Tight Closure and Its Applications 9780821804124

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Crai g H u n e k e , Tigh t closur e an d it s applications , 1 99 6 J o h n Eri k Fornaess , Dynamic s i n severa l comple x variables , 1 99 6 Sori n P o p a , Classificatio n o f subfactor s an d thei r endomorphisms , 1 99 5 Michi o J i m b o an d Tetsuj i M i w a , Algebrai c analysi s o f solvabl e lattic e models , 1 99 4 H u g h L . M o n t g o m e r y , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y and harmoni c analysis , 1 99 4 83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y value problems , 1 99 4 82 Susa n M o n t g o m e r y , Hop f algebra s an d thei r action s o n rings , 1 99 3 81 S t e v e n G . Krantz , Geometri c analysi s an d functio n spaces , 1 99 3 80 V a u g h a n F . R . J o n e s , Subfactor s an d knots , 1 99 1 79 Michae l Frazier , Bjor n J a w e r t h , an d G u i d o Weiss , Littlewood-Pale y theor y an d the stud y o f functio n spaces , 1 99 1 78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 1 99 1 77 Michae l Christ , Lecture s o n singula r integra l operators , 1 99 0 76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 1 99 0 75 F . T h o m a s Farrel l an d L . E d w i n J o n e s , Classica l aspherica l manifolds , 1 99 0 74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l equations, 1 99 0 73 Walte r A . Strauss , Nonlinea r wav e equations , 1 98 9 72 P e t e r Orlik , Introductio n t o arrangements , 1 98 9 71 Harr y D y m , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d interpolation, 1 98 9 70 Richar d F . G u n d y , Som e topic s i n probabilit y an d analysis , 1 98 9 69 Fran k D . Grosshans , Gian-Carl o R o t a , an d Joe l A . Stein , Invarian t theor y an d superalgebras, 1 98 7 68 J . W i l l i a m H e l t o n , J o s e p h A . Ball , Charle s R . J o h n s o n , an d J o h n N . P a l m e r , Operator theory , analyti c functions , matrices , an d electrica l engineering , 1 98 7 67 Haral d U p m e i e r , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics, 1 98 7 66 G . A n d r e w s , ^-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory , combinatorics, physic s an d compute r algebra , 1 98 6 65 Pau l H . R a b i n o w i t z , Minima x method s i n critica l poin t theor y wit h application s t o differential equations , 1 98 6 64 D o n a l d S . P a s s m a n , Grou p rings , crosse d product s an d Galoi s theory , 1 98 6 63 Walte r R u d i n , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n, 1986 62 B e l a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 1 98 6 61 M o g e n s F l e n s t e d - J e n s e n , Analysi s o n non-Riemannia n symmetri c spaces , 1 98 6 60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 1 98 6 59 R o g e r H o w e an d A l l e n M o y , Harish-Chandr a homomorphism s fo r p-adi c groups , 1985 58 H . B l a i n e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 1 98 5 57 Jerr y L . K a z d a n , Prescribin g th e curvatur e o f a Riemannia n manifold , 1 98 5 56 Har i Bercovici , Cipria n Foia§ , an d Car l P e a r c y , Dua l algebra s wit h application s to invarian t subspace s an d dilatio n theory , 1 98 5 55 W i l l i a m A r v e s o n , Te n lecture s o n operato r algebras , 1 98 4 54 W i l l i a m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 1 98 4 (Continued in the back of this publication)

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Tight Closure and Its Applications

http://dx.doi.org/10.1090/cbms/088

Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s Number 8 8

Tight Closur e and Its Applications Craig Hunek e

Published fo r th e Conference Boar d o f th e Mathematica l Science s by th e #^J§> America n Mathematica l Societ y Q *£!il5t" with suppor t fro m th e V\ffi& > x National Scienc e Foundatio n °^o^ °

Expository Lecture s from th e NSF-CBM S Regiona l Conferenc e held a t Nort h Dakot a Stat e University , Fargo , N D June 22-29 , 1 99 5 Research partiall y supporte d b y National Scienc e Foundatio n Gran t DM S 9301 05 3 1991 Mathematics Subject Classification. Primar y 1 3A35 , 1 3D25 ; Secondary 1 4-XX .

Library o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Huneke, C . (Craig ) Tight closur e an d it s application s / Crai g Hunek e p. cm . — (Conferenc e Boar d o f th e Mathematica l Science s regiona l conferenc e serie s i n mathematics, ISS N 01 60-7642 ; no . 88 ) Includes bibliographica l references . ISBN 0-821 8-041 2- X 1. Commutativ e rings . 2 . Complexes . I . Title . II . Series : Regiona l conferenc e serie s i n mathematics; no . 88 . QA1.R33 no . 8 8 [QA251.3] 510 s—dc2 0 [512'.4] 96-496 5 CIP

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o [email protected] .

© Copyrigh t 1 99 6 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . H Printe d o n recycle d paper . 10 9 8 7 6 5 4 3 2 1 0 1 0 0 9 9 9 8 9 7 9 6

Contents Acknowledgements i

x

Introduction Relationship Char t 4 Chapter 0 . A

Prehistor y o f Tigh t Closur e 5

Chapter 1 . Basi c Notion s

0

Chapter 2 . Tes t Element s an d th e Persistenc e o f Tigh1 t Closur e

8

Chapter 3 . Colon-Capturin g an d Direc t Summand s o f Regula r Ring s 2

4

Chapter 4 . F-Rationa l Ring s an d Rationa l Singularitie s 3

2

Chapter 5 . Integra l Closur e an d Tigh t Closur e 3

8

Chapter 6 . Th e Hilbert-Kun z Multiplicit y 4

6

Chapter 7 . Bi g Cohen-Macaula y Algebra s 5

0

Chapter 8 . Bi g Cohen-Macaula y Algebra s I I 5

6

Chapter 9 . Application s o f Bi g Cohen-Macaula y Algebra s 6 1 Chapter 1 0 . Phanto m Homolog y 6

7

Chapter 1 1 . Unifor m Artin-Ree s Theorem s 7

4

Chapter 1 2 . Th e Localizatio n Proble m 8

3

Chapter 1 3 . Regula r Bas e Chang e 8

9

Appendix 1 : Th e Notio n o f Tigh t Closur e i n Equa l Characteristi c Zer o (b y M. Hochster ) 9

4

Appendix 1 2 : Solution s t o th e Exercise s 0

7

Bibliography 3

3

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Acknowledgements Tight closur e bega n i n a conversatio n wit h Me l Hochste r a t a conferenc e a t the Universit y o f Illinois i n Novembe r o f 1 986 . Seriou s wor k o n tigh t closur e bega n early i n 1 987 , and th e subjec t quickl y too k flight . Withou t Me l Hochste r th e topi c of thi s manuscrip t woul d no t exist , an d thes e note s ar e a tribut e t o hi s beautifu l ideas an d immens e contributions . Many othe r peopl e hav e helpe d m e i n bot h m y thinkin g an d writing . I ow e special thank s t o Ia n Aberbach , Bil l Heinzer, Jo e Lipman , Kare n Smith , an d Iren a Swanson. I than k Jane t Cowde n fo r proofreadin g th e entir e manuscript . These note s ar e base d o n te n talk s I gav e a t a CBM S conferenc e a t Nort h Dakota Stat e Universit y Jun e 22-29 , 1 995 . A specia l thank s goe s t o th e organizer , Joe Brennan . I a m ver y gratefu l t o hi m fo r th e ide a o f havin g thi s conference , and fo r th e tim e an d effor t h e pu t int o i t t o mak e i t a success . Hi s effort s mad e the conferenc e bot h pleasan t an d enjoyable . I als o than k Nort h Dakot a Stat e University fo r hostin g th e conference , th e Conferenc e Boar d o f th e Mathematica l Sciences fo r choosin g thi s topic , an d th e Nationa l Scienc e Foundatio n fo r fundin g it. I woul d lik e t o than k th e mai n supportin g speakers : Ia n Aberbach , Juerge n Herzog, Me l Hochster , Jo e Lipman , Pau l Monsky , Pau l Roberts , Kare n Smith , Mark Spivakovsky , an d Kei-ich i Watanabe . Thei r talk s enhance d th e conferenc e greatly. Finally, I thank th e Universit y o f Michigan fo r suppor t durin g m y preparation s for th e conference . Craig Huneke , August , 1 99 5

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http://dx.doi.org/10.1090/cbms/088/01

Introduction As mentione d i n th e Acknowledgements , tigh t closur e bega n i n a conversatio n between Me l Hochste r an d mysel f i n 1 986 . I n it s earl y for m tigh t closur e centere d upon severa l themes. Thes e include d th e realizatio n tha t tigh t closur e gave an eas y proof o f the theore m o f Hochste r an d Robert s o n th e Cohen-Macaulaynes s o f ring s of invariant s [HR1 ] , th e recognitio n tha t tigh t closur e capture d th e obstruction s to a ring' s bein g Cohen-Macaula y (se e Chapte r 3) , th e fac t tha t i n a regula r rin g every idea l i s tightly closed , an d th e relationshi p betwee n tigh t closur e an d integra l closure, specifically throug h the theorem of Briangon-Skoda (Theore m 5.7) . Puttin g these simple properties together alread y gave quite a powerful tool . On e of our earl y realizations was the following: le t (R, m) b e a complete local equidimensional ring of positive characteristi c wit h a system o f parameters #i,... , Xd such tha t J2 i ViXi — 0 . Suppose tha t R — » S i s a n injectiv e rin g homomorphis m t o a regula r rin g S. The n for every z, r; G (x1, ...,Xi_i,x i + i, ...,x d)S. Thi s result follow s a s (yi , ...,2/») '. Vi+\ is always containe d i n th e tigh t closur e o f (yi , ...,2/i) whe n yi, ...,2/i+ i ar e parameter s in a complet e loca l equidimensiona l rin g (w e refe r t o thi s fac t a s 'tigh t closur e captures th e colon' , o r colon-capturing , fo r short) , becaus e tigh t closur e persist s from R t o 5 , an d becaus e ever y idea l i n S i s tightl y close d sinc e S i s regular . I n 1986 thi s seeme d amazin g t o me . I n retrospec t i t wa s amazin g onl y tha t n o on e had notice d i t before . There wa s som e debat e a t th e inceptio n o f tigh t closur e eve n abou t th e defi nition; shoul d a definitio n b e give n tha t wa s stabl e unde r completio n an d localiza tion? 1 Wer e ther e othe r alternativ e definition s tha t woul d b e bette r fo r developin g the theor y ? Th e searc h fo r equivalen t formulation s o f tight closur e i s still a n issue ; partly t o understand tigh t closure , partl y t o try t o find a definition i n characteristi c 0 tha t doe s no t requir e reductio n t o characteristi c p , an d mainl y t o tr y t o find a suitable definitio n fo r mixe d characteristic . A t first w e though t tha t tigh t closur e might b e th e sam e a s regular closure 2 (se e Chapte r 5 an d th e discussio n below) , but soo n discovere d thi s wa s fa r fro m th e case . Ther e i s still n o definition i n mixe d characteristic, an d n o definitio n i n equicharacteristi c 0 whic h doe s no t eventuall y refer bac k t o characteristi c p. Thi s lac k of a definition i n mixed characteristi c alon g with th e questio n o f whethe r tigh t closur e commute s wit h localizatio n ar e amon g the centra l problem s currentl y facin g us . A concep t whic h onl y seem s t o gro w i n importanc e i s the ide a o f test element s (see Chapter 2) . Thes e are elements which multiply the tight closur e of an arbitrar y 1

Even th e nam e underwen t revision . Betwee n ourselve s w e originall y calle d i t 'shar p closure ' but decide d tha t 'tight ' conveye d ou r feelin g tha t thi s closur e wa s a ver y tigh t fit t o th e ideal . 2 An elemen t x G R i s sai d t o b e i n th e regula r closur e o f a n idea l I C R if xS C IS fo r ever y regular rin g S t o whic h R maps .

