Lemma 80.12.1. 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:
$F$ is a sheaf for the étale topology,
the diagonal $\Delta : F \to F \times F$ is representable, and
there exists $U \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha /S)$ and $U \to F$ which is surjective and étale.
Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition 65.6.1.
Proof.
Note that properties (2) and (3) of the lemma and the corresponding properties (2) and (3) of Algebraic Spaces, Definition 65.6.1 are independent of the topology. This is true because these properties involve only the notion of a fibre product of presheaves, maps of presheaves, the notion of a representable transformation of functors, and what it means for such a transformation to be surjective and étale. Thus all we have to prove is that an étale sheaf $F$ with properties (2) and (3) is also an fppf sheaf.
To do this, let $R = U \times _ F U$. By (2) the presheaf $R$ is representable by a scheme and by (3) the projections $R \to U$ are étale. Thus $j : R \to U \times _ S U$ is an étale equivalence relation. Moreover $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the étale topology: (a) if $T \to F$ is a morphism, then $\{ T \times _ F U \to T\} $ is an étale covering, hence $U \to F$ is a surjection of sheaves for the étale topology, (b) if $a, b : T \to U$ map to the same section of $F$, then $(a, b) : T \to R$ hence $a$ and $b$ have the same image in the quotient of $U$ by $R$ for the étale topology. Next, let $U/R$ denote the quotient sheaf in the fppf topology which is an algebraic space by Spaces, Theorem 65.10.5. Thus we have morphisms (transformations of functors)
\[ U \to F \to U/R. \]
By the aforementioned Spaces, Theorem 65.10.5 the composition is representable, surjective, and étale. Hence for any scheme $T$ and morphism $T \to U/R$ the fibre product $V = T \times _{U/R} U$ is a scheme surjective and étale over $T$. In other words, $\{ V \to U\} $ is an étale covering. This proves that $U \to U/R$ is surjective as a map of sheaves in the étale topology. It follows that $F \to U/R$ is surjective as a map of sheaves in the étale topology. On the other hand, the map $F \to U/R$ is injective (as a map of presheaves) since $R = U \times _{U/R} U$ again by Spaces, Theorem 65.10.5. It follows that $F \to U/R$ is an isomorphism of étale sheaves, see Sites, Lemma 7.11.2 which concludes the proof.
$\square$
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