Numerically finite hereditary categories with Serre duality (Q2796089)

The main aim of the present paper is to provide a classification theorem for numerically finite nonzero indecomposable hereditary categories with Serre duality. It is proved in Theorem 1 that such a category is derived equivalent to either (1) a tube, (2) the category of finite-dimensional representations of a finite acyclic quiver, or (3) the category of coherent sheaves of a hereditary order induced by a smooth projective curve.NEWLINENEWLINEIn order to prove this theorem the author studies in Section 3 nonzero abelian hereditary Ext-finite categories with Serre duality over an algebraically closed field, and proves that these categories have an object which is either exceptional or 1-spherical (Theorem 2). Moreover, in Section 5 is proved that if such a category does not have any exceptional objects, then it is derived equivalent to either a homogeneous tube, or the category of coherent sheaves on a smooth projective curve of genus at least one (Theorem 3).