1

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 th e tigh t closur e o f a n 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 i n a regular loca l ring should b e closed unde r 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 an d 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 ca n alread y 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 exampl e th e integra l closur e o f an idea l come s close . It i s persistent, capture s th e colo n (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 o f / , an d certainl y i t ha s th e propert y tha t th e integra l closur e o f a n idea l contain s the th e integra l closur e o f the dt h powe r o f itself, wher e d i s the dimensio n 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 tr y 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 regula r ring s ar e closed . Moreover , thi s closur e does satisfy th e other propertie s i n equicharacteristic. However , th e only proof o f this for the coloncapturing propert y i s through tigh t closure : th e tight closur e sit s insid e the regula r closure. I n fac t th e questio n o f whether regula r closur e capture s th e colo n i s ope n in mixe d characteristic . I n general , regula r closur e i s ver y difficul t t o wor k with . This difficult y i s partly du e to anothe r propert y whic h a goo d closur e shoul d 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 quit e kno w thi s for tigh t closure , bu t w e do know man y case s wher e i t i s true . The connectio n wit h th e Briangon-Skod a theore m indicate d tha t tigh t closur e should b e relate d t o classificatio n o f goo d 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 i f every idea l i s 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 t o th e definitio n o f strongl y F-regula r rings , an d t o their study . From th e point o f view o f singularities, i t becam e obviou s tha t loca l ring s 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 relate d t o plus closure , the contractio n o f the expansion o f an ideal to the integral closure R+ of a ring R i n an algebraic closur e of its fractio n 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 o f big Cohen-Macaulay algebra s an d tight closur e throug h th e use of

INTRODUCTION

3

solid closure , anothe r closur e introduce d b y him . Unfortunately , thi s closur e doe s not hav e al l th e propertie s on e want s i n equicharacteristi c 0 , althoug h i t agree s with tigh t closur e i n positiv e characteristic . Th e situatio n i n mixe d characteristi c is unclear . The basi c result s o n colo n capturin g wer e pushe d i n [HH8 ] t o giv e result s o n phantom acyclicity-complexe s whos e cycle s ar e i n th e tigh t closur e o f th e bound aries. Eventuall y Ia n Aberbac h [Abl ] showe d ho w t o develo p thi s concep t int o a theory simila r t o th e existin g theor y o f module s o f finite projectiv e dimension . Other application s hav e bee n foun d t o a variet y o f problems : unifor m Arti n Rees theorems , arithmeti c Macaulayfications , ring s o f differentia l operators , an d connections wit h vanishin g theorem s fo r cohomolog y o n comple x projectiv e vari eties. I n particula r th e Kodair a vanishin g theore m i s equivalen t t o a statemen t about th e tigh t closur e o f paramete r ideals . These note s attemp t t o elucidat e man y o f thes e mai n theme s withou t gettin g lost i n details . I hav e chose n importan t theorem s whos e proof s illustrat e a rang e of th e technique s o f tigh t closure , an d I'v e include d man y furthe r result s i n th e exercises. Th e char t followin g thi s introductio n give s som e ide a o f th e curren t directions i n whic h th e subjec t i s moving . I hav e trie d t o kee p the note s faithfu l t o th e actua l talk s I gave at Fargo . How ever, I have adde d severa l chapter s t o touc h o n topic s whic h othe r peopl e spok e o n at th e conference. I n particular, Chapte r 4 on F-rational ring s and singularitie s wa s the topi c o f a talk b y Kare n Smit h an d Chapte r 6 on th e Hilbert-Kun z multiplicit y was th e topi c o f a tal k b y Pau l Monsky . I hav e als o adde d Chapte r 1 3 on regula r base change . I n additio n Me l Hochste r ha s writte n a n appendi x base d o n hi s tw o talks concernin g th e theor y o f tigh t closur e i n equicharacteristi c 0 . I'v e include d numerous exercise s that ofte n cove r topics which are important bu t whic h the note s do no t cove r directly .

4

INTRODUCTION

RELATIONSHIP CHAR T FO R TIGH T CLOSUR E A N D IT S APPLICATION S

IMPROVED HOMOLOGICAL CONJECTURES, VANISHING THEORE M FOR MAP S O F T O R

ARITHMETIC

PHANTOM

MACAULAYFICATIONS

HOMOLOGY

X

UNIFORM ARTIN-REES THEOREMS

1 BRIANgON-SKODA

BIG COHEN-MACAULAY

/ TIGH

ALGEBRAS, R+, AN

I CLOSUR E J

D

T H E O R E M S , INTEGRA L

T\

CLOSURES, AN D RATIONAL

SOLID CLOSUR E

SINGULARITIES

X VANISHING THEOREM S

i

\ CLASSIFICATION O F

D I R E C T SUMMAND S

ON HOMOLOGY :

SINGULARITIES:

OF REGULA R RING S

GRAUERT-

F - P U R E , F-RATIONAL ,

AND

RIEMENSCHNEIDER,

AND

INVARIANT THEOR Y

KODAIRA, ETC .

F - R E G U L A R RING S

X

\ DIFFERENTIAL OPERATORS AN D STRONGLY F - R E G U L A R RING S

http://dx.doi.org/10.1090/cbms/088/02

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, i n the late sixties and the seventies, used the idea of reduction t o charac teristic p an d too k advantag e o f the Frobeniu s morphis m i n positiv e characteristic . One o f th e mos t importan t fact s i n thi s proces s i s that th e Frobeniu s morphis m i s flat ove r regular ring s (althoug h perhap s 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 1 969 . Th e sam e year , Peskin e an d Szpir o [PS3 ] prove d 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 wen t o n to us e this t o prov e several of the homologica l conjecture s in characteristic p and fo r ring s essentially of finite typ e over a field o f characteristi c 0 [PS1 ] . Thei r pape r wa s th e first us e o f 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 1 973 , Hochster [Ho2 ] gav e a proo f o f the existenc e o f Bi g Cohen Macaulay modules , i.e . no t necessaril y finitely generate d module s suc h that a given system o f parameters i s a regular sequenc e o n the module . Th e proo f use d wha t 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 positiv e integer s t ,

c((x 1 , ...,x k)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 1 977 . 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 th e irreducibilit y o f cyclotomi c polynomials . Th e proo f o f Dedekin d in 1 85 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

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 referre d t o hi s proo f 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 tigh t 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 sai d 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 al l element s o f / .

The idea of using uniform annihilator s o f cohomology was used b y 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 conjecture 3 states : Let (0.1) G

: 0- G

n

- ^ G n _i ^ ..'

.- G o- 0

be a complex of finitely generate d fre e .R-module s such that th e homolog y ha s 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 . 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 condition s hold : DEFINITION

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 . Define a * = 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 comple x o f free R- modules a s in 0.1 , bi annihilates th e ( 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 wher e ai{Gi) C mGi-i b y splittin g of f extraneou s free exac t sequences . I t i s the n clea r tha t applyin g th e Frobeniu s t o th e comple x gives another comple x a s in 0.1 , but wher e the image s of the ne w maps li e in highe r and highe r power s o f th e 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 power s of the maxima l ideal , an d henc e this idea l i s 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 th e 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 i s a powerful tool , an d n o such proces s i s known i n characteristic zer o except fo r th e Koszu l complex . 3

T h e ne w intersectio n conjectur e i s no w a theore m i n al l characteristics . [Rol-3 ]

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PREHISTOR Y O F TIGH T CLOSUR E

7

The dualizing complex is acyclic exactly when R i s Cohen-Macaulay. Th e ideal s 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 zer o i f th e 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 , i n conjuctio n with th e Probeniu s morphism , shoul d b e a powerfu l tool . At approximatel y th e sam e tim e i n whic h man y o f the homologica l conjecture s were bein g solve d i n equicharacteristic , anothe r importan t theore m wa s prove d 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 regula r Noetheria n kalgebra S ha s a Cohen-Macaula y rin g of invariants. Thei r proo f als o used reductio n to characteristi c p , but ther e wa s n o apparen t connectio n betwee n th e homologica l proofs an d thei r proof . However , thei r wor k focuse d attentio n o n th e 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 i s injective s o R i s reduced). A basically equivalen t 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-pur e 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-spli t varietie s t o stud y singularities . The next importan t even t with regard to this prehistory was a CBMS conferenc e held a t Georg e Mason Universit y i n 1 979 , where Hochster wa s the principa l speake r and concentrate d o n a theore m du e t o 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 1 981 , and i n the same issue of the Michigan Math . Journal , Lipma n and Teissie r [LT ] gav e a partia l extensio n t o rationa l singularities . Th e theore m o f Lipman an d Sathay e state s T H E O R E M 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 an y cas e require s 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 completel y 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 Lipma n an d Sathaye : T H E O R E M 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 th e integra l closur e T o f a finitely generate d R- algebra S. Whil e i t i s true that th e Jacobia n idea l depend s upo n 5 , a s S varie s birationall y 4 T h e pape r o f Hochste r an d Robert s appeare d i n 1 974 . 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 1 97 6 [HR2] .

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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 wha t 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 , a s Chapte r 5 discusses. A fina l piec e o f prehistory relate d method s o f integral closure s t o tigh t closure . In 1 98 6 Itoh [It ] an d Hunek e [Hu4 ] independentl y prove d th e followin g theorem : T H E O R E M 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 nC\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 Ito h di d no t requir e th e rin g t o contai n a field and wa s based o n a study o f the loca l 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 togethe r 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 characteristi c 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 I 2 ther e exist s a n element c G R° suc h that fo r al l n, cz n G I2n. Writ e z — J2 rixi- ^ 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 tru e 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 ha s interestin g consequences , fo r instanc e a proo f o f a versio n o f th e Grauert Reimenschneider vanishin g theorem i n dimension two . Se e Exercises 5.1 2-1 3 for th e proof. Hunek e als o prove d tha t i f R i s F-pure , an d dimensio n d, the n fo r al l ideal s I o f R, 7 d + 1 C I. Thi s proo f als o was essentiall y a tigh t 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 fre e module s preserve s acyclicity . 0.1. Le t i ? b e a Noetheria n loca l rin g o f positiv e characteristi c an d le t G : 0—> G n — ^ G n-\ — • • • •— > G o — * 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 raisin g al l th e entrie s o f matrice s representin g th e map s i n the comple x t o th e pth power ) th e resultin g comple x ha s th e sam e property . 0.2. Le t R b e a regula r loca l rin g o f positiv e 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 th e associate d prime s of 1 ^ fo r al l q = p e. (Hint : us e th e Auslander-Buchsbau m formul a relatin g projective dimensio n an d depth. )

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PREHISTOR Y O F TIGH T CLOSUR E 9

0.3. Le t R b e a regula r loca l rin g o f positive characteristi c an d le t x G R, I C R, I a n ideal . Prov e tha t ( I : R x)M = I [q] : R x q fo r al l q = p e. 0.4. Le t R b e a complete local domain containing a coefficient fiel d k. Le t x\ ,.., Xd be any system of parameters, an d set A — k[[xi, ...,£ 1 , c((#i,...,#£.) # : x fc+i) S ( xi> •••5 xfc)^' i- e- thes e ar e amiabl e parameters.

http://dx.doi.org/10.1090/cbms/088/03

CHAPTER 1

Basic Notion s These lectur e note s ar e intende d a s a guideboo k t o tigh t closur e an d it s appli cations. Thi s treatment i s not intende d t o b e exhaustiv e o f this topic . Th e purpos e of thi s firs t chapte r i s t o introduc e th e basi c definitions , an d stud y ring s wit h th e property tha t ever y idea l i s tightly closed . Most o f th e result s i n thes e note s will b e vali d i n equicha r act eristic, tha t i s for Noetheria n ring s containin g a field. However , th e proof s i n th e note s wil l b e almost exclusivel y i n characteristi c p > 0 . I n particula r i n thes e note s th e phrase , "characteristic p" alway s mean s positiv e an d prim e characteristi c p. W e wil l us e c g' throughou t thes e note s t o denot e a variable powe r o f the characteristi c p. Tigh t closure i s a metho d whic h require s reductio n t o characteristi c p , althoug h muc h of th e theor y ha s no w bee n develope d fo r an y rin g containin g a field. 1 Nonetheless, all th e proof s mus t eventuall y retur n t o th e settin g o f positive characteristi c wher e the Frobeniu s ma p wil l be th e chie f weapon. 2 Th e Frobenius i s the homomorphis m taking r t o r p , wher e p i s the prim e characteristi c o f R. A basi c fac t use d i n thes e notes i s tha t th e Frobeniu s ma p i s flat i n th e cas e tha t R i s regular ; i n fac t thi s characterizes th e regularit y o f R. Defin e R° t o b e th e complemen t o f th e unio n o f all minima l prime s o f a rin g R. Th e definitio n o f tigh t closur e fo r ideal s is : DEFINITION 1 .1 . Le t R b e a Noetheria n rin g o f characteristic p > 0. Le t / b e an idea l o f R. A n element x G R i s said t o b e i n the tigh t closur e o f / i f there exist s an elemen t c G R° suc h tha t fo r al l larg e q—p e^ cx q G I^q\ wher e I™ i s th e idea l generated b y th e qth. powers o f al l element s o f / .

There i s also a definition o f the tight closur e of submodules o f finitely generate d i?-modules, whic h w e will late r need , bu t whos e discussio n wil l b e postpone d unti l Chapter 1 0 . O f particula r interes t ar e ring s i n whic h ever y idea l i s tightly closed . DEFINITION 1 .2 . A Noetheria n rin g i n whic h ever y idea l i s tightl y close d i s called weakly F-regular . A Noetheria n rin g R suc h tha t Rw i s weakl y F-regula r for ever y multiplicativ e syste m W i s calle d F-regular .

