Chapter III — Cohomological study of coherent sheaves

Sommaire

  • §1. Cohomology of affine schemes.
  • §2. Cohomological study of projective morphisms.
  • §3. The finiteness theorem for proper morphisms.
  • §4. The fundamental theorem for proper morphisms; applications.
  • §5. An existence theorem for coherent algebraic sheaves.
  • §6. Local and global Tor and Ext functors; the Künneth formula.
  • §7. Study of base change in the covariant cohomological functors of modules.
  • §8. The duality theorem on projective fibres.
  • §9. Relative and local cohomology; local duality.
  • §10. Relations between projective cohomology and local cohomology; the technique of formal completion along a divisor.
  • §11. Global and local Picard groups (¹).

This chapter establishes the fundamental theorems on the cohomology of coherent algebraic sheaves, with the exception of those that arise from the theory of residues (duality theorems), which will be treated in a later chapter. Among the present results there are essentially six fundamental theorems, which form the subject matter of the first six sections of this chapter. These results will subsequently be essential tools, even in questions that are not properly cohomological in nature; the reader will see the first examples already in §4. Section §7 gives results of a more technical nature, but of constant use in applications. Finally, in §§8 through 11 we develop certain results, tied to the duality of coherent sheaves, that are particularly important for applications and that can be set out before the general theory of residues.

(¹) Chapter IV does not depend on §§8 through 11, and will probably appear in print before them.

The content of §§1 and 2 is due to J.-P. Serre, and the reader will see that we have done little more than follow the exposition of [FAC]. Sections §§8 and 9 are likewise inspired by [FAC] (the transpositions required by the differing contexts being, however, less obvious). Finally, as we said in the Introduction, §4 should be regarded as the modern reformulation of Zariski's fundamental "invariance theorem" in the "theory of holomorphic functions".

We mention finally that the results of n° 3.4 (and the preliminary propositions of through ) will not be used in the rest of Chapter III and may therefore be omitted on a first reading.

Translator's note. As stated in the EGA III table of contents, §§8 through 11 of Chapter III were never published; the volume in print stops at §7. The unpublished sections, sketched here for the reader's orientation, were absorbed into later work on duality (Hartshorne, Residues and Duality) and on Picard schemes (Grothendieck's Bourbaki seminars 232, 236).