Acyclic Calabi-Yau categories

Abstract

We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, we show that the stable category of maximal Cohen-Macaulay modules over a certain isolated singularity of dimension three is a cluster category. As a second application, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

Introduction rewritten, references updated. 16 pages

References in corpus (4)

• On triangulated orbit categories
• Cluster algebra structures and semicanonical bases for unipotent groups
• On the structure of Calabi-Yau categories with a cluster tilting subcategory
• On the structure of triangulated category with finitely many indecomposables

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• Deformed Calabi-Yau Completions
• On two examples by Iyama and Yoshino
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• Auslander-Reiten theory revisited
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• Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions
• On Generalized Cluster Categories
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• Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras
• Crossed-products of Calabi-Yau algebras by finite groups
• A categorification of acyclic principal coefficient cluster algebras
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• The Ringel--Hall Lie algebra of a spherical object
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules
• On cluster-tilting graphs for hereditary categories
• Calabi-Yau properties of Postnikov diagrams
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• Silting reduction and Calabi--Yau reduction of triangulated categories
• Lifting preprojective algebras to orders and categorifying partial flag varieties
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• Cluster tilting for higher Auslander algebras
• Potentials and Jacobian algebras for tensor algebras of bimodules
• Rigid objects in higher cluster categories
• A new cluster character with coefficients for cluster category
• Cluster algebras generated by projective cluster variables
• The Integral Cluster Category
• Tilting objects on tubular weighted projective lines: a cluster tilting approach
• An introduction to higher cluster categories
• Cluster tilting objects in generalized higher cluster categories
• On tubular tilting objects in the stable category of vector bundles
• Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category