It i s certainl y awkwar d t o hav e th e terminolog y "weakly " F-regular . I f tigh t closure commute s wit h localization , the n thes e tw o definition s ar e equivalent . W e believe thi s t o b e true , bu t hav e bee n unabl e t o prov e i t excep t i n specia l cases . Localization i s discusse d i n Chapte r 1 1 . Howeve r w e kno w tha t weakl y F-regula r l

See Appendi x 1 for a discussio n o f tigh t closur e i n characteristi c 0 . The proces s o f reducin g t o positiv e characteristi c i s laboriou s an d depend s upo n an y on e of severa l version s o f th e Arti n Approximatio n Theorem . On e o f th e stronges t suc h version s i s called Genera l Nero n Desingularizatio n (GND) . Thi s theore m state s tha t a regula r homomorphis m A— > B betwee n Noetheria n ring s i s the filtere d inductiv e limi t o f smoot h A-algebra s o f finit e type . 2

10

1. BASI C NOTION S

11

implies F-regula r fo r Gorenstei n ring s (se e Exercis e 4.3) , fo r algebra s whic h ar e finitely generate d ove r a n uncountabl e field o f characteristi c p (du e t o Murthy , se e Theorem 1 2.2) , an d fo r Noetheria n ring s o f characteristi c p o f dimensio n a t mos t three (du e t o Lor i William s [Wil] , se e Exercis e 1 2.1 3) . Tight closur e behave s a s on e woul d wan t a closur e t o behave . T H E O R E M 1 . 3 (BASI C P R O P E R T I E S ) . [HH4 ] Let R be a Noetherian ring of characteristic p and let I be an ideal a) (/*) * = J* . Ifl x CI 2CR, then I{ C 72*. b) If R is reduced or if I has positive height, then x G R is in I* if and only if there exists c G R° such that cx q G I™ for all q—p e. c) An element x G R is in I* iff the image of x in R/P is in the tight closure of (I + P)/P for every minimal prime P of R. d) J * C I, the integral closure 2, of I. e) Further let R be a regular local ring. Then I* = I for every ideal I C R. f) If I is tightly closed, then I : J is tightly closed for every ideal J. P R O O F . Part s a) , b) , an d f ) follo w immediatel y fro m th e definitio n whil e d ) is clea r fro m severa l o f th e varian t definition s o f integra l closure . W e wil l retur n in muc h greate r detai l t o th e relationshi p o f tigh t closur e an d integra l closur e i n Chapter 5 . We wil l prov e c) . On e directio n i s clear : i f x G /* , the n thi s remain s tru e modulo ever y minima l prim e o f R sinc e c G R°. Le t P i , . . . , P n b e th e minima l primes o f R. I f c\ G R/Pi i s nonzero w e can alway s lif t c[ to a n elemen t Q G R° b y using th e Prim e Avoidanc e theorem . Suppos e tha t c[ G R/Pi i s nonzer o an d suc h that c[x q G If fo r al l larg e # , where Xi (respectivel y Ii) respresen t th e image s o f x (respectively I) i n R/P {. Choos e a liftin g c { G R° o f d {. The n ax q G I[q] + Pi fo r every i. Choos e element s U i n al l th e minima l prime s excep t Pi. Se t c = Y2i ciUIt i s eas y t o chec k tha t c G R°. Choos e q' » 0 s o tha t N^ q 1 = 0 , wher e N i s the nilradica l o f R. The n cx q G I[q] + N, an d s o c q x qq G I[qq'\ whic h prove s tha t

xel*. Condition e ) i s a crucia l propert y o f tigh t closur e an d w e wil l giv e a proo f i n a moment . Severa l proof s ar e available , bu t w e wil l d o on e whic h use s anothe r important them e durin g thes e talks , base d o n a classica l criterio n o f Buchsbau m and Eisenbu d [BE ] fo r a fre e comple x t o b e exact . DEFINITION 1 .4 . Le t R b e a Noetheria n ring , no t necessaril y o f characteristi c p, an d le t G . b e a comple x o f finitely generate d fre e i^-module s

0—> G n — • G n-\— > • • • — > Gi — -> • • • — > G\ — • G o — • 0 Denote th e ma p fro m Gi t o G%-\ b y o^ . Le t bi denot e th e ran k o f G^ , wit h th e convention bi — 0 i f i > n o r i < 0 an d le t r\ = E^ =i (—l) t-2 6t, 1 < i < n , whil e r n + i = 0 . Th e n ar e th e uniqu e integer s suc h tha t r n + i = 0 an d r^+ i + n = 3 T h e integra l closur e o f / i n a Noetheria n rin g ca n b e define d t o b e th e se t o f al l element s x such tha t ther e exist s a n elemen t c G R°, th e complemen t o f th e minima l prime s o f R, suc h tha t cxn G I n fo r infinitel y man y n . Wit h thi s definition , par t d ) o f Theore m 1 . 3 i s immediat e sinc e /i9j c I q. I n Chapte r 5 the relationshi p betwee n tigh t closur e an d integra l closur e wil l b e discusse d in greate r detail . A n exampl e o f integra l closur e i s provide d b y th e elemen t xy G (x 2,y2). Mor e generally i f x n G In fo r som e fixed n , the n x G / .

1. BASI C NOTION S

12

bi, 1 < i < n. W e sa y tha t th e comple x G m satisfie s th e standar d conditio n o n rank, if , fo r 1 < i < n, ran k a * = r^ , (equivalently , bi = ran k c^ + i + ran k c^ , 1 < i < n). W e sa y tha t a comple x a s i n (1 .4 ) satisfie s th e standar d conditio n fo r depth (respectively , th e height ) i f th e dept h (respectively , th e height ) o f th e idea l Ii = I ri{oci) i s a t leas t i , 1 < i < n. Not e tha t th e dept h (an d height ) o f th e idea l / i s +o o i f / = R whil e i f / i s prope r it s dept h i s th e lengt h o f an y maxima l Rsequence containe d i n / . Thus , th e dept h o r heigh t conditio n i s satisfied wheneve r Ii i s th e uni t ideal . Recal l tha t i f r = 0 , w e mak e th e conventio n tha t I r(a) i s th e unit ideal . With thi s notation , th e Buchsbaum-Eisenbu d criterio n state s tha t an y suc h free comple x i s acycli c if f i t satisfie s th e standar d condition s o n dept h an d rank . Later thes e note s wil l focu s o n complexe s whic h satisf y th e standar d condition s o n height an d rank . I t wil l tur n ou t tha t suc h complexe s hav e homolog y i n th e tigh t closure o f 0 . Suppose tha t R i s a regula r loca l rin g o f positiv e characteristi c p. W e clai m that th e Frobeniu s ma p F : R— > Ris flat. I t suffice s t o prov e that Tor f (M, 5) = 0 for i > 0 , wher e S — R, bu t it s i?-modul e structur e i s vi a th e Frobeniu s ma p an d where M i s a finitely generate d .R-module . M ha s a finite fre e resolutio n a s i n (1.4). Applyin g th e Frobeniu s (tha t is , tensorin g wit h S ove r R) simpl y raise s th e entries of (any ) matri x i n the resolution whic h gives a map between consecutiv e fre e modules t o th e pth power . I n particula r applyin g Frobeniu s doe s no t chang e th e ranks o f the maps , no r th e depth s o f the ideal s o f minors , sinc e i t onl y change s th e minors u p t o radical . Th e Buchsbaum-Eisenbu d criterio n the n prove s th e ensuin g complex i s exact. Henc e Torf (M, S) — 0 for i > 0 and i t follow s tha t th e Frobeniu s is flat. 4 1 .3 . e) : Le t (R,m) b e a regula r loca l rin g an d suppos e that x £ I* fo r som e idea l / o f R. Ther e exist s a nonzer o elemen t c suc h tha t cxq e /to ' fo r al l q = p e. Henc e c e f] q(l[q] '• x *0- B u t t n e flatness o f Frobenius 5 proves that / M : xq = {I : x )K I f (I : x) ^ R, the n c G f\ mq = 0 , a contradiction . Hence x e I. • PROOF O F THEORE M

Tight closur e i s ver y difficul t t o compute ; indee d tha t i s necessaril y th e case . It contain s a grea t dea l o f informatio n concernin g subtl e propertie s o f th e rin g and th e ideal . Unlik e othe r closures , suc h a s th e integra l closur e o r th e nilradical , there i s n o know n algorith m whic h ca n b e use d t o comput e tigh t closure , excep t in som e ver y specia l cases . I n particular , ther e i s n o compute r progra m whic h ca n compute th e tigh t closur e o f a n idea l i n a finitely generate d Z p -algebra. T o find such a progra m woul d b e a theoretica l breakthrough , a s wel l a s bein g extremel y 4 A mor e elementar y proo f ca n b e don e b y firs t passin g t o th e completion , an d usin g tha t it i s isomorphi c wit h /c[[xi,... , Xd]]- On e ca n furthe r reduc e t o th e cas e k i s perfec t b y usin g flat descent . The n th e flatness o f Frobeniu s i s jus t th e fac t tha t /c[[x 1 / p ,..., x d ] ] i s fre e ove r /c[[xi,..., Xd]]. O n th e othe r hand , th e proo f give n i n th e tex t abov e prove s more : applyin g th e Frobenius t o th e resolutio n o f a modul e o f finite projectiv e dimensio n preserve s exactness . Se e [Hel] fo r a converse . 5 In general , i f R — • S i s a flat homomorphism , an d / C R, x E -R , then ( / :R X)S = (IS :§ x). This follow s b y tensorin g th e exac t sequence , 0 — > R/(I : x) — • R/I —> R/(I,x) —> 0 wit h 5 . Applying thi s remar k wit h th e et h iteratio n o f th e Frobeniu s ma p yield s th e statemen t i n th e text.

1. BASI

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useful computationally . On e idea which would mak e tight closur e easier to comput e would b e th e following . I n a Noetheria n ring , ever y idea l i s th e intersectio n o f ideals primary t o maxima l ideal s (cf . Exercis e 1 .1 ) . I f tight closur e commute d wit h infinite intersection , on e could the n simpl y stud y tigh t closure s o f ideals primary t o maximal ideals . Thi s i s quit e usefu l i n general , bu t especiall y s o i n th e Gorenstei n case. On e proble m wit h infinit e intersection s i s tha t th e c i n th e definitio n varie s with th e idea l / an d th e elemen t x. A n elemen t c suc h tha t fo r al l / an d x G I*, cxq G 1^ fo r al l q i s calle d a test element . Th e existenc e o f suc h element s i s a crucial par t o f tigh t closur e theor y an d i s th e mai n topi c o f th e secon d chapter . They pla y a rol e i n par t (4 ) o f the nex t Theorem . 1 .5 . Let R be a Noetherian ring of characteristic p. 1. Let I be primary to a maximal ideal m. Then (LR m )* = (I*) m. 2. R is weakly F-regular iff every ideal primary to a maximal ideal is tightly closed. 3. R is weakly F-regular iff Rm is weakly F-regular for every maximal ideal m THEOREM

ofR. 4. Assume that R has a test element c. Then the tight closure of an arbitrary ideal I is the intersection of the tight closure of ideals primary to maximal ideals. 5. Suppose that R is local Gorenstein with maximal ideal m. Then R is weakly F-regular (in fact even F-regular) iff an ideal generated by a system of parameters is tightly closed. PROOF. Th e only point t o 1 ) is that i f c G R° an d z G R suc h that cz q G I [q]Rm then cz q G 1^ sinc e I i s ra-primary. Sinc e ever y idea l i n a Noetheria n rin g i s a n intersection o f ideal s primar y t o maxima l ideal s (se e Exercis e 1 .1 ) , Par t 2 ) follow s from 1 ) , while 3 ) i s immediate fro m 1 ) an d 2) . To prov e 4) , writ e / a s th e intersectio n o f ideal s o f th e for m / - f rn n wher e m is maximal. Clearl y I* C n n j T n (/ + ra n)* wher e th e intersectio n i s over al l maxima l m an d al l integer s n . Th e revers e inclusio n als o holds . Fo r i f x fi I* ther e exist s a fixed powe r o f p, sa y g , suc h tha t cx q £ 1 ^ . Choos e a maxima l idea l m suc h tha t the sam e equatio n hold s afte r localizin g a t m . The n lif t bac k t o R t o obtai n tha t cxq $ J M + m n fo r al l n > 0 . Henc e cx q £ (I + ra n)[c?]. Thu s x $ (I + ra n)*. We prove 5) . B y par t (2) , it suffice s t o prov e every ra-primary idea l I i s tightl y closed. Le t (a?i , ...,# 0 such tha t J = (x\, ...,x^) C I. B y part (f ) i n Theorem 1 .3 , I = J : (J : I) is tightl y closed (th e equalit y followin g sinc e R i s Gorenstein) . Thi s prove s tha t R i s weakl y F-regular. Se e Exercis e 4. 3 for th e rest . •

To paraphrase , ever y regula r rin g i s F-regular . A loca l Gorenstei n rin g i s F regular if f th e idea l generate d b y a singl e syste m o f parameter s i s tightl y closed . Of course , th e Gorenstei n cas e include s th e regula r cas e b y applyin g th e statemen t in th e Gorenstei n cas e t o a regula r syste m o f parameters , an d i t i s worthwhil e t o think abou t wha t happene d t o th e flatness o f Frobenius i n this alternat e proo f tha t regular ring s ar e F-regular .

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REMARK 1 .5.1 . On e feel s tha t i f R i s weakly F-regular , the n R[X] shoul d als o be weakl y F-regular . However , on e canno t hop e t o prov e thi s withou t provin g tha t weakly F-regula r implie s F-regular . T o se e this , not e tha t i f (R, m) i s loca l an d P is a prim e idea l o f R suc h tha t di m R/P = 1 , the n ther e ar e maxima l ideal s n o f R[x] lyin g ove r P , e.g . PR[x] + (rx — l)R[x] wher e r i s an y elemen t o f m — P. If w e kne w tha t R[x] i s weakl y F-regular , the n Theore m 1 . 5 (3 ) say s tha t R[x] n is als o weakl y F-regular . A s R[x] n i s faithfull y fla t ove r Rp i t woul d follo w tha t Rp i s weakly F-regular . Inductio n o n di m R/P woul d the n giv e thi s fo r al l prime s P C m . I t the n follow s tha t R i s F-regular (se e Exercis e 1 .1 7) . EXAMPLES1 .6 .

