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. 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 (CM) under bar (z)(S-G) of graded maximal Cohen-Macaulay modules has a tilting object. In particular, the category (CM) under bar (z)(S-G) is triangle equivalent to the derived category of a finite dimensional algebra. (C) 2016 Elsevier Inc. All rights reserved.