Elements of Algebraic Geometry

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

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

Publications mathématiques de l'IHÉS, tome 20 (1964), pp. 5–259. numdam.org/item?id=PMIHES_1964__20__5_0

© Publications mathématiques de l'I.H.É.S., 1964.

Chapter 0 (continued)

Preliminaries

This part of the running Chapter 0 (preliminaries) gathers the commutative-algebra material that EGA IV, and the rest of EGA, will use. It picks up the numbering of Chapter 0 from EGA III at §14, and continues through §23.

Sommaire (Chapter 0, continued)

  • §14. Combinatorial dimension of a topological space.
  • §15. -regular sequences and -regular sequences.
  • §16. Dimension and depth in Noetherian local rings.
  • §17. Regular rings.
  • §18. Complements on extensions of algebras.
  • §19. Formally smooth algebras and Cohen rings.
  • §20. Derivations and differentials.
  • §21. Differentials in characteristic- rings.
  • §22. Differential criteria for formal smoothness and regularity.
  • §23. Japanese rings.

Almost all of the following sections are devoted to the exposition of notions from commutative algebra that will be used in the course of Chapter IV. Although a good part of these notions already appears in several works ([1], [12], [13], [17], [30]), it seemed more convenient for the reader to have a continuous and roughly self-contained presentation. Combined with §§5, 6, 7 of Chapter IV (where the language of schemes is used), these sections constitute, within our Treatise, a small special treatise, almost independent of Chapters I to III, whose aim is to set out in a coherent form the properties of rings that "behave well" with respect to operations such as completion and integral closure, by systematically attaching these properties to a few general conceptions.1

Translator's note. As elsewhere in EGA, the symbol 0_IV (resp. ) refers to the part of Chapter 0 (resp. the list of errata) attached to Chapter IV. So means §14, paragraph 3.2 of the running Chapter 0, housed in EGA IV.

1

Most of the properties in question were discovered by Chevalley, Zariski, Nagata, and Serre. The method used here was first developed in autumn 1961, in a course taught at Harvard University by A. Grothendieck.