## Tag `01TD`

Chapter 28: Morphisms of Schemes > Section 28.19: Quasi-finite morphisms

Definition 28.19.1. Let $f : X \to S$ be a morphism of schemes.

- We say that $f$ is
quasi-finite at a point $x \in X$if there exist an affine neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and an affine open $\mathop{\rm Spec}(R) = V \subset S$ such that $f(U) \subset V$, the ring map $R \to A$ is of finite type, and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$ (see above).- We say $f$ is
locally quasi-finiteif $f$ is quasi-finite at every point $x$ of $X$.- We say that $f$ is
quasi-finiteif $f$ is of finite type and every point $x$ is an isolated point of its fibre.

The code snippet corresponding to this tag is a part of the file `morphisms.tex` and is located in lines 3129–3147 (see updates for more information).

```
\begin{definition}
\label{definition-quasi-finite}
\begin{reference}
\cite[II Definition 6.2.3]{EGA}
\end{reference}
Let $f : X \to S$ be a morphism of schemes.
\begin{enumerate}
\item We say that $f$ is {\it quasi-finite at a point $x \in X$}
if there exist an affine neighbourhood $\Spec(A) = U \subset X$
of $x$ and an affine open $\Spec(R) = V \subset S$ such that
$f(U) \subset V$, the ring map $R \to A$ is of finite type,
and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$
(see above).
\item We say $f$ is {\it locally quasi-finite} if $f$ is
quasi-finite at every point $x$ of $X$.
\item We say that $f$ is {\it quasi-finite} if $f$ is of finite type
and every point $x$ is an isolated point of its fibre.
\end{enumerate}
\end{definition}
```

## References

[EGA, II Definition 6.2.3]

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