Torsion classes, wide subcategories and localisations

TL;DR

This paper explores the relationships between torsion classes, wide subcategories, and localisations in finite dimensional algebras, providing explicit classifications and bijections especially in the representation finite case.

Contribution

It establishes new correspondences between torsion classes and wide subcategories, and connects these to ring epimorphisms and universal localisations in algebra.

Findings

Explicit bijection for representation finite algebras

Classification of universal localisations via torsion classes

Connection between support τ-tilting modules and localisations

Abstract

For a finite dimensional algebra $A$, we establish correspondences between torsion classes and wide subcategories in $mod(A)$. In case $A$ is representation finite, we obtain an explicit bijection between these two classes of subcategories. Moreover, we translate our results to the language of ring epimorphisms and universal localisations. It turns out that universal localisations over representation finite algebras are classified by torsion classes and support $\tau$-tilting modules.