The Stacks project

Lemma 65.8.4. Let $S \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Suppose given a set $I$ and algebraic spaces $F_ i$, $i \in I$. Then $F = \coprod _{i \in I} F_ i$ is an algebraic space provided $I$, and the $F_ i$ are not too “large”: for example if we can choose surjective étale morphisms $U_ i \to F_ i$ such that $\coprod _{i \in I} U_ i$ is isomorphic to an object of $(\mathit{Sch}/S)_{fppf}$, then $F$ is an algebraic space.

Proof. By construction $F$ is a sheaf. We omit the verification that the diagonal morphism of $F$ is representable. Finally, if $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ isomorphic to $\coprod _{i \in I} U_ i$ then it is straightforward to verify that the resulting map $U \to \coprod F_ i$ is surjective and étale. $\square$

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