# Split $t$-structures and torsion pairs in hereditary categories

## TL;DR

This paper characterizes when torsion pairs correspond to t-structures in hereditary categories, establishes conditions for split t-structures, and classifies such structures in hereditary algebra derived categories.

## Contribution

It provides necessary and sufficient conditions for torsion pairs to correspond to t-structures and classifies split t-structures in hereditary algebra derived categories.

## Key findings

- Torsion pairs in hereditary categories correspond to t-structures under specific conditions.
- Existence of split t-structures with nontrivial heart implies derived equivalence to hereditary categories.
- Complete classification of split t-structures in hereditary algebra derived categories.

## Abstract

We give necessary and sufficient conditions for torsion pairs in a hereditary category to be in bijection with $t$-structures in the bounded derived category of that hereditary category. We prove that the existence of a split $t$-structure with nontrivial heart in a semiconnected Krull-Schmidt category implies that this category is equivalent to the derived category of a hereditary category. We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding $t$-structures. Finally, we classify split $t$-structures in the derived category of a hereditary algebra.