On periodic stable Auslander-Reiten components containing Heller lattices over the symmetric Kronecker algebra

TL;DR

This paper investigates the structure of certain stable Auslander-Reiten components containing Heller lattices over the symmetric Kronecker algebra, revealing their shapes and properties in the context of periodic modules.

Contribution

It characterizes the shapes of stable Auslander-Reiten components containing Heller lattices of indecomposable periodic modules over the symmetric Kronecker algebra.

Findings

Determined the shapes of stable Auslander-Reiten components with Heller lattices.

Analyzed the properties of periodic modules over the symmetric Kronecker algebra.

Provided classifications of components containing Heller lattices.

Abstract

Let $\mathcal{O}$ be a complete discrete valuation ring, $\mathcal{K}$ its quotient field, and $A$ the symmetric Kronecker algebra over $\mathcal{O}$. We consider the full subcategory of the category of $A$-lattices whose objects are $A$-lattices $M$ such that $M\otimes_{\mathcal{O}}\mathcal{K}$ is projective $A\otimes_{\mathcal{O}}\mathcal{K}$-modules. In this paper, we study Heller lattices of indecomposable periodic modules over $A$. As a main result, we determine the shapes of stable Auslander--Reiten components containing Heller lattices of indecomposable periodic modules over $A$.