12 - Foundations in the graded case

Summary

If our Noetherian ring R is (ℤ-)graded, and our ideal a is graded, then it is natural to ask whether the local cohomology modules, and for a graded R-module M, also carry structures as graded R-modules. Some of the realizations of these local cohomology modules that we have obtained earlier in the book suggest that they should. For instance, if a₁, …, a_n (where n > 0) denote n homogeneous elements which generate a, then the Čech complex C*(M) of M with respect to a₁, …, a_n is composed of graded R-modules and homogeneous homomorphisms, and so (for i ∈ ℕ₀), which, by Theorem 5.1.19, is isomorphic to, inherits a grading. But is this grading independent of the choice of homogeneous generators for a?

More hopeful evidence is provided by the isomorphism

of 1.3.8. For each n ∈ ℕ, since R/aⁿ is a finitely generated graded R-module, is actually the graded K-module (see [6, pp. 32–33]) with its grading forgotten, and, for n, m ∈ ℕ with n ≥ m, the natural homomorphism is homogeneous, so that the induced homomorphism

is homogeneous; hence is graded, and inherits a grading by virtue of the above isomorphism. But is this grading the same as that which comes from the approach using the Čech complex described in the preceding paragraph?

One could take another approach to local cohomology in this graded situation, an approach which, at first sight, seems substantially different from those described in the preceding two paragraphs. Again suppose that our Noetherian ring R is (ℤ-)graded, and our ideal a is graded.