# On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras

## TL;DR

This paper classifies irreducible morphisms in the stable category of modules over repetitive algebras and describes the structure of Auslander-Reiten triangles, revealing three canonical forms and their origins from sequences.

## Contribution

It provides a detailed classification of irreducible morphisms and characterizes the shape of Auslander-Reiten triangles in the stable category of repetitive algebra modules.

## Key findings

- Irreducible morphisms fall into three canonical forms.
- Auslander-Reiten triangles are induced from sequences of finitely generated modules.
- The shape of Auslander-Reiten triangles is explicitly described.

## Abstract

Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehat{\Lambda}$ be the repetitive algebra of $\Lambda$. For the stable category of finitely generated left $\widehat{\Lambda}$-modules $\widehat{\Lambda}$-\underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $\widehat{\Lambda}$-\underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $\widehat{\Lambda}$-\underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $\widehat{\Lambda}$-modules.