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NOTES from HOMOLOGICAL ALGEBRA 1. Derived Category Let NOTES FROM HOMOLOGICAL ALGEBRA MACIEJ GAŁĄZKA 1. Derived category Let C be an abelian category. Let Kom(C ) denote the category of cochain complexes. Theorem 1.1. There is category D(C ) (the derived category of C ) and a functor Q : Kom(C ) ! D(C ) such that (1) for every quasi-isomorphism f 2 Mor(Kom C ) the morphism Q(f) is an isomorphism; (2) if any Q0 : Kom(C ) ! A (A here is any category) satisfies (1), then there is a unique functor P : D(C ) ! A such that Q Kom(C ) D(C ) Q0 P A commutes. Proof of Theorem 1.1. To construct D(C ), we need to define the localization of a category. Let B be a category, S a class of morphisms in B. The localization of B in S is a category B[S−1] and a functor L : B ! B[S−1] such that L takes any morphism in S to an isomorphism and for any functor F : B ! D which takes elements of S to isomorphisms there exists a unique functor G : B[S−1] ! D such that L B B[S−1] G F D commutes. Let us construct B[S−1]. Let Γ be a directed graph with vertices Ob B and edges Mor(B) [ fxs : s 2 Sg. If f : X ! Y , then edge f is directed from X to Y . If S 3 s : X ! Y , then edge xs is directed from Y ! X. We define HomB[S−1](X; Y ) to be equivalence classes of paths in Γ from X to Y , where the relations are generated by Date: August 8, 2015. 1 f g gf (1) U −! V −! W ∼ U −! W , id (2) U −!s V −!xs U ∼ U −−!U U, id (3) U −!xs V −!s U ∼ U −−!U U. (More precisely, we consider for each X; Y an equivalence relation ≡X;Y on the sets of paths from X to Y . We say that this family of equivalence relations is good if for any two paths p; q : X ! Y , any path u : Y ! Z, and any path v : W ! X the relation p ≡ q implies both pv ≡ qv and up ≡ uq. Then we define our family of relations ∼X;Y to be the intersection of good of families containing the three relations.) Suppose that we have any F : B ! D which takes class S to isomor- phisms. Then G : B[S−1] ! D is uniquely determined on objects. The only way to define G for morphisms is to make it map any path from X to Y into its composition in D (note that we can compose paths in D since any s 2 S goes under F to an isomorphism). It remains to prove that this agrees with ∼X;Y . For any two X; Y 2 Ob B we can define a family of equivalence relations ≡X;Y on the paths from X to Y by saying that two paths are equiv- alent if their images under F , after composing in D, are equal. But then ≡X;Y is good, so ∼⊆≡, in other words, G is well-defined. G is a functor by definition. −1 We define D(C ) = Kom(C )[(quasi-isomorphisms) ]. Definition 1.2. A class S ⊆ Mor(B) is localizing if it satisfies: (a) for all X 2 Ob(B) we have idX 2 S, (b) if s; t 2 S, then st 2 S, f g (c) for all X −! Y and Z −!s Y there are W −! Z and W −!t X such that g W Z t s f X Y g f commutes, and for all W −! Z and W −!t X there are X −! Y and Z −!s Y such that g W Z t s f X Y commutes (above we require that s; t 2 S), (d) for f; g : X ! Y we have 9s2S(sf = sg) () 9t2S(ft = gt). Remark 1.3. The class of quasi-isomorphisms in the category Kom(C ) is not localizing. 