Silting modules

TL;DR

This paper introduces silting modules, a new concept that generalizes existing tilting modules and establishes their connections with torsion classes, t-structures, and silting complexes, expanding the framework of module theory.

Contribution

The paper defines silting modules, explores their properties, and establishes bijections with silting complexes and t-structures, unifying various concepts in module and derived category theory.

Findings

Silting modules generate torsion classes with left approximations.

Partial silting modules have Bongartz complements.

Silting modules correspond bijectively to 2-term silting complexes and certain t-structures.

Abstract

We introduce the new concept of silting modules. These modules generalise tilting modules over an arbitrary ring, as well as support $\tau$-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. We show that silting modules generate torsion classes that provide left approximations, and that every partial silting module admits an analogue of the Bongartz complement. Furthermore, we prove that silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category. We also see how some of these bijections hold for silting complexes of arbitrary finite length.