The Stacks project

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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 99.14: Finiteness conditions and points

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DTK. Beware of the difference between the letter 'O' and the digit '0'.