The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

57.4 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 57.4.1. 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 25.18.2,

  2. quasi-compact, see Schemes, Lemma 25.19.3,

  3. universally closed, see Schemes, Definition 25.20.1,

  4. (quasi-)separated, see Schemes, Lemma 25.21.13,

  5. monomorphism, see Schemes, Lemma 25.23.5

  6. surjective, see Morphisms, Lemma 28.9.4,

  7. universally injective, see Morphisms, Lemma 28.10.2,

  8. affine, see Morphisms, Lemma 28.11.8,

  9. quasi-affine, see Morphisms, Lemma 28.12.5,

  10. (locally) of finite type, see Morphisms, Lemma 28.14.4,

  11. (locally) quasi-finite, see Morphisms, Lemma 28.19.13,

  12. (locally) of finite presentation, see Morphisms, Lemma 28.20.4,

  13. locally of finite type of relative dimension $d$, see Morphisms, Lemma 28.28.2,

  14. universally open, see Morphisms, Definition 28.22.1,

  15. flat, see Morphisms, Lemma 28.24.7,

  16. syntomic, see Morphisms, Lemma 28.29.4,

  17. smooth, see Morphisms, Lemma 28.32.5,

  18. unramified (resp. G-unramified), see Morphisms, Lemma 28.33.5,

  19. étale, see Morphisms, Lemma 28.34.4,

  20. proper, see Morphisms, Lemma 28.39.5,

  21. H-projective, see Morphisms, Lemma 28.41.8,

  22. (locally) projective, see Morphisms, Lemma 28.41.9,

  23. finite or integral, see Morphisms, Lemma 28.42.6,

  24. finite locally free, see Morphisms, Lemma 28.46.4,

  25. universally submersive, see Morphisms, Lemma 28.23.2,

  26. universal homeomorphism, see Morphisms, Lemma 28.43.2.

Add more as needed.

Remark 57.4.2. Of the properties of morphisms which are stable under base change (as listed in Remark 57.4.1) the following are also stable under compositions:

  1. closed, open and locally closed immersions, see Schemes, Lemma 25.24.3,

  2. quasi-compact, see Schemes, Lemma 25.19.4,

  3. universally closed, see Morphisms, Lemma 28.39.4,

  4. (quasi-)separated, see Schemes, Lemma 25.21.13,

  5. monomorphism, see Schemes, Lemma 25.23.4,

  6. surjective, see Morphisms, Lemma 28.9.2,

  7. universally injective, see Morphisms, Lemma 28.10.5,

  8. affine, see Morphisms, Lemma 28.11.7,

  9. quasi-affine, see Morphisms, Lemma 28.12.4,

  10. (locally) of finite type, see Morphisms, Lemma 28.14.3,

  11. (locally) quasi-finite, see Morphisms, Lemma 28.19.12,

  12. (locally) of finite presentation, see Morphisms, Lemma 28.20.3,

  13. universally open, see Morphisms, Lemma 28.22.3,

  14. flat, see Morphisms, Lemma 28.24.5,

  15. syntomic, see Morphisms, Lemma 28.29.3,

  16. smooth, see Morphisms, Lemma 28.32.4,

  17. unramified (resp. G-unramified), see Morphisms, Lemma 28.33.4,

  18. étale, see Morphisms, Lemma 28.34.3,

  19. proper, see Morphisms, Lemma 28.39.4,

  20. H-projective, see Morphisms, Lemma 28.41.7,

  21. finite or integral, see Morphisms, Lemma 28.42.5,

  22. finite locally free, see Morphisms, Lemma 28.46.3,

  23. universally submersive, see Morphisms, Lemma 28.23.3,

  24. universal homeomorphism, see Morphisms, Lemma 28.43.3.

Add more as needed.

Remark 57.4.3. Of the properties mentioned which are stable under base change (as listed in Remark 57.4.1) 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 34.20.19,

    2. open immersions, see Descent, Lemma 34.20.16, and

    3. quasi-compact immersions, see Descent, Lemma 34.20.21,

  2. quasi-compact, see Descent, Lemma 34.20.1,

  3. universally closed, see Descent, Lemma 34.20.3,

  4. (quasi-)separated, see Descent, Lemmas 34.20.2, and 34.20.6,

  5. monomorphism, see Descent, Lemma 34.20.31,

  6. surjective, see Descent, Lemma 34.20.7,

  7. universally injective, see Descent, Lemma 34.20.8,

  8. affine, see Descent, Lemma 34.20.18,

  9. quasi-affine, see Descent, Lemma 34.20.20,

  10. (locally) of finite type, see Descent, Lemmas 34.20.10, and 34.20.12,

  11. (locally) quasi-finite, see Descent, Lemma 34.20.24,

  12. (locally) of finite presentation, see Descent, Lemmas 34.20.11, and 34.20.13,

  13. locally of finite type of relative dimension $d$, see Descent, Lemma 34.20.25,

  14. universally open, see Descent, Lemma 34.20.4,

  15. flat, see Descent, Lemma 34.20.15,

  16. syntomic, see Descent, Lemma 34.20.26,

  17. smooth, see Descent, Lemma 34.20.27,

  18. unramified (resp. G-unramified), see Descent, Lemma 34.20.28,

  19. étale, see Descent, Lemma 34.20.29,

  20. proper, see Descent, Lemma 34.20.14,

  21. finite or integral, see Descent, Lemma 34.20.23,

  22. finite locally free, see Descent, Lemma 34.20.30,

  23. universally submersive, see Descent, Lemma 34.20.5,

  24. universal homeomorphism, see Descent, Lemma 34.20.9.

Note that the property of being an “immersion” may not be fpqc local on the base, but in Descent, Lemma 34.21.1 we proved that it is fppf local on the base.


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