Locally finitely presented and coherent hearts. (English) Zbl 1514.18009
Let \((\mathcal{T},\mathcal{F})\) be a torsion pair in a Grothendieck category \(\mathcal{G}\) and \(\mathcal{H}\) the heart of the associated HRS-tilted \(t\)-structure on \(D(\mathcal{G})\). In this paper the authors study finiteness conditions for \(\mathcal{H}\), assuming that the heart is again Grothendieck. By [C. E. Parra and M. Saorín, J. Pure Appl. Algebra 220, No. 6, 2467–2469 (2016; Zbl 1397.18027)] this is equivalent to \((\mathcal{T},\mathcal{F})\) being of finite type. In the present paper it is shown that for torsion pairs finite type is the same as quasi-cotilting, and also cosilting.
The main results are necessary and sufficient criteria for \(\mathcal{H}\) to be locally finitely presented or locally coherent, under various (often technical) assumptions on \(\mathcal{G}\) and \((\mathcal{T},\mathcal{F})\). In particular it is shown that \(\mathcal{H}\) is locally finitely presented if and only if \((\mathcal{T},\mathcal{F})\) is generated by finitely presented objects. The torsion pairs for which \(\mathcal{H}\) is locally coherent are identified as those that restrict to \(\mathrm{fp}(\mathcal{G})\) and satisfy \(\mathcal{F}\cap\mathrm{fp}(\mathcal{G})\subseteq\mathrm{fp}_\infty(\mathcal{G})\). Local coherence of \(\mathcal{H}\) is also considered in relation to local coherence for the heart associated with the restricted torsion pair \((\mathcal{T}\cap\underline{\mathcal{F}},\mathcal{F})\) in the subcategory \(\underline{\mathcal{F}}\) consisting of quotients of objects in \(\mathcal{F}\). Moreover these results are applied to the case that \(\mathcal{G}\) is a module category.
The main results are necessary and sufficient criteria for \(\mathcal{H}\) to be locally finitely presented or locally coherent, under various (often technical) assumptions on \(\mathcal{G}\) and \((\mathcal{T},\mathcal{F})\). In particular it is shown that \(\mathcal{H}\) is locally finitely presented if and only if \((\mathcal{T},\mathcal{F})\) is generated by finitely presented objects. The torsion pairs for which \(\mathcal{H}\) is locally coherent are identified as those that restrict to \(\mathrm{fp}(\mathcal{G})\) and satisfy \(\mathcal{F}\cap\mathrm{fp}(\mathcal{G})\subseteq\mathrm{fp}_\infty(\mathcal{G})\). Local coherence of \(\mathcal{H}\) is also considered in relation to local coherence for the heart associated with the restricted torsion pair \((\mathcal{T}\cap\underline{\mathcal{F}},\mathcal{F})\) in the subcategory \(\underline{\mathcal{F}}\) consisting of quotients of objects in \(\mathcal{F}\). Moreover these results are applied to the case that \(\mathcal{G}\) is a module category.
Reviewer: Lukas Bonfert (Bonn)
MSC:
| 18E10 | Abelian categories, Grothendieck categories |
| 18G80 | Derived categories, triangulated categories |
| 18E40 | Torsion theories, radicals |
Keywords:
\(t\)-structure; heart; Happel-Reiten-Smalø; locally finitely presented; locally coherent; silting; tilting; elementary cogeneratorCitations:
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