HAL : hal-00586612, version 2

arXiv : 1104.3658

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v1 (19-04-2011)

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Stable categories of Cohen-Macaulay modules and cluster categories

Claire Amiot 1, Osamu Iyama 2, Idun Reiten 3

(18/04/2011)

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

Domaine

:

Mathématiques/Théorie des représentations Mathématiques/Géométrie algébrique Mathématiques/Algèbre commutative

Mots Clés : 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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Contributeur : Claire Amiot <>

Soumis le : Mercredi 11 Juillet 2012, 13:43:20

Dernière modification le : Mercredi 11 Juillet 2012, 13:44:11