# Proof of “Hochschild-Serre” spectral sequence

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:

1. 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 $$\overset{o\ o}{\smile}$$ )

• Just to clarify, your question is about the action of $G/H$ on $H^1(...)$, and not the whole spectral sequence? You can do this explicitly using cocycles. In Hochschild and Serre's original paper has detailed discussion of this. Jun 20 at 13:40
• I would like to understand three things: the action of $G/H$; why the sequence is exact in $H^1(H,A)^{G/H}$; why H^2(K^{ur}/K, T^{I_K})=0. Jun 20 at 13:45
• It is Theorem 2.1.5 in the printed version of "Cohomology of Number Fields". In there is also one of the few places that actually shows that the differnential $d$ is the transgression map; see the theorem after. The action, inflation, restriction etc are well explained in the pages before. Jun 20 at 16:53