Relative torsion classes, relative tilting, and relative silting modules

Abstract

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

Keywords:

• Auslander–Reiten theory
• relative homological algebra
• relative tilting and relative presilting modules
• relative torsion classes

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16G10
• 18G05
• Secondary: 16E10

Acknowledgments

The authors are greatly indebted to the referee for his/her carefully reading and helpful suggestions.

Additional information

Funding

The authors thank project PAPIIT-Universidad Nacional Autónoma de México IN100520.