The Stacks project

Lemma 15.112.4. Let $A$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa $. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m$ be a maximal ideal of $B$. Then

  1. the field extension $\kappa (\mathfrak m)/\kappa $ is normal, and

  2. $D \to \text{Aut}(\kappa (\mathfrak m)/\kappa )$ is surjective.

If for some (equivalently all) maximal ideal(s) $\mathfrak m \subset B$ the field extension $\kappa (\mathfrak m)/\kappa $ is separable, then

  1. $\kappa (\mathfrak m)/\kappa $ is Galois, and

  2. $D \to \text{Gal}(\kappa (\mathfrak m)/\kappa )$ is surjective.

Here $D \subset G$ is the decomposition group of $\mathfrak m$.

Proof. Observe that $A = B^ G$ as $A$ is integrally closed in $K$ and $K = L^ G$. Thus parts (1) and (2) follow from Lemma 15.110.9. The “equivalently all” part of the lemma follows from Lemma 15.112.1. Assume $\kappa (\mathfrak m)/\kappa $ is separable. Then parts (3) and (4) follow immediately from (1) and (2). $\square$

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