1.6.1. Le t R b e a Noetherian integra l domai n wit h integra l closur e S. Th e integra l closure i s characterize d i n thi s cas e b y bein g wha t i s calle d th e complete integral closure, i.e., th e se t o f all fractions a i n the fractio n field o f R whic h satisfy c(a n) £ R fo r som e nonzer o c £ R an d fo r infinitel y man y n. Usin g this i t i s eas y t o se e tha t fo r ever y principa l idea l (x) , (x)* = xS D R. Similarly i t i s eas y t o prov e (se e Exercis e 1 .5 ) tha t a Noetheria n domai n i s normal if f ever y principa l idea l i n tightl y closed . 1.6.2. Le t (ii , m) b e a local 1 -dimensiona l Noetheria n domai n wit h infinit e residu e field. Ever y idea l I in R ha s th e sam e integra l closur e a s a principa l idea l (x) C /, a so-called reduction o f I. I n thi s case , (x)* = (x) = I D I* D (#)*, so that / * = (x)*. No w us e (1 .6.1 ) t o comput e (#)* . 1.6.3. Le t R = k[X, Y , Z]/(X3 - Y 3 - Z 3 ), wher eA : i s a field of characteristic ^ 3 . Let # , y, z denot e th e imag e o f X, Y , Z i n R. R i s easil y see n t o b e a tw o dimensional Gorenstei n norma l ring . Th e elements y, z for m a homogeneou s system o f parameters . A s w e hav e seen , i n a Gorenstei n ring , i f th e idea l generated b y a single system o f parameters i s tightly closed , the n s o is every ideal. I n particular , i f ther e exist s a n elemen t u whic h generate s th e socl e of a n idea l I generate d b y a s.o.p . suc h tha t u fi I*, the n ever y idea l i s tightly closed . I n thi s case , th e socl e o f (y, z)R i s generated b y x 2 , an d on e can se e tha t x 2 G (y, z)*. Writ e 2q = 3k + i, wher e i i s eithe r 1 or 2 . The n typica l monomia l i n th e expansio n x3-ix2q _ x 3(fc+i) _ ^ 3 _j _ z 3^fc+i ^ of th e latte r expressio n i s y 3j z 3h wit h j + h = k + 1 . I f bot h 3j < q an d 3h < q, 3(f c + 1 ) = 3( j + ft) < 2q < 3k + 2 . Thi s contradictio n prove s tha t x2G(y,z)*.6

1.6.4. Le t R = k[X, Y , Z]/(X2-Y3 Z 5 ), whereA : is a field of positive characteris tic. Le t x , 2/, z denot e th e imag e o f X, Y, Z i n R. R i s easily see n to be a tw o dimensional Gorenstei n norma l ring . Th e elements y, z for m a homogeneou s system o f parameters . I n thi s cas e th e idea l generate d b y y an d z i s tightl y closed. On e onl y need s t o chec k tha t x £ (y, z)*. A s discusse d i n 1 .6.3 , on e then obtain s tha t every idea l i n R i s tightly closed . Thi s rin g ha s a rationa l singularity an d a s i t turn s out , thi s fac t i s closel y relate d t o th e fac t ever y ideal i s tightly closed . 1.6.5 Le t R = k[X, Y , Z]/(X2 - Y 3 - Z 7 ), wher eA : i s a field of characteristi c > 5 . Let x , y, z denot e th e imag e o f X, Y , Z i n R. R i s easil y see n t o b e a tw o dimensional Gorenstei n norma l ring . Th e elements y , z for m a homogeneou s system of parameters. Althoug h the equation has changed only slightly fro m 6 An alternat e wa y of seeing that x 2 G (y, z)* i s to us e the Briangon-Skod a Theore m 5.7 . Sinc e x3 G (y,z) 3, i t follow s tha t x i s integra l ove r (y,z). Applyin g 5. 7 the n prove s tha t x 2 G (y,z)*.

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that o f 1 .6.4 , i t i s no w th e cas e tha t x £ (y, z)*. A simpl e argumen t a s i n 1.6.3 prove s this . 1.6.6 Le tA : be a fiel d o f positiv e characteristi c p, an d le t R b e th e subrin g o f th e polynomial rin g S = k[Xi, ...,Xd] generate d b y al l monomial s o f degre e n for som e fixe d n. The n R i s F-regular . Ever y idea l i s tightl y closed , an d moreover thi s i s tru e i n ever y localization . Thi s fac t i s tru e becaus e R i s a direc t summan d o f 5 a s a n i?-module . I f / C R an d x £ /* , the n sinc e R C 5 , x G (IS)* — IS, th e latte r equalit y followin g sinc e S i s regular . However IS P\ R — I a s R i s a direc t summan d o f 5 , whic h force s x E I. This ide a i s pursued i n muc h greate r detai l i n Chapte r 3 . One of the basic questions which we will try to answer in the first fe w chapters is, 'where doe s tight closur e com e from ' ? Th e example s abov e d o little t o answe r thi s question. W e kno w o f essentiall y fou r way s i n whic h tigh t closur e arises : throug h contractions fro m finit e extensions , throug h th e integra l closure s of powers of ideals (see the fifth chapter) , through the failure o f the ring to be Cohen-Macaulay (briefly , 'colon-capturing'), throug h th e persistenc e o f tigh t closur e (se e chapte r two) , an d on to p o f thes e four , othe r les s obviou s consequence s o f puttin g al l fou r together . To clos e th e firs t chapter , we'l l handl e th e firs t o f thes e ways : T H E O R E M1 . 7 ( T I G H

T CLOSUR E FRO M CONTRACTIONS) . Let S

c

T

be

a

module-finite extension of Noetherian domains of positive characteristic p. Let ICS bean ideal Then (IT)* n S C J* . P R O O F . Le t w e (IT)* n S. W e ca n choos e a n 5-linea r ma p : T— » S suc h that 0(1 ) = d G S - {0 } an d w e ca n choos e c G T - {0 } suc h tha t cw q G (IT)^ for al l q. 7 W e ca n choos e a nonzer o multipl e o f c in 5 , an d s o we may assum e tha t c G S — {0}. I t follow s tha t fo r al l q, cw q i s a T-linea r combinatio n o f element s o f J ^ . Applyin g th e ma p , w e find tha t dcw q G 1^ fo r al l g , which show s tha t w i s in th e tigh t closur e o f / . •

One immediat e corollary 8 o f thi s theore m i s that i f S i s any integra l extensio n of a Noetheria n domai n o f positiv e characteristic , R, an d / i s a n idea l o f R, the n IS D R C I*. Fo r an y elemen t i n IS H R i s i n IT D R fo r som e finit e extensio n T of R, T C 5 . The n appl y Theore m 1 .7 . For a domain R b y R+, th e absolute integral closure o f i?, we mean th e integra l closure o f R i n a n algebrai c closur e o f it s fractio n field . Th e discussio n o f th e preceding paragrap h prove s tha t fo r an y ideal , IR + D R C I*. I t i s on e o f th e 7 Hochster ha s develope d th e ide a o f thi s proo f considerabl y i n hi s pape r [Ho8 ] wher e h e introduces th e concep t o f 'soli d closure' . 8 Another usefu l corollar y o f Theore m 1 . 7 wa s notice d b y Kare n Smith . Le t R b e a complet e local domai n o f characteristi c p wit h a give n syste m o f parameters , xi, ..., x^. Choos e a coefficien t field K an d le t A = K[[x\, ...,Xd]] C R b e th e complet e subrin g generate d b y th e Xi ove r K. O f course, A i s a regula r loca l ring . Choos e an y z G ((#i , ...,Xd)R)*. Le t B = A[z\. Th e inclusion s A C B C R prov e tha t B i s a complet e loca l rin g o f th e sam e dimensio n d a s bot h A an d R. Moreover, B i s Gorenstein , i n fac t i s isomorphi c wit h J 4 [ [ Z ] ] / ( / ) fo r som e nonzer o powe r serie s / . Theorem 1 . 7 show s tha t z £ {(x\, ...,Xd)R)* n B C ((x\, ..., Xd)B)*. Thi s i s rathe r remarkabl e fo r it show s tha t th e equation s whic h caus e a n elemen t t o b e i n th e tigh t closur e o f paramete r ideal s can b e reduce d t o simpl y on e equatio n / = 0 . T o stud y th e tigh t closur e o f paramete r ideal s i t suffices t o stud y the m ove r hypersurfac e rings . Whic h powe r serie s f(Z,x±, ...,Xd) caus e z t o b e in th e tigh t closur e o f th e idea l generate d b y th e xi afte r killin g / ? Se e (1 .6 ) fo r som e examples .

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important ope n question s o f tight closur e whethe r equalit y alway s holds . Th e bes t that i s presently know n i s a resul t du e t o Kare n Smit h [Sm4] : i f / i s generated b y parameters i n a n excellen t loca l domai n o f characteristi c p , the n IR+ n R — I* . I shal l discus s thi s poin t a t muc h greate r lengt h i n Chapte r 7 . Th e rin g i? + ha s rather amazin g properties , e.g. , th e su m o f an y tw o prim e ideal s i s either prim e o r the whol e ring . Th e stud y o f suc h absolut e integra l closure s i s closel y tie d t o th e theory o f tigh t closure .

Exercises 1.1 Le t R be a, Noetheria n ring . Prov e tha t ever y idea l o f R i s a n intersectio n of ideal s primar y t o maxima l ideals . 1.2 Le t (R,m) b e a Cohen-Macaula y loca l Noetheria n rin g o f characteristi c p , and suppos e tha t x\,...,xt i s par t o f a syste m o f parameters . Assum e tha t (#1, ...,#£) i s tightl y closed . Prov e tha t (x™ , ...,x™) i s tightl y close d fo r al l n> 1 . 1.3 Prov e part s c ) an d f ) o f Theore m 1 .3 . 1.4 Le tA : b e a field o f positiv e characteristic , an d se t R = /c[[£ 5,£7, t11]]. Fin d the tigh t closur e o f th e idea l generate d b y t 7 an d t 1 1 . 1.5 Le t R b e a Noetheria n rin g o f characteristi c p suc h tha t n o prim e i s bot h minimal an d maximal . I f ever y heigh t on e principa l idea l i s tightl y closed , prove tha t R i s normal . 1.6 Le t R b e a Noetheria n domai n o f characteristi c p an d le t S b e a n extensio n ring. Suppos e ther e exist s a n .R-modul e homomorphis m : S— > R suc h that 0(1 ) = c ^ 0 . Prov e tha t fo r ever y idea l I C R, IS HRC I*. 1.7 Verif y th e statement s i n Exampl e 1 .6.5 . 1.8 Le t R = k[X,Y,Z,A,B,C]/(Xv AY P - BZ? - CY p~lZp~lX). Prov e that x G (y, z)*. (Hint : Inductivel y prov e that x q G (yq, z q, (yz) q~1 x). A s it turns out , thi s typ e o f equatio n i s crucia l i n th e stud y o f absolut e integra l closures.) 1.9 Le t RC S be Noetheria n domain s o f characteristi c p suc h tha t 5 i s weakl y F-regular an d ever y idea l o f R i s contracted fro m S. Prov e tha t R i s weakly F-regular. 1.10 Le t k be a field o f characteristi c p an d le t S = fc[xi, ...,x n ,2/i, ...,y m 1 be a polynomial rin g ove r S i n th e variable s Xi an d yi. Le t R b e th e subrin g o f S generate d ove r k b y th e nm product s xiyj . Prov e tha t R i s F-regular . 1.11 Le t R be a rin g containin g th e rationa l numbers , an d le t / b e a n idea l o f R. One migh t tr y t o generaliz e th e ide a o f tigh t closur e t o characteristi c 0 b y defining 1 ^ t o b e th e idea l generate d b y al l th e nt h power s o f element s o f / . Prov e tha t 1 ^ = J n , an d a definitio n simila r t o tigh t closur e woul d onl y give th e integra l closure . 1.12 Suppos e that R i s a Noetherian rin g of characteristic p and / C R i s a tightl y closed idea l withou t embedde d primes . Prov e tha t th e primar y component s of / ar e als o tightl y closed . W e don' t kno w i n genera l i f a tightl y close d ideal i s a finite intersectio n o f primar y tightl y close d ideals . Thi s proble m is closel y relate d t o th e proble m o f whethe r tigh t closur e commute s wit h completion.

