Lemma 37.35.2 (02LF)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.35: Étale neighbourhoods
Lemma 37.35.2 (cite)

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Lemma 37.35.2. Let $S$ be a scheme. Let $s \in S$ . Let $k/\kappa (s)$ be a finite separable field extension. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ such that the field extension $\kappa (u)/\kappa (s)$ is isomorphic to $k/\kappa (s)$ .

Proof. We may assume $S$ is affine. In this case the lemma follows from Algebra, Lemma 10.144.3. $\square$

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