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. $\square$


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