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1.13 Le t Rhea loca l Cohen-Macaulay rin g and let x\ ,..., x t b e part o f a system of parameters. Prov e th e followin g simpl e exampl e o f what wil l late r b e calle d "colon-capturing": ( x ? \ ..., A which send s d xlq t o 1 . 1

See 2.1 . 1 below . 18

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9

Next suppos e tha t A i s a regula r domai n bu t no t necessaril y local . Takin g pt h roots commute s wit h localization , s o tha t A l/q wil l b e projectiv e ove r A fo r al l q — pe. Fi x a maxima l idea l m o f A an d a nonzer o elemen t d £ A. Ther e wil l be a powe r o f p , sa y q — q(m), dependin g upo n ra, suc h tha t d £ m\ q\ an d s o dl/q $. mA l/q. On e want s a unifor m q working fo r al l maxima l ideal s (se e Exercis e 11.4 for a mor e genera l result) . B y th e paragrap h above , ther e i s a homomorphis m from ArL q t o A m sendin g d 1 ^ t o 1 . Clearin g denominator s on e sees that ther e i s an element r m £ m suc h tha t ther e i s a n ^-linea r ma p m : ArJ? ~^ A rm sendin g dl'q t o 1 . Th e idea l generated b y al l such r m i s not containe d i n any maxima l idea l so tha t ther e ar e finitely man y o f them , sa y r m i , ...,r mfc , whic h generat e th e uni t ideal. Se t q = max{g(rai)} . Le t m b e a n arbitrar y maxima l idea l of A. Som e r mi i s not containe d i n ra, sa y r = r mi. A s there i s an ^4 r-linear ma p fro m j4 r m — > Ar sending d 1 /^™1 ^ t o 1 , a fortiori ther e i s suc h a ma p fro m Am — • Am. I n particular, d £ ra^, whic h prove s th e existenc e o f a unifor m q. The existenc e o f suc h a q prove s tha t fo r eac h maxima l idea l m o f i , ther e is a n elemen t r = r m an d a n ^4 r-linear ma p fro m A r— > A r sendin g d 1 ^ t o 1 . For eac h suc h r , ther e i s a powe r r Nr, suc h tha t ther e i s a n A linea r ma p r fro m Al/q— > A sendin g d llq t o r Nr. Ther e exist s a finite numbe r o f suc h r generatin g the uni t ideal , s o w e ma y expres s 1 = ^$irf^ , wher e N i s take n large r tha n al l Nri. Takin g = X ^ 5 ^ n give s a n ^4-linea r ma p takin g d l/q t o 1 . T H E O R E M 2.1 . [HH3 , T h e o r e m 3.4 ] Let R be an F-finite reduced ring of characteristic p. Let c be any nonzero element of R such that R c is regular. Then c has a power which is a test element.

P R O O F . Sinc e R c i s regular, th e discussion preceedin g this theorem prove s tha t for ever y nonzer o elemen t d G R ther e i s a sufficientl y hig h powe r o f p, sa y Q , suc h that ther e exist s a n j? c-linear ma p fro m R c t o R c sendin g d 1 ^ t o 1 . Liftin g 1 back t o R, on e obtain s a n ^-linea r ma p fro m R ^ t o R sendin g d 1 /® t o a powe r of c. Takin g d = 1 yields a n i£-linea r ma p fro m R 1 /® t o R sendin g 1 to c N fo r some N. Th e embedding o f R 1 ^ int o R 1 ^ compose d wit h thi s R linea r ma p yield s an ^-linea r ma p fro m R 1 ^ t o R sendin g 1 to c N. Relabe l thi s powe r o f c a s c. Then ther e i s an i?-linea r ma p 0 fro m R}/ p t o R sendin g 1 to c. W e clai m tha t fo r any suc h c , c 2 i s a tes t element , excep t i n characteristi c 2 , wher e c 3 wil l b e a tes t element. Let J b e a n arbitrar y idea l o f R, an d le t z G I*. Ther e i s a n elemen t d G R, not i n an y minima l prim e o f R suc h tha t fo r al l g , dz q G 1^. Fro m th e result s o f the paragraph s above , ther e i s a powe r o f p , sa y q', an d a n i?-linea r ma p a fro m Rl/q - * R sendin g d xlq t o c N fo r some N. I n this case, c Nzq G I{q] fo r all q. Simpl y take g't h root s o f th e equatio n dz qq G I [qq] t o obtai n tha t d xlq z q G I^R 1 ^'. Applying a yield s tha t c Nzq G J^ fo r al l q. Th e proble m i s w e mus t prov e tha t this powe r N ca n b e chose n independentl y o f the elemen t z an d th e idea l / . Choose N leas t wit h th e propert y tha t c Nzq G I [q] fo r al l q. Writ e N = p([N/p\) + i. Takin g pth root s yield s tha t c ^N^^l^zq G I^R1^ fo r al l q. Henc e j[q] Ri/P for a l l q Applyin g 0 w e obtain tha t c ^N/p^2zq G I[q] fo r al l clN/P]+izq e q. A s N wa s chose n least , w e must hav e tha t [N/p\ + 2 > N. I t easil y follow s tha t in od d characteristics ,T V < 2 and i f p — 2 , N < 3. •

20 2

. TES T ELEMENT S AND TH E PERSISTENC E O F TIGH T CLOSUR E

A ring R i s F-finite i f R i s essentially o f finite typ e over a perfect fiel d K , o r if R is complete wit h perfec t residu e fiel d K. I n fact , al l on e need s i n bot h case s i s tha t Kl/p b e finit e ove r K. Thus , thi s i s no t a ver y restrictiv e hypothesis . Th e mor e general theore m o f th e existenc e o f tes t element s fo r excellen t loca l ring s follow s from Theore m 2. 1 by passin g t o th e complet e cas e an d expandin g th e residu e fiel d to mak e i t finit e ove r it s pth powers . However , thi s i s difficul t sinc e on e mus t b e able t o contro l th e fiber s o f thi s bas e chang e wel l enoug h t o contro l wha t happen s to tigh t closures . 2.1 .1 . Th e proof show s that i f there i s an .R-linea r ma p / : Rl/p— > R sending 1 to c , where R c i s regular an d R i s F-finite, the n c 2 i s a test elemen t unles s the characteristi c i s 2 , i n whic h cas e c 3 i s a tes t element . I n particula r th e sam e is tru e fo r Rw fo r a n arbitrar y multiplicativel y close d se t W i n R; Rw i s stil l F finite, (Rw)c i s regular, an d the map fw provide s the required map as the Frobeniu s commutes wit h localization . I t follow s tha t ever y elemen t c such tha t R c i s regula r has a power whic h i s a stable test element , i n the sens e that i t i s also a test elemen t for ever y localization . Moreover , Kun z [Kul ] ha s show n tha t i f R i s F-finit e the n it i s excellent . Henc e i f R i s F-finit e an d loca l an d S i s it s completion , the n S i s also F-finit e an d i f R c i s regular, s o i s S c. Th e completio n o f th e ma p / the n give s a ma p fro m S 1 ^ t o S sendin g 1 to c k fo r som e fixe d powe r o f c a s i n th e proo f of Theore m 2.1 . , and the n th e appropriat e powe r o f c will b e a tes t elemen t fo r S as well . Thi s (fixed ) powe r o f c then work s fo r ever y localizatio n o f R an d fo r th e completion o f ever y localizatio n o f R. Suc h tes t element s w e call completely stable. REMARK

It i s convenien t t o hav e a concep t o f tes t element s whic h work s equall y wel l when th e ring s ar e no t necessaril y reduced . Th e existenc e o f a tes t elemen t force s the rin g to hav e no nilpotents sinc e nilpotents ar e in the tigh t closur e of every ideal . The followin g definitio n cover s th e non-reduce d case : 2.2 . 2 Le t R b e a Noetheria n rin g o f characteristi c p an d le t q' — p fo r som e intege r e ' £ N . W e say tha t c £ R° i s a q'-weak test element i f fo r every idea l / C R, a n elemen t x £ R i s i n I* i f an d onl y i f cx q £ / M fo r al l q>q'. DEFINITION e

Suppose tha t c £ R° i s a tes t elemen t fo r R red- Choos e q' suc h tha t N^ q 1 = 0 where N i s the nilradical of R. The n c is a (/-weak tes t elemen t fo r R. I n particular , a loca l excellent rin g always ha s a wea k tes t element . W e will particularly nee d th e concept o f wea k tes t element s i n Chapte r 1 2 . The theor y o f tes t element s als o ha s a n importan t applicatio n i n provin g th e property o f persistenc e o f tigh t closure . Le t R b e a rin g an d suppos e tha t I i s an idea l wit h z £ /* . I f w e ma p R t o anothe r rin g S vi a a homomorphis m , is (j){z) £ (IS)* ? I n th e definitio n o f tigh t closure , th e tes t elemen t c migh t b e mapped int o som e minima l prim e o f 5 , o r eve n b e mappe d t o 0 . Fo r instance , i f J is th e idea l generate d b y tes t elements , the n th e persistenc e o f tigh t closur e fro m R t o R/J i s unclear. Th e nex t theore m [HH9] , Theore m 6.24 , prove s tha t i n mos t cases persistenc e holds . T H E O R E M 2. 3 (PERSISTENC E O F T I G H T C L O S U R E ) . Let

(j> : R - > S

be

a

homomorphism of Noetherian rings of characteristic p. Let I be an ideal of R and 2 In th e definition s i n thi s section , w e ar e definin g tes t element s an d varian t notion s fo r ideals . In fact , the y ar e properl y define d fo r module s also . However , unde r th e assumptio n tha t ou r ring s are locall y approximatel y Gorenstein , an y tes t elemen t fo r ideal s i s also a tes t elemen t fo r module s [ H H 4 , (8.1 5)] .

2. TES T ELEMENT S AND TH E PERSISTENC E O F TIGH T CLOSUR E 2 1

let w G R be an element in I*. Assume either that R is essentially of finite type over an excellent local ring, or that R red is F-finite. Then (w) is in the tight closure of IS. P R O O F . I f ther e i s a counterexampl e t o th e theorem , ther e i s on e i n whic h S is a domain , fo r i f (/>(w) i s not i n th e tigh t closur e o f IS, thi s wil l remain tru e whe n S i s replace d b y S/P fo r a suitabl e minima l prim e P. Thus , w e ma y assum e tha t S i s a domain . Le t Q = Ker (R— > S). Then w e may replac e S b y R/Q a s well . Fo r i f tigh t closur e i s preserved whe n we pas s t o R/Q, i t wil l als o b e preserve d whe n w e pas s t o 5 sinc e R/Q embed s in S. (Th e definitio n o f tigh t closur e immediatel y prove s ther e i s no proble m wit h Theorem 2. 3 as long as a tes t elemen t doe s no t g o into a minima l prim e o f S unde r 0, whic h i s certainl y th e cas e i f th e ma p i s a n injectiv e ma p o f domains) . Thus , there i s n o los s o f generalit y i n supposin g tha t S = R/Q fo r a suitabl e prim e Q of R. Le t Q = Qh 3 Qh-i 3 • • • 2 Qo b e a saturate d chai n o f prim e ideal s of R descendin g fro m Q suc h tha t Q o i s a minima l prim e o f R. W e shal l prov e by inductio n o n i tha t tigh t closur e i s preserve d whe n w e pas s fro m R t o R/Qi, 0 < i < h. Fo r i = 0 thi s i s clear , fo r tigh t closur e i s alway s preserve d whe n on e kills a minima l prime . T o carr y throug h th e inductiv e step , w e ma y replac e R b y

R/Qi-iTo complet e th e proo f o f i) , i t suffice s t o sho w tha t i f R i s a domai n an d Q is a heigh t on e prim e idea l o f R, the n tigh t closur e i s preserve d whe n w e pas s t o S = R/Q. T o see this, le t R f b e the integra l closur e o f R i n it s fraction field (whic h is module-finit e ove r R, sinc e R i s excellent) , an d le t Q' b e a prim e idea l o f R! which lie s over Q, s o that R/Q — » R'/Q' i s injective an d module-finite . Now , tigh t closure i s obviousl y preserve d whe n w e pas s fro m R t o R' D R. Moreover , sinc e R' i s excellen t norma l an d Q' i s heigh t on e (s o tha t RQ, i s regular) , ther e i s a n element c € R' — Q' suc h tha t R' c i s regular . Afte r replacin g c by a powe r w e se e that w e ma y assum e tha t c is a tes t elemen t fo r R' no t i n Q'. I t follow s tha t tigh t closure wil l b e preserve d whe n w e pass fro m R' t o T = R f/Q'. Thus , th e imag e o f w i n T i s i n th e tigh t closur e o f IT. T o finish th e proof , w e mus t sho w tha t thi s implies tha t th e imag e o f w i n S i s i n th e tigh t closur e o f IS. (Here , 5 C T i s a module-finite extensio n o f domains. ) Thi s follow s fro m Theore m 1 .7 . • As I noted a t th e beginnin g o f this section , ever y reduce d rin g o f characteristi c p which i s essentiall y o f finite typ e ove r a n excellen t loca l rin g ha s abundan t tes t elements i n th e sens e tha t i f Rd i s regular , the n d ha s a powe r whic h i s a tes t element. A n ope n questio n is : QUESTION 2. 4 (EXISTENC E O F T E S T ELEMENTS) . Le t R be a reduce d ex cellent rin g o f finite Krul l dimensio n an d o f characteristi c p. Doe s R hav e a tes t element? 3 Eve n more , i f Rd i s regular, doe s d have a power whic h is a test element ?

Ian Aberbac h [Ab2 ] ha s show n tha t i f R i s a n excellen t domai n an d c i s a nonzero elemen t o f R suc h tha t R c i s regula r an d suc h tha t c(IR}/ p D R) C I fo r 3 T h e existenc e o f suc h a tes t elemen t ha s implication s outsid e o f tigh t closure . Fo r instance , their existenc e woul d impl y tha t R satisfie s th e unifor m Artin-Ree s condition . Se e Chapte r 1 1 fo r details.

