Cohomology continuity at closed subgroups ($H^q(P,M)={\underrightarrow{\lim }}_{U\supseteq P}{H}^q(U,M)$)

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 $P$.