The Stacks project

Definition 29.20.1. Let $f : X \to S$ be a morphism of schemes.

1. 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).

2. We say $f$ is locally quasi-finite if $f$ is quasi-finite at every point $x$ of $X$.

3. We say that $f$ is quasi-finite if $f$ is of finite type and every point $x$ is an isolated point of its fibre.