Generalizing Morita Duality: A Homological Approach

Let R and S be arbitrary associative rings. Given a bimodule RWS, we denote by Δ? and Γ? the functors Hom?(−, W) and Ext1?(−, W), where ? = R or S. The functors ΔR and ΔS are right adjoint with the evaluation maps δ as unities. A module M is Δ-reflexive if δM is an isomorphism. In this paper we give, for a weakly cotilting bimodule RWS, the notion of Γ-reflexivity. We construct large Abelian subcategories R and S where the functors ΓR and ΓS are left adjoint and a “cotilting theorem” holds.