Elements of Algebraic Geometry

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

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

Publications mathématiques de l'IHÉS, tome 24 (1965), pp. 5–231. numdam.org/item?id=PMIHES_1965__24__5_0

© Publications mathématiques de l'I.H.É.S., 1965.

Chapter IV (continued)

Local study of schemes and morphisms of schemes

This second part of Chapter IV develops the "Noetherian" half of the chapter: base change and flatness (§2), associated prime cycles and primary decompositions (§3), base field change in preschemes (§4), the dimension and depth machinery for preschemes (§5), flat morphisms of locally Noetherian preschemes (§6), and the fine theory of Noetherian local rings and their completion, including the notion of excellent ring (§7).

Sommaire (Chapter IV, continued)

• §2. Base change and flatness.
• §3. Associated prime cycles and primary decompositions.
• §4. Base field change in preschemes.
• §5. Dimension and depth in preschemes.
• §6. Flat morphisms of locally Noetherian preschemes.
• §7. Application to the relations between a Noetherian local ring and its completion. Excellent rings.

The decimal numbering continues that of Part 1: cross-references like (IV, 2.3.7) and (IV, 7.8.1) point at this part, while (IV, 1.4.2) continues to point at §IV.1 in Part 1, and $(0_{IV},\mathrm{17.1.1})$ continues to point at the Chap 0_IV preliminaries.