Cohomology of Sheaves

Abstract

Let A be an abelian category. Definition 1.1. A complex in A, A•, is a collection of objects A, i ∈ Z and boundary morphisms d : A → A such that d ◦ d = 0 for all i ∈ Z. If A• and B• are complexes, a map f : A• → B• is a collection morphisms f i : A → B commuting with the boundary morphisms. Two maps f, g : A• → B• are said to be homotopic if there are morphisms k : A → Bi−1 such that f i − g = di−1 B ◦ k + kdA. Two complexes are homotopy equivalent if there exist maps f : A• → B• and g : B• → A• such that the compositions are homotopic to the appropriate identity map.