HAL: hal-00586612, version 2

arXiv: 1104.3658

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v1 (2011-04-19)

v2 (2012-07-11)

Stable categories of Cohen-Macaulay modules and cluster categories

Claire Amiot 1, Osamu Iyama 2, Idun Reiten 3

(2011-04-18)

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.

1:	Institut de Recherche Mathématique Avancée (IRMA)
	CNRS : UMR7501 – Université de Strasbourg
2:	Nagoya University
	Nagoya University
3:	Institutt for matematiske fag (IMF)
	Trondheim University

Subject

:

Mathematics/Representation Theory Mathematics/Algebraic Geometry Mathematics/Commutative Algebra

Keyword(s): Cohen-Macaulay modules – stable categories – Calabi-Yau categories – cluster categories – cluster tilting – Auslander algebras – preprojective algebras – Calabi-Yau algebras – dimer models

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From: Claire Amiot <>

Submitted on: Wednesday, 11 July 2012 13:43:20

Updated on: Wednesday, 11 July 2012 13:44:11