On the stable Auslander-Reiten components of certain monomorphism categories

TL;DR

This paper investigates the structure of stable Auslander-Reiten components in a specific subcategory of monomorphism categories over Artin algebras, revealing a three-periodicity pattern and connections to auto-equivalences.

Contribution

It characterizes the shape and properties of certain Auslander-Reiten components, including their periodicity and relation to auto-equivalences in Gorenstein projective modules.

Findings

Certain components have cardinalities divisible by 3.

The shape of components is explicitly described.

A three-periodicity phenomenon is identified.

Abstract

Let $\Lambda$ be an Artin algebra and let $\rm{Gprj}\mbox{-}\Lambda$ denote the class of all finitely generated Gorenstein projective $\Lambda$-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category $\mathcal{S}({\rm Gprj}\mbox{-}\Lambda)$ containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category $\underline{\rm{Gprj}}\mbox{-}\Lambda$. In particular, for the finite components, we show that under certain mild conditions their cardinalities are divisible by $3$. We see that this three-periodicity phenomenon reoccurs several times in the paper.