4 - Change of rings

Summary

The main results of this chapter concern a homomorphism of commutative Noetherian rings f : R → R′. More precisely, we shall prove two fundamental comparison results for local cohomology modules in this context. The first of these, which we shall call the ‘Independence Theorem’, compares, for an R′ module M′ and an i ∈ ℕ₀, the local cohomology modules and: to form the first of these, we consider M′ as an R-module by restriction of scalars using f; also, aR′ denotes the extension of a to R′ under f. Our second main result, which we shall refer to as the ‘Flat Base Change Theorem’, compares the local cohomology modules and for i ∈ ℕ₀ and an arbitrary R-module M under the additional assumption that the ring homomorphism f is flat.

Our main results rely on the fact that certain modules are acyclic with respect to torsion functors. We say that an R-module A is Γ_a-acyclic precisely when for all i > 0. As was explained in 1.2.2, the most basic method for calculation, for an R-module M and an i ∈ ℕ₀, of is to take an injective resolution I* of M, apply Γ_a to I* to obtain the complex Γ_a(I*), and take the i-th cohomology module of this complex: we have. However, it is an easy exercise in homological algebra to show that a resolution of M by Γ_a-acyclic R-modules will serve this purpose just as well.