Chapter I — The Language of Schemes

Summary

§1. Affine schemes. §2. Preschemes and morphisms of preschemes. §3. Products of preschemes. §4. Subpreschemes and immersion morphisms. §5. Reduced preschemes; separation condition. §6. Finiteness conditions. §7. Rational maps. §8. Chevalley schemes. §9. Complements on quasi-coherent sheaves. §10. Formal schemes.

Paragraphs §1 to §8 are devoted essentially to developing a language — the one used throughout the rest of the treatise. Note however that, in keeping with the general spirit of this treatise, §§7 and 8 will be used less than the others, and less essentially; we have discussed Chevalley schemes only to make the connection with the language of Chevalley [1] and Nagata [9]. §9 gives definitions and results on quasi-coherent sheaves; some of these go beyond the mere "geometric" translation of notions from commutative algebra and are already global in nature, and they will be indispensable, beginning with the following chapters, in the global study of morphisms. Finally, §10 introduces a generalization of the notion of scheme, which will serve us in Chapter III as an intermediary for stating and proving the fundamental results of the cohomological study of proper morphisms; we also note that the notion of formal scheme seems indispensable for expressing certain facts of "moduli theory" (problems of classification of algebraic varieties). The results of §10 will not be used before §3 of Chapter III, and we recommend omitting that section until then.

Contents