Cluster-tilted algebras are Gorenstein and stably Calabi–Yau

Abstract

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.

Toda Systems, Cluster Characters, and Spectral Networks

“…This amounts to advancing n`1 steps in the A n Q-system [Ked08], hence the associated automorphism is discrete integrable in the sense that it preserves the Hamiltonians H i [DK10,GSV11]. On the other hand, by results of [KR07] this automorphism has the property that, after composing with a specific permutation of the vertices of Q n , it takes CCpM q to CCpτ M q (this is the unique permutation inducing an isomorphism of the principally-framed quiver and the result of applying the maximal green sequence to the principally-framed quiver [BDP14]) 2 . In our case, this permutation coincides with the one induced by the Nakayama involution ν n , which can be shown by studying the induced maximal green sequences of the A n subquivers of Q n .…”

The minimal representation-infinite algebras which are special biserial

“…Other similarities with cluster tilted algebras (see [14]) should be mentioned: For our example 2, here are the minimal injective resolutions of the indecomposable projective modules:…”

Auslander–Reiten theory revisited

“…When T is the stable category CM(Λ) of an R-order Λ, then End T (M ) is a factor algebra of the 2-Auslander algebra End Λ (M ) which enjoys nice properties (see Theorem 3.14, Proposition 3.24). The following result [39,94] shows that 2-CY tilted algebras also enjoy nice properties. (c) We have the following equivalence of categories…”

“…A class of functorially finite subcategories is introduced in [85,86] (see also [94]). Definition 2.1.…”