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Tight Closure and Its Applications
Craig Huneke, Purdue University, West Lafayette, IN
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1996; 137 pp; softcover
Number: 88
ISBN-10: 0-8218-0412-X
ISBN-13: 978-0-8218-0412-4
List Price: US$36
Member Price: US$28.80
All Individuals: US$28.80
Order Code: CBMS/88
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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
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