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

摘要

Tilting objects play a key role in the study of triangulated categories. Afamous result due to Iyama and Takahashi asserts that the stable categories ofgraded maximal Cohen-Macaulay modules over quotient singularities have tiltingobjects. This paper proves a noncommutative generalization of Iyama andTakahashi's theorem using noncommutative algebraic geometry. Namely, if $S$ isa 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 stablecategory ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ of graded maximalCohen-Macaulay modules has a tilting object. In particular, the category${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ is triangle equivalent to thederived category of a finite dimensional algebra.