Cluster-tilted Algebras Are Gorenstein and Stably Calabi-yau

  title={Cluster-tilted Algebras Are Gorenstein and Stably Calabi-yau},
  author={Bernhard Keller},
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-CalabiYau. 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… CONTINUE READING
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