Tight Closure and Its Applications
About this Title
Craig L. Huneke, Purdue University, West Lafayette, IN
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 1996: Volume 88
ISBNs: 978-0-8218-0412-4 (print); 978-1-4704-2448-0 (online)
MathSciNet review: MR1377268
MSC: Primary 13-02; Secondary 13A35, 13C14, 13H10
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.
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.
Graduate students and research mathematicians interested in commutative rings and algebras.
Table of Contents
- 1. Introduction
- 2. A prehistory of tight closure (Chapter 0)
- 3. Basic Notions (Chapter 1)
- 4. Test elements and the persistence of tight closure (Chapter 2)
- 5. Colon-capturing and direct summands of regular rings (Chapter 3)
- 6. F-Rational rings and rational singularities (Chapter 4)
- 7. Integral closure and tight closure (Chapter 5)
- 8. The Hilbert-Kunz multiplicity (Chapter 6)
- 9. Big Cohen-Macaulay algebras (Chapter 7)
- 10. Big Cohen-Macaulay algebras II (Chapter 8)
- 11. Applications of big Cohen-Macaulay algebras (Chapter 9)
- 12. Phantom homology (Chapter 10)
- 13. Uniform Artin-Rees theorems (Chapter 11)
- 14. The localization problem (Chapter 12)
- 15. Regular base change (Chapter 13)
- 16. The notion of tight closure in equal characteristic zero (Appendix 1)
- 17. Solutions to exercises (Appendix 2)