Hereditary triangulated categories

TL;DR

This paper characterizes hereditary triangulated categories, which are equivalent to bounded derived categories of hereditary abelian categories, providing intrinsic criteria and applications to piecewise hereditary algebras.

Contribution

It offers two intrinsic characterizations of hereditary triangulated categories and applies these to the study of piecewise hereditary algebras.

Findings

Intrinsic characterizations using subcategories and path conditions

Equivalence to bounded derived categories of hereditary abelian categories

Application to classifying piecewise hereditary algebras

Abstract

We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the triangulated category is algebraical, we may replace the equivalence by a triangle equivalence. We give two intrinsic characterizations of hereditary triangulated categories using a certain full subcategory and the non-existence of certain paths. We apply them to piecewise hereditary algebras.