Lemma 80.3.6. Let S be a scheme contained in \mathit{Sch}_{fppf}. Let F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}. Let a : F \to G be representable by algebraic spaces. If G is an algebraic space, then so is F.
Proof. We have seen in Lemma 80.3.4 that F is a sheaf.
Let U be a scheme and let U \to G be a surjective étale morphism. In this case U \times _ G F is an algebraic space. Let W be a scheme and let W \to U \times _ G F be a surjective étale morphism.
First we claim that W \to F is representable. To see this let X be a scheme and let X \to F be a morphism. Then
W \times _ F X = W \times _{U \times _ G F} U \times _ G F \times _ F X = W \times _{U \times _ G F} (U \times _ G X)
Since both U \times _ G F and G are algebraic spaces we see that this is a scheme.
Next, we claim that W \to F is surjective and étale (this makes sense now that we know it is representable). This follows from the formula above since both W \to U \times _ G F and U \to G are étale and surjective, hence W \times _{U \times _ G F} (U \times _ G X) \to U \times _ G X and U \times _ G X \to X are surjective and étale, and the composition of surjective étale morphisms is surjective and étale.
Set R = W \times _ F W. By the above R is a scheme and the projections t, s : R \to W are étale. It is clear that R is an equivalence relation, and W \to F is a surjection of sheaves. Hence R is an étale equivalence relation and F = W/R. Hence F is an algebraic space by Spaces, Theorem 65.10.5. \square
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