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Tag 02NU

Chapter 28: Morphisms of Schemes > Section 28.42: Integral and finite morphisms

Lemma 28.42.9. A finite morphism is quasi-finite.

Proof. This is implied by Algebra, Lemma 10.121.4 and Lemma 28.19.9. Alternatively, all points in fibres are closed points by Lemma 28.42.8 (and the fact that a finite morphism is integral) and use Lemma 28.19.6 (3) to see that $f$ is quasi-finite at $x$ for all $x \in X$. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 10335–10338 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-quasi-finite}
    A finite morphism is quasi-finite.
    \end{lemma}
    
    \begin{proof}
    This is implied by Algebra, Lemma \ref{algebra-lemma-quasi-finite}
    and Lemma \ref{lemma-quasi-finite-locally-quasi-compact}.
    Alternatively, all points in fibres are closed points by
    Lemma \ref{lemma-integral-fibres} (and the fact that a finite
    morphism is integral) and use
    Lemma \ref{lemma-quasi-finite-at-point-characterize} (3) to
    see that $f$ is quasi-finite at $x$ for all $x \in X$.
    \end{proof}

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