8 0. A PREHISTOR Y O F TIGH T CLOSUR E there ar e ofte n commo n element s i n th e relativ e Jacobia n ideals . Indeed , thei r proof o f the stronge r Briangon-Skod a theore m depend s upo n thi s fact. I t turn s ou t that Theore m 5 gives a ric h sourc e o f what w e call 'test ' elements . Th e discussio n in th e abov e paragraph s suggest s tha t applyin g th e Frobeniu s t o thi s proble m i n characteristic p migh t b e valuable. Thi s i s indeed th e case , as Chapte r 5 discusses. A final piec e of prehistory relate d method s o f integral closure s to tight closure . In 198 6 Itoh [It ] an d Hunek e [Hu4 ] independentl y prove d th e followin g theorem : THEOREM 6 . Let R be a Cohen-Macaulay local ring containing a field, and let xi,...,Xd be a system of parameters generating an ideal I. Then I n C\In~1 = I n ~lI for all n 1 . This wa s late r generalize d b y Ito h t o th e sam e conclusio n fo r formall y equidi - mensional loca l rings . Th e proo f o f Itoh di d no t requir e th e rin g t o contai n a field and was based on a study o f the local cohomology o f Rees algebras. Huneke' s proo f was a reductio n t o characteristi c p , an d use d a multiplie r c together wit h Frobe - nius t o analyz e th e intersection . Essentiall y th e proo f wa s a tigh t closur e proo f related t o colo n capturing . Th e simpl e cas e i n whic h n = 2 gives th e flavor o f th e argument. Le t R b e a Noetheria n rin g o f characteristic p an d le t z G I2 f ! I wher e / = (xi,...,#d ) i s generate d b y a regula r sequence . Sinc e z G I2 ther e exist s a n element c G suc h that fo r all n, cz n G I2n. Writ e z J2 r ixi- ^ suffice s t o prov e that Ti G I. Th e equatio n cz q = ^ cr^xf G I2q implie s tha t th e coefficient s cr\ ar e in I q (sinc e th e xi for m a regula r sequence) . Thi s i s true fo r al l larg e power s o f p, and i t follow s tha t n i s i n th e integra l closur e o f I a s claimed . Eve n thi s simpl e case has interestin g consequences , fo r instanc e a proof o f a version o f the Grauert - Reimenschneider vanishin g theorem i n dimension two. Se e Exercises 5.12-13 for th e proof. Hunek e als o proved tha t i f R i s F-pure, an d dimensio n d, then fo r al l ideal s I o f R, 7 d + 1 C I. Thi s proo f als o was essentially a tight closur e proo f (cf . Exercis e 5.10). Exercises Exercises 0.1)-3 ) belo w al l us e tha t applyin g th e Frobeniu s t o a finite acycli c complex o f free module s preserve s acyclicity . 0.1. Le t i ? b e a Noetheria n loca l ring o f positive characteristi c an d le t G : 0 G n ^ G n -\ Go—* 0 be a comple x o f finitely generate d fre e i2-module s suc h tha t al l positiv e homology ha s finit e length . Prov e tha t afte r applyin g th e Frobeniu s (whic h has th e affec t o f raising al l th e entrie s o f matrice s representin g th e map s i n the comple x to the pth power ) th e resultin g comple x ha s the sam e property . 0.2. Le t R b e a regular loca l ring of positive characteristi c an d le t / b e a n ideal . Prove tha t th e associate d prime s o f / ar e th e sam e a s the associate d prime s of 1^ fo r al l q = pe. (Hint : us e th e Auslander-Buchsbau m formul a relatin g projective dimensio n an d depth. )
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