Characterising the bounded derived category of an hereditary abelian category

TL;DR

This paper establishes conditions under which a triangulated category containing an admissible hereditary abelian subcategory is equivalent to the bounded derived category of that subcategory, linking t-structures to category equivalences.

Contribution

It proves that the inclusion of an admissible hereditary abelian subcategory can be lifted to a fully faithful functor, characterizing when a triangulated category is equivalent to a bounded derived category.

Findings

A triangulated category with a split, bounded t-structure is equivalent to a bounded derived category.

The inclusion of an admissible hereditary abelian subcategory can be extended to a fully faithful functor.

Provides criteria for when a triangulated category is equivalent to the derived category of an hereditary abelian category.

Abstract

We show that if a (not necessarily algebraic) triangulated category T contains an admissible hereditary abelian subcategory H, then we can lift the inclusion of H into T to a fully faithful triangle functor from the whole of the bounded derived category of H to T. This allows us prove, for example, that a triangulated category T is triangle equivalent to the bounded derived category of an hereditary abelian category if and only if T admits a split, bounded t-structure.