Stable categories of Cohen-Macaulay modules and cluster categories

Abstract

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is 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 $\Lambda$. 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 $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

accepted for publication in Amer. Journ. Math

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• Tensor products of n-complete algebras
• Singularity categories of deformations of Kleinian singularities
• Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities
• Relative Singularity Categories
• Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities
• Higher zigzag algebras
• A1-homotopy invariants of dg orbit categories
• The Nori-Hilbert scheme is not smooth for 2-Calabi Yau algebras
• Higher APR tilting preserves $n$-representation infiniteness
• On the tilting complexes for the Auslander algebra of the truncated polynomial ring
• Tilting objects in singularity categories and levelled mutations
• Preprojective algebras and Calabi-Yau duality
• On cluster categories of weighted projective lines with at most three weights