# The Stacks Project

## Tag 01TJ

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 3372–3377 (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}

There are no comments yet for this tag.

## Add a comment on tag 01TJ

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?