## Tag `02LP`

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

Lemma 36.36.6. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

- $f$ is locally of finite type,
- $f$ is separated, and
- $X_s$ has at most finitely many isolated points.
Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition $$ U \times_S X = W \amalg V $$ into open and closed subschemes such that the morphism $V \to U$ is finite, and the fibre $W_u$ of the morphism $W \to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_u = \emptyset$.

Proof.This is clear from Lemma 36.36.4 by choosing $x_1, \ldots, x_n$ the complete set of isolated points of $X_s$ and setting $V = \bigcup V_i$. $\square$

The code snippet corresponding to this tag is a part of the file `more-morphisms.tex` and is located in lines 10528–10546 (see updates for more information).

```
\begin{lemma}
\label{lemma-etale-splits-off-quasi-finite-part}
Let $f : X \to S$ be a morphism of schemes.
Let $s \in S$. Assume that
\begin{enumerate}
\item $f$ is locally of finite type,
\item $f$ is separated, and
\item $X_s$ has at most finitely many isolated points.
\end{enumerate}
Then there exists an elementary \'etale neighbourhood $(U, u) \to (S, s)$
and a decomposition
$$
U \times_S X = W \amalg V
$$
into open and closed subschemes such that the morphism
$V \to U$ is finite, and the fibre $W_u$ of the
morphism $W \to U$ contains no isolated points.
In particular, if $f^{-1}(s)$ is a finite set, then $W_u = \emptyset$.
\end{lemma}
\begin{proof}
This is clear from
Lemma \ref{lemma-etale-splits-off-quasi-finite-part-technical}
by choosing $x_1, \ldots, x_n$ the complete set of
isolated points of $X_s$ and setting $V = \bigcup V_i$.
\end{proof}
```

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