20 - Links with sheaf cohomology

Summary

In this last chapter we shall develop the links between local cohomology and the cohomology of quasi-coherent sheaves over certain Noetherian schemes. Here we shall assume for the first time that the reader has some basic knowledge about schemes and sheaves: our reference for these topics is Hartshorne's book [30]. The central idea in this chapter is to extend our earlier relations in 2.3.2 and 15.1.3 between ideal transforms and rings of regular functions on varieties to quasi-coherent sheaves over certain Noetherian schemes. We shall be very concerned with a generalization of the ‘Deligne Isomorphism’ (see [30, Chapter III, Exercise 3.7, p. 217]) which links the group of sections (over an open subset) of an induced sheaf on an affine scheme with an ideal transform. More precisely, let denote the sheaf induced by an R-module M on the affine scheme Spec(R), and let U = Spec(R) \ Var(a), where a ⊂ R; then the group of sections Γ (U,) is isomorphic to the ideal transform D_a(M). We shall use standard techniques involving negative strongly connected sequences of functors to extend this Deligne Isomorphism, and our generalization of it, to produce the Deligne Correspondence 20.3.11. This correspondence provides connections between higher cohomology groups of induced sheaves on the one hand, and local cohomology modules on the other.

We shall also examine the case when R is graded in some detail. Here the central result for us is the Serre–Grothendieck Correspondence 20.3.15, which we shall also derive from the Deligne Isomorphism by standard ‘connected sequence’ arguments.