The Stacks project

Lemma 69.5.16. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume

  1. $Y$ quasi-compact and quasi-separated,

  2. $X_ i$ quasi-compact and quasi-separated,

  3. the transition morphisms $X_{i'} \to X_ i$ are closed immersions,

  4. $X_ i \to Y$ locally of finite type

  5. $X \to Y$ is a closed immersion.

Then $X_ i \to Y$ is a closed immersion for $i$ large enough.

Proof. Choose an affine scheme $W$ and a surjective ├ętale morphism $W \to Y$. Then $X \times _ Y W$ is a closed subspace of $W$ and it suffices to check that $X_ i \times _ Y W$ is a closed subspace $W$ for some $i$ (Morphisms of Spaces, Lemma 66.12.1). By Lemma 69.5.11 this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 32.4.20. $\square$


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Your email address will not be published.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 0A0T. Beware of the difference between the letter 'O' and the digit '0'.