The Stacks project

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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