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Let $G$ be a (profinite) group. It is known that if $H^n(G_p,A) = 0$ for all $p$, $S_p$ the Sylow subgroups of $G$, then $H^n(G,A) = 0$.

Are there other local-global principles for different sets of subgroups?

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One I see commonly in my area is when $G$ is the absolute Galois group of a global field (for example a number field) and the subgroups are the absolute Galois groups of the completions of the field (these are subgroups of $G$). In many instances local-global principles do not hold, and their failure is measured by a "class group" or "Tate-Schaferevich group". –  Kevin Buzzard Jan 19 '10 at 13:09
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The local-global principle you are citing comes from the fact that for any open subgroup $H\leq G$, $H^n(G,A)\stackrel{\text{Res}}{\longrightarrow}H^n(H,A)\stackrel{\text{Cor}}{\longrightarrow}H^n(G,A)$ is multiplication by $[G:H]$. So from that you can derive lots of local-global principles. E.g. as a generalisation of the one you cite, you can deduce that if $H_1$ and $H_2$ are two open subgroups of co-prime index such that $H^n(H_i,A)=0$ for $i=1,2$, then $H^n(G,A)=0$.

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