Lemma 65.7.1. Let S be a scheme contained in \mathit{Sch}_{fppf}. Let F, G be algebraic spaces over S. Then F \times G is an algebraic space, and is a product in the category of algebraic spaces over S.
Proof. It is clear that H = F \times G is a sheaf. The diagonal of H is simply the product of the diagonals of F and G. Hence it is representable by Lemma 65.3.4. Finally, if U \to F and V \to G are surjective étale morphisms, with U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf}), then U \times V \to F \times G is surjective étale by Lemma 65.5.7. \square
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