The Stacks project

Definition 67.23.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $f$ locally of finite type if the equivalent conditions of Lemma 67.22.1 hold with $\mathcal{P} = \text{locally of finite type}$.

2. Let $x \in |X|$. We say $f$ is of finite type at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally of finite type.

3. We say $f$ is of finite type if it is locally of finite type and quasi-compact.