2 Lemma 1.4. If S is localizing in B, then we can present any morphism X ! Y in B[S−1] as a triangle X0 f s X Y with equivalence X000 h u X0 X00 f s g t X Y . Pairs (s; f) and (t; g) are equivalent if and only if there is a pair (u; h) such that the two squares (or quadrangles) commute. This also applies to left fractions. Theorem 1.5. In any of K(C ), K−(C ),K+(C ),Kb(C ) the class of quasi- isomorphisms is localizing. Definition 1.6. Given a cochain complex X, define the complex X[a] by X[a]i = Xa+i, a dX[a] = (−1) dX . Definition 1.7. For a morphism f : X ! Y , where X; Y 2 Kom(C ), define the cone C(f) 2 Kom(C ) as C(f)i = X[1]i ⊕ Y i, dC(f) = (− dX ◦ pr1; f[1] ◦ pr1 + dY ◦ pr2). Define the cylinder Cyl(f) 2 Kom(C ) as Cyl(f)i = Xi ⊕ X[1]i ⊕ Y i, dCyl(f) = (dX −idX[1] + 0; 0 − dX +0; 0 + f[1] + dY ). Proposition 1.8. The cone and the cylinder are cochain complexes. Proof. For the cone, it is enough to check that − d 2 X = 0. f[1] dY 2 But this follows from the fact that dX = 0 and that f is a morphism of chain complexes. For the cylinder, it is enough to check that 0 12 dX − idX[1] @ − dX A = 0. f[1] dY 3 It is true for similar reasons. Proposition 1.9. For any f : X ! Y there is the following diagram with exact rows 0 Y π C(f) X[1] 0 αf 0 X i Cyl(f) C(f) 0 βf =f+0+idY f 0 X Y where the most of the maps are inclusions of direct summands (in particular i is the inclusion of X into the cylinder!) or projections onto them. The maps αf ; βf are quasi-isomorphisms. Proof. We have βα = idY . Enough to prove that αβ ∼ idCyl(f). Note that αβ : X ⊕ X[1] ⊕ Y ! X ⊕ X[1] ⊕ Y is given by the matrix 00 0 0 1 @0 0 0 A f 0 idY We define the homotopy s : X ⊕X[1]⊕Y ! X ⊕X[1]⊕Y by (0; id +0+0; 0), or in other words, the matrix 0 0 0 01 @idX[1] 0 0A 0 0 0 We calculate: 0 1 0 1 0 1 0 1 0 0 0 dX − idX[1] dX − idX 0 0 0 @idX[1] 0 0A·@ − dX A+@ − dX A·@idX 0 0A 0 0 0 f[1] dY f dY 0 0 0 0 1 0 1 0 1 − idX 0 0 0 0 0 idX 0 0 = @ 0 − idX 0A = @ 0 0 0 A − @ 0 idX 0 A . f[−1] 0 0 f[−1] 0 idY 0 0 idY Notice that we wrote dX and dY instead of dX[−1] and dY [−1], because we don’t want to change the sign of the coboundary map as in Definition 1.6. Let f; g : X ! Y . If we take a chain homotopy from f to g, then it gives a map Cyl(h) : Cyl(f) ! Cyl(g) given by the matrix 0 1 idX 0 0 @ 0 idX[1] 0 A 0 h idY 4 and also a map C(h) : C(f) ! C(g) given by id 0 X[1] . h idY Exercise 1.10. The maps Cyl(h) and C(h) are chain maps. Proposition 1.11. The maps Cyl(h) and C(h) are isomorphisms in the category Kom(C ). Proof. The inverses are just given by Cyl(−h) and C(−h) (note that −h is a homotopy from g to f). Definition 1.12. Let K(C ) be the homotopy category of Kom(C ) (i.e. we mod out the morphisms by the chain homotopy relation). Definition 1.13. In any category of complexes (like Kom(C ; K(C ), or D(C )) any sequence of the form X u Y v Z w X[1] is called a triangle. We say that it is distinguished if it is isomorphic (in the corresponding category, or to be more precise, in the category of triangles in the corresponding category) to a triangle X0 ! Cyl(f) ! C(f) ! X0[1] for some f : X0 ! Y 0. Proposition 1.14. Every exact sequence in Kom(C ) is quasi-isomorphic to a sequence 0 ! X ! Cyl(f) ! C(f) ! 0. From the proposition if follows that if 0 ! A ! B ! C ! 0 is an exact w sequence of complexes, then there is a morphism C −! A[1] in D(C ) such w that A ! B ! C −! A[1] is distinguished in D(C ). Theorem 1.15. Let S be the class of quasi-isomorphisms in K(C ). Then K(C )[S−1] is canonically isomorphic to D(C ). This applies to any version of Kom(C ). Proof. We have the following diagram Q Kom(C ) D(C ) Q¯ K(C )[S−1] Then Q¯ is a bijection on objects. Note that Q¯ takes paths of morphisms of complexes into paths of corresponding classes (to see this, check that first f for paths of length one of the form X −! Y , and then for paths of the form X −!xs Y ). It follows that Q¯ is a surjection on morphisms. We will show the following 5 Lemma 1.16. Assume that f; g : X ! Y are chain homotopic in Kom(C ).
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