The Stacks project

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 29.44). 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. $\square$

Comments (3)

Comment #4018 by Davide Lombardo on

Small typo: "the fibre over ", not over , 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.

There are also:

  • 2 comment(s) on Section 41.17: Étale local structure of unramified morphisms

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