6 0. A PREHISTOR Y O F TIGH T CLOSUR E ((x\,...,xtk)R : £*. +1 ) = (x*,...,#£.)/? . Whe n R i s no t Cohen-Macaulay , i t i s no t difficult t o prov e tha t an y c such tha t R c i s Cohen-Macaula y ha s a power , sa y c n , such tha t c n ((x\, ..^xfyR : x^.+1) C (x* , ...,x^.)i?. Howeve r th e powe r n depend s upon t. Th e fac t tha t a singl e c works i s what I mea n b y "uniforml y annihilates" . Hochster ha s referred t o hi s proof o f the existenc e o f big Cohen-Macaula y module s as the first 'tigh t closure ' proof. I t i s useful t o compar e th e definitio n o f amiable t o the definitio n o f tight closure , a t leas t fo r ideals : DEFINITION 2 . Le t R b e a Noetherian rin g of characteristic p 0 . Le t I b e a n ideal o f R, an d x G R. A n elemen t x i s said t o b e i n th e tigh t closur e o f / i f ther e exists a n elemen t c , no t i n an y minima l prim e o f R, suc h tha t fo r al l larg e q = p e , cxq /M , wher e I™ i s the idea l generate d b y th e qth power s o f all elements o f / . The idea of using uniform annihilator s of cohomology was used by Paul Robert s (1976) [Rol ] i n hi s proo f o f th e ne w intersectio n theorem . Th e ne w intersectio n conjecture3 states : Let (0.1) G : 0 - G n - ^ G n _i ^ ..' . - G o - 0 be a complex of finitely generated free .R-module s such that th e homology has finite length, an d i s not al l zero. I f n dim(li), the n G i s exact . Roberts' proo f o f thi s i n characteristi c p use s anothe r resul t whic h concern s dualizing complexes . DEFINITION 3 . Le t R b e a Noetherian ring. A complex D# o f injective module s is called a dualizing complex fo r R i f the followin g tw o conditions hold : ii) H l (D°) i s finitely generate d fo r al l i. Any homomorphi c imag e o f a Gorenstei n loca l rin g ha s a dualizin g complex . Any Noetheria n loca l rin g wit h a dualizin g comple x als o ha s a canonica l module , namely th e initia l nonzer o homolog y modul e o f the dualizin g complex . An y homo - morphic imag e o f a rin g wit h a dualizin g comple x als o ha s a dualizin g complex . Definea * = Ann(i/i(D*)) , an d se t bi equa l t o th e produc t o f Ooai...Oj . Robert s proves that fo r an y complex of free R- modules a s in 0.1, bi annihilates the ( n z)th homology an d furthermor e dim(R/bi) i. Roberts' proo f o f th e ne w intersectio n conjectur e i n characteristi c p goe s a s follows: first reduc e t o th e cas e where ai{Gi) C mGi-i b y splittin g of f extraneou s free exac t sequences . I t i s then clea r tha t applyin g th e Frobeniu s t o th e comple x gives another comple x as in 0.1, but wher e the images of the new maps lie in higher and highe r power s o f the maxima l idea l time s th e fre e modul e i n whic h th e imag e sits. Sinc e b n annihilate s th e 0t h homology , w e obtai n tha t b n mus t li e i n highe r and highe r powers of the maximal ideal, and hence this ideal is 0. Bu t the n dim(ii ) = dim(R/b n ) n. The Frobeniu s allow s u s t o replac e fre e complexe s b y othe r fre e complexe s where th e entrie s o f the matrice s givin g th e map s ar e i n highe r an d highe r power s of th e maxima l idea l i n suc h a wa y tha t th e ideal s o f minor s onl y chang e u p t o radical. Thi s is a powerful tool , and n o such process is known in characteristic zer o except fo r th e Koszu l complex . 3 The ne w intersectio n conjectur e i s no w a theore m i n al l characteristics . [Rol-3 ]
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