Auslander-Reiten theory via Nakayama duality in abelian categories

TL;DR

This paper uses Nakayama duality to provide a unified approach to Auslander-Reiten theory in abelian categories, establishing the existence of almost split sequences in various module categories.

Contribution

It offers a new, concise framework for Auslander-Reiten dualities via Nakayama functors, extending results to categories with different finiteness conditions.

Findings

Existence of almost split sequences in categories of finitely presented modules

Existence of almost split sequences in categories of finitely copresented modules

Conditions for almost split sequences in module categories over quivers with relations

Abstract

Using the Nakayama duality induced by a Nakayama functor, we provide a novel and concise account of the existence of Auslander-Reiten dualities and almost split sequences in abelian categories with enough projective objects or enough injective objects. As an example, we establish the existence of almost split sequences ending with finitely presented modules and those starting with finitely copresented modules in the category of all modules over a small endo-local Hom-reflexive category. Specializing to algebras given by (not necessarily finite) quivers with relations, we further investigate when the categories of finitely presented modules, finitely copresented modules and finite dimensional modules have almost split sequences on either or both sides.