Determination of some almost split sequences in morphism categories

Abstract

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.