On the stable radical of the module category for special biserial algebras

Abstract

Suppose that $\Lambda$ is a special biserial algebra over an algebraically closed field. Schröer showed that if $\Lambda$ is domestic, then the radical of the category of finitely generated (left) $\Lambda$-modules is nilpotent, and the least ordinal, denoted as $\mathrm{st}\left(\Lambda \right)$, where the decreasing sequence of powers of the radical stabilizes satisfies . With Gupta and Sardar, the third author conjectured that if $\Lambda$ has at least one band, then even when $\Lambda$ is nondomestic. In this paper, we settle this conjecture in the affirmative. We also describe an algorithm to compute $\mathrm{st}\left(\Lambda \right)$ up to a finite error. We also show that for each , there is a finite-dimensional tame representation-type algebra $\Gamma$ with $\mathrm{st}\left(\Gamma \right)=\alpha$.