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

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.