11 - Local duality

Summary

Suppose, temporarily, that (R, m) is local, and that M is a finitely generated jR-module. In Theorem 7.1.3, we showed that is Artinian for all i ∈ ℕ₀. When R is complete, Matlis duality (see 10.2.12) provides a very satisfactory correspondence between the category of Artinian R-modules and the category of Noetherian R-modules, and so it is natural to ask, in this situation, which Noetherian R-modules correspond to the local cohomology modules. Local duality provides an answer to this question, and also provides a fundamental tool for the study of local cohomology modules with respect to the maximal ideal of a local ring.

For a Gorenstein local ring (R, m) of dimension n, the Local Duality Theorem tells us that, if M is a finitely generated R-module, then, for each i ∈ ℕ₀, the local cohomology module is the Matlis dual of the finitely generated R-module and, as R is Gorenstein, quite a lot is known about these ‘Ext’ modules. The Independence Theorem 4.2.1 then allows us to extend the Local Duality Theorem to any local ring which is a homomorphic image of a Gorenstein local ring, and the class of such local rings includes the local rings of points on affine and quasi-affine varieties, and, by the structure theorems for complete local rings (see [35, Theorem 29.4(ii)], for example), all complete local rings. Furthermore, the special case of the Local Duality Theorem for a Cohen-Macaulay local ring R which is a homomorphic image of a Gorenstein local ring can be formulated in terms of the canonical module ω_Rfor R.