Decomposition group - Knowlpedia

Definition

Let $L/K$ be a finite Galois extension of global fields, let $v$ be a nonarchimedean place of $K$ , and choose a place $w$ of $L$ above $v$ . The decomposition group at $w$ is the stabilizer

D_w=\{\sigma\in\operatorname{Gal}(L/K):\sigma w=w\}.

Restriction to the completion identifies $D_w\simeq\operatorname{Gal}(L_w/K_v)$ . Its action on residue fields gives a surjection

D_w\longrightarrow\operatorname{Gal}(k_w/k_v)

whose kernel is the inertia subgroup $I_w$ .

Dependence on the chosen place

The Galois group acts transitively on the places above $v$ . Replacing $w$ by $\tau w$ replaces $D_w$ by the conjugate $\tau D_w\tau^{-1}$ . Consequently the decomposition group attached to $v$ is canonical only up to conjugacy.

For a separable closure $\overline K/K$ and an extension $\overline v$ of $v$ , the same stabilizer construction gives an absolute decomposition group $D_{\overline v}\subset\operatorname{Gal}(\overline K/K)$ , canonically isomorphic to the absolute Galois group of $K_v$ after the choice of $\overline v$ .

References

The Stacks Project Authors, “Fundamental Groups of Schemes,” §58.13, “Ramification theory.” Stacks Project.
Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, 1979, Chapter I, §7.