We study the projective dimension of finitely generated modules over clustertilted algebras End_C(T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End_C(T)-modules are of the formHom_C(T,M) for some objectM in C, and since End_C(T) is Gorenstein of dimension 1, the projective dimension of Hom_C(T,M) is either zero, one or infinity. We define in this article the ideal I_M of End_C(T 1) given by all endomorphisms that factor throughM, and show that the End_C(T)-moduleHom_C(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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