0. A PREHISTOR Y O F TIGH T CLOSUR E 7 The dualizing complex is acyclic exactly when R i s Cohen-Macaulay. Th e ideals bi pla y th e rol e o f a unifor m annihilato r o f homolog y whic h shoul d b e zer o whe n the rin g i s Cohen-Macaulay , jus t a s th e c in th e definitio n o f amiabilit y uniforml y annihilates certai n homolog y whic h i s zero i f the rin g i s Cohen-Macaulay . I n fact , most o f th e homologica l conjecture s ar e nontrivia l onl y whe n th e bas e rin g i s no t Cohen-Macaulay, s o it is not surprising that suc h uniform annihilation , in conjuctio n with th e Probeniu s morphism , shoul d b e a powerful tool . At approximatel y th e sam e time i n which many o f the homologica l conjecture s were bein g solve d i n equicharacteristic, anothe r importan t theore m wa s proved b y Hochster an d Joe l Robert s [HR1 ] .4 The y prove d tha t a linearl y reductiv e affln e linear algebrai c grou p ove r a field k actin g A:-rationall y o n a regular Noetheria n k- algebra S ha s a Cohen-Macaulay rin g of invariants. Thei r proof also used reductio n to characteristi c p , but ther e wa s no apparen t connectio n betwee n th e homologica l proofs an d thei r proof . However , thei r work focuse d attentio n o n the propert y tha t the Frobeniu s homomorphis m b e pure . Thi s mean s tha t tensorin g th e Probeniu s map R — » R wit h a n arbitrar y i?-modul e give s a n injectio n (i n particula r th e Frobenius is injective s o R i s reduced). A basically equivalent formulatio n o f purit y is t o sa y tha t wheneve r x q £ l' 9 ', the n x e I whe n / C R i s a n ideal . Kei-ich i Watanabe an d Richar d Fedde r develope d th e theor y o f F-pure ring s considerably , while Meht a an d Ramanatha n [MR ] an d Meht a an d Sriniva s [MS ] hav e use d th e idea o f F-split varietie s t o stud y singularities . The next important even t with regard to this prehistory was a CBMS conferenc e held at Georg e Mason University in 1979, where Hochster was the principal speake r and concentrate d o n a theorem du e to Briango n an d Skod a [BS] . At tha t tim e th e only proo f o f thi s theore m wa s analytic , an d Hochste r challenge d th e algebraist s present t o find a n algebrai c proof . Thi s wa s don e successfull y b y Lipma n an d Sathaye [LS ] in 1981, and in the same issue of the Michigan Math. Journal , Lipma n and Teissie r [LT ] gave a partial extensio n t o rationa l singularities . Th e theore m of Lipman an d Sathay e state s THEOREM 4 . [LS ] Let (R,m) be a regular local ring. Let I be any ideal of R generated by I elements. Then for any w 0 Jl+w Q jw+l Although i t i s only clea r i n retrospect, an d i n any cas e requires a reintrepreta - tion o f th e results , th e proo f o f thi s theore m agai n use s a typ e o f unifor m annihi - lation, bu t o f a completely differen t sor t tha n tha t use d b y Hochster , Roberts , etc . above. Th e mai n technica l resul t use d i n th e proo f o f Theore m 4 i s th e followin g theorem o f Lipman an d Sathaye : THEOREM 5 . Let R be a regular Noetherian domain with quotient field K. Let L be a finite separable field extension of K, and let S be a finitely generated R-subalgebra of L. Set JS/R = J = 0 th Fitting ideal of the S-module of Kdhler R-differentials QS/R- Let T be the integral closure of S. Then JT C S. The unifor m annihilato r i s th e relativ e Jacobia n ideal , an d i t uniforml y anni - hilates th e quotien t T/S o f the integra l closur e T o f a finitely generate d R- algebra S. Whil e it i s true that th e Jacobia n idea l depends upo n 5 , a s S varie s birationall y 4 The pape r o f Hochste r an d Robert s appeare d i n 1974 . A late r pape r b y th e sam e author s on th e purit y o f Frobeniu s appeare d i n 197 6 [HR2] .

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