The Stacks project

Lemma 80.12.2. Denote the common underlying category of $\mathit{Sch}_{fppf}$ and $\mathit{Sch}_{\acute{e}tale}$ by $\mathit{Sch}_\alpha $ (see Topologies, Remark 34.11.1). Let $S$ be an object of $\mathit{Sch}_\alpha $. Let

\[ F : (\mathit{Sch}_\alpha /S)^{opp} \longrightarrow \textit{Sets} \]

be a presheaf with the following properties:

  1. $F$ is a sheaf for the étale topology,

  2. there exists an algebraic space $U$ over $S$ and a map $U \to F$ which is representable by algebraic spaces, surjective, and étale.

Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition 65.6.1.

Proof. Set $R = U \times _ F U$. This is an algebraic space as $U \to F$ is assumed representable by algebraic spaces. The projections $s, t : R \to U$ are étale morphisms of algebraic spaces as $U \to F$ is assumed étale. The map $j = (t, s) : R \to U \times _ S U$ is a monomorphism and an equivalence relation as $R = U \times _ F U$. By Theorem 80.10.1 the fppf quotient sheaf $F' = U/R$ is an algebraic space. The morphism $U \to F'$ is surjective, flat, and locally of finite presentation by Lemma 80.11.6. The map $R \to U \times _{F'} U$ is surjective as a map of fppf sheaves by Groupoids in Spaces, Lemma 78.19.5 and since $j$ is a monomorphism it is an isomorphism. Hence the base change of $U \to F'$ by $U \to F'$ is étale, and we conclude that $U \to F'$ is étale by Descent on Spaces, Lemma 74.11.28. Thus $U \to F'$ is surjective as a map of étale sheaves. This means that $F'$ is equal to the quotient sheaf $U/R$ in the étale topology (small check omitted). Hence we obtain a canonical factorization $U \to F' \to F$ and $F' \to F$ is an injective map of sheaves. On the other hand, $U \to F$ is surjective as a map of étale sheaves and hence so is $F' \to F$. This means that $F' = F$ and the proof is complete. $\square$


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