Stable categories of Cohen-Macaulay modules and cluster categories

Details Stable categories of Cohen-Macaulay modules and cluster categories Stable categories of Cohen-Macaulay modules and cluster categories

2011-04-19

arxiv.org/abs/1104.3658v1

Mathematics

Representation Theory

35 pages

Scientific paper

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 singularity $R$ and the generalized 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 $\Lambda$.

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