Derived equivalences via HRS-tilting

TL;DR

This paper establishes criteria for when a Happel-Reiten-Smal{}o} tilt of an abelian category is derived equivalent to the original, highlighting the role of torsion pairs and the properties of realization functors.

Contribution

It provides necessary and sufficient conditions for derived equivalences induced by HRS-tilting, including the impact of splitting torsion pairs and properties of realization functors.

Findings

Splitting torsion pairs induce derived equivalences.

Denseness of the realization functor implies full faithfulness.

Criteria for derived equivalences via HRS-tilting are established.

Abstract

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be the Happel-Reiten-Smal{\o} tilt of $\mathcal{A}$ with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between $\mathcal{B}$ and $\mathcal{A}$, which is compatible with the inclusion of $\mathcal{B}$ into the derived category of $\mathcal{A}$. In particular, any splitting torsion pair induces a derived equivalence. We prove that for the realization functor of any bounded $t$-structure, its denseness implies its fully-faithfulness.