2020/01/14 by Jorge Vitória, Vitória, Jorge
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2001.04730
openalex publication_date 2020/01/14 · openalex created_date 2020/01/23 · openalex updated_date 2026/07/28
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.