The Stacks project

Lemma 100.7.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent:

  1. $f$ is quasi-compact (as in Properties of Stacks, Section 99.3), and

  2. for every quasi-compact algebraic stack $\mathcal{Z}$ and any morphism $\mathcal{Z} \to \mathcal{Y}$ the algebraic stack $\mathcal{Z} \times _\mathcal {Y} \mathcal{X}$ is quasi-compact.

Proof. Assume (1), and let $\mathcal{Z} \to \mathcal{Y}$ be a morphism of algebraic stacks with $\mathcal{Z}$ quasi-compact. By Properties of Stacks, Lemma 99.6.2 there exists a quasi-compact scheme $U$ and a surjective smooth morphism $U \to \mathcal{Z}$. Since $f$ is representable by algebraic spaces and quasi-compact we see by definition that $U \times _\mathcal {Y} \mathcal{X}$ is an algebraic space, and that $U \times _\mathcal {Y} \mathcal{X} \to U$ is quasi-compact. Hence $U \times _ Y X$ is a quasi-compact algebraic space. The morphism $U \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z} \times _\mathcal {Y} \mathcal{X}$ is smooth and surjective (as the base change of the smooth and surjective morphism $U \to \mathcal{Z}$). Hence $\mathcal{Z} \times _\mathcal {Y} \mathcal{X}$ is quasi-compact by another application of Properties of Stacks, Lemma 99.6.2

Assume (2). Let $Z \to \mathcal{Y}$ be a morphism, where $Z$ is a scheme. We have to show that the morphism of algebraic spaces $p : Z \times _\mathcal {Y} \mathcal{X} \to Z$ is quasi-compact. Let $U \subset Z$ be affine open. Then $p^{-1}(U) = U \times _\mathcal {Y} \mathcal{Z}$ and the algebraic space $U \times _\mathcal {Y} \mathcal{Z}$ is quasi-compact by assumption (2). Hence $p$ is quasi-compact, see Morphisms of Spaces, Lemma 66.8.8. $\square$


Comments (0)


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Your email address will not be published.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 050T. Beware of the difference between the letter 'O' and the digit '0'.