Lemma 70.5.16 (0A0T)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.5: Descending properties
Lemma 70.5.16 (cite)

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Lemma 70.5.16. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ . Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume

$Y$ quasi-compact and quasi-separated,
$X_ i$ quasi-compact and quasi-separated,
the transition morphisms $X_{i'} \to X_ i$ are closed immersions,
$X_ i \to Y$ locally of finite type
$X \to Y$ is a closed immersion.

Then $X_ i \to Y$ is a closed immersion for $i$ large enough.

Proof. Choose an affine scheme $W$ and a surjective étale morphism $W \to Y$ . Then $X \times _ Y W$ is a closed subspace of $W$ and it suffices to check that $X_ i \times _ Y W$ is a closed subspace $W$ for some $i$ (Morphisms of Spaces, Lemma 67.12.1). By Lemma 70.5.11 this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 32.4.20. $\square$

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