The Stacks project

Remark 101.34.4. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target and stable under base change. Then the property of DM morphisms of algebraic stacks defined in Definition 101.34.2 is stable under arbitrary base change. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks and $g : \mathcal{Y}' \to \mathcal{Y}$ be a morphism of algebraic stacks and assume $f$ has property $\mathcal{P}$. Then the base change $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ is a DM morphism by Lemma 101.4.4. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \to \mathcal{Y}' \times _\mathcal {Y} V$. Then the morphism $U \to V$ has property $\mathcal{P}$ by definition. Whence $V' \times _ V U \to V'$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under base change. Considering the diagram

\[ \xymatrix{ V' \times _ V U \ar[r] \ar[d] & \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ V' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

we see that the left top horizontal arrow is surjective and

\[ V' \times _ V U \to V' \times _\mathcal {Y} (\mathcal{Y}' \times _{\mathcal{Y}'} \mathcal{X}) = V' \times _ V (\mathcal{X} \times _\mathcal {Y} V) \]

is étale as a base change of $U \to \mathcal{X} \times _\mathcal {Y} V$, whence by definition we see that the projection $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ has property $\mathcal{P}$.

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