Definition 66.23.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

We say $f$

*locally of finite type*if the equivalent conditions of Lemma 66.22.1 hold with $\mathcal{P} = \text{locally of finite type}$.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.We say $f$ is

*of finite type*if it is locally of finite type and quasi-compact.

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