Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity R and the generalized (higher) cluster category of a finite dimensional algebra Λ. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of R as well as the higher preprojective algebra of an extension of Λ. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay R-modules and the derived category of Λ. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.