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CLUSTER-TILTED ALGEBRAS ARE GORENSTEIN AND STABLY CALABI-YAU

### Idun Reiten and 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-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 Geiss-Leclerc-Schroer. 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 by Geiss-Leclerc-Schroer for simple modules. Finally, we
generalize the results obtained from 2-Calabi-Yau to d-Calabi-Yau categories
using techniques similar to those of Iyama. We show how to produce many
examples of d-cluster tilting algebras.

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

Bernhard Keller, le 20 décembre 2005

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