Lemma 100.9.17 (05WF)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 100: Properties of Algebraic Stacks
Section 100.9: Immersions of algebraic stacks
Lemma 100.9.17 (cite)

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

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