## Tag `02LN`

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

Lemma 36.36.4. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that

- $f$ is locally of finite type,
- $f$ is separated, and
- $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ U \times_S X = W \amalg V_1 \amalg \ldots \amalg V_n $$ into open and closed subschemes such that the morphisms $V_i \to U$ are finite, the fibres of $V_i \to U$ over $u$ are singletons $\{v_i\}$, each $v_i$ maps to $x_i$ with $\kappa(x_i) = \kappa(v_i)$, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_i$.

Proof.Choose $(U, u) \to (S, s)$ and $V_i \subset X_U$ as in Lemma 36.36.2. Since $X_U \to U$ is separated (Schemes, Lemma 25.21.13) and $V_i \to U$ is finite hence proper (Morphisms, Lemma 28.42.11) we see that $V_i \subset X_U$ is closed by Morphisms, Lemma 28.39.7. Hence $V_i \cap V_j$ is a closed subset of $V_i$ which does not contain $v_i$. Hence the image of $V_i \cap V_j$ in $U$ is a closed set (because $V_i \to U$ proper) not containing $u$. After shrinking $U$ we may therefore assume that $V_i \cap V_j = \emptyset$ for all $i, j$. This gives the decomposition as in the lemma. $\square$

The code snippet corresponding to this tag is a part of the file `more-morphisms.tex` and is located in lines 10176–10195 (see updates for more information).

```
\begin{lemma}
\label{lemma-etale-splits-off-quasi-finite-part-technical}
Let $f : X \to S$ be a morphism of schemes.
Let $s \in S$. Let $x_1, \ldots, x_n \in X_s$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $x_1, \ldots, x_n$ are pairwise distinct isolated points of $X_s$.
\end{enumerate}
Then there exists an elementary \'etale neighbourhood $(U, u) \to (S, s)$
and a decomposition
$$
U \times_S X = W \amalg V_1 \amalg \ldots \amalg V_n
$$
into open and closed subschemes such that the morphisms
$V_i \to U$ are finite, the fibres of $V_i \to U$ over $u$ are
singletons $\{v_i\}$, each $v_i$ maps to $x_i$ with
$\kappa(x_i) = \kappa(v_i)$, and the fibre of $W \to U$
over $u$ contains no points mapping to any of the $x_i$.
\end{lemma}
\begin{proof}
Choose $(U, u) \to (S, s)$ and $V_i \subset X_U$ as in
Lemma \ref{lemma-etale-makes-quasi-finite-finite-multiple-points}.
Since $X_U \to U$ is separated
(Schemes, Lemma \ref{schemes-lemma-separated-permanence})
and $V_i \to U$ is finite hence proper
(Morphisms, Lemma \ref{morphisms-lemma-finite-proper})
we see that $V_i \subset X_U$ is closed by
Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}.
Hence $V_i \cap V_j$ is a closed subset of $V_i$ which
does not contain $v_i$. Hence the image of $V_i \cap V_j$
in $U$ is a closed set (because $V_i \to U$ proper) not
containing $u$. After shrinking $U$ we may therefore assume
that $V_i \cap V_j = \emptyset$ for all $i, j$. This gives the
decomposition as in the lemma.
\end{proof}
```

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