The Stacks project

Remark 64.4.3. Of the properties mentioned which are stable under base change (as listed in Remark 64.4.1) 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.

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