The Stacks project

Lemma 87.19.5. Let $S$ be a scheme. Let $Y$ be an affine 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. Write $Y = \mathop{\mathrm{colim}}\nolimits Y_\lambda $ as in Definition 87.9.1. For each $\lambda $ the fibre product $X \times _ Y Y_\lambda $ is an algebraic space. Hence $X = \mathop{\mathrm{colim}}\nolimits X \times _ Y Y_\lambda $ is a formal algebraic space by Lemma 87.13.1. $\square$

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