Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities

摘要

In this paper, we define a notion of noncommutative graded isolated singularity, and study AS-Gorenstein isolated singularities and the categories of graded maximal Cohen-Macaulay modules over them. In particular, for an AS-Gorenstein algebra A of dimension d ≥ 2, we show that A is a graded isolated singularity if and only if the stable category of graded maximal Cohen-Macaulay modules over A has the Serre functor. Using this result, we also show the existence of cluster tilting objects in the categories of graded maximal Cohen-Macaulay modules over Veronese subalgebras of certain AS-regular algebras.