Addendum to “Direct limits in the heart of a t-structure: the case of a torsion pair”. (English) Zbl 1397.18027

Parra, Carlos E.; Saorín, Manuel

J. Pure Appl. Algebra 220, No. 6, 2467-2469 (2016).

Summary: Let \(\mathcal{G}\) be a Grothendieck category, let \(\mathbf{t} = (\mathcal{T}, \mathcal{F})\) be a torsion pair in \(\mathcal{G}\) and let \((\mathcal{U}_{\mathbf{t}}, \mathcal{W}_{\mathbf{t}})\) be the associated Happel-Reiten-Smalø t-structure in the derived category \(\mathcal{D}(\mathcal{G})\). We prove that the heart of this t-structure is a Grothendieck category if, and only if, the torsionfree class \(\mathcal{F}\) is closed under taking direct limits in \(\mathcal{G}\).
Addendum to the authors’ paper [ibid. 219, No. 9, 4117–4143 (2015; Zbl 1333.18017)].

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

Keywords:

\(\mathrm{t}\)-structure; direct limit; torsion pair; Grothendieck category; tilting theory; triangular category

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