Cluster-tilted algebras are Gorenstein and stably Calabi–Yau

By B. Keller, I. Reiten. 2005

435 citations recorded by the source corpus.

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 Geis, Bernard Leclerc, Jan Schroer, 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 Geis, Bernard Leclerc, Jan Schroer, 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.

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

View Paper: https://www.semanticscholar.org/paper/3f6892a96903c2b0086bb6390bce37d79b28a35f

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Citations in the corpus

Derived equivalence classification for cluster-tilted algebras of type An (2007, background): “In fact they are 1-Gorenstein and in particular they have finitistic dimension 1, by [ KR ].”
Linear independence of cluster monomials for skew-symmetric cluster algebras (2012, background): “A result of [24] implies the following proposition ifH0Γ is finite-dimensional.”
Calabi-Yau algebras (2006, neutral): “rem 3.4.3, Theorem 3.6.4, Theorem 4.4.6, and Theorem 5.3.1. Thepoint of view on Calabi-Yau algebras advocated in this paperisnot the only possible one. The reader is referred to [Bo], [CR], [IR], and [KR] for alternative approaches. 1.2. Acknowledgements. I am very much indebted to Maxim Kontsevich ”
Representations of Algebras and Related Topics (2011, background): “jects is very close to a module category over a 2-CY-tilted algebra. Such a 2-CY category can be seen as a “recollement” of module categories over 2-CY tilted algebras. Proposition 2.4 (Keller-Reiten [70]). Let Cbe a 2-CY triangulated category with a cluster-tilting object T. Then the functor FT = H”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “There is also an analogy for the one-one correspondence between Morita equivalence classes of finite dimensional algebras of finite representation type and classes of Auslander algebras. See [35, 49 , 31, 43] for further results.”
Generalized cluster complexes via quiver representations (2006, background): “It is the automorphism induced by Auslander-Reiten translation τ in C(H) (compare [Z1, Z2]).”
A geometric description of m-cluster categories (2006, background): “We remark that such Calabi-Yau categories have also been studied in [ KR ].”
On algebras of finite Cohen-Macaulay type (2011, methodology): “Since T is rigid, by a result of Keller-Reiten [49], the functor H induces an equivalence between the stable category `”
Stable categories of higher preprojective algebras (2009, background): “property” (see 3.5) which generalizes n-representation-ﬁniteness. By deﬁnition this condition is always satisﬁed in the cases n = 1 or 2. We generalize Theorem 1.1 and the main result of KellerReiten [KR] for the case n = 2 by showing the following. Theorem 1.2. Let Λ be an algebra satisfying the vo”
Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories (2008, neutral): “This paper is organized as follows: The first section is dedicated to notation and necessary background from [8], [9], [17], [19].”