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.

Readership

Graduate students and research mathematicians interested in commutative rings and algebras.

Reviews

"The book [is] easily readable by a person who wants to study tight closure in depth as well as by a person who wants to read lightly and still gain some understanding."

-- Zentralblatt MATH

Table of Contents

• Acknowledgements
• Introduction
• Relationship chart
• A prehistory of tight closure
• Basic notions
• Test elements and the persistence of tight closure
• Colon-capturing and direct summands of regular rings
• F-rational rings and rational singularities
• Integral closure and tight closure
• The Hilbert-Kunz multiplicity
• Big Cohen-Macaulay algebras
• Big Cohen-Macaulay algebras II
• Applications of big Cohen-Macaulay algebras
• Phantom homology
• Uniform Artin-Rees theorems
• The localization problem
• Regular base change
• Appendix 1: The notion of tight closure in equal characteristic zero (by M. Hochster)
• Appendix 2: Solutions to the exercises
• Bibliography