Laura string algebras

TL;DR

This paper provides a combinatorial criterion to identify Laura algebras among special biserial algebras and proves a conjecture relating finiteness of certain indecomposable modules to the Laura property.

Contribution

It introduces a simple criterion for recognizing Laura algebras and proves a conjecture linking the Laura property to finiteness of specific indecomposable modules.

Findings

A combinatorial criterion for Laura algebras is established.

A special biserial algebra is Laura iff it has finitely many indecomposables with certain dimension properties.

The conjecture by Skowronski is confirmed for special biserial algebras.

Abstract

We give a simple combinatorial criterion allowing to recognize whether a string (or, more generally, a special biserial) algebra is a laura algebra or not. We also show that a special biserial algebra is laura if and only if it has a finite number of isomorphism classes of indecomposable modules which have projective dimension and injective dimension greater than or equal to two, solving a conjecture ok Skowronski for special biserial algebras.