Loading [MathJax]/extensions/tex2jax.js

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. $\square$


Comments (1)


Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.