The geometry of uniserial representations of finite dimensional algebras III: Finite uniserial type

TL;DR

This paper characterizes finite uniserial types over finite dimensional algebras by describing sequences of simple modules with finitely many uniserial modules, providing necessary and sufficient conditions for finiteness using algebraic varieties.

Contribution

It offers a complete characterization of sequences of simple modules with finitely many uniserial modules and establishes criteria for algebras to have finitely many uniserial module types.

Findings

Identifies sequences with finitely many uniserial modules.

Provides necessary and sufficient conditions for finite uniserial type.

Uses algebraic varieties to parametrize uniserial modules.

Abstract

A description is given of those sequences ${\Bbb S}= (S(0),S(1),\dots,S(l))$ of simple modules over a finite dimensional algebra for which there are only finitely many uniserial modules with consecutive composition factors $S(0),\dots,S(l)$. Necessary and sufficient conditions for an algebra to permit only a finite number of isomorphism types of uniserial modules are derived. The main tools in this investigation are the affine algebraic varieties parametrizing the uniserial modules with composition series ${\Bbb S}$.