The Stacks project

Lemma 93.13.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$.

  1. A category fibred in groupoids $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ which is representable by an algebraic space is a Deligne-Mumford stack.

  2. If $F$ is an algebraic space over $S$, then the associated category fibred in groupoids $p : \mathcal{S}_ F \to (\mathit{Sch}/S)_{fppf}$ is a Deligne-Mumford stack.

  3. If $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then $(\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ is a Deligne-Mumford stack.

Proof. It is clear that (2) implies (3). Parts (1) and (2) are equivalent by Lemma 93.12.4. Hence it suffices to prove (2). First, we note that $\mathcal{S}_ F$ is stack in sets since $F$ is a sheaf (Stacks, Lemma 8.6.3). A fortiori it is a stack in groupoids. Second the diagonal morphism $\mathcal{S}_ F \to \mathcal{S}_ F \times \mathcal{S}_ F$ is the same as the morphism $\mathcal{S}_ F \to \mathcal{S}_{F \times F}$ which comes from the diagonal of $F$. Hence this is representable by algebraic spaces according to Lemma 93.9.4. Actually it is even representable (by schemes), as the diagonal of an algebraic space is representable, but we do not need this. Let $U$ be a scheme and let $h_ U \to F$ be a surjective étale morphism. We may think of this as a surjective étale morphism of algebraic spaces. Hence by Lemma 93.10.3 the corresponding $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ is surjective and étale. $\square$

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