Common use of Lemma Clause in Contracts

Lemma. Let A be a hereditary abelian category, i.e. an abelian category such that Exti (A, B) = 0 for all A, B ∈ A, i > 1 (when A = R-Mod, this condition is equivalent to R being a hereditary ring; in particular, Z and any principal ideal domain is hereditary). 1) In the derived category D(A ) every complex A• is isomorphic to the complex · · · → Hi−1(A•) 0 Hi(A•) 0 Hi+1(A•) → · · · A• =∼ M Hi(A•)[−i] ∼= ∏ Hi(A•)[−i]. 2) The morphisms in D(A ) are given by HomD(A )(A•, B•) ∼= i∈Z ∏ HomA (Hi(A•), Hi(B•)) ⊕ ∏ Ext1 (Hi(A•), Hi−1(B•)). i∈Z i∈Z