gr.group theory - local-global principles in group cohomology

The local-global principle you are citing comes from the fact that for any open subgroup $Hleq G$, $H^n(G,A)stackrel{text{Res}}{longrightarrow}H^n(H,A)stackrel{text{Cor}}{longrightarrow}H^n(G,A)$ 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 $H^n(G,A)=0$.