following construction. For each n 0, consider allformal linear combinations of n-simplexes; that is, form the free abelian groupCn (X ) with basis all n-simplexes [vi0 , . . . , vin ], and call such linear combinations simplicial n-chains; define C 1 (X ) {0}. Some n-chains ought to be

1.1 Simplicial Homology5boundaries of some union of (n 1)-simplexes; call them simplicial n-cycles.For example, adding the three edges of a triangle (with appropriate choice ofsigns) is a 1-cycle, as is the sum of the four outer edges of a rectangle. Certainsimplicial n-chains actually are boundaries, and these are called simplicial nboundaries. For example, if is a triangle in the punctured plane X , then thealternating sum of the edges of is a 1-cycle; this 1-cycle is a 1-boundary ifand only if the origin does not lie in the interior of . Here are the precisedefinitions.Definition.Let X be a finite simplicial complex. If n 1, define n : Cn (X ) Cn 1 (X )by n [v0 , . . . , vn ] n ( 1)i [v0 , . . . , vi , . . . , vn ]i 0(the notation vi means that vi is omitted). Define 0 : C0 (X ) C 1 (X ) tobe identically zero [since C 1 (X ) {0}, this definition is forced on us]. Asevery simplicial n-chain has a unique expression as a linear combination ofsimplicial n-simplexes, n extends by linearity3 to a function defined on all ofCn (X ). The homomorphisms n are called simplicial boundary maps.The presence of signs gives the following fundamental result.Proposition 1.1.For all n 0, n 1 n 0.iProof. Each term of n [x 0 , . . . , x n ] has the form ( 1) [x 0 , . . . , x i , . . . , x n ].Hence, n [x0 , . . . , xn ] i ( 1)i [x0 , . . . , xi , . . . , xn ], and n 1 [x0 , . . . , xi , . . . , xn ] [x0 , . . . , xi , . . . , xn ] · · · ( 1)i 1 [x0 , . . . , xi 1 , xi , . . . , xn ] ( 1)i [x0 , . . . , xi , xi 1 . . . , xn ] · · · ( 1)n 1 [x0 , . . . , xi , . . . , xn ]3 Proposition 2.34 says that if F is a free abelian group with basis Y , and if f : Y Gis any function with values in an abelian group G, then there exists a unique homomorphism f : F G with f (y) f (y) for all y Y . The map f is obtained from f byextending by linearity: if u m 1 y1 · · · m p y p , then f (u) m 1 f (y1 ) · · · m p f (y p ).

6IntroductionCh. 1(when k i 1, the sign of [x0 . . . . , xi , . . . , xk , . . . , xn ] is ( 1)k 1 , for thevertex xk is the (k 1)st term in the list x0 , . . . , xi , . . . , xk , . . . , xn ). Thus, n 1 [x0 , . . . , xi , . . . , xn ] i 1 ( 1) j [x0 , . . . , x j , · · · , xi , . . . , xn ]j 0 n ( 1)k 1 [x0 , . . . , xi , . . . , xk , . . . , xn ].k i 1Now [x0 , . . . , xi , . . . , x j , . . . , xn ] occurs twice in n 1 n [x0 , . . . , xn ]: from n 1 [x0 , . . . , xi , . . . , xn ] and from n 1 [x0 , . . . , x j , . . . , xn ]. Therefore, thefirst term has sign ( 1)i j , while the second term has sign ( 1)i j 1 . Thus,the (n 2)-tuples cancel in pairs, and n 1 n 0. Definition. For each n 0, the subgroup ker n Cn (X ) is denoted byZ n (X ); its elements are called simplicial n-cycles. The subgroup im n 1 Cn (X ) is denoted by Bn (X ); its elements are called simplicial n-boundaries.Corollary 1.2. For all n,Bn (X ) Z n (X ).Proof. If α Bn , then α n 1 (β) for some (n 1)-chain β. Hence, n (α) n n 1 (β) 0, so that α ker n Z n . We have defined a sequence of abelian groups and homomorphisms inwhich composites of consecutive arrows are 0: 3 2 1 0· · · C3 (X ) C2 (X ) C1 (X ) C0 (X ) 0.The interesting group is the quotient group Z n (X )/Bn (X ).Definition.plex X isThe nth simplicial homology group of a finite simplicial comHn (X ) Z n (X )/Bn (X ).What survives in the quotient group Z n (X )/Bn (X ) are the n-dimensionalholes; that is, those n-cycles that are not n-boundaries; Hn (X ) {0} meansthat X has no n-dimensional holes.4 For example, if X is the punctured plane,4 Eventually, homology groups will be defined for mathematical objects other than topo-logical spaces. It is always a good idea to translate Hn (X ) {0} into concrete terms, ifpossible, as some interesting property of X . One can then interpret the elements of Hn (X )as being obstructions to whether X enjoys this property.

1.2 Categories and Functors7then H1 (X ) {0}: if [a, b, c] is a triangle in X having the origin in itsinterior, then α [b, c] [a, c] [a, b] is a 1-

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 results are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand- Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfand