Symmetries and connected components of the AR-quiver

TL;DR

This paper investigates the structure and symmetries of the AR-quiver of certain Gorenstein singularities, revealing the existence of infinite families of indecomposable modules with specific properties and symmetries.

Contribution

It demonstrates that properties of maximal Cohen-Macaulay modules define unions of connected components in the AR-quiver and uncovers symmetries in the stable quiver of these singularities.

Findings

Properties define unions of connected components in the AR-quiver.

Existence of infinite families of indecomposable modules satisfying certain properties.

Presence of symmetries in the stable AR-quiver for Gorenstein isolated singularities.

Abstract

Let $(A,\mathfrak{m})$ be a commutative complete equicharacteristic Gorenstein isolated singularity of dimension $d $ with $k = A/\mathfrak{m}$ algebraically closed. Let $\Gamma(A)$ be the AR (Auslander-Reiten) quiver of $A$. Let $\mathcal{P}$ be a property of maximal Cohen-Macaulay $A$-modules. We show that some naturally defined properties $\mathcal{P}$ define a union of connected components of $\Gamma(A)$. So in this case if there is a maximal Cohen-Macaulay module satisfying $\mathcal{P}$ and if $A$ is not of finite representation type then there exists a family $\{ M_n \}_{n \geq 1}$ of maximal Cohen-Macaulay indecomposable modules satisfying $\mathcal{P}$ with multiplicity $e(M_n) > n$. Let $\underline{\Gamma(A)}$ be the stable quiver. We show that there are many symmetries in $\underline{\Gamma(A)}$. As an application we show that if $(A,\mathfrak{m})$ is a two dimensional Gorenstein isolated singularity with multiplicity $e(A) \geq 3$ then for all $n \geq 1$ there exists an indecomposable self-dual maximal Cohen-Macaulay $A$-module of rank $n$.