Universal localisations via silting

TL;DR

This paper establishes a deep connection between silting modules and ring localisations, showing that every universal localisation corresponds to a partial silting module and vice versa, using the morphism category and finite-type classification.

Contribution

It demonstrates that silting modules are of finite type and provides a classification linking silting modules with universal localisations of rings.

Findings

Every partial silting module induces a localisation at a set of maps.

Every universal localisation arises from a silting module.

Silting modules are of finite type.

Abstract

We show that silting modules are closely related with localisations of rings. More precisely, every partial silting module gives rise to a localisation at a set of maps between countably generated projective modules and, conversely, every universal localisation, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.