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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:
 1.8e+41 Torsion theories, radicals 1.8e+11 Abelian categories, Grothendieck categories
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References:
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