Norm map of ideles and injectivity of corresponding map on Galois groups

I'm referring to another question of mine, since it made me think about this:

Kronecker-Weber false for non-trivial number fields

What I'm essentially asking for is given two number fields $L/K$ and the norm map $N_{L/K}:C_L\to C_K$ between the idele class groups, then how will noninjectivity transfer to the profinite completion? In general given a map $f:G\to G'$ between some groups will that always induce a map between the profinite completions (e.g. is it functorial) and how/when will injectivity/surjectivity transfer?

It seems like my books on number theory leave out these parts of the theory of profinite groups.

• Regarding your last question, any group homomorphism $f:H\to G$ induces a map on profinite completions $\hat{f}:\widehat{H}\to\widehat{G}$ by general nonsense. If $f$ is surjective then $\hat{f}$ is too, obviously. It may be the case that $f$ is injective but $\hat{f}$ is not, even if $H$ and $G$ are residually finite.
– HJRW
Jan 16, 2012 at 16:13
• What about $f$ noninjective but $\hat{f}$ injective? That's really what's needed for the number theoretic problem.
– asdf
Jan 16, 2012 at 16:18
• @asdf: there are easy examples with $f$ non-injective but $\hat{f}$ injective: Let $f: Q \to Z$ be the zero map. Then $\hat{f}: \hat{Q} \to \hat{Z}$ is injective since $\hat{Q} = 0$.
– SGP
Jan 16, 2012 at 17:46
• @SGP: Is a reason for why this is not injective for the maps between idele class groups without invoking the result that I'm trying to show?
– asdf
Jan 16, 2012 at 18:21
• @SGP: The question makes more sense when both $H$ and $G$ are residually finite, so that these groups embed in their profinite completions.
– user6976
Jan 23, 2012 at 14:03

1 Answer

You should look at the Grothendieck problem solved by Bridson and Grunewald. Also this paper may be relevant to what you want: Coulbois, Thierry; Sapir, Mark; Weil, Pascal A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups. Publ. Mat. 47 (2003), no. 2, 477–487.