22

2. TES T ELEMENT S AN D TH E PERSISTENC E O F TIGH T CLOSUR E

all ideal s primar y t o a maxima l ideal , the n c 3 i s a tes t element . H e prove s tha t excellent domain s o f dimensio n 2 have abundan t tes t elements . Exercises 2.0 Le t R b e a Noetherian rin g of characteristic p. Le t / b e an idea l of R. Prov e that a n elemen t c £ R° i s a tes t elemen t if f cl* C I fo r ever y idea l I C R (use Exercis e 1 .1 4) . 2.1 Le t (R,m) b e a Noetheria n loca l rin g o f characteristi c p. Assum e tha t R has a tes t element . Prov e fo r ever y idea l / tha t J * = C) n(I + ra n)*. 2.2 Le t (R,m) b e a Noetheria n loca l rin g o f characteristi c p. Assum e tha t R has a test element . Le t / b e an idea l of R an d suppos e tha t u £ R suc h tha t for som e deR°,du q £ (/M) * fo r infinitel y man y q. Prov e tha t u G I*. The nex t fe w exercise s dea l wit h th e existenc e o f tes t element s i n a stronge r form. Le t R b e a complet e loca l domai n o f characteristic p. Sa y a nonzer o elemen t c £ R i s a flattener i f ther e exist s a complet e regula r loca l rin g A C R suc h tha t R i s module-finit e an d genericall y smoot h ove r A (th e latte r conditio n mean s tha t the fractio n field o f R i s separabl e ove r th e fractio n field o f A), an d fo r al l g , cRl^qCA^q[R}. 2.3 Le t R b e a complet e loca l Noetheria n domai n whic h i s a module-finit e an d generically smoot h extensio n o f a complet e loca l regula r domai n A. Prov e that A^ q[R] ^ A xlq 1 . B y (5.8.2 ) thi s latte r idea l i s contained i n (af , . . ^ a ^ ) ™ * 1 / ^ - 1 ) . Pu tA T = g = p e . W e obtai n tha t cz * G (j™+i)M b y ignorin g th e ter m / i V ( n x ) i n the containmen t o f the abov e line . Henc e z G (J^+1)* a s claimed . D The proo f abov e i s amazin g i n it s simplicity . I f R i s regula r the n ever y idea l is tightly close d an d Theore m 5. 7 become s th e usua l Briangon-Skod a theorem . O f course somethin g i s los t sinc e an y tigh t closur e proo f mus t assum e tha t th e rin g contains a field. However , th e proo f o f Theorem 5. 7 suggests severa l improvements . In th e proo f w e simply "forgot " abou t th e extr a ter m / M ^ - i ) whic h occur s a s th e coefficients o f wha t w e want . Notic e tha t J9( n _ 1 )/9 C /M . B y usin g thi s extr a information a bette r resul t ca n b e obtained . Thi s bette r resul t become s iterativel y stronger a s on e neve r seem s t o los e th e extr a coefficient s / A r ( n - 1 ) . Thi s analysi s leads t o th e followin g definition . DEFINITION 5.9 . [AH2 ] Le t R b e an y commutativ e Noetheria n rin g an d le t J C I b e tw o ideal s o f R. Th e coefficient ideal o f I relativ e t o J , a(I , J ), i s th e largest idea l b of R fo r whic h lb — Jb. Whe n th e ideal s I an d J ar e understoo d we will writ e simpl y a . REMARK 5.1 0 . Th e idea l a(7 , J) exists . Furthermore , provide d a(7 , J) con tains a nonzerodivisor , i t follow s tha t / an d J hav e th e sam e integra l closur e (se e Exercise 5.20) .

When / i s an m-primar y idea l w e can comput e a algorithmically. Thi s compu tation i s base d o n th e observatio n tha t a(/ , J) C J : I. Le t a\ = (J : I). Defin e a n + i inductivel y a s a n + 1 = Ja n : /. Befor e statin g th e proposition , w e recal l tha t the reduction number o f / wit h respec t t o J , rj(I), i s th e leas t intege r r = rj(I) such tha t / r + 1 = P J . PROPOSITION 5.1 1 . Let I be an ideal, let J C I be a reduction and let a n be defined as above. Let r = rj(I). Then

1. F c

a

,

2. a C a n and a n + i C a n for all n, 3. If R is local and I is m-primary then the sequence a\ D ci2 2 • * • stabilizes and a = a n for n ^> 0 . PROOF. (1 ) i s clea r sinc e I rI = V J. (2 ) follow s b y inductio n sinc e a n + i / C J&n Q J a n _ i , henc e a n +i C a n , an d ai — a J C a n J implie s a C a n +i. We no w prov e (3) . Sinc e I i s primar y t o th e maxima l ideal , R/I r i s Artinian , hence th e descendin g sequenc e a\/I r 2 ^2IP 2 • * * stops, s o 0 1 5 ci 2 2 • • • doe s too. Suppos e tha t a n = a n +i. The n Ia n = / a n + i C Ja n C 7a n . Henc e Ia n — Ja n, so a n C a . Bu t a C a n b y (2) , s o 0 = a n . •

5. INTEGRAL CLOSURE A N D TIGHT CLOSURE

42

REMARK 5.1 2 . Propositio n 5.1 1 show s tha t whe n / i s m-primary the n calcu lating a(7 , J) ca n be done easil y usin g a computer progra m suc h a s MACAULAY .

The idea l a(7 , J) ha s another nic e propert y relativ e t o /, whic h wil l be crucial. T H E O R E M 5.1 3 . Let (R, m) be a regular ring of characteristic p and dimension d having an infinite residue field, and let I be an m-primary ideal with minimal reduction J. Then for all q = p e

lW-i)q C o ( I , J )

[


—1 , jd+w c r + 1 a ( I , J ) . The proo f wil l onl y b e given i n characteristic p > 0. P R O O F I N CHARACTERISTI C p. Th e case w = — 1 follow s immediatel y fro m Theorem 5.1 3 with q = p° — 1. (Her e w e interpret J° = R.) Therefor e w e assume for th e remainder o f the proof i n characteristic p tha t w > 0. Let x e I d+W = J d+W. Choos e d G R° suc h tha t dx m G j( 0. Then fo r q—p e

dxq e j( d+w^q c (jM)™+ij(d-i) g c (jW)

u,+1

a(J, J ) w ,

using Remar k 5.8(2 ) an d Theore m 5.1 3 . Sinc e (jM)™* 1 = ( J x G ( Jw + 1a ) * - J™ +1 a. D

w+1

) ^ W e hav e

EXAMPLE 5.1 5 . W e restrict t o the case i n which R i s a 2-dimensiona l regula r local ring . Usin g wor k o f either Lipma n [L2 ] or Swanson an d Huneke [HSw] 4 , one obtains tha t i f I i s a n integrall y close d m-primar y idea l wit h minima l reductio n J, the n I (J : /) = J (J : I). I n particular , i n th e notatio n o f Propositio n 5.11, J : / = d i = ei 2 = .. . = a(7 , J ). Eve n mor e precisely , I ca n be written a s the n by 4 This wor k studie s th e core of an ideal, namel y th e intersection o f all reductions o f a given ideal. Thi s concept wa s introduced b y Judith Sall y and Rees [RS ] in their stud y of the Briangon Skoda theorem .

5. INTEGRAL CLOSURE A N D TIGHT CLOSURE

43

n minor s o f a n b y n + 1 matrix A b y th e theore m o f Hilber t an d Burch . Fo r an y minimal reductio n J o f / , th e coefficien t idea l i s th e idea l generate d b y th e n — 1 by n — 1 minors o f A. I n particula r th e coefficien t idea l a(7 , J) i s independen t o f J. A simpl e exampl e i s provide d b y th e nt h powe r m n o f th e maxima l idea l o f R. In thi s case , th e coefficien t idea l wit h respec t t o an y minima l reductio n i s exactl y ran_1. Thes e result s i n dimensio n tw o sugges t tha t a(7 , J) coul d b e independen t of th e minima l reductio n J , a t leas t i n th e cas e tha t / i s integrall y closed . REMARK 5.1 6 . Recentl y Lipma n [L2 ] ha s prove d (usin g completel y differen t methods) closel y relate d result s fo r regula r loca l ring s b y introducin g th e adjoin t of a n ideal . Hi s method s sho w th e existenc e o f a n ideal , calle d th e adjoin t o f / , denote d adj(i) , suc h tha t adj(/ d _ 1 ) C a(7 , J) fo r an y integrall y close d idea l / with minima l reductio n J havin g d generators, an d suc h tha t I d+W C adj(/ d+ ™) = J™ + 1 adj(/ d _ 1 ). I t is not known whether adj(/ d _ 1 ) = a(7 , J) i n general for integrall y closed ideal s i n regula r rings . Lipma n need s t o b e i n a situatio n wher e h e ca n apply certai n vanishin g theorem s whic h ar e no w know n onl y fo r ring s essentiall y of finit e typ e ove r th e comple x numbers , fo r analyti c loca l rings , o r i n dimensio n two. However , Lipma n ha s pointe d ou t tha t th e adjoin t behave s wel l wit h respec t to completio n o f excellen t regula r loca l ring s provide d enoug h desingularization s exist: on e need s tha t fo r ever y idea l I C R ther e exist s a desingularizatio n o f Spec(i^) o n whic h / become s invertible . Henc e Lipman' s result s probabl y wor k fo r any excellen t loca l rin g containin g a fiel d o f characteristi c zero . If R i s a regular loca l ring o f dimension two , then Lipma n prove s tha t adj(7 ) = J : / fo r an y minima l reductio n J o f an integrall y close d idea l I. B y Exampl e 5.1 5 , it follow s tha t i n thi s case , J : I — adj(7 ) = a(7 , J ).

One coul d hop e tha t Theore m 5.1 4 remain s tru e fo r arbitrar y loca l ring s i f one replace s th e idea l J w+la(I,J) b y it s tigh t closure . However , thi s i s no t true . For instanc e i n th e exampl e x 3 — y 3 — z 3 = 0 , w e sa w tha t x G (y,z) = (x,y,z). We coul d hop e tha t x 2 G ((t/, z)a((x, y, z), (y , z)))*. A calculatio n o n MACAULA Y proves tha t i n th e recursiv e calculatio n o f a((x , y, z), (y , z)) on e get s a i = (x 2 , y, z) and ci 2 = CI 3 = .. . = &n = (x,y,z) 2. Bu t x 2 £ ((?/ , z)(x,y, z) 2)*, sinc e x 2 i s not eve n in th e integra l closur e o f th e latte r ideal . There ar e man y ope n question s concernin g th e coefficien t ideals . I f I i s inte grally closed , i s a(7 , J) independen t o f th e minima l reductio n J o f / ? I s a(I , J) — adj(/ d _ 1 ) fo r integrall y close d ideal s / an d minima l reduction s J i n regula r loca l rings ? Doe s the construction o f the coefficien t idea l a s in Proposition 5.1 1 (3 ) wor k if / i s no t m-primary ? I f J i s integrall y close d an d J i s a reductio n o f / i s a(7 , J ) also integrall y close d ? Exercises 5.1 Le t R = C[[Xi , ...,X n ]], an d le t / € R. Le t d(f) denot e th e idea l generate d by th e partia l derivativesua f / . Prov e tha t / € md(f). 5.2 Le t R — C[[Xi,... , Xn ]], and le t / G R. A n evolution o f R i s a prime idea l P of R togethe r wit h a minimal generato r / o f P suc h that d(f) C P. Suppos e that / G md(f) fo r ever y nonuni t / G R. Prov e tha t R ha s n o evolutions . It i s a n ope n proble m whethe r / G md(f) fo r ever y nonuni t i n R. 5.3 Le t R = k[X, Y, Z]/(X 3 + Y 3 + Z 3 ). Prov e tha t th e regula r closur e o f (y , z) contains x , bu t th e tigh t closur e o f (y , z) doe s no t contai n x.

