The Stacks project

Lemma 41.18.1. 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 étale at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_ U$, $i = 1, \ldots , n$, $j = 1, \ldots , m_ i$ such that

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

  2. $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and

  3. any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.1. Now $V_{i, j} \to U$ is a closed immersion and étale. Hence it is an open immersion, for example by Theorem 41.14.1. Replace $U$ by the intersection of the images of $V_{i, j} \to U$ to get the lemma. $\square$

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