Direct limits in the heart of a t-structure: The case of a torsion pair

Abstract

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.

Silting Modules over Commutative Rings

“…Therefore, there is a map h : The correspondence is given as follows: In [16], it was asked whether any tilting torsion pair (T , F ) of finite type is classical (that is, T is generated by a finitely presented tilting module). The answer turned out to be negative for general rings, but positive for commutative rings ( [7]).…”

Torsion pairs over n-hereditary rings

“…The importance of torsion pairs is highlighted by the theorem of Popescu and Gabriel [Ste75, X, §4] which reduces the theory of Grothendieck categories to the study of categories of modules of quotients by a hereditary torsion pairs. All this have made the theory of torsion pairs a valuable toolkit and an active research area on its own; see [BP16], [CGM07], [Hrb16], [PS15].…”

Properties of abelian categories via recollements

“…For a general TTF triple in an abelian category, we have the following theorem, whose proof will also rely on generation by finite reduction. This argument is inspired by a similar construction in [26,Proposition 4.7].…”

“…Since Mod-A is AB5 (and thus (T , F ) is a directed torsion pair), if F were closed for direct limits, then Hom A (A/AeA, −) would commute with direct limits, which is not the case since A/AeA is not finitely presented. Thus, F is not closed for direct limits and, therefore, by [26,Theorem 4.8], the heart H = {X ∈ D(A) : H −1 (X) ∈ F , H 0 (X) ∈ T , H k (X) = 0, ∀k = −1, 0} of the t-structure in D(A) associated with the torsion pair (T , F ) (see [12, Proposition 2.1] for the construction of this t-structure) is not an AB5 abelian category. Using the techniques explored in detail in [20,Section 6], there is a recollement of hearts of the form…”

“…Equivalently, we may think of A as the matrix ring where the outer terms are AB5 (even Grothendieck) categories and the middle term is not an AB5 abelian category. It is known from [26,Proposition 3.3] that H is AB4 and AB4*, and it also follows from Proposition 3.2 that it is well-powered. For more details on constructing recollements of hearts from recollements of abelian categories in the way exemplified above, we refer to [30, Section 10].…”