Chapter 2 - ADIC TOPOLOGIES AND COMPLETIONS

Summary

The homological conjectures which are discussed in this book are local and, what is more, need only be proved over a complete noetherian local ring. The structure of such rings, expressed in the Cohen Structure Theorems, is heavily used in their solution. Apart from these, we only need results from this chapter incidentally.

There are in the literature several excellent accounts of the topics in the title [AM], [Bo 61b, Ch. 3], [ZS], [Ma 86], but these do tend to concentrate all too soon on finitely generated modules over noetherian rings. Nevertheless, there exists a quite attractive theory at least for arbitrary modules over noetherian rings, and in some cases we may even waive the noetherian condition. Since this book features the “construction” of certain infinitely generated complete modules with good properties, cf. Theorems 5.2.3 and 9.1.1, and complete modules are drawing an increasing measure of attention [Ba], [Si], we have chosen to present an outline of this theory. In doing so we recall several standard results without proof, and for the less standard ones give either a proof, hints for a proof or a reference.

In the first section we show how the notion of purity can serve in the realm of adic topologies when the Artin-Rees Lemma is unavailable. On the other hand, a pure submodule is a poor man's direct summand, and this paves the way to our proof of Hochster's Direct Summand Theorem in equal characteristic, 10.3.5.