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

Abstract

Suppose $\Lambda$ is a special biserial algebra over an algebraically closed field. Schr\"oer 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 $\mathrm{st}(\Lambda)$, where the decreasing sequence of powers of the radical stabilizes satisfies $\mathrm{st}(\Lambda)<\omega^2$. With Gupta and Sardar, the third author conjectured that if $\Lambda$ has at least one band then $\omega\le\mathrm{st}(\Lambda)<\omega^2$ even when $\Lambda$ is non-domestic. In this paper we settle this conjecture in the affirmative. We also describe an algorithm to compute $\mathrm{st}(\Lambda)$ up to a finite error. We also show that for each $\omega\leq\alpha<\omega^2$ there is a finite-dimensional tame representation type algebra $\Gamma$ with $\mathrm{st}(\Gamma)=\alpha$.