The Stacks project

Remark 63.4.1. 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. ├ętale, 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.


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