The Stacks project

Lemma 66.17.7. 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 ├ętale localization, see Lemmas 66.16.3 and 66.16.4. Hence this lemma follows from the case of schemes, see Morphisms, Lemma 29.7.7. $\square$

Comments (2)

Comment #1682 by Matthieu Romagny on

Beginning of proof: In both cases the formation of

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