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Projective Dimension of Modules over Cluster-Tilted Algebras

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Abstract

We study the projective dimension of finitely generated modules over cluster-tilted algebras End đť’ž (T) where T is a cluster-tilting object in a cluster category đť’ž. It is well-known that all End đť’ž (T)-modules are of the form Hom đť’ž (T, M) for some object M in đť’ž, and since End đť’ž (T) is Gorenstein of dimension 1, the projective dimension of Hom đť’ž (T, M) is either zero, one or infinity. We define in this article the ideal I M of End đť’ž (T[1]) given by all endomorphisms that factor through M, and show that the End đť’ž (T)-module Hom đť’ž (T, M) has infinite projective dimension precisely when I M is non-zero. Moreover, we apply the results above to characterize the location of modules of infinite projective dimension in the Auslander-Reiten quiver of cluster-tilted algebras of type A and D.

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Authors and Affiliations

  1. NSERC of Canada, University of Sherbrooke, Sherbrooke, Canada

    Louis Beaudet & Gordana Todorov

  2. Bishop’s University, Sherbrooke, Canada

    Thomas Brüstle & Gordana Todorov

Authors
  1. Louis Beaudet
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  2. Thomas BrĂĽstle
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  3. Gordana Todorov
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Correspondence to Thomas BrĂĽstle.

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Presented by Raymundo Bautista.

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Beaudet, L., Brüstle, T. & Todorov, G. Projective Dimension of Modules over Cluster-Tilted Algebras. Algebr Represent Theor 17, 1797–1807 (2014). https://doi.org/10.1007/s10468-014-9472-0

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  • DOI: https://doi.org/10.1007/s10468-014-9472-0

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