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ACYCLIC CALABI-YAU CATEGORIES ARE CLUSTER CATEGORIES

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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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