On components of stable Auslander-Reiten quivers that contain Heller lattices: the case of truncated polynomial rings

TL;DR

This paper investigates the structure of stable Auslander-Reiten quivers for categories of lattices over truncated polynomial rings, specifically identifying the shape of components containing Heller lattices.

Contribution

It characterizes the shape of components in the stable Auslander-Reiten quiver for lattices over truncated polynomial rings, focusing on those containing Heller lattices.

Findings

Determined the shape of components containing Heller lattices.

Provided a classification of stable Auslander-Reiten quiver components.

Enhanced understanding of lattice categories over truncated polynomial rings.

Abstract

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring $\mathcal{O}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes \mathcal{K}$ is projective as an $A\otimes \mathcal{K}$-module, where $\mathcal{K}$ is the fraction field of $\mathcal{O}$. Then, we may define the stable Auslander-Reiten quiver of the category. We determine the shape of the components of the stable Auslander-Reiten quiver that contain Heller lattices.