Serre functor and torsion pairs
Zhe Han, Ping He
TL;DR
The paper investigates when the Happel-Reiten-Smalø (HRS) tilt $\mathcal{B}$ of an abelian category $\mathcal{A}$ with torsion pair $(\mathcal{T},\mathcal{F})$ yields a derived-equivalence $D^b(\mathcal{B})\to D^b(\mathcal{A})$, showing this holds precisely for effaceable torsion pairs in Ext-finite categories with Serre duality. It proves one direction: effaceable implies the HRS aisle $\mathcal{U}_{\mathcal{T}}$ is closed under the Serre functor $\mathbb{S}$, and in the hereditary module case it proves the converse via recollement and compatibility results for HRS-tilts. The approach relies on tilting theory, perpendicular categories, and exceptional objects to construct recollements that preserve the tilting structure, enabling a reduction from general torsion classes to finitely generated or Ext-projective-free cases. The results connect the Serre-closure of aisles with effective obstructions to non-equivalences of realization functors, offering a framework to understand when HRS-tilts reproduce the ambient derived category. These insights have implications for understanding when tilted hearts faithfully realize the ambient triangulated category and for potential generalizations beyond hereditary or finite-type settings.
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 $D^b({\mathcal A})$.