On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras

TL;DR

This paper studies Gorenstein algebras called dimer tree algebras and their skew group algebras, showing how group actions change their Cohen-Macaulay type and providing geometric models for their syzygy categories.

Contribution

It demonstrates that skew group algebras of dimer tree algebras can have Cohen-Macaulay type D, extending known results from type A, and offers a geometric model for their syzygy categories.

Findings

Stable Cohen-Macaulay category of $A$ is a 2-cluster category of type A.

Skew group algebra $AG$ has Cohen-Macaulay type D.

Provides geometric model using punctured polygon with checkerboard pattern.

Abstract

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra $A$ is a 2-cluster category of Dynkin type $\mathbb{A}$. Here we show that, if $A$ has an admissible action by the group $G$ with two elements, then the stable Cohen-Macaulay category of the skew group algebra $AG$ is a 2-cluster category of Dynkin type $\mathbb{D}$. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible $G$-action on the path algebra of type $\mathbb{A}$ the resulting skew group algebra is of type $\mathbb{D}$. Moreover, we provide a geometric model of the syzygy category of $AG$ in terms of a punctured polygon $\mathcal{P}$ with a checkerboard pattern in its interior, such that the 2-arcs in $\mathcal{P}$ correspond to indecomposable syzygies in $AG$ and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type $\mathbb{A}$ and $\mathbb{D}$ respectively. We also provide examples of types $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$.