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Tag 01WN

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

Lemma 28.42.10. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. $f$ is finite, and
  2. $f$ is affine and proper.

Proof. This follows formally from Lemma 28.42.7, the fact that a finite morphism is integral and separated, the fact that a proper morphism is the same thing as a finite type, separated, universally closed morphism, and the fact that an integral morphism of finite type is finite (Lemma 28.42.4). $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 10332–10339 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-proper}
    Let $f : X \to S$ be a morphism of schemes. The following are equivalent
    \begin{enumerate}
    \item $f$ is finite, and
    \item $f$ is affine and proper.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This follows formally from
    Lemma \ref{lemma-integral-universally-closed},
    the fact that a finite morphism is integral and separated,
    the fact that a proper morphism is the same thing as
    a finite type, separated, universally closed morphism,
    and the fact that an integral morphism of finite type is
    finite (Lemma \ref{lemma-finite-integral}).
    \end{proof}

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