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
- $f$ is finite, and
- $f$ 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}Comments (0)
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