Universal localisations and tilting modules for finite dimensional algebras

TL;DR

This paper classifies universal localisations for finite dimensional algebras, establishing connections with tilting modules in hereditary and Nakayama cases, thus advancing the understanding of their module categories.

Contribution

It provides a complete classification of universal localisations for certain classes of finite dimensional algebras and links them to support tilting and τ-tilting modules.

Findings

Complete classification for hereditary algebras

Correspondence between localisations and tilting modules

Extension of results to Nakayama and local algebras

Abstract

We study universal localisations, in the sense of Cohn and Schofield, for finite dimensional algebras and classify them by certain subcategories of our initial module category. A complete classification is presented in the hereditary case as well as for Nakayama algebras and local algebras. Furthermore, for hereditary algebras, we establish a correspondence between finite dimensional universal localisations and finitely generated support tilting modules. In the Nakayama case, we get a similar result using $\tau$-tilting modules, which were recently introduced by Adachi, Iyama and Reiten.