I'm looking for a detailed proof of the Hochschild-Serre spectral sequence for Galois Cohomology:

If we have $H$ a normal subgroup of $G$ (profinite group) and $T$ is a $G-$module, we get:

$$ 0\rightarrow H^1(G/H, A^H)\overset{inf}{\rightarrow} H^1(G, A) \overset{res}{\rightarrow} H^1(H, A)^{G/H}\overset{d}{\rightarrow}H^2(G/H, A^H)\overset{inf}{\rightarrow} H^2(G, A) $$

Just from the statement I don't understand one thing:

- How $G/H$ acts on $H^1(H, A)$;

I'm studying this to understand: $$ 0\rightarrow H^1(K^{ur}/K, T^{I_K})\overset{inf}{\rightarrow}H^1(K,T)\overset{res}{\rightarrow} H^1(I_K,T)^{G^{ur}_K}\rightarrow 0 $$ (if there is a simper way to see this case please let me know $\overset{o\ o}{\smile}$ )

Thank you in advance!

Local fields, ch. VII, §6. $\endgroup$ – abx Jun 20 at 13:572more comments