Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities

TL;DR

This paper extends the understanding of stable categories of graded maximal Cohen-Macaulay modules over noncommutative quotient singularities, showing they possess tilting objects and are equivalent to derived categories of finite-dimensional algebras.

Contribution

It generalizes Iyama and Takahashi's theorem to noncommutative settings using noncommutative algebraic geometry, establishing tilting objects in these categories.

Findings

Stable categories have tilting objects.

Categories are equivalent to derived categories of finite-dimensional algebras.

Applicable to Gorenstein isolated singularities in noncommutative algebra.

Abstract

Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting objects. This paper proves a noncommutative generalization of Iyama and Takahashi's theorem using noncommutative algebraic geometry. Namely, if $S$ is a noetherian AS-regular Koszul algebra and $G$ is a finite group acting on $S$ such that $S^G$ is a "Gorenstein isolated singularity", then the stable category ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ of graded maximal Cohen-Macaulay modules has a tilting object. In particular, the category ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ is triangle equivalent to the derived category of a finite dimensional algebra.