The Stacks project

Lemma 63.11.6. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Let $a : F \to G$ be a morphism. Given any $V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective étale morphism $q : V \to G$ there exists a $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ F \ar[r]^ a & G } \]

with $p$ surjective and étale.

Proof. First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ with surjective étale morphism $W \to F$. Next, put $U = W \times _ G V$. Since $G$ is an algebraic space we see that $U$ is isomorphic to an object of $(\mathit{Sch}/S)_{fppf}$. As $q$ is surjective étale, we see that $U \to W$ is surjective étale (see Lemma 63.5.5). Thus $U \to F$ is surjective étale as a composition of surjective étale morphisms (see Lemma 63.5.4). $\square$


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