INTRODUCTION 3 solid closure , anothe r closur e introduce d b y him . Unfortunately , thi s closur e doe s not hav e al l th e propertie s on e want s i n equicharacteristi c 0 , althoug h i t agree s with tigh t closur e i n positiv e characteristic . Th e situatio n i n mixe d characteristi c is unclear . The basi c result s o n colo n capturin g wer e pushe d i n [HH8 ] t o giv e result s o n phantom acyclicity-complexe s whos e cycle s ar e i n th e tigh t closur e o f th e bound - aries. Eventuall y Ia n Aberbac h [Abl ] showe d ho w t o develo p thi s concep t int o a theory simila r t o th e existin g theor y o f modules o f finite projectiv e dimension . Other application s hav e bee n foun d t o a variet y o f problems : unifor m Arti n Rees theorems , arithmeti c Macaulayfications , ring s o f differentia l operators , an d connections wit h vanishin g theorem s fo r cohomolog y o n comple x projectiv e vari - eties. I n particula r th e Kodair a vanishin g theore m i s equivalen t t o a statemen t about th e tigh t closur e o f parameter ideals . These note s attemp t t o elucidat e man y o f thes e mai n theme s withou t gettin g lost i n details . I hav e chose n importan t theorem s whos e proof s illustrat e a rang e of th e technique s o f tigh t closure , an d I'v e include d man y furthe r result s i n th e exercises. Th e char t followin g thi s introductio n give s som e ide a o f th e curren t directions i n which th e subjec t i s moving . I have tried t o keep the note s faithful t o the actua l talk s I gave at Fargo . How - ever, I have adde d severa l chapter s t o touch o n topics which othe r peopl e spok e o n at th e conference. I n particular, Chapte r 4 on F-rational ring s and singularities was the topi c of a talk b y Karen Smit h an d Chapte r 6 on the Hilbert-Kun z multiplicit y was th e topi c o f a tal k b y Pau l Monsky . I hav e als o adde d Chapte r 1 3 on regula r base change . I n additio n Me l Hochste r ha s writte n a n appendi x base d o n hi s tw o talks concernin g th e theor y o f tigh t closur e i n equicharacteristi c 0 . I'v e include d numerous exercises that ofte n cove r topics which are important bu t whic h the note s do no t cove r directly .
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