10 - Matlis duality

Summary

Prior to this point in the book, we have not made use of the decomposition theory (due to E. Matlis [34]) for injective modules over our (Noetherian) ring R. However, our work in the next Chapter 11 on local duality will involve use of the structure of the terms in the minimal injective resolution of a Gorenstein local ring, and so we can postpone no longer use of the decomposition theory for injective modules. Our discussion of local duality in Chapter 11 will also involve Matlis duality.

Our purpose in this chapter is to prepare the ground for Chapter 11 by reviewing, sometimes in detail, those parts of Matlis's theories that we shall need later in the book. Sometimes we simply refer to [35, Section 18] for proofs; in other cases, we provide alternative proofs for the reader's convenience. An experienced reader who is familiar with this work of Matlis should omit this chapter and progress straight to the discussion of local duality in Chapter 11: for one thing, there is no local cohomology theory in this chapter! However, graduate students might find this chapter helpful.

Indecomposable injective modules

Reminders. Let M be a submodule of the R-module L.

1. (i) We say that L is an essential extension of M precisely when B ∩ M ≠ 0 for every non-zero submodule B of L.

2. (ii) We say that L is an injective envelope (or injective hull) of M precisely when L is an injective R-module which is also an essential extension of M.