ACYCLIC CALABI-YAU CATEGORIES ARE CLUSTER CATEGORIES

joint with Idun Reiten, with an Appendix by Michel Van den Bergh

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.



http://www.math.jussieu.fr/~keller/publ/AcyclicCalabiYauAbstract.html

Bernhard Keller, le 20 octobre 2006.
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