Inertia subgroup - Knowlpedia

Definition

Let $L_w/K_v$ be a finite Galois extension of nonarchimedean local fields. The inertia subgroup is

I_w=\ker\!\left( \operatorname{Gal}(L_w/K_v)\longrightarrow \operatorname{Gal}(k_w/k_v) \right).

Equivalently, it consists of the automorphisms acting trivially on the residue field. In the global situation it is the kernel of the residue-field action of the decomposition group.

For a nonarchimedean local field $F$ , let $F^{\mathrm s}$ be the separable closure inside a fixed algebraic closure. Then passing through all finite Galois extensions gives an exact sequence

1\longrightarrow I_F\longrightarrow \operatorname{Gal}(F^{\mathrm s}/F)\longrightarrow \operatorname{Gal}(\overline{k}_F/k_F)\simeq\widehat{\mathbb Z} \longrightarrow1.

Here $\operatorname{Gal}(F^{\mathrm s}/F)$ is the absolute Galois group of $F$ .

Tame and wild inertia

If the residue characteristic is $p$ , inertia contains a distinguished pro- $p$ subgroup $P_F$ , the wild inertia group. The quotient $I_F/P_F$ is tame inertia; after compatible choices it is isomorphic to $\prod_{\ell\ne p}\mathbb Z_\ell(1)$ . A representation is tamely ramified when $P_F$ acts trivially and unramified when all of $I_F$ acts trivially.

The local Weil group contains the same $I_F$ as an open compact subgroup. This is why “trivial on inertia” is the common unramified condition for Galois, Weil, and Langlands parameters.

References

Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979, Chapter IV.
The Stacks Project Authors, “Fundamental Groups of Schemes,” §58.13, “Ramification theory.” Stacks Project.