The Stacks project

65.4 Lists of useful properties of morphisms of schemes

For ease of reference we list in the following remarks the properties of morphisms which possess some of the properties required of them in later results.

Remark 65.4.1. Here is a list of properties/types of morphisms which are stable under arbitrary base change:

  1. closed, open, and locally closed immersions, see Schemes, Lemma 26.18.2,

  2. quasi-compact, see Schemes, Lemma 26.19.3,

  3. universally closed, see Schemes, Definition 26.20.1,

  4. (quasi-)separated, see Schemes, Lemma 26.21.12,

  5. monomorphism, see Schemes, Lemma 26.23.5

  6. surjective, see Morphisms, Lemma 29.9.4,

  7. universally injective, see Morphisms, Lemma 29.10.2,

  8. affine, see Morphisms, Lemma 29.11.8,

  9. quasi-affine, see Morphisms, Lemma 29.13.5,

  10. (locally) of finite type, see Morphisms, Lemma 29.15.4,

  11. (locally) quasi-finite, see Morphisms, Lemma 29.20.13,

  12. (locally) of finite presentation, see Morphisms, Lemma 29.21.4,

  13. locally of finite type of relative dimension $d$, see Morphisms, Lemma 29.29.2,

  14. universally open, see Morphisms, Definition 29.23.1,

  15. flat, see Morphisms, Lemma 29.25.8,

  16. syntomic, see Morphisms, Lemma 29.30.4,

  17. smooth, see Morphisms, Lemma 29.34.5,

  18. unramified (resp. G-unramified), see Morphisms, Lemma 29.35.5,

  19. étale, see Morphisms, Lemma 29.36.4,

  20. proper, see Morphisms, Lemma 29.41.5,

  21. H-projective, see Morphisms, Lemma 29.43.8,

  22. (locally) projective, see Morphisms, Lemma 29.43.9,

  23. finite or integral, see Morphisms, Lemma 29.44.6,

  24. finite locally free, see Morphisms, Lemma 29.48.4,

  25. universally submersive, see Morphisms, Lemma 29.24.2,

  26. universal homeomorphism, see Morphisms, Lemma 29.45.2.

Add more as needed.

Remark 65.4.2. Of the properties of morphisms which are stable under base change (as listed in Remark 65.4.1) the following are also stable under compositions:

  1. closed, open and locally closed immersions, see Schemes, Lemma 26.24.3,

  2. quasi-compact, see Schemes, Lemma 26.19.4,

  3. universally closed, see Morphisms, Lemma 29.41.4,

  4. (quasi-)separated, see Schemes, Lemma 26.21.12,

  5. monomorphism, see Schemes, Lemma 26.23.4,

  6. surjective, see Morphisms, Lemma 29.9.2,

  7. universally injective, see Morphisms, Lemma 29.10.5,

  8. affine, see Morphisms, Lemma 29.11.7,

  9. quasi-affine, see Morphisms, Lemma 29.13.4,

  10. (locally) of finite type, see Morphisms, Lemma 29.15.3,

  11. (locally) quasi-finite, see Morphisms, Lemma 29.20.12,

  12. (locally) of finite presentation, see Morphisms, Lemma 29.21.3,

  13. universally open, see Morphisms, Lemma 29.23.3,

  14. flat, see Morphisms, Lemma 29.25.6,

  15. syntomic, see Morphisms, Lemma 29.30.3,

  16. smooth, see Morphisms, Lemma 29.34.4,

  17. unramified (resp. G-unramified), see Morphisms, Lemma 29.35.4,

  18. étale, see Morphisms, Lemma 29.36.3,

  19. proper, see Morphisms, Lemma 29.41.4,

  20. H-projective, see Morphisms, Lemma 29.43.7,

  21. finite or integral, see Morphisms, Lemma 29.44.5,

  22. finite locally free, see Morphisms, Lemma 29.48.3,

  23. universally submersive, see Morphisms, Lemma 29.24.3,

  24. universal homeomorphism, see Morphisms, Lemma 29.45.3.

Add more as needed.

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.

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