Lemma 100.14.1 (0DTK)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 100: Properties of Algebraic Stacks
Section 100.14: Finiteness conditions and points
Lemma 100.14.1 (cite)

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Lemma 100.14.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent

some morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ in the equivalence class of $x$ is quasi-compact, and
any morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ in the equivalence class of $x$ is quasi-compact.

Proof. Let $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be in the equivalence class of $x$ . Let $k'/k$ be a field extension. Then we have to show that $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ is quasi-compact if and only if $\mathop{\mathrm{Spec}}(k') \to \mathcal{X}$ is quasi-compact. This follows from Morphisms of Spaces, Lemma 67.8.6 and the principle of Algebraic Stacks, Lemma 94.10.9. $\square$

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