The Stacks project

Lemma 63.8.3. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $F$ be an algebraic space over $S$. Given a set $I$ and sheaves $F_ i$ on $\mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, if $F \cong \coprod _{i\in I} F_ i$ as sheaves, then each $F_ i$ is an algebraic space over $S$.

Proof. The representability of $F \to F \times F$ implies that each diagonal morphism $F_ i \to F_ i \times F_ i$ is representable (immediate from the definitions and the fact that $F \times _{(F \times F)} (F_ i \times F_ i) = F_ i$). Choose a scheme $U$ in $(\mathit{Sch}/S)_{fppf}$ and a surjective ├ętale morphism $U \to F$ (this exist by hypothesis). The base change $U \times _ F F_ i \to F_ i$ is surjective and ├ętale by Lemma 63.5.5. On the other hand, $U \times _ F F_ i$ is a scheme by Lemma 63.8.1. Thus we have verified all the conditions in Definition 63.6.1 and $F_ i$ is an algebraic space. $\square$

Comments (0)

There are also:

  • 10 comment(s) on Section 63.8: Glueing algebraic spaces

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Your email address will not be published.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).

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02WP. Beware of the difference between the letter 'O' and the digit '0'.