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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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