Stable categories of Cohen-Macaulay modules and cluster categories

TL;DR

This paper develops a systematic method to establish triangle equivalences between stable Cohen-Macaulay modules over Gorenstein singularities and higher cluster categories, extending Auslander's correspondence to broader classes of singularities.

Contribution

It introduces a new construction linking stable Cohen-Macaulay categories with higher cluster categories via bimodule Calabi-Yau algebras, generalizing known results for simple singularities.

Findings

Constructs triangle equivalences for Gorenstein isolated singularities.

Connects stable Cohen-Macaulay modules with higher cluster categories.

Applies to cyclic quotient singularities and toric threefolds.

Abstract

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e. the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.