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Direct limits in the heart of a t-structure: the case of a torsion pair. (English) Zbl 1333.18017
J. Pure Appl. Algebra 219, No. 9, 4117-4143 (2015); addendum ibid. No. 6, 2467-2469 (2016).
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.

MSC:
18E30 Derived categories, triangulated categories (MSC2010)
18E15 Grothendieck categories (MSC2010)
18E40 Torsion theories, radicals
16E05 Syzygies, resolutions, complexes in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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