Noetherian hereditary abelian categories satisfying Serre duality

Abstract

In this paper we classify Ext \operatorname {Ext} -finite noetherian hereditary abelian categories over an algebraically closed field k k satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories. As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.