Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 101.43.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper, and

  2. $f$ satisfies both the uniqueness and existence parts of the valuative criterion.

Proof. A proper morphism is the same thing as a separated, finite type, and universally closed morphism. Thus this lemma follows from Lemmas 101.41.2, 101.41.3, 101.42.1, and 101.42.2. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 101.43: Valuative criterion for properness

Your email address will not be published.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.