## Tag `02LM`

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

Lemma 36.36.3. 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 étale neighbourhood $(U, u) \to (S, s)$,
- (b) for each $i$ an integer $m_i$ and open subschemes $V_{i, j} \subset X_U$, $j = 1, \ldots, m_i$
such that we have

- (\romannumeral1) each $V_{i, j} \to U$ is a finite morphism,
- (\romannumeral2) there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\kappa(u) \subset \kappa(v_{i, j})$ finite purely inseparable,
- (\romannumeral4) if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and
- (\romannumeral3) the points $v_{i, j}$ map to $x_i$ in $X$ and no other points of $(X_U)_u$ map to $x_i$.

Proof.This proof is a variant of the proof of Algebra, Lemma 10.141.23 in the language of schemes. By Morphisms, Lemma 28.19.6 the morphism $f$ is quasi-finite at each of the points $x_i$. Hence $\kappa(s) \subset \kappa(x_i)$ is finite for each $i$ (Morphisms, Lemma 28.19.5). For each $i$, let $\kappa(s) \subset L_i \subset \kappa(x_i)$ be the subfield such that $L_i/\kappa(s)$ is separable, and $\kappa(x_i)/L_i$ is purely inseparable. Choose a finite Galois extension $\kappa(s) \subset L$ such that there exist $\kappa(s)$-embeddings $L_i \to L$ for $i = 1, \ldots, n$. Choose an étale neighbourhood $(U, u) \to (S, s)$ such that $L \cong \kappa(u)$ as $\kappa(s)$-extensions (Lemma 36.31.2).Let $y_{i, j}$, $j = 1, \ldots, m_i$ be the points of $X_U$ lying over $x_i \in X$ and $u \in U$. By Schemes, Lemma 25.17.5 these points $y_{i, j}$ correspond exactly to the primes in the rings $\kappa(u) \otimes_{\kappa(s)} \kappa(x_i)$. This also explains why there are finitely many; in fact $m_i = [L_i : \kappa(s)]$ but we do not need this. By our choice of $L$ (and elementary field theory) we see that $\kappa(u) \subset \kappa(y_{i, j})$ is finite purely inseparable for each pair $i, j$. Also, by Morphisms, Lemma 28.19.13 for example, the morphism $X_U \to U$ is quasi-finite at the points $y_{i, j}$ for all $i, j$.

Apply Lemma 36.36.2 to the morphism $X_U \to U$, the point $u \in U$ and the points $y_{i, j} \in (X_U)_u$. This gives an étale neighbourhood $(U', u') \to (U, u)$ with $\kappa(u) = \kappa(u')$ and opens $V_{i, j} \subset X_{U'}$ with the properties (\romannumeral1), (\romannumeral2), and (\romannumeral3) of that lemma. We claim that the étale neighbourhood $(U', u') \to (S, s)$ and the opens $V_{i, j} \subset X_{U'}$ are a solution to the problem posed by the lemma. We omit the verifications. $\square$

The code snippet corresponding to this tag is a part of the file `more-morphisms.tex` and is located in lines 10101–10127 (see updates for more information).

```
\begin{lemma}
\label{lemma-etale-makes-quasi-finite-finite-multiple-points-var}
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 \'etale neighbourhood $(U, u) \to (S, s)$,
\item[(b)] for each $i$ an integer $m_i$ and
open subschemes $V_{i, j} \subset X_U$, $j = 1, \ldots, m_i$
\end{enumerate}
such that we have
\begin{enumerate}
\item[(\romannumeral1)] each $V_{i, j} \to U$ is a finite morphism,
\item[(\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$
mapping to $u$ in $U$ with $\kappa(u) \subset \kappa(v_{i, j})$
finite purely inseparable,
\item[(\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and
$j = j'$, and
\item[(\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and
no other points of $(X_U)_u$ map to $x_i$.
\end{enumerate}
\end{lemma}
\begin{proof}
This proof is a variant of the proof of
Algebra, Lemma \ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}
in the language of schemes.
By Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}
the morphism $f$ is quasi-finite at each of the points $x_i$.
Hence $\kappa(s) \subset \kappa(x_i)$ is finite for each $i$
(Morphisms, Lemma \ref{morphisms-lemma-residue-field-quasi-finite}).
For each $i$, let $\kappa(s) \subset L_i \subset \kappa(x_i)$
be the subfield such that $L_i/\kappa(s)$ is separable, and
$\kappa(x_i)/L_i$ is purely inseparable. Choose a finite Galois
extension $\kappa(s) \subset L$ such that there exist
$\kappa(s)$-embeddings $L_i \to L$ for $i = 1, \ldots, n$.
Choose an \'etale neighbourhood $(U, u) \to (S, s)$ such that
$L \cong \kappa(u)$ as $\kappa(s)$-extensions
(Lemma \ref{lemma-realize-prescribed-residue-field-extension-etale}).
\medskip\noindent
Let $y_{i, j}$, $j = 1, \ldots, m_i$ be the points of $X_U$
lying over $x_i \in X$ and $u \in U$. By
Schemes, Lemma \ref{schemes-lemma-points-fibre-product}
these points $y_{i, j}$ correspond exactly to the primes in the rings
$\kappa(u) \otimes_{\kappa(s)} \kappa(x_i)$. This also
explains why there are finitely many; in fact
$m_i = [L_i : \kappa(s)]$ but we do not need this.
By our choice of
$L$ (and elementary field theory)
we see that $\kappa(u) \subset \kappa(y_{i, j})$ is
finite purely inseparable for each pair $i, j$.
Also, by Morphisms, Lemma \ref{morphisms-lemma-base-change-quasi-finite}
for example, the morphism
$X_U \to U$ is quasi-finite at the points $y_{i, j}$ for
all $i, j$.
\medskip\noindent
Apply Lemma \ref{lemma-etale-makes-quasi-finite-finite-multiple-points}
to the morphism $X_U \to U$, the point $u \in U$
and the points $y_{i, j} \in (X_U)_u$. This gives an \'etale neighbourhood
$(U', u') \to (U, u)$ with $\kappa(u) = \kappa(u')$ and
opens $V_{i, j} \subset X_{U'}$ with the properties
(\romannumeral1), (\romannumeral2), and (\romannumeral3)
of that lemma. We claim that the \'etale neighbourhood
$(U', u') \to (S, s)$ and the opens $V_{i, j} \subset X_{U'}$
are a solution to the problem posed by the lemma.
We omit the verifications.
\end{proof}
```

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