Lemma 41.17.1 (04HH)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 41: Étale Morphisms of Schemes
Section 41.17: Étale local structure of unramified morphisms
Lemma 41.17.1 (cite)

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Lemma 41.17.1. 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 unramified 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

$V_{i, j} \to U$ is a closed immersion passing through $u$ ,
$u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$ , and
any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$ .

Proof. By Morphisms, Definition 29.35.1 there exists an open neighbourhood of each $x_ i$ which is locally of finite type over $S$ . Replacing $X$ by an open neighbourhood of $\{ x_1, \ldots , x_ n\}$ we may assume $f$ is locally of finite type. Apply More on Morphisms, Lemma 37.41.3 to get the étale neighbourhood $(U, u)$ and the opens $V_{i, j}$ finite over $U$ . By Lemma 41.7.3 after possibly shrinking $U$ we get that $V_{i, j} \to U$ is a closed immersion. $\square$

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