Direct limits in the heart of a t-structure: the case of a torsion pair (Q2349946)

The authors study the behavior of direct limits in the heart of a \(\mathrm{t}\)-structure. They prove that, for any compactly generated \(\mathrm{t}\)-structure in a triangulated category with coproducts, countable direct limits are exact in its heart. For any Grothendieck category \(\mathcal{G}\) and a torsion pair \(\mathrm{t} =(\mathcal{T}, \mathcal{F})\) in \(\mathcal{G}\), one can associate a \(\mathrm{t}\)-structure in the derived category \(D(\mathcal{G})\), whose heart is denoted by \(\mathcal{H}_{\mathrm{t}}\). A sufficient and necessary condition for \(\mathcal{H}_{\mathrm{t}}\) to be AB5, or equivalently, to be Grothendieck, is given. Moreover, for some special classes of torsion pairs, such as hereditary ones, those for which \(\mathcal{T}\) is a cogenerating class and those for which \(\mathcal{F}\) is a generating class, this condition can be simplified. More precisely, it is proved in such cases that the heart \(\mathcal{H}_{\mathrm{t}}\) is a Grothendieck category if and only if \(\mathcal{F}\) is closed under taking direct limits in \(\mathcal{G}\). As applications of these results, some known results, including the classical results on tilting and cotilting theory of module categories, can be improved or extended to more general Grothendieck categories.