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. $\square$

There are also:

• 2 comment(s) on Section 41.18: Étale local structure of étale morphisms

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).