Direct limits in the heart of a t-structure: The case of a torsion pair☆
By C. Parra, Manuel Saor'in. 2013
54 citations recorded by the source corpus.
We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t=(T,F) in G, we show that the heart Ht of the associated t-structure in the derived category D(G) is AB5 if, and only if, it is a Grothendieck category. If this is the case, then F is closed under taking direct limits. The reverse implication is true for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend results by Buan–Krause and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tilting theory of modules.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2ca084354c1d28de724377aa5d0177cdcbb9dbd9
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Citations in the corpus
- Silting Objects (2018, methodology): “Then hearts of t-structures also have (co)products: the (co)product of a family of objects in the heart is obtained by applying the functor H0 t to the corresponding (co)product of the same family in T , see [85].”
- Silting modules over commutative rings (2016, background): “y Proposition 4.4, it is clear that Θ(Ξ(D)) = Dfor any silting class D. That Ξ(Θ(G)) = Gfor any Gabriel topology of finite type follows from [17, Chapter VI, Theorem 5.1], and by character duality. In [16], it was asked whether any tilting torsion pair (T,F) of finite type is classical (that is, Tis g”
- One-tilting classes and modules over commutative rings (2015, background): “([PS13]) Let R be a ring and T a 1-tilting module such that the induced torsion-free class F = Ker HomR(T,−) is closed under direct limits.”
- Torsion pairs in silting theory (2016, unknown): “First, recall from [Parra and Saorín 2015] that hearts of t-structures in a triangulated category with products and coproducts also have products and coproducts.”