A simple homological characterization of string algebras of finite representation type

TL;DR

This paper characterizes string algebras of finite representation type using a homological property related to extensions of indecomposable modules, distinguishing them from non-domestic string algebras.

Contribution

It provides a homological criterion that precisely characterizes finite representation type string algebras among all finite dimensional algebras.

Findings

String algebras of finite representation type have extensions with at most two direct factors.

Non-domestic string algebras do not satisfy this property.

The property serves as a homological characterization of these algebras.

Abstract

We prove that among the finite dimensional algebras of finite representation type those that are string algebras are precisely the ones that have the property that the middle term of an arbitrary extension of indecomposable modules has at most two direct factors. On the other hand, we show that non-domestic string algebras are very far from having that property.