Definition 15.112.6. With assumptions and notation as in Lemma 15.112.5.

1. The wild inertia group of $\mathfrak m$ is the subgroup $P$.

2. The tame inertia group of $\mathfrak m$ is the quotient $I \to I_ t$.

We denote $\theta : I \to \mu _ e(\kappa (\mathfrak m))$ the surjective map (15.112.5.1) whose kernel is $P$ and which induces the isomorphism $I_ t \to \mu _ e(\kappa (\mathfrak m))$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).