**Proof.**
The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 90.13.1. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists a sheaf $F$ on $(\mathit{Sch}/S)_{fppf}$ and an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$. By Lemma 90.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme, and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective étale morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 76.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 90.10.4 we conclude that $h_ U \to F$ is surjective and étale. Finally, we apply Bootstrap, Theorem 76.6.1 to see that $F$ is an algebraic space.
$\square$

## Comments (0)