Let $\Gamma$ be a pro-$p$ abelian group. Is it true that the Krull dimension of the completed group algebra $\mathbb F_p[[\Gamma]]$ is equal to the $\mathbb Z_p$-rank of $\Gamma$?
If $\Gamma\simeq \mathbb Z_p^r$, then the completed group algebra is isomorphic to the ring of power series: $\mathbb Z_p[[\Gamma]]\simeq \mathbb Z_p[[T_1,\ldots,T_r]]$ (cf. e.g. 5.3.5. of Neukirch–Schmidt–Wingberg). Quotienting out by $p$, we get that $\mathbb F_p[[\Gamma]]$ is isomorphic to $\mathbb F_p[[T_1,\ldots,T_r]]$, which has Krull dimension $r$, which is what we wanted.
What happens if $\Gamma$ also has a torsion part?