Chapter IV — Local study of schemes and morphisms of schemes

Sommaire

• §1. Relative finiteness conditions. Constructible sets in preschemes.
• §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.
• §8. Projective limits of preschemes.
• §9. Constructible properties.
• §10. Jacobson preschemes.
• §11.¹ 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.
• §16. Differential invariants. Differentially smooth morphisms.
• §17. Smooth, unramified, étale morphisms.
• §18. Complements on étale morphisms. Henselian local rings.
• §19. Regular and transversally regular immersions.
• §20. Hyperplane sections; generic projections.
• §21. Infinitesimal extensions.

Translator's note. The sommaire above reproduces the 1964 announcement printed in Part 1. As the footnote warned, the program was revised during the later parts. §11 was never published — Part 3 (1966) prints §§8, 9, 10, 12, 13, 14, 15, skipping §11. §§20 and 21 were redrawn: in Part 4 (1967) §20 became Meromorphic functions; pseudo- morphisms and §21 became Divisors, replacing the announced "hyperplane sections" and "infinitesimal extensions" respectively. The translation files use the published §20 and §21 titles, and there is no §11 file.

The subjects treated in this chapter call for the following remarks.

a) Their common character is to bear on local properties of preschemes or of morphisms, i.e. properties considered at a point, or at the points of a fibre, or in a neighbourhood (unspecified) of a point or of a fibre. These properties are generally topological, differential, or dimensional in nature (i.e. they bring into play the notions of dimension and depth), and they are tied to the properties of the local rings at the points in question. A typical problem is to relate, for a given morphism $f:X\to Y$ and a point $x\in X$, the properties of $X$ at $x$ to those of $Y$ at $y=f(x)$ and of the fibre $X_y=f^{-1}(y)$ at $x$. Another is to determine the topological nature (for example, constructibility, or being open or closed) of the set of points $x\in X$ at which $X$ has a given property, or for which the fibre $X_y$ through $x$ has a given property at $x$. Similarly, one is interested in the topological nature of the set of points $y\in Y$ such that $X$ has a given property at every point of the fibre $X_y$, or such that this fibre has a given property.

b) The notions of greatest importance for the chapters that follow are those of flat morphism of finite presentation and of smooth morphism and étale morphism (which are special cases of the former). Their detailed study (together with that of related questions) really begins at §11.

c) §§1 through 10 may be regarded as preliminary in nature; they develop three types of technique, used not only in the other sections of the chapter but, naturally, in the chapters that follow as well.

$c_1$) In §§1 through 4 we consider various aspects of the notion of base change, especially in relation with finiteness or flatness conditions; we initiate, at its most elementary level, the technique of descent (the "effectiveness" questions tied to that technique will be studied in Chapter V).

$c_2$) §§5 through 7 are centred on what we may call Noetherian techniques: the preschemes considered are always supposed locally Noetherian, while no finiteness conditions are imposed on the morphisms; this is essentially because the notions of dimension and depth are not really workable except for Noetherian local rings. Let us recall that §7 constitutes a "fine" theory of Noetherian local rings, used relatively little in the rest of the chapter.

$c_3$) §§8 through 10 provide, among other things, the means to eliminate the Noetherian hypotheses on the preschemes under study, replacing them by suitable finiteness hypotheses ("finite presentation") on the morphisms considered: the advantage of this substitution is that the latter hypotheses are stable under base change, which is not the case for Noetherian hypotheses on preschemes.

The technique permitting this passage rests, on the one hand, on the use of the notion of projective limit of preschemes, by means of which one may reduce a question to the same question under Noetherian hypotheses; on the other hand, on the systematic use of constructible sets, which have the double advantage of being preserved under inverse image (by an arbitrary morphism) and under direct image (by a morphism of finite presentation), and of having manageable topological properties in locally Noetherian preschemes. The same techniques often allow further reduction to more particular Noetherian rings, for example finitely generated $k$-algebras ($k$ a field), and this is where the properties of "excellent" rings (studied in §7) play a decisive role. Independently of the question of eliminating Noetherian hypotheses, the techniques of §§8 through 10, of very elementary nature, are in constant use in almost all applications.