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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

  1. $f$ is locally of finite type,
  2. $f$ is separated, and
  3. $x_1, \ldots, x_n$ 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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