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

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 coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category GG and a torsion pair t=(T,F)t=(T,F) in GG, we show that the heart HtHt of the associated t-structure in the derived category D(G)D(G) is AB5 if, and only if, it is a Grothendieck category. If this is the case, then FF is closed under taking direct limits. The reverse implication is true for a wide class of torsion pairs which include the hereditary ones, those for which TT is a cogenerating class and those for which FF is a generating class. The results allow to extend results by Buan–Krause and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tilting theory of modules.