Serre functor and torsion pairs

Abstract

Given a torsion pair $(\mathcal{T},\mathcal{F})$ in an abelian category $\mathcal{A}$ and its Happel-Reiten-Smal{\o} tilt $\mathcal{B}$, the equivalence of the realization functor $D^b({\mathcal B})\to D^b({\mathcal A})$ is determined by some properties of the torsion pair [9]. We call $(\mathcal{T},\mathcal{F})$ satisfying such a property effaceable. If $\mathcal{A}$ is an Ext-finite abelian category with Serre duality, we prove that $(\mathcal{T},\mathcal{F})$ is effaceable implies that $\mathcal{U}_{\mathcal T}$ is closed under Serre functor. Conversely, when $\mathcal A$ is the module category of a finite-dimensional hereditary algebra, we prove that the torsion pair $(\mathcal{T},\mathcal{F})$ is effaceable if and only if $\mathcal{U}_\mathcal{T}$ is closed under the Serre functor via a recollement of $D^b({\mathcal A})$.