The Stacks project

Lemma 99.21.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Then

  1. For every DM algebraic stack $\mathcal{Z}$ and morphism $\mathcal{Z} \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

  2. For every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} Z$.

Proof. Proof of (1). As $f$ is DM we see that the base change $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ is DM by Lemma 99.4.4. Since $\mathcal{Z}$ is DM this implies that $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is DM by Lemma 99.4.11. Hence there exists a scheme $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$, see Theorem 99.21.6. Part (2) is a special case of (1) since an algebraic space (when viewed as an algebraic stack) is DM by Lemma 99.4.3. $\square$


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