Lemma 87.9.11. Let S be a scheme. Let f : X \to Y be a map of presheaves on (\mathit{Sch}/S)_{fppf}. If X is an affine formal algebraic space and f is representable by algebraic spaces and locally quasi-finite, then f is representable (by schemes).
Proof. Let T be a scheme over S and T \to Y a map. We have to show that the algebraic space X \times _ Y T is a scheme. Write X = \mathop{\mathrm{colim}}\nolimits X_\lambda as in Definition 87.9.1. Let W \subset X \times _ Y T be a quasi-compact open subspace. The restriction of the projection X \times _ Y T \to X to W factors through X_\lambda for some \lambda . Then
W \to X_\lambda \times _ S T
is a monomorphism (hence separated) and locally quasi-finite (because W \to X \times _ Y T \to T is locally quasi-finite by our assumption on X \to Y, see Morphisms of Spaces, Lemma 67.27.8). Hence W is a scheme by Morphisms of Spaces, Proposition 67.50.2. Thus X \times _ Y T is a scheme by Properties of Spaces, Lemma 66.13.1. \square
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