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 Briancon-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.It 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; and, numerous exercises with solutions.
Reihe
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Gewicht
ISBN-13
978-0-8218-0412-4 (9780821804124)
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Schweitzer Klassifikation
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.
