  Let C be a connected noetherian hereditary abelian Ext-finite category with
Serre functor over an algebraically closed field k, with finite dimensional
homomorphism and extension spaces. Using the classification of such categories
from math.RT/9911242, 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 has a
tilting object.
