Lemma 86.9.11. 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 86.9.1. 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 66.27.8). Hence $W$ is a scheme by Morphisms of Spaces, Proposition 66.50.2. Thus $X \times _ Y T$ is a scheme by Properties of Spaces, Lemma 65.13.1. $\square$

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