Lemma 58.12.5 (0BSU)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.12: Group actions and integral closure
Lemma 58.12.5 (cite)

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Lemma 58.12.5. Let $A$ be a normal domain with fraction field $K$ . Let $L/K$ be a (possibly infinite) Galois extension. Let $G = \text{Gal}(L/K)$ and let $B$ be the integral closure of $A$ in $L$ . Let $\mathfrak q \subset B$ . Set

$I = \{ \sigma \in G \mid \sigma (\mathfrak q) = \mathfrak q \text{ and } \sigma \bmod \mathfrak q = \text{id}_{\kappa (\mathfrak q)}\}$

Then $(B^ I)_{B^ I \cap \mathfrak q}$ is a filtered colimit of étale $A$ -algebras.

Proof. We can write $L$ as the filtered colimit of finite Galois extensions of $K$ . Hence it suffices to prove this lemma in case $L/K$ is a finite Galois extension, see Algebra, Lemma 10.154.3. Since $A = B^ G$ as $A$ is integrally closed in $K = L^ G$ the result follows from Lemma 58.12.4. $\square$

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