  We study the projective dimensions of the restriction of functors Hom(-,X) to
a contravariantly finite rigid subcategory T of a triangulated category C. We
show that the projective dimension of Hom(-,X)|T is at most one if and only if
there are no non-zero morphisms between objects in T[1] factoring through X,
when the object X belongs to a suitable subcategory of C. As a consequence, we
obtain a characterisation of the objects of infinite projective dimension in
the category of finitely presented contravariant functors on a cluster-tilting
subcategory of C.
