Introduction As mentioned i n the Acknowledgements , tigh t closur e bega n i n a conversatio n between Me l Hochste r an d mysel f i n 1986 . I n it s earl y for m tigh t closur e centere d upon severa l themes. Thes e included th e realization tha t tigh t closur e gave an eas y proof o f the theore m o f Hochster an d Robert s o n the Cohen-Macaulaynes s o f rings of invariant s [HR1] , th e recognitio n tha t tigh t closur e capture d th e obstruction s to a ring' s bein g Cohen-Macaula y (se e Chapte r 3) , th e fac t tha t i n a regula r rin g every ideal is tightly closed , an d th e relationship betwee n tight closur e and integra l closure, specifically throug h the theorem of Briangon-Skoda (Theore m 5.7). Puttin g these simple properties together already gave quite a powerful tool . On e of our early realizations was the following: le t (R, m) b e a complete local equidimensional ring of positive characteristic wit h a system of parameters #i,... , Xd such that J2 i ViXi 0. Suppose tha t R » S i s an injectiv e rin g homomorphis m t o a regular rin g S. The n for every z, r G (x1, ...,Xi_i,x i+ i, ...,xd)S. Thi s result follow s as (yi, ...,2/») '. Vi+\ is always containe d i n the tigh t closur e o f (yi , ...,2/i) whe n yi, ...,2/i+ i ar e parameter s in a complet e loca l equidimensiona l rin g (w e refe r t o thi s fac t a s 'tigh t closur e captures th e colon' , o r colon-capturing , fo r short) , becaus e tigh t closur e persist s from R t o 5 , an d becaus e ever y idea l i n S i s tightl y close d sinc e S i s regular . I n 1986 thi s seeme d amazin g t o me . I n retrospec t i t wa s amazin g onl y tha t n o on e had notice d i t before . There wa s som e debat e a t th e inceptio n o f tigh t closur e eve n abou t th e defi - nition shoul d a definitio n b e give n tha t wa s stable unde r completio n an d localiza - tion?1 Wer e there othe r alternativ e definition s tha t woul d b e better fo r developin g the theory ? Th e search for equivalen t formulation s o f tight closur e is still an issue partly to understand tigh t closure , partly to try to find a definition i n characteristi c 0 tha t doe s no t requir e reductio n t o characteristi c p , an d mainl y t o tr y t o find a suitable definitio n fo r mixe d characteristic . A t first w e thought tha t tigh t closur e might b e th e sam e a s regular closure 2 (se e Chapte r 5 an d th e discussio n below) , but soo n discovered thi s was far fro m th e case . Ther e i s still no definition i n mixe d characteristic, an d n o definitio n i n equicharacteristi c 0 which doe s no t eventuall y refer bac k to characteristic p. Thi s lack of a definition i n mixed characteristi c alon g with th e questio n o f whethe r tigh t closur e commute s wit h localizatio n ar e amon g the centra l problem s currentl y facin g us . A concept whic h onl y seems to gro w i n importance i s the ide a o f test element s (see Chapter 2) . Thes e are elements which multiply the tight closur e of an arbitrar y 1 Even th e nam e underwen t revision . Betwee n ourselve s w e originally calle d i t 'shar p closure ' but decide d tha t 'tight ' conveye d ou r feelin g tha t thi s closur e wa s a ver y tigh t fit t o th e ideal . 2 An elemen t x G R i s said t o b e i n th e regula r closur e o f a n idea l I C R if xS C IS fo r ever y regular rin g S t o whic h R maps . 1 http://dx.doi.org/10.1090/cbms/088/01
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