44

5. INTEGRAL CLOSURE A N D TIGHT CLOSURE

5.4 Le t R b e a Noetheria n rin g le t J — (ai,..., an) b e a n idea l o f R. Prov e tha t for al l h > 0, J n^+wh c ( a ^ ...a£)™ +1 j M " - i ). 5.5 Giv e a n exampl e o f a n idea l / i n a Noetheria n loca l rin g R whic h i s tightl y closed, bu t whos e squar e i s no t tightl y closed . 5.6 Le t (i? , m) b e a d-dimensiona l regula r loca l ring . Prov e tha t th e coefficien t ideal o f m n wit h respec t t o a n arbitrar y minima l reductio n o f m n contain s m (d-i)(n-i) j n £a c t - ^ is e q U a j t o ^jg icl eal5 a n d i s consequently independen t of th e minima l reduction . 5.7 Le t (R,m) b e a d-dimensiona l regula r loca l ring . Le t i " = (#i , ...,# n ) wit h n > d + 1 . Afte r changin g th e generator s w e ma y assum e tha t th e first d element s for m a reductio n o f / . I t i s well-know n tha t th e loca l coho mology Hj'(R) = 0 . Thi s loca l cohomolog y ca n b e though t o f a s th e di rect limi t o f th e cycli c modules , R/(x t1 ,...,xtn). I n particular , an y elemen t u G R/(x\ J...,xtn) eventuall y map s t o zer o i n th e direc t limit . Usin g th e Briangon-Skoda theorem , prov e tha t an y suc h elemen t u map s t o zer o i n a t most \dt/(n — d)~\ steps. 5.8 Le t R b e a Noetheria n loca l rin g o f characteristi c p, an d le t / b e a n idea l generated b y a regula r sequenc e o f lengt h d. Suppos e tha t / * i s containe d in th e idea l generate d b y al l th e tes t element s o f R. Prov e fo r al l k > 0 , Jd+k+t Q I k+l. 5.9 Le t (R,m) b e a Noetheria n reduce d d-dimensiona l rin g o f characteristi c p wit h a n m-primar y tes t idea l (e.g. , i f R i s F-finit e an d ha s a n isolate d singularity, the n on e ca n appl y Theore m 2. 1 t o obtai n this) . Assum e tha t the residu e field o f R i s infinite . Prov e tha t ther e exist s a n intege r t suc h that i f J C m*, the n I n+d C I n fo r al l n . (Us e Exercis e 5.8. ) 5.10 Mimi c th e proo f o f Theore m 5. 7 t o prov e th e following : Le t R b e a Noe therian Cohen-Macaula y loca l ^-dimensiona l rin g o f characteristi c p whic h is F-pure (i.e . R i s pure i n R 1 /?). The n fo r al l ideal s / , I n+d C I n fo r al l n . 5.11 Le t R b e a Noetheria n loca l d-dimensiona l rin g o f characteristi c p whic h i s F-rational. Assum e tha t th e residu e field o f R i s infinite . Fo r ever y idea l I o f JR , prov e tha t I n+d~l C I n fo r al l n. (Hint : reduc e t o th e m-primar y case b y replacin g / b y I + m 1 .) The nex t fe w exercise s dea l wit h usin g th e theore m o f Ito h an d Hunek e t o prove a versio n o f Grauert-Riemenschneide r vanishin g [GraR ] i n dimensio n two . This version , du e t o Sanch o d e Sala s [SS] , states th e following : T H E O R E M . Let (R,m) be a Cohen-Macaulay reduced local ring which is essentially of finite type over an algebraically closed field of characteristic 0 . Let I be an ideal of R such that Xj, the blowup of I, is regular. Then there exists an integer n ^ > 0 such that G — grin (R) is Cohen-Macaulay.

The blowup of I i s th e projectiv e schem e Pro$(R[It]). Th e assumptio n tha t i t is regula r mean s tha t ther e i s a generatin g se t (x\, ...,x n ) fo r / suc h tha t th e ring s R[I/xi] ar e regular . Th e schem e Xj i s th e sam e i f w e "twist " / b y replacin g i t b y a powe r o f itself , o r b y replacin g / b y it s integra l closur e (ou r assumptio n tha t Xi is regula r an d i n particula r i s norma l i s necessar y fo r th e latte r statement) . Th e graded rin g of an ideal / i s the ring R/I@I/I 2(&' • • . I f J i s m-primary wit h minima l reduction J generate d b y x\,..., xa wher e d is the dimension of R, a theorem o f Valla

5. INTEGRAL CLOSURE A N D TIGHT CLOSURE

45

and Valabreg a [VV ] says that G is Cohen-Macaulay i f f J n P 1 = JI n~1 fo r all n > 2 (this use s tha t P i s Cohen-Macaulay) . 5.12 Le t (P , m) b e a Noetherian loca l rin g an d let / b e an ra-primary idea l o f of R. Prov e tha t fo r al l larg e £ , I 1 ha s a minima l reductio n wit h a reductio n number a t mos t d , th e dimensio n o f R. 5.13 Le t (R,m) b e a two-dimensiona l norma l Noetheria n loca l rin g whic h i s excellent. Le t / b e an arbitrary m-primar y ideal . Prov e that fo r t ^ > 0, setting J = 7* , the associated grade d rin g G = R/J 0 J/J 2 0 .. . 0 J n/Jn+1 0 .. . is Cohen-Macaulay. (Thi s i s a for m o f th e Grauert-Riemenschneide r vanish ing theorem . T o prov e it , us e th e fac t tha t fo r t ^ > 0, th e idea l J wil l b e normal, i.e. , all powers o f J wil l be integrally closed . Choos e a power wher e one ca n appl y Exercis e 5.1 2 . A minima l reductio n o f J wil l b e a regula r sequence x, y o f tw o generators , an d J n = (x,y)J n~l fo r al l n > 3 . G i s Cohen-Macaulay if f (x,y) n J N = (x,y)J N-x fo r al l TV . Us e Theorem 6 of It oh an d Huneke , i n Chapte r 0 , to finish th e proof. ) The nex t exercise s us e the idea o f a reductio n an d the flatness o f Frobenius t o prove tha t regula r loca l ring s o f characteristic p ar e UFDs. 5.14 Le t R b e a regula r loca l rin g o f characteristi c p wit h infinit e residu e field. Let P b e a heigh t 1 prime idea l o f R. Prov e tha t fo r al l q = p e, P ^ i s unmixed. (Hint : us e the fac t tha t th e Frobenius i s flat.) 5.15 Adop t th e notatio n o f Exercis e 5.1 4 . Prov e tha t p M = P q = P^\ th e symbolic qth powe r fo r al l q. (Us e that regula r loca l ring s ar e normal. ) 5.16 Adop t th e notation o f Exercise 5.1 4 . Prov e tha t th e analyti c sprea d o f P i s 1, and us e this t o prov e tha t P i s principal. Conclud e tha t R i s a UFD. The nex t tw o exercise s ar e t o prov e th e followin g theore m o f Aberbac h an d Huneke [AHl ] i n the cas e i n whic h th e rin g ha s characteristi c p. I f I i s an ideal , we let J m m denot e th e intersectio n o f the minimal primar y component s o f I. T H E O R E M . Let (P , ra) be a regular local ring containing a field. Let I be an ideal of R having analytic spread £ and let J be an arbitrary reduction of I. Set h = bight(I). For all n>0, T^ 1 C jn+i (ji-h^min t

5.17 Le t P, /, J, £, h be as in the statement o f the theorem above . Furthe r assum e that P ha s characteristi c p. Prov e ther e exist s a n elemen t / ^ 0 such tha t for al l q = p e , fj^' 1 ^ C (J e~h)M. (Hint : us e Exercise 0.2. ) 5.18 Prov e th e theorem abov e i n the cas e th e residue field o f P i s infinite an d P has characteristi c p. The next exercis e illustrates ho w knowlegde of the test idea l can give Brian^on Skoda typ e theorems . 5.19 Le t P = k[X, Y, Z]/(X3 + F 3 + Z 3) wher e th e characteristic o f k i s at leas t 5. Prov e tha t fo r ever y idea l / i n P , I 2+k C I k fo r al l k > 1 . (Hint : Us e Exercises 2.1 6 and 5.8. ) 5.20 Prov e tha t th e coefficien t idea l exists . Suppos e tha t th e coefficien t idea l a(7, J) contain s a nonzerodivisor. Prov e that I an d J hav e the same integra l closure. I f P i s a domain, prov e that th e coefficient idea l of / an d J i s nonzero iff / an d J hav e th e sam e integra l closure .

http://dx.doi.org/10.1090/cbms/088/08

CHAPTER 6

The Hilbert-Kun z Multiplicit y In thi s chapte r w e prove som e basi c result s concernin g th e Hilbert-Kun z func tion, base d o n Monsky' s pape r [Mo] . Th e mai n resul t i n thi s chapte r i s tha t th e Hilbert-Kunz multiplicit y i s define d an d i s a positiv e rea l number . Firs t recal l th e definitions o f the las t chapter : i f / i s an m-primar y idea l in (R, m), a Noetherian local ring, then th e multiplicity o f /, denote d e(I) , i s e(J) = li m X(R/In)(d\/nd). Th e Hilbert-Kunz multiplicity , denote d CHK(I), i s CHK(I) = \im X(R/I^)/p de, wher e d i s th e dimensio n o f R, provide d thi s limi t exists . W e wil l prov e tha t i n fac t th e limit doe s alway s exist s an d i s a positiv e rea l number . B y definitio n th e Hilbert Kunz multiplicit y o f R i s CHK{R) = e//x(ra) . Conjecturall y th e Hilbert-Kun z multiplicity i s alway s rational , bu t ther e ar e fe w positiv e result s i n thi s direction . Monsky ha s don e remarkabl e calculation s o n th e behaviou r o f th e Hilbert-Kun z function fo r man y hypersurfaces ; th e behaviou r o f thi s functio n seem s t o b e enor mously ric h an d mysterious . Th e first lemm a give s a relationshi p betwee n thes e two multiplicities . LEMMA 6.1 . Let (R, m) be a local Noetherian ring of characteristic p. Set d = dim(R), and let I be an m-primary ideal. Then (q = p e),

e(I)/d\ < liminf X(R/I [q])/qd


X(R/I

q

).

For larg e q, th e righ t han d lengt h i s give n b y a polynomia l i n q o f degre e d wit h leading coefficien t e(I)/d\. Dividin g b y q d give s on e inequality . Fo r th e other , us e Lech's lemm a [Mat , 1 4.1 2] : lim X(R/J^)/q d = e(J) . Sinc e J i s a reductio n o f / , e(J) = e(I). ~^ • COROLLARY 6.2 . Let (R,m) be a local Noetherian ring of characteristic p. Assume that the dimension of R is one. Let I be an m-primary ideal. Then e(I) = eHK(I)- In particular, the Hilbert-Kunz multiplicity exists and is an integer. PROOF.

Simpl y se t d — 1 in th e formul a o f Lemm a 6.1 . •

46

6. T H

E HILBERT-KUN Z MULTIPLICIT Y

47

EXAMPLE 6.3 . Le t R — k[[x,y,z]]/(f) wher e k i s a field o f characteristi c p , and / ha s orde r 2 . The n R i s not F-rationa l if f enK(R) — 2 . To se e thi s w e ma y assum e withou t los s o f generalit y tha t (x,y)R for m a reduction o f th e maxima l idea l m. Sinc e / ha s orde r two , i t the n follow s tha t m2 Q (# , y)R, an d R i s F-rationa l if f (x , y)* = (x, y) ifi m =fi (x , y)*. Suppose first tha t R i s F-rational . The n th e Hilbert-Kun z multiplicit y o f m and (x,y) ar e different , b y Theore m 5.4 . Bu t th e Hilbert-Kun z multiplicit y o f a regula r sequenc e i s jus t th e usua l multiplicit y o f i t (cf . Exercis e 6.1 ) , henc e eHK(R) + e HK((x,y)) = \(R/(x,y)) = 2. Conversely, suppos e tha t R i s not F-rational . The n (x,y) * = m , an d Theore m 5.4 then prove s that th e Hilbert-Kunz multiplicit y o f m i s the same as that o f (x , y), and therefor e i s 2. Lemma 6. 1 prove s i n an y cas e tha t th e Hilbert-Kun z multiplicit y o f m i s a t least 1 , an d i t i s eas y t o se e tha t i t canno t b e exactl y 1 . S o 1 < CHK(R) < 2 , wit h equality a t th e to p if f R i s not F-rational .

To further analys e the behavior o f the Hilbert-Kunz multiplicity , i t is importan t to exten d th e definitio n t o modules . DEFINITION 6.4 . Le t (i? , m) b e a Noetherian loca l ring of positive characteris tic, and let M b e a finitely generated i^-modul e an d / a n m-primary ideal . W e define the Hilbert-Kun z multiplicit y o f I o n M t o b e e HK(I, M) = lim \(M/I^M)/q d , i f this limi t exists .

It ma y b e bette r t o defin e th e limi t i n term s o f th e dimensio n o f M rathe r than th e dimensio n o f R, bu t fo r ou r purpose s here , i t i s convenien t t o hav e th e multiplicity b e 0 when the dimensio n o f M i s less than th e dimensio n o f R. W e will prove th e limi t abov e alway s exists , s o that thi s multiplicit y i s defined . PROPOSITION 6.5 . Let (R, m) be a local Noetherian ring of characteristic p and dimension d. Let M , N be finitely generated R-modules, and let I be an m-primary ideal. 1. / / dim(M) < d, then X(M/I^M) = 0(q d~l) and thus e KK(I, M) = 0 . 2. Let W be the complement of the set of minimal primes Q such that dim(R/Q) — d. IfM w^Nw,

\X(M/I[q]M) -

X(N/I^N)\ =

0(q

1 d

- ).

In particular, eHK(I,M) = CHK{J->N), provided these limits exist. If CHK(I,M) exists then so does CHK{I^N) and vice versa. PROOF. Choos e an element f £ R whic h annihilates M suc h that dim(R/(f)) < d. Se t A — R/(f). Ma p a finite fre e A-modul e ont o M . Tensorin g wit h R/I^ preserves thi s surjectio n an d prove s tha t M/I^M i s a homomorphi c imag e o f a fixed numbe r o f copie s o f A/I™A. Th e proo f o f Lemm a 6. 1 the n show s tha t th e lengths o f M/I^M ar e 0 ( ^ - 1 ) , an d i n particula r CHK{I, M) = 0 . The last statement s o f this part follo w from th e displayed formula . Sinc e Mw — Nw, w e ca n find a ma p fro m M t o N suc h tha t th e cokerne l C i s kille d b y som e element o f W. Tensorin g th e exac t sequenc e M - • N - > C - > 0 with R/lM yield s a shor t exac t sequenc e whic h show s tha t th e lengt h o f N/I^N i s at mos t th e su m of th e length s o f M/I^M an d C/I^C. Th e first par t o f thi s Propositio n prove s

48

6. T H E H I L B E R T - K U N Z M U L T I P L I C I T Y

that X(C/I^C) = 0(q d~l). Reversin g th e role s o f M an d N give s th e opposit e inequality an d proves th e proposition. • PROPOSITION 6.6 . Let (R,m) be a Noetherian local ring of dimension d, and letO—>N—>M^>K-^Obea short exact sequence of finitely generated Rmodules. Then,

X(N/I^N) +

X(M/I^M) =

X(K/I^ q]K) +

0(q

d l

~ ).

