Relative Torsion Classes, relative tilting and relative silting modules

TL;DR

This paper generalizes the concept of torsion classes in module categories over Artin algebras by introducing F-torsion classes and F-presilting modules, extending the framework of τ-tilting theory.

Contribution

It introduces F-torsion classes and F-presilting modules, broadening the scope of τ-tilting theory and characterizing their properties in module categories.

Findings

F-torsion classes are characterized in terms of preenveloping properties.

F-presilting modules generalize τ-rigid and F-tilting modules.

The paper provides criteria for when F-torsion classes are preenveloping.

Abstract

Let $\Lambda$ be an Artin algebra. In 2014, T. Adachi, O. Iyama and I. Reiten proved that the torsion funtorially finite classes in $\mathrm{mod}\,(\Lambda)$ can be described by the $\tau$-tilting theory. The aim of this paper is to introduce the notion of $F$-torsion class in $\mathrm{mod}\,(\Lambda)$, where $F$ is an additive subfunctor of $\mathrm{Ext}^1_\Lambda,$ and to characterize when these clases are preenveloping and $F$-preenveloping. In order to do that, we introduce the notion of $F$-presilting $\Lambda$-module. The latter is both a generalization of $\tau$-rigid and $F$-tilting in $\mathrm{mod}(\Lambda).$