On the abundance of silting modules

TL;DR

This paper discusses the widespread nature of silting modules and their role in classifying various torsion classes and localisations in algebra, highlighting recent developments in the field.

Contribution

It provides an overview of silting and cosilting modules and explains their use in classifying torsion classes and localisations in algebraic structures.

Findings

Silting modules parametrize definable torsion classes over noetherian rings.

Silting modules classify hereditary torsion pairs of finite type over commutative rings.

Silting modules correspond to universal localisations of hereditary rings and finite-dimensional algebras.

Abstract

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or of a finite dimensional algebra of finite representation type, can be parametrised by silting modules. In these notes, we give a brief introduction to the fairly recent concepts of silting and cosilting module, and we explain the classification results mentioned above.