Determination of some almost split sequences in morphism categories

TL;DR

This paper investigates the structure of almost split sequences in the morphism category of Artin algebras, providing new insights into their translation, midterms, and connections to Dynkin diagrams.

Contribution

It interprets Auslander-Reiten translations in morphism categories, calculates almost split sequences, and links representation-finite morphism categories to Dynkin diagrams.

Findings

Auslander-Reiten translations in morphism categories are expressed via module category translations

Midterms of almost split sequences in morphism categories are characterized

A structural theorem links representation-finite morphism categories to Dynkin diagrams

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 $\Lambda$. 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 $\Lambda$-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the midterms of almost split sequences. As such, another part of the paper is devoted to discuss the midterm of certain almost split sequences in the morphism category of $\Lambda$. 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.