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 $A$ and $D$.