Definition 67.28.1 (03XP)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.28: Morphisms of finite presentation
Definition 67.28.1 (cite)

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Definition 67.28.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ .

We say $f$ is locally of finite presentation if the equivalent conditions of Lemma 67.22.1 hold with $\mathcal{P} =$ “locally of finite presentation”.
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 presentation ¹.
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.

Comments (2)

Comment #4863 by Heejong Lee on December 30, 2019 at 15:58

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

Comment #5149 by Johan on June 08, 2020 at 16:29

Thanks and fixed here.

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