The Stacks project

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

  1. $V_{i, j} \to U$ is a closed immersion passing through $u$,

  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. 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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