The Stacks project

Lemma 101.22.1. Let $\mathcal{X}$ be an algebraic stack. There exist open substacks

\[ \mathcal{X}'' \subset \mathcal{X}' \subset \mathcal{X} \]

such that $\mathcal{X}''$ is DM, $\mathcal{X}'$ is quasi-DM, and such that these are the largest open substacks with these properties.

Proof. All we are really saying here is that if $\mathcal{U} \subset \mathcal{X}$ and $\mathcal{V} \subset \mathcal{X}$ are open substacks which are DM, then the open substack $\mathcal{W} \subset \mathcal{X}$ with $|\mathcal{W}| = |\mathcal{U}| \cup |\mathcal{V}|$ is DM as well. (Similarly for quasi-DM.) Although this is a cheat, let us use Theorem 101.21.6 to prove this. By that theorem we can choose schemes $U$ and $V$ and surjective ├ętale morphisms $U \to \mathcal{U}$ and $V \to \mathcal{V}$. Then of course $U \amalg V \to \mathcal{W}$ is surjective and ├ętale. The quasi-DM case is proven by exactly the same method using Theorem 101.21.3. $\square$

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