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

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.