The Stacks project

Lemma 41.18.2. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is étale at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition

\[ X_ U = W \amalg \coprod \nolimits _{i, j} V_{i, j} \]

of schemes such that

  1. $V_{i, j} \to U$ is an isomorphism,

  2. the fibre $W_ u$ contains no point mapping to any $x_ i$.

In particular, if $f^{-1}(\{ s\} ) = \{ x_1, \ldots , x_ n\} $, then the fibre $W_ u$ is empty.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.2. As in the proof of Lemma 41.18.1 the morphisms $V_{i, j} \to U$ are open immersions and we win after replacing $U$ by the intersection of their images. $\square$

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