Torsion Pairs in Recollements of Abelian Categories

TL;DR

This paper investigates how torsion pairs in abelian categories relate within recollements, establishing conditions under which torsion pairs can be transferred or induced across the categories involved.

Contribution

It provides new insights into the relationship between torsion pairs in abelian categories connected by recollements, including conditions for their transfer.

Findings

Torsion pairs in outer categories induce torsion pairs in the middle category.

Under certain conditions, torsion pairs in the middle category induce torsion pairs in outer categories.

The paper clarifies the conditions needed for the converse transfer of torsion pairs.

Abstract

For a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of abelian categories, we show that torsion pairs in $\mathcal{A}$ and $\mathcal{C}$ can induce torsion pairs in $\mathcal{B}$; and the converse holds true under certain conditions.