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Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 3, June 2015
- pp. 813-857
- 10.1353/ajm.2015.0019
- Article
- Additional Information
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 $\Lambda$. 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.