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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

  1. $f$ is locally of finite type,
  2. $f$ is separated, and
  3. $X_s$ 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 10300–10318 (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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