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

By C. Amiot, Osamu Iyama, I. Reiten. 2011

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d127cb865164ba0f755f414f63d9f696fd8295a4

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Citations in the corpus

Representations of Algebras and Related Topics (2011, methodology): “= 0,...,n −1, the graded SG-modules Tj = L i∈ZSi+ j n . Then we have T0 = SG and T = Ln−1 j=0 Tj ≃S as S G-modules. The following result is also proved in [90] using diﬀerent techniques. Theorem 4.8 ([4]). Let S and G be as above. Let A be the stable endomorphism algebra A := End GrSG(T). Then the f”
n-Representation infinite algebras (2012, methodology): “23 Proof. By Lemma 5.3, there is a subgroupH of SLn+1(K) such that Γ/B ∼= S∗H. This implies that Γ/B has a canonical projective (Γ/B)e-resolution that we now recall following [BSW] (compare also with [AIR]). Let R be the semisimple subalgebra K(Q0/B) of Γ/B. We consider Γ/B and K(Q/B) as Re-modules.”
Representation Theory of Geigle-Lenzing Complete Intersections (2014, background): “er algebra EndL/Zω~ R (V ) (see Corollary 7.20): Λ preprojective algebra Π(Λ) ≃EndL/Zω~ R (V ) R. d-Auslander algebra 8 HERSCHEND, IYAMA, MINAMOTO, AND OPPERMANN A similar picture already appeared in [AIR] in a diﬀerent setting. We are expecting that d-tilting objects in CMLR always lift to d-tiltin”
An introduction to higher Auslander–Reiten theory (2016, background): “In addition to representation theory [15, 23, 31], it has exhibited connections to commutative algebra, commutative and non-commutative algebraic geometry, and combinatorics, see for example [1, 16, 17, 24, 32].”