Tilting and cluster tilting for quotient singularities

We shall show that the stable categories of graded Cohen-Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander-Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen-Macaulay modules.