Functoriality of the local Artin reciprocity

Question

Let $k$ be a finite extension of $\mathbb{Q}_p$. Then the local Artin reciprocity tells us that $$\text{Gal}(\bar{k}/k)^{\text{ab}}\cong \hat{k^\times}.$$ If we have a finite extension $\bar{k}/l/k$, then we have a natural inclusion $\text{Gal}(\bar{k}/l)^{\text{ab}}\to \text{Gal}(\bar{k}/k)^{\text{ab}}$. I would like to understand the corresponding map $\hat{l^\times}\to \hat{k^\times}$. For any extension $l'/l$ the map $$\text{Gal}(l'/l)\cong l^\times/N_{l'/l}(l'^\times)\longrightarrow \text{Gal}(l'/k)\cong k^\times/N_{l'/k}(l'^\times)$$ is induced by the norm $N_{l/k}$. It would seem natural that the continuous map $\hat{l^\times}\to \hat{k^\times}$ is given by the completion of the norm $l^\times \to k^\times$. But that doesn't make sense, no? Because this map isn't injective, and the inclusion is. Clearly I'm missing something stupid...

Answer 1

There is a natural homomorphism $\text{Gal}(\bar{l}/l)^{\text{ab}}\to \text{Gal}(\bar{k}/k)^{\text{ab}}$ it is not injective, the natural guess is that it corresponds to $N_{l/k}: l^\times\to k^\times$