Estimate of the number of one-parameter families of modules over a tame algebra

TL;DR

This paper investigates the classification of modules over tame algebras, providing bounds on the number of parametric families and finite modules, thereby advancing understanding of their structure and complexity.

Contribution

It introduces bounds on the number of canonical parametric matrices and estimates the count of finite modules over tame algebras.

Findings

Bound of 4^s on the number of canonical parametric matrices

Estimate of the number of finite modules f(d,A)

Modules are classified into finite sets and one-parameter series

Abstract

The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most d divides into a finite number f(d,A) of modules and one-parameter series of modules. We prove that the number of m-by-n canonical parametric block matrices with a given partition into blocks is bounded by 4^s, where s is the number of free entries (which is at most mn), and estimate the number f(d,A).