CHAPTER 0 A Prehistor y o f Tigh t Closur e The root s o f tigh t closur e ca n b e foun d i n th e wor k o f severa l authors , mos t notably Christia n Peskin e an d Lucie n Szpiro , Melvi n Hochster , an d Pau l Roberts . This work, in the late sixties and the seventies, used the idea of reduction to charac- teristic p an d too k advantag e o f the Frobeniu s morphis m i n positive characteristic . One of the mos t importan t fact s i n this proces s i s that th e Frobeniu s morphis m i s flat ove r regular rings (although perhaps the single most importan t poin t i s that th e Frobenius i s a n endomorphism) . Thi s fac t an d it s convers e wer e prove d b y Kun z [Ku2] i n 1969 . Th e sam e year, Peskin e an d Szpir o [PS3 ] proved tha t applyin g th e Frobenius t o a finit e acycli c sequenc e o f finitely generate d fre e module s preserve s exactness. The y went o n to use this to prove several of the homological conjecture s in characteristic p and for rings essentially of finite type over a field of characteristi c 0 [PS1] . Thei r pape r wa s the first us e of the metho d o f reduction t o characteristi c p in commutativ e algebra, 1 althoug h th e metho d ha d bee n use d fo r man y year s i n other areas. 2 I n 1973 , Hochster [Ho2 ] gav e a proof o f the existenc e o f Big Cohen - Macaulay modules, i.e. no t necessaril y finitely generate d module s such that a given system o f parameters i s a regular sequenc e on the module . Th e proo f use d what h e called 'amiable ' system s o f parameters. Her e i s the definitio n o f amiable : DEFINITION 1 . A syste m o f parameter s xi,...,Xd o f a Noetheria n loca l rin g (R,m) o f dimensio n d i s sai d t o b e amiable i f ther e i s a n elemen t c G R, no t nilpotent, suc h that , fo r al l k, 0 k d, an d fo r al l positive integer s t , c((x1, ...,xk)R : x kJtl ) C (a^ , ...,x k )R. Hochster prove d tha t i f (R,m) i s a n integrall y close d Cohen-Macaula y loca l domain, an d S i s a module-finit e loca l extensio n domai n whos e fractio n field i s separable ove r th e fractio n field o f R, the n an y syste m o f parameter s o f R i s a n amiable syste m o f parameter s i n S. Toda y w e kno w tha t an y syste m o f param - eters o f a loca l equidimensiona l rin g whic h ha s a dualizin g comple x i s amiable . The elemen t c i n th e definitio n o f amiabilit y play s th e rol e o f a tes t elemen t i n tight closur e theory . Suc h c ar e unifor m annihilators—i n thi s cas e suc h c uni - formly annihilat e th e failur e o f th e rin g t o b e Cohen-Macaula y i n th e followin g sense. I f R wer e Cohen-Macaulay , the n fo r ever y syste m o f parameter s xi,...,Xd, x At th e sam e time , Robi n Hartshorn e wa s on e o f th e pioneer s i n applyin g th e Frobeniu s to th e stud y o f loca l cohomology , an d hi s in-dept h stud y wit h Rober t Speise r o n th e actio n o f Frobenius o n loca l cohomolog y appeare d i n th e Annal s i n 1977 . 2 Where reductio n t o characteristi c p wa s first use d i s a myster y t o me , bu t on e gues s comin g from Hochste r i s in a proo f o f the irreducibilit y o f cyclotomic polynomials . Th e proo f o f Dedekin d in 185 7 use d reductio n t o characteristi c p. However , i t ma y b e tha t eve n Gaus s wa s th e first t o use thi s technique . 5 http://dx.doi.org/10.1090/cbms/088/02
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