Lemma 76.37.1 (0876)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.37: Extending properties from an open
Lemma 76.37.1 (cite)

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Lemma 76.37.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $V \subset Y$ be an open subspace. Assume

$f$ is locally of finite presentation,
$\mathcal{F}$ is of finite type and flat over $Y$ ,
$V \to Y$ is quasi-compact and scheme theoretically dense,
$\mathcal{F}|_{f^{-1}V}$ is of finite presentation.

Then $\mathcal{F}$ is of finite presentation.

Proof. It suffices to prove the pullback of $\mathcal{F}$ to a scheme surjective and étale over $X$ is of finite presentation. Hence we may assume $X$ is a scheme. Similarly, we can replace $Y$ by a scheme surjective and étale and over $Y$ (the inverse image of $V$ in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 67.17). Thus we reduce to the case of schemes which is More on Flatness, Lemma 38.11.1. $\square$

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