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

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.