2 INTRODUCTION ideal int o th e ideal. Thu s on e can think o f them a s a typ e o f uniform annihilator , a poin t o f vie w whic h turn s ou t t o b e ver y useful . Thei r existenc e make s lif e much easier . Indeed , i t wa s not unti l thei r existenc e wa s proved i n a stron g for m [HH3,9] tha t th e important propert y o f persistence wa s proved. Se e Theorem 2.3. Persistence mean s tha t a n elemen t i n the tigh t closur e o f an idea l remain s i n th e tight closur e o f the expanded idea l afte r arbitrar y bas e change . One ca n conceptualiz e wha t propertie s a closur e operatio n fo r ideal s shoul d have t o giv e i t muc h o f th e sam e punc h a s tigh t closure . Th e closur e shoul d b e persistent. Ever y idea l in a regular loca l ring should b e closed under th e operation. The operatio n shoul d captur e th e colon . Th e closur e o f a n idea l shoul d contai n the integra l closur e o f th e dth powe r o f itself , wher e d i s th e dimensio n o f th e ring. Th e expansion and contraction o f an ideal in a module-finite extensio n shoul d be i n th e closur e o f th e ideal . Give n a closur e operatio n wit h thes e propertie s one can already prov e tha t direc t summand s o f regular ring s ar e Cohen-Macaulay , and prov e man y o f th e homologica l conjectures . I t i s fa r fro m clea r tha t suc h a closure operatio n exists . Fo r example th e integra l closur e o f an idea l come s close . It i s persistent, capture s th e colon (unde r som e mil d assumptions) , th e expansio n and contractio n o f / t o a module-finit e extensio n land s i n th e integra l closur e of /, an d certainl y i t ha s the propert y tha t th e integra l closur e o f an idea l contain s the th e integra l closur e o f the dth power o f itself, wher e d is the dimension o f the ring. However , integra l closur e fail s badl y t o hav e th e importan t propert y tha t ideals i n regula r ring s shoul d b e close d unde r th e closur e operation . On e can try to rectif y thi s b y lookin g a t th e regula r closur e o f a n ideal . Certainl y i t i s now true tha t ideal s i n regular ring s ar e closed. Moreover , thi s closur e does satisfy th e other propertie s in equicharacteristic. However , the only proof of this for the colon- capturing propert y i s through tigh t closure : th e tight closur e sits inside the regular closure. I n fac t th e question o f whether regula r closur e capture s th e colon i s open in mixe d characteristic . I n general , regula r closur e i s very difficul t t o wor k with . This difficult y i s partly du e to another propert y whic h a good closur e should have , which relate s t o th e theor y o f tes t elements : i f c £ R i s suc h tha t ever y idea l i n Rc i s close d unde r th e operation , the n ther e shoul d b e a fixed powe r o f c whic h multiplies th e closure o f every idea l / C R bac k int o / . W e do not quite kno w thi s for tigh t closure , bu t w e do know man y case s where i t i s true. The connectio n wit h th e Briangon-Skoda theore m indicate d tha t tigh t closur e should b e relate d t o classificatio n o f good singularities , e.g . rationa l singularities . This wa s alread y implici t i n th e wor k o f Fedder , Watanab e an d other s throug h their wor k o n F-purity . A pantheo n o f singularitie s wer e ther e t o study weakly F-regular if every ideal is tightly closed , F-rational i f parameters ar e tightly closed , and correspondin g notion s stabl e unde r localization . Th e attemp t t o prov e tha t weakly F-regula r localize s le d to th e definitio n o f strongly F-regula r rings , an d t o their study . From the point o f view of singularities, i t became obviou s that loca l rings wit h isolated singularit y coul d b e studie d usin g tigh t closur e method s becaus e ther e will b e a n m-primar y idea l o f tes t elements . Th e stud y o f suc h ring s le d t o th e understanding tha t tigh t closur e i s related t o plus closure , the contractio n o f the expansion of an ideal to the integral closure R+ of a ring R i n an algebraic closure of its fraction field. Th e philosophy o f tight closur e eventuall y le d to a proof tha t R+ is Cohen-Macaulay (se e Theorem 7.1) . Hochste r showe d a deep connection betwee n the propertie s of big Cohen-Macaulay algebra s and tight closur e through th e use of
Previous Page Next Page