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Tag 01TJ

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

Lemma 28.19.9. Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-finite if and only if $f$ is locally quasi-finite and quasi-compact.

Proof. Assume $f$ is quasi-finite. It is quasi-compact by Definition 28.14.1. Let $x \in X$. We see that $f$ is quasi-finite at $x$ by Lemma 28.19.6. Hence $f$ is quasi-compact and locally quasi-finite.

Assume $f$ is quasi-compact and locally quasi-finite. Then $f$ is of finite type. Let $x \in X$ be a point. By Lemma 28.19.6 we see that $x$ is an isolated point of its fibre. The lemma is proved. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 3375–3380 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-finite-locally-quasi-compact}
    Let $f : X \to S$ be a morphism of schemes.
    Then $f$ is quasi-finite if and only if $f$ is
    locally quasi-finite and quasi-compact.
    \end{lemma}
    
    \begin{proof}
    Assume $f$ is quasi-finite. It is quasi-compact by Definition
    \ref{definition-finite-type}. Let $x \in X$.
    We see that $f$ is quasi-finite at $x$ by
    Lemma \ref{lemma-quasi-finite-at-point-characterize}.
    Hence $f$ is quasi-compact and locally quasi-finite.
    
    \medskip\noindent
    Assume $f$ is quasi-compact and locally quasi-finite.
    Then $f$ is of finite type. Let $x \in X$ be a point.
    By Lemma \ref{lemma-quasi-finite-at-point-characterize}
    we see that $x$ is an isolated point of its fibre.
    The lemma is proved.
    \end{proof}

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