Lemma 41.18.3 (04HN)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 41: Étale Morphisms of Schemes
Section 41.18: Étale local structure of étale morphisms
Lemma 41.18.3 (cite)

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Lemma 41.18.3. Let $f : X \to S$ be a finite étale morphism of schemes. Let $s \in S$. There exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition

\[ X_ U = \coprod \nolimits _ j V_ j \]

of schemes such that each $V_ j \to U$ is an isomorphism.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.3. As in the proof of Lemma 41.18.1 we see that $V_{i, j} \to U$ is an open immersion and we win after replacing $U$ by the intersection of their images. $\square$

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