Lemma 101.27.10. Let \mathcal{P} be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and fppf local on the target. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let \mathcal{Z} \to \mathcal{Y} be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change \mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z} has \mathcal{P}, then f has \mathcal{P}.
Proof. Assume \mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z} has \mathcal{P}. Choose an algebraic space W and a surjective smooth morphism W \to \mathcal{Z}. Observe that W \times _\mathcal {Z} \mathcal{Z} \times _\mathcal {Y} \mathcal{X} = W \times _\mathcal {Y} \mathcal{X}. Thus by the very definition of what it means for \mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z} to have \mathcal{P} (see Definition 101.16.2 and Lemma 101.16.1) we see that W \times _\mathcal {Y} \mathcal{X} \to W has \mathcal{P}. On the other hand, W \to \mathcal{Z} is surjective, flat, and locally of finite presentation (Morphisms of Spaces, Lemmas 67.37.7 and 67.37.5) hence W \to \mathcal{Y} is surjective, flat, and locally of finite presentation (by Properties of Stacks, Lemma 100.5.2 and Lemmas 101.25.2 and 101.27.2). Thus we may replace \mathcal{Z} by W.
Choose an algebraic space V and a surjective smooth morphism V \to \mathcal{Y}. Choose an algebraic space U and a surjective smooth morphism U \to V \times _\mathcal {Y} \mathcal{X}. We have to show that U \to V has \mathcal{P}. Now we base change everything by W \to \mathcal{Y}: Set U' = W \times _\mathcal {Y} U, V' = W \times _\mathcal {Y} V, \mathcal{X}' = W \times _\mathcal {Y} \mathcal{X}, and \mathcal{Y}' = W \times _\mathcal {Y} \mathcal{Y} = W. Then it is still true that U' \to V' \times _{\mathcal{Y}'} \mathcal{X}' is smooth by base change. Hence by Lemma 101.16.1 used in the definition of \mathcal{X}' \to \mathcal{Y}' = W having \mathcal{P} we see that U' \to V' has \mathcal{P}. Then, since V' \to V is surjective, flat, and locally of finite presentation as a base change of W \to \mathcal{Y} we see that U \to V has \mathcal{P} as \mathcal{P} is local in the fppf topology on the target. \square
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