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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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