The Stacks project

Lemma 99.27.13. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then for every scheme $U$ and object $y$ of $\mathcal{Y}$ over $U$ there exists an fppf covering $\{ U_ i \to U\} $ and objects $x_ i$ of $\mathcal{X}$ over $U_ i$ such that $f(x_ i) \cong y|_{U_ i}$ in $\mathcal{Y}_{U_ i}$.

Proof. We may think of $y$ as a morphism $U \to \mathcal{Y}$. By Properties of Stacks, Lemma 98.5.3 and Lemmas 99.27.3 and 99.25.3 we see that $\mathcal{X} \times _\mathcal {Y} U \to U$ is surjective, flat, and locally of finite presentation. Let $V$ be a scheme and let $V \to \mathcal{X} \times _\mathcal {Y} U$ smooth and surjective. Then $V \to \mathcal{X} \times _\mathcal {Y} U$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 65.37.7 and 65.37.5). Hence also $V \to U$ is surjective, flat, and locally of finite presentation, see Properties of Stacks, Lemma 98.5.2 and Lemmas 99.27.2, and 99.25.2. Hence $\{ V \to U\} $ is the desired fppf covering and $x : V \to \mathcal{X}$ is the desired object. $\square$

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