Intervals of torsion pairs and generalized Happel-Reiten-Smalø tilting (2512.13273v2)

Abstract: Let $\mathcal{A}$ be an abelian category with a torsion pair $(\mathcal{T},\mathcal{F})$. Happel-Reiten-Smalo tilting provides a method to construct a new abelian category $\mathcal{B}$ with a torsion pair associated to $(\mathcal{T},\mathcal{F})$, which is exactly the heart of a certain $t$-structure on the bounded derived category $D^b(\mathcal{A})$. In this paper, we mainly study generalized HRS tilting. We first show that an interval of torsion pairs in extriangulated categories with negative first extensions is bijectively associated with torsion pairs in the corresponding heart, which yields several new observations in triangulated categories. Then we obtain a generalization of HRS tilting by replacing hearts of $t$-structures with extended hearts. As an application, we show that certain $t$-structures on triangulated subcategories can be extended to $t$-structures on the whole triangulated categories.

• The paper establishes mutually inverse, order-preserving bijections between torsion pairs in extriangulated categories and corresponding structures in their hearts.
• It generalizes classical HRS tilting by replacing standard hearts with extended hearts via m-fold extensions, unifying various tilting approaches.
• The study extends t-structures from triangulated subcategories to ambient categories, providing a systematic framework for tilting and mutation in homological algebra.

Intervals of Torsion Pairs and Generalized Happel-Reiten-Smalø Tilting

Overview

This paper develops a structural theory of torsion pairs and their intervals in extriangulated categories with negative first extensions, establishing deep correspondences to torsion pairs and $t$-structures in associated hearts. The authors generalize classical Happel-Reiten-Smalø (HRS) tilting, replacing the heart of a $t$-structure by extended hearts, and derive new bijections for related structures. The framework is further leveraged to address the extension problem for $t$-structures from triangulated subcategories to ambient triangulated categories.

Torsion Pairs and Intervals in Extriangulated Categories

Let $C$ be an extriangulated category with a negative first extension $\mathbb{E}^{-1}$. The authors define a partial order on torsion pairs $(U, V)$, denoted $t_1 \preccurlyeq t_2$, via the vanishing of morphisms and negative first extensions between the torsion and torsionfree classes. An interval $\tors[t_1, t_2]$ is the collection of torsion pairs ordered between $t_1$ and $t_2$.

The main technical achievement here is a set of mutually inverse, order-preserving bijections between:

• the set of torsion pairs (resp. $s$-torsion pairs -- those with vanishing negative first extensions) in an interval of $C$,
• and the set of torsion pairs in the intersection (or heart) $U_2 \cap V_1$, subject to vanishing conditions on negative first extensions.

Formally, Theorem 3.8 proves the strong assertion that such bijections exist between

$\tors[t_1, t_2] \overset{\Phi}{\underset{\Psi}{\rightleftarrows}} \widetilde{\tors}H_{[t_1, t_2]}$

with explicit formulas.

For triangulated categories, this formalism translates to order-preserving correspondences for torsion pairs and $t$-structures within bounded intervals, with shift functor respect conditions. These represent nontrivial organizational control over the posets of admissible torsion and $t$-structure classes.

Generalization: HRS Tilting and Extended Hearts

The classical HRS tilting procedure constructs new abelian hearts in the derived category via a torsion pair in an abelian category, producing specific $t$-structures. This work generalizes this mechanism in two significant steps:

1. By replacing hearts arising from $t$-structures with extended hearts, $m$-fold extensions of an original heart (Zhou's "extended heart").
2. By introducing binary relations among these extended hearts, capturing the notion of tilting in broader settings (including proper abelian subcategories as per Jørgensen).

The central result (Theorem 4.8) establishes order-preserving, mutually inverse bijections between the poset of $s$-torsion pairs in an $m$-extended heart $\mathcal{H}$ and the poset of $m$-extended hearts ordered within $[\mathcal{H}[m], \mathcal{H}]$.

This construction unifies various recent generalizations of HRS tilting, showing that tilting along intervals or more exotic subcategories yields bijective control over associated categorical structures.

Extensions of $t$-Structures: From Subcategory to Category

The theory is applied to study when $t$-structures on triangulated subcategories extend naturally to $t$-structures on the parent category. Given a triangulated subcategory $S$ containing the heart $H$ of a $t$-structure $U$ on $D$, and $U_S$ the restriction to $S$, the authors prove (Theorem 5.3) that there exist order-preserving, mutually inverse bijections between the $t$-structures in intervals $[U[m], U]$ on $D$ and corresponding intervals $[U_S[m], U_S]$ on $S$.

Concretely, the bijections are realized via intersections and tilting with $m$-fold extensions, providing a detailed specification for reconstructing $t$-structures between subcategory and category in homological algebra, notably for derived categories with unbounded vs. bounded contexts.

Numerical and Structural Results

• Explicit enumeration: The theory is illustrated with detailed examples for hereditary algebras and derived categories, including Auslander-Reiten quivers (Figures 1–7) and explicit count of torsion (and $s$-torsion) pairs in given intervals.
• Strong bijective assertions: Every torsion (or $s$-torsion) pair within an admissible interval corresponds precisely to combinatorial and categorical data in the heart or extended heart.
• Generalized construction: The framework captures prior results (AET, Jørgensen, Zhou) as special cases, including those for Serre subcategories and quasi-abelian settings.

Implications and Future Directions

The categorical control afforded by intervals of torsion pairs and extended hearts underlies a systematic approach to tilting theory and mutation in triangulated and extriangulated categories. Practically, these results facilitate modular design of derived categories, hearts, and related structures in algebraic geometry, representation theory, and homological algebra.

The extension framework may catalyze further results on stratifications of derived categories, stability conditions, and moduli spaces. Future work may focus on computational aspects for large or wild categories, connections with Bridgeland stability, or applications to noncommutative and derived algebraic geometry.

Conclusion

This paper rigorously generalizes HRS tilting to intervals of torsion pairs and extended hearts, providing precise categorical correspondences and extension theory for $t$-structures. The work synthesizes recent advances and offers a foundational toolset for further categorical and homological explorations in representation theory and beyond (2512.13273).