Local Artin map and the Weil group

Let $K$ be a $p$-adic local field, $\Gamma = \operatorname{Gal}(\overline{K}/K)$, $I = \operatorname{Gal}(K^{\textrm{ur}}/K)$ the inertia group, and $\Gamma/I = \operatorname{Gal}(K^{\textrm{ur}}/K) = \operatorname{Gal}(\overline{k}/k)$, where $k$ is the residue field of $k$. Let $M$ be the subgroup of $\sigma \in \Gamma/I$ which act on $\overline{k}$ by $x \mapsto x^{q^n}$ for some $n \in \mathbb Z$ ($q = |k|$). Then $M$ is cyclic and generated by the Frobenius. Let $W$ be the preimage of $M$ in $\Gamma$. It is called the local Weil group. Then we have an exact sequence of groups

$$1 \rightarrow I \rightarrow W \rightarrow M \rightarrow 1$$

which necessarily splits as a semidirect product, since $M \cong \mathbb Z$. Choosing a splitting $W \cong I \rtimes M$, give $W$ the product topology (this will be independent of the choice of splitting). Then one can show that

• $W$ is a locally profinite topological group.

• The inclusion $W \subset \Gamma$ is continuous with dense image, although $W$ does not have the induced topology.

I'm trying to reinterpret local class field theory using the Weil group. Are the following statements true?

• Let $W^{\textrm{ab}}$ be $W$ modulo its commutator subgroup. The induced continuous homomorphism $W^{\textrm{ab}} \rightarrow \Gamma/\overline{\Gamma}_{\textrm{der}} = \operatorname{Gal}(K^{\operatorname{ab}}/K)$ is injective with dense image.

• The image of the local Artin map $\theta_K: K^{\ast} \rightarrow \operatorname{Gal}(K^{\textrm{ab}}/K)$ is contained in $W^{\operatorname{ab}}$.

• The local Artin map defines an isomorphism of topological groups $K^{\ast} \rightarrow W^{\operatorname{ab}}$.

• The homomorphism $W^{\operatorname{ab}} \rightarrow \operatorname{Gal}(K^{\textrm{ab}}/K)$ makes $\operatorname{Gal}(K^{\textrm{ab}}/K)$ the profinite completion of $W^{\operatorname{ab}}$.

• I believe the answer here (math.stackexchange.com/questions/38799/…) should for the most part answer this question. – D_S May 22 '18 at 16:39
• If you change the definition of $W^{ab}$ by $W$ modulo closure of its commutator subgroup, all you claimed seems true. Artin and Tate's "Class field theory" ch.15 is one of excellent reference. – MiRi_NaE Jun 21 '18 at 6:25