On Irreducible Morphisms and Auslander-Reiten Triangles in the Stable Category of Modules over Repetitive Algebras

Abstract

Let ${\Bbbk }$ be an algebraically closed field, let Λ be a finite dimensional ${\Bbbk }$ -algebra, and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. For the stable category of finitely generated left $\widehat {\Lambda }$ -modules $\widehat {\Lambda }$ -mod, we prove that every Auslander-Reiten triangle in $\widehat {\Lambda }$ -mod is induced from an Auslander-Reiten sequence of finitely generated left $\widehat {\Lambda }$ -modules. We use this fact to show that the irreducible morphisms in $\widehat {\Lambda }$ -mod 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. Finally, we use this classification to describe the shape of the Auslander-Reiten triangles in $\widehat {\Lambda }$ -mod.