Tight closure and its applications : Expository lectures from the NSF-CBMS regional conference held at North Dakota State University, Fargo, ND, June 22-29, 1995

This monograph deals with the theory of tight closure and its applications. The contents are based on ten talks given at a CBMS conference held at North Dakota State University in June 1995. Tight closure is a method to study rings of equicharacteristic by using reduction to positive characteristic. In this book, the basic properties of tight closure are covered, including various types of singularities, e.g. F-regular and F-rational singularities. Basic theorems in the theory are presented including versions of the Briancon-Skoda theorem, various homological conjectures, and the Hochster-Roberts/Boutot theorems on invariants of reductive groups. Several applications of the theory are given. These include the existence of big Cohen-Macaulay algebras and various uniform Artin-Rees theorems. Features: The existence of test elements. A study of F-rational rings and rational singularities. Basic information concerning the Hilbert-Kunz function, phantom homology, and regular base change for tight closure. Numerous exercises with solutions. This monograph deals with the theory of tight closure and its applications. The contents are based on ten talks given at a CBMS conference held at North Dakota State University in June 1995. Tight closure is a method to study rings of equicharacteristic by using reduction to positive characteristic. In this book, the basic properties of tight closure are covered, including various types of singularities, e.g. F-regular and F-rational singularities. Basic theorems in the theory are presented including versions of the Briançon-Skoda theorem, various homological conjectures, and the Hochster-Roberts/Boutot theorems on invariants of reductive groups. Several applications of the theory are given. These include the existence of big Cohen-Macaulay algebras and various uniform Artin-Rees theorems. Features: The existence of test elements. A study of F-rational rings and rational singularities. Basic information concerning the Hilbert-Kunz function, phantom homology, and regular base change for tight closure. Numerous exercises with solutions. Cover 1 Title 6 Copyright 7 Contents 8 Acknowledgements 10 Introduction 12 Relationship Chart 15 Chapter 0. A Prehistory of Tight Closure 16 Chapter 1. Basic Notions 21 Chapter 2. Test Elements and the Persistence of Tight Closure 29 Chapter 3. Colon-Capturing and Direct Summands of Regular Rings 35 Chapter 4. F-Rational Rings and Rational Singularities 43 Chapter 5. Integral Closure and Tight Closure 49 Chapter 6. The Hilbert-Kunz Multiplicity 57 Chapter 7. Big Cohen-Macaulay Algebras 61 Chapter 8. Big Cohen-Macaulay Algebras II 67 Chapter 9. Applications of Big Cohen-Macaulay Algebras 72 Chapter 10. Phantom Homology 78 Chapter 11. Uniform Artin-Rees Theorems 85 Chapter 12. The Localization Problem 94 Chapter 13. Regular Base Change 100 Appendix 1: The Notion of Tight Closure in Equal Characteristic Zero 105 Appendix 2: Solutions to the Exercises 118 Bibliography 144 Back Cover 151 Deals with the theory of tight closure and its applications. This book covers the basic properties of tight closure including various types of singularities such as F-regular and F-rational singularities. It presents basic theorems in the theory including versions of the Briancon-Skoda theorem, and various homological conjectures.