Serre functor and torsion pairs

Abstract

Given a torsion pair $(\mathcal{T},\mathcal{F})$ in an abelian category $\mathcal{A}$ and its Happel-Reiten-Smalø 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 $D^b(\mathcal{A})$.

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