Lemma 101.27.14 (07AN)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.27: Morphisms of finite presentation
Lemma 101.27.14 (cite)

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Lemma 101.27.14. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ , $j \in J$ be a family of morphisms of algebraic stacks which are each flat and locally of finite presentation and which are jointly surjective, i.e., $|\mathcal{X}| = \bigcup |f_ j|(|\mathcal{X}_ j|)$ . Then for every scheme $U$ and object $x$ of $\mathcal{X}$ over $U$ there exists an fppf covering $\{ U_ i \to U\} _{i \in I}$ , a map $a : I \to J$ , and objects $x_ i$ of $\mathcal{X}_{a(i)}$ over $U_ i$ such that $f_{a(i)}(x_ i) \cong y|_{U_ i}$ in $\mathcal{X}_{U_ i}$ .

Proof. Apply Lemma 101.27.13 to the morphism $\coprod _{j \in J} \mathcal{X}_ j \to \mathcal{X}$ . (There is a slight set theoretic issue here – due to our setup of things – which we ignore.) To finish, note that a morphism $x_ i : U_ i \to \coprod _{j \in J} \mathcal{X}_ j$ is given by a disjoint union decomposition $U_ i = \coprod U_{i, j}$ and morphisms $U_{i, j} \to \mathcal{X}_ j$ . Then the fppf covering $\{ U_{i, j} \to U\}$ and the morphisms $U_{i, j} \to \mathcal{X}_ j$ do the job. $\square$

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