Contents Acknowledgements i x Introduction 1 Relationship Char t 4 Chapter 0 . A Prehistory o f Tight Closur e 5 Chapter 1 . Basi c Notion s 1 0 Chapter 2 . Tes t Element s an d th e Persistenc e o f Tight Closur e 1 8 Chapter 3 . Colon-Capturin g an d Direc t Summand s o f Regular Ring s 2 4 Chapter 4 . F-Rationa l Ring s an d Rationa l Singularitie s 3 2 Chapter 5 . Integra l Closur e an d Tigh t Closur e 3 8 Chapter 6 . Th e Hilbert-Kun z Multiplicit y 4 6 Chapter 7 . Bi g Cohen-Macaulay Algebra s 5 0 Chapter 8 . Bi g Cohen-Macaula y Algebra s I I 5 6 Chapter 9 . Application s o f Big Cohen-Macaulay Algebra s 6 1 Chapter 10 . Phanto m Homolog y 6 7 Chapter 11 . Unifor m Artin-Ree s Theorem s 7 4 Chapter 12 . Th e Localizatio n Proble m 8 3 Chapter 13 . Regula r Bas e Chang e 8 9 Appendix 1 : Th e Notio n o f Tigh t Closur e i n Equa l Characteristi c Zer o (b y M. Hochster ) 9 4 Appendix 2 : Solution s t o th e Exercise s 10 7 Bibliography 13 3
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