Stable-rank characterization of Drozd–Crawley-Boevey representation-type trichotomy

Let $\Lambda$ be a finite-dimensional algebra over a field $\mathcal{K}$, let $\mathrm{st}(\Lambda)$ be the stable rank at which the descending powers of the module-category radical stabilize, and let $\operatorname{rad}_\Lambda^{\mathrm{st}(\Lambda)}$ be the corresponding transfinite radical power. Stable-rank trichotomy conjecture. One has

$$\mathrm{st}(\Lambda)<\omega^\omega.$$

Moreover:

• $\mathrm{st}(\Lambda)<\omega$ if and only if $\Lambda$ is of finite representation type;
• $\omega\leq\mathrm{st}(\Lambda)<\omega^2$ if and only if $\Lambda$ is of tame representation type but not of finite representation type, and $\Lambda$ is of domestic representation type if and only if $\operatorname{rad}_\Lambda^{\mathrm{st}(\Lambda)}=0$;
• $\omega^2\leq\mathrm{st}(\Lambda)<\omega^\omega$ if and only if $\Lambda$ is of wild representation type.

This is proposed as a stable-rank formulation of the Drozd–Crawley-Boevey trichotomy, extending the finite- and domestic-type results discussed immediately beforehand; the source presents it as a conjectural classification.