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Tag 02NG

Chapter 28: Morphisms of Schemes > Section 28.19: Quasi-finite morphisms

Lemma 28.19.7. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

  1. $f$ is locally of finite type, and
  2. $f^{-1}(\{s\})$ is a finite set.

Then $X_s$ is a finite discrete topological space, and $f$ is quasi-finite at each point of $X$ lying over $s$.

Proof. Suppose $T$ is a scheme which (a) is locally of finite type over a field $k$, and (b) has finitely many points. Then Lemma 28.15.10 shows $T$ is a Jacobson scheme. A finite Jacobson space is discrete, see Topology, Lemma 5.18.6. Apply this remark to the fibre $X_s$ which is locally of finite type over $\mathop{\rm Spec}(\kappa(s))$ to see the first statement. Finally, apply Lemma 28.19.6 to see the second. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 3318–3328 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-fibre}
    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, and
    \item $f^{-1}(\{s\})$ is a finite set.
    \end{enumerate}
    Then $X_s$ is a finite discrete topological space, and
    $f$ is quasi-finite at each point of $X$ lying over $s$.
    \end{lemma}
    
    \begin{proof}
    Suppose $T$ is a scheme which (a) is locally of finite type
    over a field $k$, and (b) has finitely many points. Then
    Lemma \ref{lemma-ubiquity-Jacobson-schemes} shows $T$ is a
    Jacobson scheme. A finite Jacobson space is discrete, see
    Topology, Lemma \ref{topology-lemma-finite-jacobson}.
    Apply this remark to the fibre $X_s$ which is locally of finite type over
    $\Spec(\kappa(s))$ to see the first statement. Finally, apply
    Lemma \ref{lemma-quasi-finite-at-point-characterize} to see the second.
    \end{proof}

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