Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

Abstract

Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday.Abstract. 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 Λ. 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 Λ. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay R-modules and the derived category of Λ. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

Higher APR tilting preserves n-representation infiniteness

“…Then the truncated Jacobian algebra Λ of (Q, W, C) is a 2-representation infinite algebra (see [AIR,section 6]), whose quiver is the left upper one in the picture below. We can consider the 1-APR tilting Λ-module T associated with vertex 1.…”

“…[AIR,Ke2,MM] There is a one-to-one correspondence between isomorphism classes of n-representation infinite algebras Λ and isomorphism classes of graded bimodule (n + 1)-CY algebras A of Gorenstein parameter 1. The correspondence is given by…”

Untitled

“…We refer to [Auslander 1978;Curtis and Reiner 1981;Leuschke and Wiegand 2012;Simson 1992;Yoshino 1990] for a general background on this subject. We also refer to [Amiot et al 2015;Araya 1999;Demonet and Luo 2016a;2016b;Herschend et al 2014;de Thanhoffer de Völcsey and Van den Bergh 2010;Iyama and Takahashi 2013;Kajiura et al 2007;Keller and Reiten 2008] for recent results about connections with tilting theory and cluster categories.…”

Introduction to τ-tilting theory

“…These 2-Calabi-Yau triangulated categories have quite different origin from those in (2) above, but it was shown in ref. 46 that they are often equivalent.…”