Elements of Algebraic Geometry

IV. Local study of schemes and morphisms of schemes — Part three

A. Grothendieck (with the collaboration of J. Dieudonné).

Publications mathématiques de l'IHÉS, tome 28 (1966), pp. 5–255. numdam.org/item?id=PMIHES_1966__28__5_0

© Publications mathématiques de l'I.H.É.S., 1966.

Chapter IV (continued)

Local study of schemes and morphisms of schemes

This third part of Chapter IV develops the non-Noetherian techniques of the chapter: projective limits of preschemes (§8), constructible properties (§9), Jacobson preschemes (§10), the study of the fibres of flat morphisms of finite presentation (§12), equidimensional morphisms (§13), universally open morphisms (§14), and the study of the fibres of a universally open morphism (§15). The combination of projective-limit techniques with the constructibility tools of EGA III §0_III.9 and EGA IV §0_IV.9 allows one to eliminate Noetherian hypotheses on preschemes in exchange for hypotheses of finite presentation on the morphisms — hypotheses that, unlike Noetherianness, are stable under base change.

Sommaire (Chapter IV, continued)

  • §8. Projective limits of preschemes.
  • §9. Constructible properties.
  • §10. Jacobson preschemes.
  • §11. Topological properties of flat morphisms of finite presentation; local criteria of flatness.
  • §12. Study of the fibres of flat morphisms of finite presentation.
  • §13. Equidimensional morphisms.
  • §14. Universally open morphisms.
  • §15. Study of the fibres of a universally open morphism.

Translator's note. §11 of Chapter IV was originally announced in the 1964 sommaire of Part 1 with a footnote warning that its order and content might be modified before publication. Part 3 (1966) of EGA IV did publish §11 alongside §§8-10 and §§12-15. (Older secondary literature occasionally treated §11 as unpublished; we follow the primary source.) §11 develops the topological properties of flat morphisms of finite presentation, including the opens of flatness, the local criteria of flatness, the descent of flatness by arbitrary morphisms, and Raynaud-style separating-family criteria.