Lemma 41.17.3. Let $f : X \to S$ be a finite unramified morphism of schemes. Let $s \in S$. There exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition

$X_ U = \coprod \nolimits _ j V_ j$

such that each $V_ j \to U$ is a closed immersion.

Proof. Since $X \to S$ is finite the fibre over $s$ is a finite set $\{ x_1, \ldots , x_ n\}$ of points of $X$. Apply Lemma 41.17.2 to this set (a finite morphism is separated, see Morphisms, Section 29.42). The image of $W$ in $U$ is a closed subset (as $X_ U \to U$ is finite, hence proper) which does not contain $u$. After removing this from $U$ we see that $W = \emptyset$ as desired. $\square$

Comment #4018 by Davide Lombardo on

Small typo: "the fibre over $s$", not over $S$, I guess.

Comment #4021 by Laurent Moret-Bailly on

Of course this is trivial, but I believe the statement should say that the sum is finite.

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).