The Stacks project

Lemma 92.15.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. Let $V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Let $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$ be surjective and smooth. Then there exists an object $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a $2$-commutative diagram

\[ \xymatrix{ (\mathit{Sch}/U)_{fppf} \ar[r]_ a \ar[d]_ x & (\mathit{Sch}/V)_{fppf} \ar[d]^ y \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]

with $x$ surjective and smooth.

Proof. First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective smooth $1$-morphism $z : (\mathit{Sch}/W)_{fppf} \to \mathcal{X}$. As $\mathcal{Y}$ is an algebraic stack we may choose an equivalence

\[ j : \mathcal{S}_ F \longrightarrow (\mathit{Sch}/W)_{fppf} \times _{f \circ z, \mathcal{Y}, y} (\mathit{Sch}/V)_{fppf} \]

where $F$ is an algebraic space. By Lemma 92.10.6 the morphism $\mathcal{S}_ F \to (\mathit{Sch}/W)_{fppf}$ is surjective and smooth as a base change of $y$. Hence by Lemma 92.10.5 we see that $\mathcal{S}_ F \to \mathcal{X}$ is surjective and smooth. Choose an object $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective ├ętale morphism $U \to F$. Then applying Lemma 92.10.5 once more we obtain the desired properties. $\square$


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