Let $K$ be a finite extension of $Q_p$ and $H$ is the inertia group of $G_K$, then we have the well-known local Artin isomorphism $\widehat{K^\times}\cong G_K^{ab}$ where the completion is for the norm groups. But we also have $\widehat{K^\times}\cong O_K^*\times \hat{Z}$, so we have the isomorphism $O_K^*\times \hat{Z}\cong G_K^{ab}$.
Question: Define the composition map $H\hookrightarrow G_K\rightarrow G_K^{ab}\rightarrow O_K^*\times \hat{Z}$ by $f$, then what is the image $f(H)$ in $O_K^*\times \hat{Z}$ ? Do we have $f(H)\subseteq O_K^* \times \{1\} $ ?
Thanks for any answers!