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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).
18E40 Torsion theories, radicals
18E10 Abelian categories, Grothendieck categories
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