Noetherian hereditary abelian categories satisfying Serre duality

Abstract

In this paper we classify <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E x t"> <mml:semantics> <mml:mi>Ext</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {Ext}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite noetherian hereditary abelian categories over an algebraically closed field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.