Lemma 80.5.4 (0AHV)—The Stacks project

Lemma 80.5.4. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $X$ be a scheme and let $X \to F$ be representable by algebraic spaces and locally quasi-finite. Then $X \to F$ is representable (by schemes).

Proof. Let $T$ be a scheme and let $T \to F$ be a morphism. We have to show that the algebraic space $X \times _ F T$ is representable by a scheme. Consider the morphism

$X \times _ F T \longrightarrow X \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} T$

Since $X \times _ F T \to T$ is locally quasi-finite, so is the displayed arrow (Morphisms of Spaces, Lemma 67.27.8). On the other hand, the displayed arrow is a monomorphism and hence separated (Morphisms of Spaces, Lemma 67.10.3). Thus $X \times _ F T$ is a scheme by Morphisms of Spaces, Proposition 67.50.2. $\square$

The Stacks project

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