= Trivial coefficients detect the cohomological dimension of a pro-p group
{title2=$H^{n+1}(P,\mathbb F_p)=0\ \Longleftrightarrow\ \operatorname{cd}_p(P)\le n$}
For a <pro-p group> $P$, vanishing of $H^{n+1}(P,\mathbb F_p)$ implies vanishing in that degree for every discrete $p$-primary module. Finite such modules have composition factors equal to the trivial module $\mathbb F_p$, so the long exact sequence gives the result by induction. General modules are unions of finite stable submodules, and continuous cohomology commutes with filtered unions. Dimension shifting gives $\operatorname{cd}_p(P)\le n$. The converse is immediate from the definition.
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