Serre functor and torsion pairs

2024/03/12 by Han, Zhe, He, Ping
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2403.07252

Abstract

Given a torsion pair (T,F) in an abelian category A and its Happel-Reiten-Smalø tilt B, the equivalence of the realization functor D^b(\mathcal B)→ D^b(\mathcal A) is determined by some properties of the torsion pair [9]. We call (T,F) satisfying such a property effaceable. If A is an Ext-finite abelian category with Serre duality, we prove that (T,F) is effaceable implies that 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 (T,F) is effaceable if and only if U_T is closed under the Serre functor via a recollement of D^b(\mathcal A).

Related

• Derived equivalences via HRS-tilting
• Intervals of torsion pairs and generalized Happel-Reiten-Smalø tilting
• Torsion pairs and filtrations in abelian categories with tilting objects
• Classification of abelian hereditary directed categories satisfying Serre duality
• Relative Serre functor for comodule algebras