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Tight Closure and Its Applications
 
Craig L. Huneke Purdue University, West Lafayette, IN
A co-publication of the AMS and CBMS
Front Cover for Tight Closure and Its Applications
Available Formats:
Electronic ISBN: 978-1-4704-2448-0
Product Code: CBMS/88.E
137 pp 
List Price: $38.00
Individual Price: $30.40
Front Cover for Tight Closure and Its Applications
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  • Front Cover for Tight Closure and Its Applications
  • Back Cover for Tight Closure and Its Applications
Tight Closure and Its Applications
Craig L. Huneke Purdue University, West Lafayette, IN
A co-publication of the AMS and CBMS
Available Formats:
Electronic ISBN:  978-1-4704-2448-0
Product Code:  CBMS/88.E
137 pp 
List Price: $38.00
Individual Price: $30.40
  • Book Details
     
     
    CBMS Regional Conference Series in Mathematics
    Volume: 881996
    MSC: Primary 13; Secondary 14;



    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.

  • Table of Contents
     
     
    • Chapters
    • 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)
  • 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
  • Request Review Copy
Volume: 881996
MSC: Primary 13; Secondary 14;



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.

  • Chapters
  • 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)
  • 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
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