Quotient Morita Theory with Applications

We study equivalences between quotient categories of module categories associated with Gabriel topologies. We establish necessary and sufficient conditions for a Morita context to induce an equivalence between quotient categories and characterize such equivalences in terms of Morita contexts after passing to rings of quotients. Motivated by Artin–Zhang’s noncommutative Serre theorem, we introduce a notion of ampleness relative to a Gabriel topology and use it to characterize quotient categories of finitely generated modules under suitable noetherian hypotheses. As an application, we establish a sufficient condition under which the noncommutative Auslander theorem holds for actions of finite-dimensional Hopf algebras on AS-regular algebras without assuming that the Hopf algebras are semisimple.