Definition 29.20.1. 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 $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ 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-finite*if $f$ is quasi-finite at every point $x$ of $X$.We say that $f$ is

*quasi-finite*if $f$ is of finite type and every point $x$ is an isolated point of its fibre.

## Comments (2)

Comment #2714 by Ariyan Javanpeykar on

Comment #2751 by Takumi Murayama on