On the non-periodic stable Auslander--Reiten Heller component for the Kronecker algebra over a complete discrete valuation ring

TL;DR

This paper investigates the structure of a specific non-periodic component of the stable Auslander--Reiten quiver for the Kronecker algebra over a complete discrete valuation ring, revealing new restrictions and classifications.

Contribution

It determines the non-periodic component of the stable Auslander--Reiten quiver for certain lattices over the Kronecker algebra and establishes strong restrictions on such quivers for symmetric orders.

Findings

Identified the non-periodic component containing Heller lattices.

Provided classifications of stable Auslander--Reiten quivers for symmetric orders.

Revealed restrictions on the structure of these quivers.

Abstract

We consider the Kronecker algebra $A=\mathcal{O}[X, Y]/(X^2, Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A \otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one can compute a complete list of indecomposable $A\otimes\kappa$-modules. For each indecomposable $A\otimes\kappa$-module, we obtain a special kind of $A$-lattices called "Heller lattices". In this paper, we determine the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contains "Heller lattices". Moreover, we give the strong restrictions on stable Auslander--Reiten quivers for symmetric orders over a complete discrete valuation ring.