The Stacks project

Lemma 65.11.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$, $G$ be algebraic spaces over $S$. Let $G \to F$ be a representable morphism. Let $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and $q : U \to F$ surjective and ├ętale. Set $V = G \times _ F U$. Finally, let $\mathcal{P}$ be a property of morphisms of schemes as in Definition 65.5.1. Then $G \to F$ has property $\mathcal{P}$ if and only if $V \to U$ has property $\mathcal{P}$.

Proof. (This lemma follows from Lemmas 65.5.5 and 65.5.6, but we give a direct proof here also.) It is clear from the definitions that if $G \to F$ has property $\mathcal{P}$, then $V \to U$ has property $\mathcal{P}$. Conversely, assume $V \to U$ has property $\mathcal{P}$. Let $T \to F$ be a morphism from a scheme to $F$. Let $T' = T \times _ F G$ which is a scheme since $G \to F$ is representable. We have to show that $T' \to T$ has property $\mathcal{P}$. Consider the commutative diagram of schemes

\[ \xymatrix{ V \ar[d] & T \times _ F V \ar[d] \ar[l] \ar[r] & T \times _ F G \ar[d] \ar@{=}[r] & T' \\ U & T \times _ F U \ar[l] \ar[r] & T } \]

where both squares are fibre product squares. Hence we conclude the middle arrow has property $\mathcal{P}$ as a base change of $V \to U$. Finally, $\{ T \times _ F U \to T\} $ is a fppf covering as it is surjective ├ętale, and hence we conclude that $T' \to T$ has property $\mathcal{P}$ as it is local on the base in the fppf topology. $\square$

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