## 99.14 Finiteness conditions and points

This section is the analogue of Decent Spaces, Section 67.4 for points of algebraic stacks.

Lemma 99.14.1. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point. The following are equivalent

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

2. 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 66.8.6 and the principle of Algebraic Stacks, Lemma 93.10.9. $\square$

Sometimes people say that a point $x \in |\mathcal{X}|$ satisfying the equivalent conditions of Lemma 99.14.1 is a “quasi-compact point”.

Comment #3394 by Matthieu Romagny on

Remove "for" in statement of condition (1).

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