Tight Closure Theory and Characteristic p Methods

Summary

Abstract. We give an introductory overview of the theory of tight closure, which has recently played a primary role among characteristic-p methods. We shall see that such methods can be used even when the ring contains a field of characteristic 0.

Introduction

The theory of tight closure has recently played a primary role among commutative algebraic methods in characteristic p. We shall see that such methods can be used even when the ring contains a field of characteristic 0.

Unless otherwise specified, the rings that we consider here will be Noetherian rings R containing a field. Frequently, we restrict, for simplicity, to the case of domains finitely generated over a field K. The theory of tight closure exists in much greater generality. For the development of the larger theory and its applications, and for discussion of related topics such as the existence of big Cohen–Macaulay algebras, we refer the reader to the joint works by Hochster and Huneke listed in the bibliography, to [Hochster 1994a; 1994b; 1996], to the expository accounts [Bruns 1996; Huneke 1996; 1998], and to the appendix to this paper by Graham Leuschke.

Here, in reverse order, are several of the most important reasons for studying tight closure theory, which gives a closure operation on ideals and on submodules. We focus mostly on the case of ideals here, although there is some discussion of modules. We shall elaborate on the themes brought forth in the list below in the sequel.