Projective dimension of modules over cluster-tilted algebras
Abstract
We study the projective dimension of finitely generated modules over cluster-tilted algebras End(T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End(T)-modules are of the form Hom(T,M) for some object M in C, and since End(T) is Gorenstein of dimension 1, the projective dimension of Hom(T,M) is either zero, one or infinity. We define in this article the ideal I_M of End(T[1]) given by all endomorphisms that factor through M, and show that the End(T)-module Hom(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 and .
Cite
@article{arxiv.1111.2013,
title = {Projective dimension of modules over cluster-tilted algebras},
author = {Louis Beaudet and Thomas Brustle and Gordana Todorov},
journal= {arXiv preprint arXiv:1111.2013},
year = {2013}
}
Comments
Version 2 contains a new section studying in detail the cluster types A and D