Numerically finite hereditary categories with serre duality

Abstract

Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.