Definition 65.28.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $f$ is locally of finite presentation if the equivalent conditions of Lemma 65.22.1 hold with $\mathcal{P} =$“locally of finite presentation”.

2. Let $x \in |X|$. We say $f$ is of finite presentation at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally of finite presentation1.

3. A morphism of algebraic spaces $f : X \to Y$ is of finite presentation if it is locally of finite presentation, quasi-compact and quasi-separated.

[1] It seems awkward to use “locally of finite presentation at $x$”, but the current terminology may be misleading in the sense that “of finite presentation at $x$” does not mean that there is an open neighbourhood $X' \subset X$ such that $f|_{X'}$ is of finite presentation.

Comment #4863 by Heejong Lee on

At the beginning of the definition, $f$ is omitted in "$X\xrightarrow{} Y$ be a morphism of algebraic spaces over $S$".

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