The stable Auslander-Reiten components of certain monomorphism categories

Let A be an Artin algebra and let Gprj-Lambda denote the class of all the 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 S(Gprj-Lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{S}}(\text{Gprj-}\Lambda)$$\end{document} 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 Gprj-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.

Automated summary

In this paper, we investigate the components of the stable Auslander-Reiten quiver of a certain subcategory of monomorphism category S(Gprj-Lambda) that contains boundary vertices. We describe the shape of these components and demonstrate that certain components are connected to the orbits of an auto-equivalence on the stable category Gprj-Lambda. Moreover, we show that under certain conditions, the cardinalities of finite components are divisible by 3, suggesting a recurring three-periodicity phenomenon.