4 - LOCALIZATION, PRIMARY DECOMPOSITION, AND THE SECOND LAYER

Summary

This chapter is the first of four chapters in which we attempt to develop a new approach to localization in Noetherian rings. The impetus for our attempt is provided by the desire to find a way out of the stymie mentioned in the Preface and in section 3.4.

The focal point of our attempt is a new concept: the second layer of an S-tame module. To state it roughly, if S is a semi-prime ideal in a Noetherian ring R and if M is an R-module that can be embedded in a direct sum of copies of E(R/S)_R then M is called S-tame and the module M/ann_M S is called the second layer of M.

The aim of this chapter is to show that the study of the second layer may be viewed as a synthesis of the ‘primary decomposition’ of modules over Noetherian rings and of the usual approach to localization in Noetherian rings. This view allows one to interpret the usual approach to localization as but a traditionally preferred means to an end, the end being a grip on the second layer. Intriguingly, this interpretation leaves open the possibility that other means may be available when the traditionally preferred one – localization – fails. Such means are developed in chapters 5 and 6, and are exploited in chapter 7.

This chapter is organized as follows. The first section contains some background material about injectives. The second section is devoted to primary decomposition. The third section interprets (classical) localizability in terms of injectives. The last section concerns the second layer and the synthesis hinted above.