Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes.Let X and Y be two classes of R-modules with Ext ≥1(X,Y) = 0 for each object X ∈X and each object Y ∈Y.We show that if A,B ∈C■(R) are X-complexes and U,V ∈ C■(R) are Y-complexes,then U■V■Hom(A,U)■Hom(A,V);A■B■Hom(B,U)■Hom(A,U).As an application,we give a sufficient condition for the Hom evaluation morphism being invertible.     

Self-Small Abelian Groups as Modules over Their Endomorphism Rings

Abstract

The strongly essentially indecomposable self-small abelian groups, of finite torsion free rank, which are (almost-)projective as modules over their endomorphism rings are characterized.