Elements of Algebraic Geometry

II. Elementary global study of some classes of morphisms

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

Publications mathématiques de l'IHÉS, tome 8 (1961), pp. 5–222. numdam.org/item?id=PMIHES_1961__8__5_0

© Publications mathématiques de l'I.H.É.S., 1961.

Chapter II — Elementary global study of some classes of morphisms

Sommaire

• §1. Affine morphisms.
• §2. Homogeneous prime spectra.
• §3. Homogeneous spectrum of a sheaf of graded algebras.
• §4. Projective bundles. Ample sheaves.
• §5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms.
• §6. Integral morphisms and finite morphisms.
• §7. Valuative criteria.
• §8. Blow-ups; projecting cones; projective closure.

The various classes of morphisms studied in this chapter are treated without significant use of cohomological methods; a deeper study, drawing on those methods, will be carried out in Chapter III, which will rely above all on §§2, 4 and 5 of Chapter II. Section §8 may be omitted on first reading: it presents some complements to the formalism developed in §§1–3, reducing to easy applications of that formalism, and we shall make less constant use of its results than of those of the other sections of this chapter.