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

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