Intervals of torsion pairs and generalized Happel-Reiten-Smal{\o} tilting

TL;DR

This paper explores the structure of torsion pairs in abelian and triangulated categories, generalizing HRS tilting by using extended hearts, and shows how certain t-structures can be extended across categories.

Contribution

It introduces a bijective correspondence between intervals of torsion pairs and torsion pairs in hearts, generalizes HRS tilting with extended hearts, and demonstrates extension of t-structures.

Findings

Interval of torsion pairs bijectively corresponds to torsion pairs in hearts.

Generalized HRS tilting replaces hearts with extended hearts.

Certain t-structures can be extended to larger triangulated categories.

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.