= Cohomology continuity at closed subgroups
{title2=$H^q(P,M)=\varinjlim_{U\supseteq P}H^q(U,M)$}
For a closed subgroup $P$ of a <profinite group> $G$ and a discrete $G$-module $M$, restriction gives $H^q(P,M)=\varinjlim_{U\supseteq P,\ U\ \mathrm{open}}H^q(U,M)$. 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 $P$.
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