Copure semisimple categories and almost split maps

Abstract

We study copure semisimple Grothendieck categories, i.e. Krull–Schmidt categories satisfying the artinian condition on morphisms between finitely presented indecomposable objects. It is shown that these categories have many attractive properties, e.g. the endofiniteness of finitely presented objects and the existence of left almost split maps, among others. Applications are given to pure semisimple Grothendieck categories, and categories of locally finite representation type. In particular, we prove the existence of almost split sequences over Grothendieck categories of locally finite representation type with enough projectives. Our methods are based on functor categories and a module theory over Krull–Schmidt rings with enough idempotents.