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. 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 $\mathrm{st}(\Lambda )$, 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 $\omega \le \alpha <{\omega }^2$ there is a finite-dimensional tame representation type algebra $\Gamma$ with $\mathrm{st}(\Gamma )=\alpha$.

36 pages, 3 Figures