The Stacks project

Lemma 86.7.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism from an algebraic space over $S$ to a formal algebraic space over $S$. Then $f$ is representable by algebraic spaces.

Proof. Let $Z \to Y$ be a morphism where $Z$ is a scheme over $S$. We have to show that $X \times _ Y Z$ is an algebraic space. Choose a scheme $U$ and a surjective ├ętale morphism $U \to X$. Then $U \times _ Y Z \to X \times _ Y Z$ is representable surjective ├ętale (Spaces, Lemma 64.5.5) and $U \times _ Y Z$ is a scheme by Lemma 86.7.2. Hence the result by Bootstrap, Theorem 79.10.1. $\square$

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