What is the usual norm argument?

Question

I'm currently reading this paper (PDF page 5/page 111) and the following WLOG statement has me stumped:

Main Lemma: Let $K/\mathbb{Q}$ be a field extension of transcendence degree $1$ and $\ell\neq 2$ be a prime. The map $$\mathrm{H}^3(K,\mathbb{Z}/\ell)\longrightarrow \prod_{w}\mathrm{H}^3({K_w},\mathbb{Z}/\ell)$$ induced by the restrictions is injective.

Proof: Since the degree of the extension $K(\mu_\ell,\sqrt{-1})/K$ is prime to $\ell$ the assertion may be reduced to the case that $\mu_\ell\subset K$ and $K$ is not formally real by the usual sort of norm argument. Hence we may replace $\mathbb{Z}/\ell$ by $\mathbb{Z}/\ell(2) = \mu_\ell^{\otimes 2}$ in the statement.

[...]

As usual, $\mathrm{H}^n(K,-) = \mathrm{H}^n(G_K,-)$. Here, the product ranges over all valuations $w$ of $K$, such that its residue class field $k(w)$ has positive characteristic and is algebraic over its prime field. $K_w$ denotes a henselization of $K$ with respect to $w$.

In this context, what is the usual sort of norm argument?

Answer 1

Sorry for answering late. Here is my clue on the "sort of norm argument". It is actually a quite general process. For any field $F$ and any $G_F$- module $A$ of $l$-primary order, let $E/F$ be an extension of degree $d$ not divisible by $l$. Then $G_E$ is a subgroup of $G_F$ of index $d$, and one has two cohomology maps $Res : H^n (F, A) \to H^n (E, A)$ and $Cor : H^n (E, A) \to H^n (F, A)$ (the corestriction is a "sort of norm") such that $Cor . Res = d$ . Since the $H^n$ here are killed by a power of $l$ and $(l, d) = 1$ , it follows immediately that the restriction must be injective . Then one just have to take $F = K$ or $K_w$ and $E = K(\sqrt -1, \mu_l)$ or $K_w(\sqrt -1, \mu_l)$.

NB: I was confused at first because I read "completion" in place of "henselization". If the $K_w$'s were the completions of $K$, then the local groups $H^3$ would be $0$ because $cd_l (G_{K_w}) \le 2$ .