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 10
Chapter 2 . Tes t Element s an d th e Persistenc e o f Tight Closur e 18
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 107
Bibliography 133
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