Numerically finite hereditary categories with Serre duality

TL;DR

This paper classifies numerically finite hereditary categories with Serre duality, extending understanding of their structure and relation to categories like coherent sheaves on smooth projective varieties.

Contribution

It provides a classification of numerically finite hereditary categories with Serre duality up to derived equivalence under specific finiteness conditions.

Findings

Classification of such categories achieved

Connection established with categories of coherent sheaves

Framework for understanding derived equivalences

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.