# 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}

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