The Stacks project

Lemma 101.27.9. Let $P$ be a property of morphisms of algebraic spaces which is fppf local on the target and preserved by arbitrary base change. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\mathcal{Z} \to \mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Set $\mathcal{W} = \mathcal{Z} \times _\mathcal {Y} \mathcal{X}$. Then

\[ (f\text{ has }P) \Leftrightarrow (\text{the projection }\mathcal{W} \to \mathcal{Z}\text{ has }P). \]

For the meaning of this statement see Properties of Stacks, Section 100.3.

Proof. Choose an algebraic space $W$ and a morphism $W \to \mathcal{Z}$ which 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 the composition $W \to \mathcal{Y}$ is also surjective, flat, and locally of finite presentation. Denote $V = W \times _\mathcal {Z} \mathcal{W} = V \times _\mathcal {Y} \mathcal{X}$. By Properties of Stacks, Lemma 100.3.3 we see that $f$ has $\mathcal{P}$ if and only if $V \to W$ does and that $\mathcal{W} \to \mathcal{Z}$ has $\mathcal{P}$ if and only if $V \to W$ does. The lemma follows. $\square$

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