Quantity vs. size in representation theory
Abstract
In this note, we survey two instances in the representation theory of finite-dimensional algebras where the quantity of a type of structures is intimately related to the size of those same structures. More explicitly, we review the fact that (1) a finite-dimensional algebra admits only finitely many indecomposable modules up to isomorphism if and only if every indecomposable module is finite-dimensional; (2) the category of modules over a finite-dimensional algebra admits only finitely many torsion classes if and only if every torsion class is generated by a finite-dimensional module.
Cite
@article{arxiv.2001.04730,
title = {Quantity vs. size in representation theory},
author = {Jorge Vitória},
journal= {arXiv preprint arXiv:2001.04730},
year = {2020}
}
Comments
Invited survey to a special issue of the "Boletim da Sociedade Portuguesa de Matem\'atica", aimed at a general audience. This work does not contain any new results