Lemma 96.19.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

1. Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object $y$ of $\mathcal{Y}$ there exists an fppf covering $\{ y_ i \to y\}$ and objects $x_ i$ of $\mathcal{X}$ such that $f(x_ i) \cong y_ i$ in $\mathcal{Y}$.

2. Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then for any object $y$ of $\mathcal{Y}$ there exists an étale covering $\{ y_ i \to y\}$ and objects $x_ i$ of $\mathcal{X}$ such that $f(x_ i) \cong y_ i$ in $\mathcal{Y}$.

Proof. Proof of (1). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times _\mathcal {Y} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective, flat, and locally of finite presentation. Choose a scheme $U$ and a surjective étale morphism $U \to W$. Then $U \to V$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 67.39.7, 67.39.8, 67.5.4, 67.28.2, and 67.30.3). Hence $\{ U \to V\}$ is an fppf covering. Denote $x$ the object of $\mathcal{X}$ over $U$ corresponding to the $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$. Then $\{ f(x) \to y\}$ is the desired fppf covering of $\mathcal{Y}$.

Proof of (2). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times _\mathcal {Y} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective and smooth. Choose a scheme $U$ and a surjective étale morphism $U \to W$. Then $U \to V$ is also surjective and smooth (see Morphisms of Spaces, Lemmas 67.39.6, 67.5.4, and 67.37.2). Hence $\{ U \to V\}$ is a smooth covering. By More on Morphisms, Lemma 37.38.7 there exists an étale covering $\{ V_ i \to V\}$ such that each $V_ i \to V$ factors through $U$. Denote $x_ i$ the object of $\mathcal{X}$ over $V_ i$ corresponding to the $1$-morphism

$(\mathit{Sch}/V_ i)_{fppf} \to (\mathit{Sch}/U)_{fppf} \to \mathcal{X}.$

Then $\{ f(x_ i) \to y\}$ is the desired étale covering of $\mathcal{Y}$. $\square$

Comment #5139 by Dario Weißmann on

typo: the second proof of (1) should be a proof of (2)

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