Injectivity of idele norm map

Question

Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb A_K^\times\to\Bbb A_F^\times, \qquad N_{K/F}((a_\mu)_\mu)=(\prod_{\mu|\nu}N_{K_\mu/F_\nu}(a_\mu))_\nu$$ via the local norm maps. It is known that local norms are not necessarily injective. My question is: Is the idele norm map injective, that is, if $N_{K/F}(a)=1$ do we have $a=1$? If it is not the case, what is the kernel of $N_{K/F}$?

Answer 1

In cohomological terms, the answer is easy. Suppose that the extension K/F is Galois, with Galois group G. I prefer to denote the idèle group of K by $J_K$. For any place v of F, pick an arbitrary place w of K over v. It follows readily from Shapiro's lemma that for any r in Z, $H^r$(G, $J_K$) is isomorphic to the direct sum of the local groups $H^r$($K_w/F_v$, $K_w$*) for all v (Tate cohomology). In particular $H^1$(G, $J_K$)= $0$ (analog of Hilbert 90). If moreover G is cyclic, generated by s, any idèle a of norm $1$ is of the form $b^{-1}$. s(b). Note that this is the best result you can get, because it is already so for local fields. If G is not cyclic, one can replace $H^1$ by $H^{-1}$. The global $H^{-1}$ is by definition Ker N /$I_G$.$J_K$, where $I_G$ is the "augmentation ideal" of Z[G], generated by all the elements $1$ - s, s running through G. An absolutely analogous description holds for the local $H^{-1}$'s, but class field theory gives a "better" result, namely $H^{-1}$($K_w/F_v$, $K_w$*) is isomorphic to $H^{-3}$($K_w/F_v$, Z) (trivial Galois action on Z) .