HOMOLOGICAL ALGEBRA Our goal will be to get far enough to develop some fluency working with spectral sequences. We will follow Weibel. You should get errata for Weibel's book because it has many errors. Contents 1. Complexes of modules 1 2. Abelian categories 2 3. Algebraic constructions 5 4. Topologically-motivated constructions 7 5. Derived functors 10 1. Complexes of modules Let R be a ring (perhaps non-commutative). Definition 1.1. A chain complex of R-modules is C = C• = (fCig; fdig)i2Z with fCigi a set of R-modules and di : Ci ! Ci−1 satisfying di ◦ di−1 = 0. Remark 1.2. It is common in homological algebra to omit indices when they are clear from context. For example, we will usually write the condition that di ◦di−1 = 0 as \d2 = 0". Definition 1.3. The cycles of C are Zi = Zi(C) := ker di, the boundaries of C are Bi := Bi(C) = im(di+1), and the homology of C is Hi(C) := Zi=Bi. Example 1.4. Set Ci = Z= h8i for i ≥ 0 and Ci = 0 for i < 0. Set di : n 7! 4n for all i. One can check that 8 = h2i ; i ≥ 1 <>Z Hi(C) = Z= h4i ; i = 0 :>0; else } Definition 1.5. A morphism of complexes C ! D is a set of R-linear maps fi : Ci ! Di such that the following commutes: di Ci Ci−1 fi fi−1 di Di Di−1 A morphism is an isomorphism if fi is an isomorphism for all i. We will denote the category of chain complexes of R-modules by Ch(R). 1 2 HOMOLOGICAL ALGEBRA Exercise 1.6. Show that a morphism of complexes sends cycles to cycles and bound- aries to boundaries and therefore induces a map on homology. Show that each homology group is a functor Hi : Ch(R) ! Mod(R). Definition 1.7. A morphism of chain complexes is a quasi-isomorphism if it induces isomorphisms on all homology groups. Exercise 1.8. Check that 2 2 2 ··· Z=4 Z=4 Z=4 0 ··· ··· 0 0 Z=2 0 ··· is a quasi-isomorphism. Definition 1.9. If B; C 2 Ch(R), then B is a subcomplex of C if Bn is a submodule of Cn for each n 2 Z, and the differential of B is the restriction of the differential of C. If B is a subcomplex of C, then the quotient C=B is the complex with (C=B)n = Cn=Bn with differential given by the induced maps on the quotients.. Definition 1.10. If f : C ! C0 is a morphism of complexes, then the kernel ker(f) is the subcomplex (ker f)n = ker(fn) with differential the restriction of that of C. The image and cokernel complexes, im(f) and coker(f), are defined similarly. • i i i Definition 1.11. A cochain complex is C = C = (fC g; fd g)i2Z with C R-modules and di : Ci ! Ci+1 such that di+1di = 0 for all i. Definition 1.12. Let C be a cochain complex. The cocycles of C are Zi = ker di, the coboundaries are Bi = im di−1, and the cohomology is Hi = Zi=Bi. Chain complexes and complexes are equivalent: given a chain complex C•, we i • obtain a cochain complex by setting C := C−i, and given a cochain complex C , −i we obtain a chain complex C• by Ci = C . 2. Abelian categories Abelian categories are the natural setting for homological algebra. Let Ab be the category of abelian groups. Definition 2.1. A category A is enriched in Ab if for all objects X; Y 2 A, Hom(X; Y ) is equipped with an abelian group structure such that composition is Z-bilinear. Example 2.2. Both Mod(R) and Ch(R) are enriched in Ab (addition of morphisms in Ch(R) is done degree-wise). } Definition 2.3. An additive functor is F : A!B between categories enriched in Ab is a functor whose induced maps on morphism sets are group homomorphisms. Definition 2.4. An additive category is an Ab-enriched category A with a zero object that is both initial and terminal, and a product X × Y 2 A for any objects X; Y 2 A. Exercise 2.5. Show that initial and terminal objects are unique up to unique iso- morphism. HOMOLOGICAL ALGEBRA 3 Example 2.6. The zero module 0 2 Mod(R) is both initial and terminal. } Example 2.7. Both Mod(R) and Ch(R) are additive. } While the definition only requires an additive category to have finite products, additive categories also have finite coproducts. Proposition 2.8. In an additive category, finite