Relative Torsion Classes, relative tilting and relative silting modules

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 ${\mathrm{Ext}}_{\Lambda }^1,$ 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 $\mathrm{mod}(\Lambda ).$