## Tag `02LL`

Chapter 36: More on Morphisms > Section 36.36: Étale localization of quasi-finite morphisms

Lemma 36.36.2. 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

- $f$ is locally of finite type, and
- $x_i \in X_s$ is isolated for $i = 1, \ldots, n$.
Then there exist

- (a) an elementary étale neighbourhood $(U, u) \to (S, s)$,
- (b) for each $i$ an open subscheme $V_i \subset X_U$,
such that for each $i$ we have

- (\romannumeral1) $V_i \to U$ is a finite morphism,
- (\romannumeral2) there is a unique point $v_i$ of $V_i$ mapping to $u$ in $U$, and
- (\romannumeral3) 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 36.36.1 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 36.36.1 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. $\square$

The code snippet corresponding to this tag is a part of the file `more-morphisms.tex` and is located in lines 10330–10352 (see updates for more information).

```
\begin{lemma}
\label{lemma-etale-makes-quasi-finite-finite-multiple-points}
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
\begin{enumerate}
\item $f$ is locally of finite type, and
\item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$.
\end{enumerate}
Then there exist
\begin{enumerate}
\item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$,
\item[(b)] for each $i$ an open subscheme $V_i \subset X_U$,
\end{enumerate}
such that for each $i$ we have
\begin{enumerate}
\item[(\romannumeral1)] $V_i \to U$ is a finite morphism,
\item[(\romannumeral2)] there is a unique point $v_i$ of $V_i$
mapping to $u$ in $U$, and
\item[(\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and
$\kappa(x_i) = \kappa(v_i)$.
\end{enumerate}
\end{lemma}
\begin{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 \ref{lemma-etale-makes-quasi-finite-finite-at-point} there
exists an elementary \'etale 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 \ref{lemma-etale-makes-quasi-finite-finite-at-point}
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.
\end{proof}
```

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