The Stacks project

Lemma 100.9.17. Let $\mathcal X$ be an algebraic stack. Let $\mathcal{X}_ i$, $i \in I$ be a set of open substacks of $\mathcal{X}$. Assume

  1. $\mathcal{X} = \bigcup _{i \in I} \mathcal{X}_ i$, and

  2. each $\mathcal{X}_ i$ is a scheme

Then $\mathcal{X}$ is a scheme.

Proof. By Lemma 100.9.16 we see that $\mathcal{X}$ is an algebraic space. Since any algebraic space has a largest open subspace which is a scheme, see Properties of Spaces, Lemma 66.13.1 we see that $\mathcal{X}$ is a scheme. $\square$

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