Abstract
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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