3-manifold Groups and Nonpositive Curvature by Kapovich M.

By Kapovich M.

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Precisely, we have the following statement: If s ∈ S, then Q(s) is an isomorphism. 4 Derived Categories 43 X 1 s X s Y 1 s 1 (observe here that Y ← X → Y is equivalent to Y ← Y → Y ). 3 Let K and L be categories and S a multiplicative system in K. Any functor F : K → L such that F (s) is an isomorphism for all s ∈ S factors uniquely through Q : K → KS . g. [GM99, Chap. 8]). 4 Let K be a triangulated category and S a multiplicative system in K. We call S compatible with the triangulation if the following hold: (S4): s ∈ S if and only if T (s) ∈ S (where T is the translation functor).

14 For an arbitrary functor F : C(A) → C(B) (A, B abelian categories) it is by no means clear whether it induces a functor D(A) → D(B). An analysis of when and how this is possible leads to the notion of “derived functors,” see Sect. 3. In Sect. 3, we have discussed the problem of associating a distinguished triangle to a short exact sequence in C(A). Now that we have access to the derived category, the matter can be settled in a beautiful way: Every short exact sequence in C(A) induces a distinguished triangle in D(A).

P∈Z We shall describe the differentials d n : Homn (A• , B• ) → Homn+1 (A• , B• ) by their operation on sections dn Γ (U ; Homn (A• , B• )) −→ Γ (U ; Homn+1 (A• , B• )) (U ⊂ X open). 3 Complexes of Sheaves 25 Note that Γ (U ; Homn (A• , B• )) = Γ U ; p ∼ = ∼ = Hom(Ap , Bp+n ) Γ (U ; Hom(Ap , Bp+n )) p Hom(Ap |U , Bp+n |U ), p so we need a map dn Hom(Ap |U , Bp+n |U ) → p If {f p }p∈Z ∈ Hom(Ap |U , Bp+n+1 |U ). p p Hom(Ap |U , Bp+n |U ), let p+n d n (f p ) = dB • p ◦ f p + (−1)n+1 f p+1 ◦ dA . 6 Hom Homp,q = Hom(A−p , Bq ) and appropriate differentials, but in taking the simple complex one must use direct products instead of direct sums (the “second total complex”).

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