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 SG is a “Gorenstein isolated singularity”, then the stable category CM_Z(SG) of graded maximal Cohen–Macaulay modules has a tilting object. In particular, the category CM_Z(SG) is triangle equivalent to the derived category of a finite dimensional algebra.