The Stacks project

Lemma 93.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent

  1. $\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids,

  2. $\mathcal{X}$ is a Deligne-Mumford stack such that the canonical $1$-morphism $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is an equivalence, and

  3. $\mathcal{X}$ is representable by an algebraic space.

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 93.13.1. 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 93.9.5 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 ├ętale 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 79.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 93.10.4 we conclude that $h_ U \to F$ is surjective and ├ętale. Finally, we apply Bootstrap, Theorem 79.6.1 to see that $F$ is an algebraic space. $\square$


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