The Stacks project

Lemma 87.19.6. Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$. Let $f : X \to Y$ be a map of sheaves on $(\mathit{Sch}/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space.

Proof. Let $\{ Y_ i \to Y\} $ be as in Definition 87.11.1. Then $X \times _ Y Y_ i \to X$ is a family of morphisms representable by algebraic spaces, étale, and jointly surjective. Thus it suffices to show that $X \times _ Y Y_ i$ is a formal algebraic space, see Lemma 87.15.1. This follows from Lemma 87.19.5. $\square$

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