Are special biserial algebras homologically tame?

TL;DR

This paper investigates the homological tameness of special biserial algebras, demonstrating that certain algebras previously thought not to be homologically tame are indeed tame, based on finitistic dimension analysis.

Contribution

The paper proves that a class of algebras previously believed to be non-tame are actually homologically tame by analyzing their finitistic dimensions.

Findings

All studied algebras are homologically tame.

Finitistic dimensions are equal and finite for these algebras.

Counterexamples to previous assertions are provided.

Abstract

Birge Huisgen-Zimmermann calls a finite dimensional algebra homologically tame provided the little and the big finitistic dimension are equal and finite. The question formulated in the title has been discussed by her in the paper "Representation-tame algebras need not be homologically tame", by looking for any $r > 0$ at a sequence of algebras $\Lambda_m$ with big finitistic dimension $r+m$. As we will show, also the little finitistic dimension of $\Lambda_m$ is r+m. It follows that contrary to her assertion, all her algebras $\Lambda_m$ are homologically tame.