The Quiver of Projectives in Hereditary Categories with Serre Duality

TL;DR

This paper characterizes hereditary categories with Serre duality generated by preprojective objects, showing they are derived equivalent to representations of certain strongly locally finite quivers, using novel distance concepts.

Contribution

It introduces light cone and round trip distances on quivers to analyze the structure of hereditary categories with Serre duality and preprojective generation.

Findings

Hereditary categories with Serre duality and no infinite radicals are derived equivalent to quiver representations.

Introduction of light cone and round trip distances for quiver analysis.

Classification of such categories via strongly locally finite quivers.

Abstract

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.