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\(t\mathrm{CG}\) torsion pairs. (English) Zbl 1428.18013
A t-structure, \((\mathcal{U}, \mathcal{V})\) in a triangulated category is compactly generated if its aisle, \(\mathcal{U}\), is compactly generated. A torsion pair in an abelian category, \(\mathcal{A}\) is called tCG if its associated Happel-Reiten-Smalø t-structure in \(\mathcal{D}(\mathcal{A})\) [D. Happel et al., Tilting in abelian categories and quasitilted algebras. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0849.16011)] is compactly generated. Such torsion pairs are the main objects of study in this article with an aim being to answer the question ‘when is the heart of a compactly generated t-structure a Grothendieck category?’ for tCG torsion pairs.
For a commutative Noetherian ring, \(R\), the tCG torsion pairs of \(R\)-Mod are precisely the hereditary torsion pairs. For an arbitrary ring, \(R\), it is shown that (see Theorem 3.3), in \(R\)-Mod, a torsion pair \((\mathcal{T}, \mathcal{F})\) is tCG if and only if there exists a set \(\{ T_\lambda \}_{\lambda \in \Lambda}\) of finitely presented \(R\)-modules in \(\mathcal{T}\) such that \(\mathcal{F} = \bigcap_{\lambda \in \Lambda} \text{Ker(Hom}_R(T_\lambda, ?)\). Every tCG torsion pair \((\mathcal{T}, \mathcal{F})\) is of finite type (that is, \(\mathcal{F}\) is closed under taking direct limits), but the converse is not true in general. For the case of Noetherian rings and regular Von Neumann rings, the authors present a precise description of all tCG torsion pairs.
For any ring, the tCG torsion pairs have associated t-structures with Grothendieck heart and for left Noetherian rings these are precisely the torsion pairs with this property (see Corollary 3.5 and Theorem 3.10).
MSC:
18E40 Torsion theories, radicals
18E10 Abelian categories, Grothendieck categories
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