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 étale morphism $U \to F$ (this exist by hypothesis). The base change $U \times _ F F_ i \to F_ i$ is surjective and étale 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.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: