Lemma 100.9.12 (06FJ)—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 100.9.12 (cite)

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Lemma 100.9.12. Let $\mathcal{X}$ be an algebraic stack. The rule $\mathcal{U} \mapsto |\mathcal{U}|$ defines an inclusion preserving bijection between open substacks of $\mathcal{X}$ and open subsets of $|\mathcal{X}|$.

Proof. Choose a presentation $[U/R] \to \mathcal{X}$, see Algebraic Stacks, Lemma 94.16.2. By Lemma 100.9.11 we see that open substacks correspond to $R$-invariant open subschemes of $U$. On the other hand Lemmas 100.4.5 and 100.4.7 guarantee these correspond bijectively to open subsets of $|\mathcal{X}|$. $\square$

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