## 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 10317–10320 (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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