Common use of Lemma Clause in Contracts

Lemma. 1) If C• and C′• are almost perfect, then the group HomD(Ab)(C•, C′• ) has no nontrivial divisible subgroups. 2) If A• is a complex such that Hi (A• ) are finite dimensional Q-vector spaces and C• is a complex such that Hi (C• ) are finitely generated abelian groups, then the group HomD(Ab)(A•, C• ) is divisible. Proof. By 0.3.1 we have HomD(Ab)(C•, C′•) ∼= ∏ Hom(Hi(C•), Hi(C′•)) ⊕ ∏ Ext(Hi(C•), Hi−1(C′•)). i∈Z i∈Z Note that by our assumptions, both groups ∏i Z Hom(Hi(C•), Hi(C′•)) and