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Tag 02UP

Chapter 36: More on Morphisms > Section 36.38: Zariski's Main Theorem

Lemma 36.38.5. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that $f$ is proper and $f^{-1}(\{s\})$ is a finite set. Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{s\})$ by Morphisms, Lemma 28.19.7. By Morphisms, Lemma 28.52.2 the set of points at which $f$ is quasi-finite is an open $U \subset X$. Let $Z = X \setminus U$. Then $s \not \in f(Z)$. Since $f$ is proper the set $f(Z) \subset S$ is closed. Choose any open neighbourhood $V \subset S$ of $s$ with $Z \cap V = \emptyset$. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 28.19.9), hence has finite fibres (Morphisms, Lemma 28.19.10), hence is finite by Lemma 36.38.4. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 10911–10918 (see updates for more information).

    \begin{lemma}
    \label{lemma-proper-finite-fibre-finite-in-neighbourhood}
    Let $f : X \to S$ be a morphism of schemes.
    Let $s \in S$.
    Assume that $f$ is proper and $f^{-1}(\{s\})$ is a finite set.
    Then there exists an open neighbourhood $V \subset S$ of $s$
    such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.
    \end{lemma}
    
    \begin{proof}
    The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{s\})$
    by Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}.
    By Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-points-open} the
    set of points at which $f$ is quasi-finite is an open $U \subset X$.
    Let $Z = X \setminus U$. Then $s \not \in f(Z)$. Since $f$ is proper
    the set $f(Z) \subset S$ is closed. Choose any open neighbourhood
    $V \subset S$ of $s$ with $Z \cap V = \emptyset$. Then
    $f^{-1}(V) \to V$ is locally quasi-finite and proper.
    Hence it is quasi-finite
    (Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-locally-quasi-compact}),
    hence has finite fibres
    (Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}), hence
    is finite by Lemma \ref{lemma-characterize-finite}.
    \end{proof}

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