Grothendieck groups and tilting objects

Let C be a connected Noetherian hereditary Abelian category with a Serre fu nctor over an algebraically closed field k, with finite-dimensional homomor phism and extension spaces, Using the classification of such categories fro m our 1999 preprint, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C ha s a tilting object.