Lemma 15.112.4 (09ED)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.112: Galois extensions and ramification
Lemma 15.112.4 (cite)

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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

the field extension $\kappa (\mathfrak m)/\kappa$ is normal, and
$D \to \text{Aut}(\kappa (\mathfrak m)/\kappa )$ is surjective.

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

$\kappa (\mathfrak m)/\kappa$ is Galois, and
$D \to \text{Gal}(\kappa (\mathfrak m)/\kappa )$ 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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