The Stacks project

Remark 65.4.3. Of the properties mentioned which are stable under base change (as listed in Remark 65.4.1) the following are also fpqc local on the base (and a fortiori fppf local on the base):

  1. for immersions we have this for

    1. closed immersions, see Descent, Lemma 35.23.19,

    2. open immersions, see Descent, Lemma 35.23.16, and

    3. quasi-compact immersions, see Descent, Lemma 35.23.21,

  2. quasi-compact, see Descent, Lemma 35.23.1,

  3. universally closed, see Descent, Lemma 35.23.3,

  4. (quasi-)separated, see Descent, Lemmas 35.23.2, and 35.23.6,

  5. monomorphism, see Descent, Lemma 35.23.31,

  6. surjective, see Descent, Lemma 35.23.7,

  7. universally injective, see Descent, Lemma 35.23.8,

  8. affine, see Descent, Lemma 35.23.18,

  9. quasi-affine, see Descent, Lemma 35.23.20,

  10. (locally) of finite type, see Descent, Lemmas 35.23.10, and 35.23.12,

  11. (locally) quasi-finite, see Descent, Lemma 35.23.24,

  12. (locally) of finite presentation, see Descent, Lemmas 35.23.11, and 35.23.13,

  13. locally of finite type of relative dimension $d$, see Descent, Lemma 35.23.25,

  14. universally open, see Descent, Lemma 35.23.4,

  15. flat, see Descent, Lemma 35.23.15,

  16. syntomic, see Descent, Lemma 35.23.26,

  17. smooth, see Descent, Lemma 35.23.27,

  18. unramified (resp. G-unramified), see Descent, Lemma 35.23.28,

  19. étale, see Descent, Lemma 35.23.29,

  20. proper, see Descent, Lemma 35.23.14,

  21. finite or integral, see Descent, Lemma 35.23.23,

  22. finite locally free, see Descent, Lemma 35.23.30,

  23. universally submersive, see Descent, Lemma 35.23.5,

  24. universal homeomorphism, see Descent, Lemma 35.23.9.

Note that the property of being an “immersion” may not be fpqc local on the base, but in Descent, Lemma 35.24.1 we proved that it is fppf local on the base.


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Your email address will not be published.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 02WH. Beware of the difference between the letter 'O' and the digit '0'.