Lemma 37.41.2. 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 that

1. $f$ is locally of finite type, and

2. $x_ i \in X_ s$ is isolated for $i = 1, \ldots , n$.

Then there exist

1. an elementary étale neighbourhood $(U, u) \to (S, s)$,

2. for each $i$ an open subscheme $V_ i \subset X_ U$,

such that for each $i$ we have

1. $V_ i \to U$ is a finite morphism,

2. there is a unique point $v_ i$ of $V_ i$ mapping to $u$ in $U$, and

3. the point $v_ i$ maps to $x_ i$ in $X$ and $\kappa (x_ i) = \kappa (v_ i)$.

Proof. We will use induction on $n$. Namely, suppose $(U, u) \to (S, s)$ and $V_ i \subset X_ U$, $i = 1, \ldots , n - 1$ work for $x_1, \ldots , x_{n - 1}$. Since $\kappa (s) = \kappa (u)$ the fibre $(X_ U)_ u = X_ s$. Hence there exists a unique point $x'_ n \in X_ u \subset X_ U$ corresponding to $x_ n \in X_ s$. Also $x'_ n$ is isolated in $X_ u$. Hence by Lemma 37.41.1 there exists an elementary étale neighbourhood $(U', u') \to (U, u)$ and an open $V_ n \subset X_{U'}$ which works for $x'_ n$ and hence for $x_ n$. By the final assertion of Lemma 37.41.1 the open subschemes $V'_ i = U'\times _ U V_ i$ for $i = 1, \ldots , n - 1$ still work with respect to $x_1, \ldots , x_{n - 1}$. Hence we win. $\square$

Comment #2348 by Eric Ahlqvist on

space after \romannumeral

Comment #2417 by on

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