Continuous cohomology of a profinite group ($H_{\mathrm{cont}}^q(G,M)$)

For a profinite group and a discrete module with continuous action, continuous cohomology is computed by continuous cochains. On a compact domain these cochains have finite image. Positive-degree classes become zero on a sufficiently small open subgroup and hence are torsion by restriction followed by corestriction.

For a closed subgroup $P$ of a profinite group $G$ and a discrete $G$-module $M$, restriction gives $H^q(P,M)={\underrightarrow{\lim }}_{U\supseteq P,U\mathrm{open}}{H}^q(U,M)$. Thus a class restricting to zero on $P$ already restricts to zero on one such open subgroup. The property follows by extending continuous finite-image cochains and their identities to a sufficiently small open neighborhood of $P$.

The value ${\mathrm{cd}}_p(G)$ is the least $n$ such that all cohomology in degrees greater than $n$ vanishes for every discrete $p$-primary torsion module. Vanishing in degree $n+1$ for all such modules suffices by dimension shifting. Dimension does not increase on closed subgroups. For a field, use its absolute Galois group.

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 ${\mathrm{cd}}_p(P)\le n$. The converse is immediate from the definition.

A field of characteristic $p$ satisfies ${\mathrm{cd}}_p(K)\le 1$. The Artin-Schreier sequence and vanishing of positive-degree cohomology of the additive separable-closure module yield the bound. It concerns discrete $p$-primary modules; it does not assert that the $p$-primary Brauer group vanishes.