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This is a stupid question about group cohomology, but it confuses me a lot. Basically I think that the problem is the fact that I do not really understand Shapiro's lemma.

Say we take a profinite group $G$ and some finite index normal subgroup $H$. Consider the map of $G$-modules $\mathbb{Z} \to \mathbb{Z}[G/H]$ given by $1 \mapsto N_{G/H} = \sum_{\overline{\sigma} \in G/H} \overline{\sigma}$. This map induces a maps in group cohomology $H^i(G,\mathbb{Z}) \to H^i(G,\mathbb{Z}[G/H]) \cong H^i(H,\mathbb{Z})$, where the isomorphism is Shapiro's lemma.

Now my question is: is this just the map obtained when "restricting" from $G$ to the subgroup $H$?

For example if $i = 2$, then the map can be identified with a map $$\text{Hom}_{\text{continuous}}(G,\mathbb{Q}/\mathbb{Z}) \to \text{Hom}_{\text{continuous}}(H,\mathbb{Q}/\mathbb{Z}).$$ Is this just the map obtained by precomposing $G \to \mathbb{Q}/\mathbb{Z}$ with the inclusion $H \to G$?

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The answer to your question is yes, and it is true in wider generality. Have a look at Benson's Representations and Cohomology vol 1, the exercise at the end of ch2 (p.48 in my edition). –  mt_ Dec 12 '11 at 17:16

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Yes, the restriction map and the corestriction map on group cohomology can both be interpreted in terms of the isomorphism provided by Shapiro's lemma. Details can be found on pp 60-61 of 'Cohomology of Number Fields' by Neukirch/Schmidt/Wingberg. You may also find the discussion on pp 67-68 of Milne's notes on Class Field Theory (available at his website) to be helpful.

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