I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that
The cohomological dimension $cd \ G$ of a pro-finite group $G$ is the smallest integer $n$ such that $$H^q(G,A) = 0 \text{ for all } q > n$$ and all $G$ torsion modules $A$. Analogously, the cohomological $p$-dimension $cd_p \ G$ is the smallest integer $n$ such that $$H^q(G,A)(p) = 0 \text{ for all } q > n$$ an all $G$ torsion modules $A$. (Here, $A(p)$ denotes the $p$-primary part for an abelian group $A$).
My question is really simple I guess.
For a pro-$p$ group $G$, how are $cd \ G$ and $cd_p \ G$ related?
For an arbitrary profinite group $G$, we have $$cd \ G = \sup_p cd_p \ G$$ But this does not really help me. My guess is that they're in fact equal. Do you have any hints for me? Thank you!
If $G$ is a pro-$p$ group and we require our cocycles to be continuous, then $$ H^q(G, A) $$ will already be $p$-primary for any $G$-torsion module, and so your guess is right. (Proof: cohomology commutes with direct limits, so we may assume $A$ is finite. Decompose $A$ as an abelian group. Any summand of $A$ of order prime to $p$ has trivial continuous cohomology, and the remaining cohomology is killed by any power of $p$ killing the $p$ part of $A$.)