Direct limits in the heart of a t-structure: the case of a torsion pair

TL;DR

This paper investigates how direct limits behave in the heart of a t-structure, establishing conditions under which they are exact and characterizing when the heart is a Grothendieck category, extending known module theory results.

Contribution

It extends the understanding of direct limits in the heart of t-structures within Grothendieck categories, generalizing previous module theory results.

Findings

Countable direct limits are exact in the heart of compactly generated t-structures.

If the heart is AB5, the torsion-free class is closed under direct limits.

The heart is a Grothendieck category under certain torsion pair conditions.

Abstract

We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with exact coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t = (T ;F) in G, we show that if the heart of the associated t-structure in the derived category D(G) is AB5, then F is closed under taking direct limits. The reverse implication is true, even implying that the heart is a Grothendieck category, for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend well-known results by Buan-Krause, Bazzoni and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tlting theory of modules.