The Stacks project

Lemma 64.11.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $G \to F$ be a transformation of presheaves on $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of schemes. Assume

  1. $\mathcal{P}$ is preserved under any base change, fppf local on the base, and morphisms of type $\mathcal{P}$ satisfy descent for fppf coverings, see Descent, Definition 35.36.1,

  2. $G$ is a sheaf,

  3. $F$ is an algebraic space,

  4. there exists a $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective étale morphism $U \to F$ such that $V = G \times _ F U$ is representable, and

  5. $V \to U$ has $\mathcal{P}$.

Then $G$ is an algebraic space, $G \to F$ is representable and has property $\mathcal{P}$.

Proof. Let $R = U \times _ F U$, and denote $t, s : R \to U$ the projection morphisms as usual. Let $T$ be a scheme and let $T \to F$ be a morphism. Then $U \times _ F T \to T$ is surjective étale, hence $\{ U \times _ F T \to T\} $ is a covering for the étale topology. Consider

\[ W = G \times _ F (U \times _ F T) = V \times _ F T = V \times _ U (U \times _ F T). \]

It is a scheme since $F$ is an algebraic space. The morphism $W \to U \times _ F T$ has property $\mathcal{P}$ since it is a base change of $V \to U$. There is an isomorphism

\begin{align*} W \times _ T (U \times _ F T) & = (G \times _ F (U \times _ F T)) \times _ T (U \times _ F T) \\ & = (U \times _ F T) \times _ T (G \times _ F (U \times _ F T)) \\ & = (U \times _ F T) \times _ T W \end{align*}

over $(U \times _ F T) \times _ T (U \times _ F T)$. The middle equality maps $((g, (u_1, t)), (u_2, t))$ to $((u_1, t), (g, (u_2, t)))$. This defines a descent datum for $W/U \times _ F T/T$, see Descent, Definition 35.34.1. This follows from Descent, Lemma 35.39.1. Namely we have a sheaf $G \times _ F T$, whose base change to $U \times _ F T$ is represented by $W$ and the isomorphism above is the one from the proof of Descent, Lemma 35.39.1. By assumption on $\mathcal{P}$ the descent datum above is representable. Hence by the last statement of Descent, Lemma 35.39.1 we see that $G \times _ F T$ is representable. This proves that $G \to F$ is a representable transformation of functors.

As $G \to F$ is representable, we see that $G$ is an algebraic space by Lemma 64.11.3. The fact that $G \to F$ has property $\mathcal{P}$ now follows from Lemma 64.11.4. $\square$

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