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

Abstract

Suppose $Λ$ is a special biserial algebra over an algebraically closed field. Schröer showed that if $Λ$ is domestic then the radical of the category of finitely generated (left) $Λ$-modules is nilpotent, and the least ordinal, denoted $\mathrm{st}(Λ)$, where the decreasing sequence of powers of the radical stabilizes satisfies $\mathrm{st}(Λ)<ω^2$. With Gupta and Sardar, the third author conjectured that if $Λ$ has at least one band then $ω\le\mathrm{st}(Λ)<ω^2$ even when $Λ$ is non-domestic. In this paper we settle this conjecture in the affirmative. We also describe an algorithm to compute $\mathrm{st}(Λ)$ up to a finite error. We also show that for each $ω\leqα<ω^2$ there is a finite-dimensional tame representation type algebra $Γ$ with $\mathrm{st}(Γ)=α$.