13 - Graded versions of basic theorems

Summary

We have now laid the foundations of graded local cohomology theory in Chapter 12. Indeed, in the case where R is graded and the ideal a is graded, we now know that, for a graded R-module M, there is a natural way in which to define gradings on the local cohomology modules; furthermore, whenever f: M → N is a morphism in, then is a homogeneous homomorphism for all i ∈ ℕ₀; also, whenever

is an exact sequence in the ‘graded’ category, then all the homomorphisms, including the connecting homomorphisms, in the induced long exact sequence of local cohomology modules (with respect to a) are homogeneous.

We shall also need in the following chapters the refinements available in the graded case of such fundamental results as the Independence Theorem 4.2.1 and the Flat Base Change Theorem 4.3.2, and the theory of local duality of Chapter 11. This chapter is concerned with such refinements. In fact, one could take the view that part of this chapter is a rapid retracing of steps through earlier chapters, revisiting many of the highlights in order to ‘add graded frills’! Often, but not always, we shall present graded versions by means of exercises.

Fundamental theorems

Notation and Terminology. We shall employ the notation and terminology concerning graded rings and modules described in 12.1.1. In addition, when R is graded and the ideal a is graded, and M is a graded K-module, we use to denote the n-th component of the graded R-module (for i ∈ ℕ₀ and n ∈ ℤ).