Lemma 79.5.4. 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
Since $X \times _ F T \to T$ is locally quasi-finite, so is the displayed arrow (Morphisms of Spaces, Lemma 66.27.8). On the other hand, the displayed arrow is a monomorphism and hence separated (Morphisms of Spaces, Lemma 66.10.3). Thus $X \times _ F T$ is a scheme by Morphisms of Spaces, Proposition 66.50.2. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (1)
Comment #1613 by Jonathan Wise on