Tilting modules and tilting torsion pairs - filtrations induced by tilting modules

TL;DR

This paper surveys how tilting modules create equivalences in module categories and explores the extent to which these equivalences reveal information about the entire category, including new insights.

Contribution

It provides a comprehensive survey of tilting modules and torsion pairs, with new insights into their role in understanding module categories.

Findings

Summarizes key results on tilting modules and torsion pairs.

Introduces new perspectives on filtrations induced by tilting modules.

Highlights the extent of information transfer between parts and the whole category.

Abstract

Tilting modules, generalising the notion of progenerator, furnish equivalences between pieces of module categories. This paper is dedicated to study how much these pieces say about the whole category. We will survey the existing results in the literature, introducing also some new insights.