Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

Details Cluster-tilted algebras are Gorenstein and stably Calabi-Yau Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

2005-12-20

arxiv.org/abs/math/0512471v4

Mathematics

Representation Theory

missing conditions added in section 3.2, thanks to Xiao-Wu Chen

Scientific paper

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 on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.

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