CBMS Regional Conference Series in Mathematics 1996; 137 pp; softcover Number: 88 ISBN10: 082180412X ISBN13: 9780821804124 List Price: US$38 Member Price: US$30.40 All Individuals: US$30.40 Order Code: CBMS/88
 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. Fregular and Frational singularities. Basic theorems in the theory are presented including versions of the BriançonSkoda theorem, various homological conjectures, and the HochsterRoberts/Boutot theorems on invariants of reductive groups. Several applications of the theory are given. These include the existence of big CohenMacaulay algebras and various uniform ArtinRees theorems. Features:  The existence of test elements.
 A study of Frational rings and rational singularities.
 Basic information concerning the HilbertKunz 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
 Coloncapturing and direct summands of regular rings
 Frational rings and rational singularities
 Integral closure and tight closure
 The HilbertKunz multiplicity
 Big CohenMacaulay algebras
 Big CohenMacaulay algebras II
 Applications of big CohenMacaulay algebras
 Phantom homology
 Uniform ArtinRees 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
