The Grothendieck-semigroup criterion for semiperfectness of localized group algebras

Let $G$ be a finite group, let $p$ be a prime, and let $bbZ_{(p)}G$ be the group algebra over the localization of the integers at $p$. For a ring $R$, let $bbG(R)$ and $bbK(R)$ denote the Grothendieck rings over $bbZ$ of finitely generated $R$-modules and finitely generated projective $R$-modules, respectively, and let $bbG^+(R)$ and $bbK^+(R)$ be their positive Grothendieck semigroups. Consider the decomposition map and Cartan homomorphism

$$\mathbf{G}(\mathbb{Q} G)\xrightarrow{\alpha}\mathbf{G}(\mathbb{F}_p G)\xleftarrow{\beta}\mathbf{K}(\mathbb{F}_p G).$$

Semiperfectness criterion. The ring $\mathbb{Z}_{(p)}G$ is semiperfect if and only if

$$\alpha\bigl(\mathbf{G}^+(\mathbb{Q} G)\bigr)\supseteq\beta\bigl(\mathbf{K}^+(\mathbb{F}_p G)\bigr).$$

This conjecture is proposed as a workable criterion for semiperfectness in the modular case $p\mid |G|$, where the lifting of idempotents over the non-complete local ring $\mathbb{Z}_{(p)}$ is harder to characterize. In the ordinary case $p\nmid |G|$, the paper gives a complete criterion in terms of Schur indices and inertness of $p$ in character-value fields.