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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:
 1.8e+31 Derived categories, triangulated categories (MSC2010) 1.8e+16 Grothendieck categories (MSC2010) 1.8e+41 Torsion theories, radicals 1.6e+06 Syzygies, resolutions, complexes in associative algebras 1.6e+31 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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