Are special biserial algebras homologically tame?

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.