Grothendieck groups and tilting objects

Abstract

Let C be a connected Noetherian hereditary Abelian category with a Serre functor over an algebraically closed field k, with finite-dimensional homomorphism and extension spaces, Using the classification of such categories from 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 has a tilting object.