Lemma 67.17.7 (088G)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.17: Scheme theoretic closure and density
Lemma 67.17.7 (cite)

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Lemma 67.17.7. Let $S$ be a scheme. Let $h : Z \to X$ be an immersion of algebraic spaces over $S$. Assume either $Z \to X$ is quasi-compact or $Z$ is reduced. Let $\overline{Z} \subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \to \overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subspace of $\overline{Z}$. Moreover, $Z$ is topologically dense in $\overline{Z}$.

Proof. In both cases the formation of $\overline{Z}$ commutes with étale localization, see Lemmas 67.16.3 and 67.16.4. Hence this lemma follows from the case of schemes, see Morphisms, Lemma 29.7.7. $\square$

Comments (2)

Comment #1682 by Matthieu Romagny on October 31, 2015 at 18:22

Beginning of proof: In both cases the formation of $\overline{Z}$

Comment #1731 by Johan on December 15, 2015 at 18:42

Thanks, fixed here.

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