products and coproducts exist and are the same. Proof. Let A be an additive category and X; Y 2 A. Let πX : X × Y ! X and πY : X × Y ! Y be the projections from the product. We have a commuting diagram X X × Y πY Y πX ιX Id X 0 where ιX exists by the universal property of the product. You should now check that ιX and the similarly-defined map ιY make X ×Y satisfy the universal property of the coproduct. Exercise 2.9. Show that arbitrary products and finite coproducts commute with taking homology in Ch(R). Definition 2.10. A kernel of a morphism f : B ! C in an additive category A is a map i : A ! B such that (i) f ◦ i = 0 (ii) i is universal with respect to this property; i.e. for all morphisms i0 : A0 ! B such that f ◦i0 = 0, there exists a unique g : A0 ! A such that i0 = i◦g. Dually, we have Definition 2.11. A cokernel of a morphism f : B ! C is a map π : C ! D such that π ◦ f = 0 and π is universal with respect to this property. We will sometimes be sloppy with notation and write only the object rather than the map for a kernel or cokernel. Example 2.12. Kernals and cokernels in Mod(R) and Ch(R) are are categorical kernels and cokernels. } Definition 2.13. A morphism i : A ! B in an additive category is monic if for all g : A0 ! A, ig = 0 =) g = 0. Dually, π : B ! C is epic if for all h : C ! C0, hπ = 0 =) h = 0. Exercise 2.14 (Tedious but healthy exercise). In an additive category, kernels are the same as monics, and cokernels are the same as epics. Definition 2.15. An additive category A is abelian if (i) Every map has both a kernel and a cokernel (ii) Every monic is the kernel of its cokernel (iii) Every epic is the cokernel of its kernel. Example 2.16. Mod(R) is abelian } 4 HOMOLOGICAL ALGEBRA In fact, Mod(R) is in some sense the only example. Theorem 2.17 (Freyd-Mitchell embedding). If A is a small abelian category, then there is a fully faithful embedding of Mod(R) into Mod(R) for some R that sends exact sequences to exact sequences. Definition 2.18. If A is an abelian category and f : A ! B is a morphism, then the image is im(f) := ker coker f. f g Definition 2.19. A sequence A ! B ! C in an abelian category is exact if ker g = im f. The category of complexes in an abelian category A is defined as it is for the category of modules and is denoted by Ch(A). The proof of the following appears in Weibel as Theorem 1.2.3. We omit its proof here. Proposition 2.20. If A is an abelian category, then Ch(A) is abelian. Definition 2.21. Let A be an abelian category. A bicomplex or double com- h plex in A is a family fCp;qgp;q2Z of objects in A with maps d : Cp;q ! Cp−1;q and v d : Cp;q ! Cp;q−1 satisfying (i)( dh)2 = 0 and (dv)2 = 0 (ii) dvdh + dhdv = 0 The superscripts h and v on the differentials stand for \`horizontal" and \verti- cal". We will usually depict a bicomplex by a diagram with horizontal arrows going left and vertical arrows going down. dv dv ··· Cp−1;q+1 Cp;q+1 ··· dh dh dh dv dv ··· Cp−1;q Cp;q ··· dh dh dh Remark 2.22. A bicomplex is not a complex in Ch(A), since the squares in a bicomplex must anticommute, rather than commute. Suppose that A has infinite products and coproducts. Definition 2.23. Given a bicomplex C in A, we can define two complexes: TotΠ(C) and Tot⊕(C), both called total complexes and defined by Π Y ⊕ M Tot (C)n = Cp;q Tot (C)n = Cp;q p+q=n p+q=n with differential dv + dh. Definition 2.24. A bicomplex C is bounded if C has only finitely many nonzero terms along each diagonal p + q = n. When C is bounded, TotΠ(C) = Tot⊕(C). Exercise 2.25. Let C be a bounded bicomplex with exact rows (or exact columns). Prove that TotΠ(C) = Tot⊕(C) is exact. HOMOLOGICAL ALGEBRA 5 3. Algebraic constructions 3.1. Truncations. There are two common ways to convert an unbounded chain complex into a bounded one. Definition 3.1. Let C be a chain complex. Fix n 2 Z and let σ≥nC be the subcomplex ( Ci; i ≥ n (σ≥nC)i := 0; else Define σ≥nC similarly. The complexes σ≥nC and σ≤nC are the brutal trunca- tions of C.