PROOF. Firs t suppos e tha t R i s reduced. The n M an d N(BK hav e isomorphi c localizations a t eac h minima l prim e o f R, an d the claim follow s fro m Propositio n 6.5. If R i s not reduced , choos e q f suc h tha t (nilrad(i?)) ^ ' = 0 , an d consider th e same exac t sequenc e a s a sequenc e o f R q -modules . Thi s rin g i s reduced an d applying th e reduced cas e wit h th e ideal I^ q ' fl Rq yield s tha t

X(M/I^M) =

X(N/I^N) +

Since 0(q d~1 ) = 0((qq') d~l), th

X(K/I^K) +

0(g d _ 1 ).

e Proposition i s proved. •

T H E O R E M 6. 7 ( T H E HILBERT-KUN Z MULTIPLICIT Y E X I S T S ) . Let (R,m) be

a d-dimensional Noetherian local ring of characteristic p. Let M be a finitely generated R-module, and let I be an m-primary ideal There is a positive real constant e = e HK(I,M) such that X(M/I^M) = eq d + 0(q d~l). If 0-> N - • M -+K - > 0 is a short exact sequence of finitely generated R-modules, then eHK(I, M) = eHK(I, K) + e HK(I, N). P R O O F . B y making a faithfull y fla t extensio n ther e i s no loss o f generality i n assuming tha t R i s a complete loca l ring wit h algebraicall y close d residu e field . B y taking a prime filtratio n o f M an d using Propositio n 6. 6 it suffice s t o do the case in whic h M = R/P fo r som e prim e P o f R. Thu s ther e i s no loss o f generality i n assuming tha t R i s a domain an d M = R i n proving the first assertion . Th e second assertion follow s immediatel y fro m th e first assertio n an d Proposition 6.6. Let S = R l/p. Sinc e R i s F-finite , S i s a finitel y generate d torsion-fre e Rmodule, an d the rank o f S ove r R i s pd b y Exercise 6.2 . Ther e i s an exact sequence ,

0->G^S->C-+0, where G i s a fre e jR-modul e o f ran k p d an d dim(C ) < d. Defin e c q b y c q = X(R/I^)/qd. Propositio n 6.6 applied t o the above exac t sequenc e yields , \pdX(R/lM) -

X{S/I^S)\
S. I t suffice s t o solv e th e problem fo r eac h ma p separately . I n th e cas e o f a n inclusion , R °- > 5, not e tha t a n algebraic closur e o f th e fractio n field o f S wil l contai n a n algebrai c closur e o f th e fraction field o f R, whic h give s th e resul t a t once . I n th e cas e o f a surjection , w e may assum e tha t S = R/q wit h q prime. Sinc e R^~ is integral ove r R i t ha s a prim e ideal Q lyin g ove r q. The n S — R/q inject s int o R+ jQ = (R/q)+ . I n Chapte r 9 we will retur n t o thi s poin t i n equicharacteristic . Thi s weakl y functoria l propert y ha s many applications . The proo f o f Theore m 7. 1 involve s severa l idea s whic h als o aris e i n th e theor y of tigh t closure , no t th e leas t o f whic h i s the ide a o f unifor m annihilatio n o f coho mology. Th e mai n resul t w e need ca n b e summed u p i n a theorem du e originall y t o Schenzel [Sch] . Als o see [Wi ] an d [Fa2] . Le t (R,m) b e a d-dimensional Noetheria n local ring . Se t (7.3) 9

* = f^Ann((xi, ...,Xi-i)

:

R

Xi)/(xi,...,Xi-i))

r

where th e intersectio n run s ove r al l parameters #i,... , X{ oiR. Observ e tha t 9\ — R exactly whe n R i s Cohen-Macaula y . Set (\i — Ann(H^(R)), an d le t q = qoq i • • • qd-i b e th e produc t o f th e annihi lators o f th e loca l cohomolog y module s o f R (no t includin g th e to p nonvanishin g one). T H E O R E M 7. 4 (UNIFOR M ANNIHILATIO N O F COHOMOLOGY) . Let (R,m) be

a

d-dimensional Noetherian local ring with d> 1 . Then *R d C q C 91. We will us e thi s theore m b y th e followin g consequence : COROLLARY 7.5 . Let (R, m) be a d-dimensional Noetherian local ring which is the homomorphic image of a Gorenstein local ring (S , n) of dimension D. Assume further that R is equidimensional. If c € R is such that R c is Cohen-Macaulay , then there is a power of c in %\, defined by 7.3 above. P R O O F . Th e proo f rest s o n Theore m 7. 4 an d loca l duality. B y killing a regula r sequence o f S i f necessary, w e may assum e tha t d — D. Loca l dualit y say s tha t th e modules H^R) an d Ext^~ l(R, S) ar e Matli s dual , i.e. , on e i s obtaine d fro m th e other b y dualizing int o the injectiv e hul l of the residu e clas s field of 5. I n particula r these tw o module s hav e th e sam e annihilator . A s R c i s Cohen-Macaula y i t follow s that ( E x t ^ ( # , 5)) c = 0 fo r d - i > 1 . Her e w e us e th e equidimensionalit y o f R. In particula r th e finite generatio n o f these Ex t module s prove s tha t ther e i s a fixed power o f c which annihilate s the m i n th e rang e d — i > 1 . I t the n follow s tha t th e product o f thes e power s o f c is in q which b y Theore m 7. 4 i s containe d i n 91 . •

Corollary 7. 5 is one of three main ingredient s neede d t o prove (7.1 ) . Th e ide a is to trivializ e relation s o n parameter s b y passin g t o finite integra l extensio n domain s of R. Althoug h ne w nontrivia l relation s ma y appear , w e wil l trivializ e the m b y passing t o a furthe r finite extension . Whil e i t ma y b e impossible , i n general , t o trivialize al l relation s o n parameter s i n a finite extensio n (suc h a n extensio n woul d then b e a smal l Cohen-Macaula y algebra) , b y passin g t o i? + w e wil l trivializ e al l of them . Serre' s criterio n show s tha t t o trivializ e relation s o n tw o parameter s th e

52

7. BI G C O H E N - M A C A U L A Y A L G E B R A S

integral closur e o f the ring suffices . Ther e wer e n o known "natural " construction s to increas e dept h pas t 2 . There ar e two major problem s i n th e proo f o f Theore m 7.1 . Th e first i s to understand i n any example ho w one can trivialize a relatio n o n parameters. Le t xi, ...,#fc+ i b e parameters o f an excellent loca l domai n R. Suppos e tha t the y are not a regular sequence , s o there i s a (nontrivial ) relatio n (7.6) ]

Tr

xx%

= 0.

1 k. Furthermor e assum e tha t there i s a Cohen-Macaulay multiplie r y such tha t x\ y...,Xt,y ar e parameters. The n junm Q junm g j n c e DO th o f these ideal s ar e respectively equa l t o 1 : x an d J : x. Since 1 C J, the result follows . I'l l be using this fact implicitl y in the proof below. I t is not true in general. Fo r instance let I = 0 and J b e any full syste m of parameters. The unmixe d par t o f J i s just J itself , an d intersecting ove r al l possible J give s us the zer o ideal . I f R i s not unmixed , the n th e unmixed par t o f 0 is strictly large r than 0 .

The followin g lemm a i s helpful i n the proof o f Theorem 8.1 .

8. BI

G COHEN-MACAULA Y ALGEBRA S I I

59

LEMMA 8.7 . Let (R,m) be an excellent local domain of characteristic p. Let x,xi,...,Xh,y, Zh+2, ---,Zk be a sequence of elements in m which is part of a system of parameters such that 1. x is a CM multiplier for R, and 2. y is a CM multiplier for R/I unrn, where I = (xi , ...,XH)RLet V denote the ideal generated by the z 's. Let a be a positive integer. Then (7 + y aR + I') unrn =

( 7 + I') unrn +

y a-\l +

yR + l')

unm

.

P R O O F . I t is easy t o verify tha t eac h of the tw o ideal s in the sum o n the right is containe d i n the ideal o n the left. Suppos e tha t u G (I + yaR + I f>)unrn. Sinc e x i s a C M multiplier, i t follow s tha t ther e i s an equation xu — i -\- yar + i' wher e i G 7 and i' G 7'. Passin g t o the ring R — R/Iunrn (an d using a ba r to indicat e images i n R), on e obtains u G (ya, I')R : x. Th e images of x, y and the zi still for m part o f a system o f parameters i n R. Sinc e th e image o f y in R i s assumed t o be a CM mutliplier , w e obtain tha t yu G yaTl + VR S O that yu G Iunrn + yaR + 7'. Thi s means tha t ther e exist s a n element r G R suc h tha t yu — yar G Iunrn + 7 ' . The assumptio n o n x the n show s tha t x(yu — yar) G 7 + 7' , an d hence (u — a l y ~ r) G (7 + 7' ) : xy whic h the n implie s tha t (u - y a~lr) G (7 + J')™*™ . T o finish th e argumen t i t onl y remain s t o prov e tha t r G (7 + yR + 7 / ) w n m . Writ e ya-ir — u + v^ w here v G (7 + j ^ u n r n . Th e choice o f w shows the n tha t y a~lr G (J + ^ ii + i7)™1™- Th e assumptio n o n x then give s r G (I + yaR+r) : a ^ "1. The n a l a there i s an element s £ R suc h tha t xy ~ r — sy G J + 7' , from whic h i t follow s that x r - sy G (J + 7') : ya~x C (7 + i 7 ) : x. Thi s yield s tha t x 2 r G (7 + yR + 7'), and finally tha t r G (7 + yR + 7 ' ) u n m , a s required. • P R O O F O F T H E O R E M 8.1

.

By repeate d application s o f Lemma 8.7 , we obtain tha t (x? 1 , ...,x

a k unm k )

yx2 ,...,x /~a2 o

fc

; fe

^x 2 ,...,x + X- ^ X

C

k\unm ,

(~.a,2 a

fc 2

(applying th e result wit h y = x x) a i -1

/a

r X]_ ^ x i , x

\unm, o i - l

; -

/ a

r xx ^ x i , x

(Xi,X2,X

2

2

3

3

3

a

,...,x

k\unm r

fc

a

,...,x

; ^

_

fc\unm

fc

;

,...,X^ . j

Continuing thi s way , a t th e hth stag e w e have bot h th e sum of the h 'stable ' terms whic h wil l not be expanded further , namely , 0 < l < / l - l1 95(5 ) b y Theorem 9.3. A balance d bi g Cohen-Macaula y algebr a ove r a regula r loca l rin g i s faithfull y flat as i n Theore m 9.1 , and henc e S i s pur e i n 95(5 ) (9.5.1 ) . Thu s R i s pur e i n 95(5) . It the n follow s tha t R i s pur e i n 9 5 (i?), an d henc e mus t b e Cohen-Macaulay . b y (9.5.5). •

Another applicatio n o f Theore m 9. 3 i s a stron g vanishin g theore m fo r map s o f Tor whic h b y itself implie s Theore m 9.6 , the direc t summan d conjecture , an d othe r homological conjecture s [HH1 1 , T h e o r e m 4.1 ] . T H E O R E M 9. 7 (VANISHIN

G T H E O R E M FO R M A P S O F T O R ) . Let A

be

an

equicharacteristic regular Noetherian domain, let R be a ring that is a module-finite

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extension of A and torsion-free as an A-module, and let R —> S be any homomorphism from R to a regular Noetherian ring 5 . Then for every finitely generated A-module M and all i > 1 , the map Tor?(M,R)—*Tor?(M,S) is zero. PROOF. I f the displayed ma p is nonzero then i t remain s nonzero after replacin g 5 b y a suitabl e localization , s o w e ma y assum e tha t 5 i s local . W e ma y furthe r assume that 5 i s complete. The n localiz e A (an d M , R) a t th e prime ideal of A tha t is th e contractio n o f th e maxima l idea l o f 5 . The n replac e A, M, an d R b y thei r completions. Thus , w e may assum e tha t A an d 5 ar e complet e regula r loca l rings . The kerne l of the ma p R — > 5 mus t contai n a minimal prim e p of i?, and p does no t meet A. W e the n hav e a factorizatio n o f th e ma p o n Tor s throug h Tor f (M, R/p) and i t suffice s t o replac e R b y R/p an d assum e tha t R i s a domain . The ma p R — > 5 extend s t o a ma p o f balance d bi g Cohen-Macaula y algebras , 05(R) -+95(5) . Fo r simplicity , denot e 3 5 (ii) b y B an d 05(5 ) b y C . Sinc e th e ma p from 5 t o C i s faithfull y fla t (a s i n th e proo f o f (9.6) ) th e ma p Torf(M , 5 ) — > Tor^(M, C) i s injective. I t therefor e suffice s t o prov e tha t th e induce d ma p Torf (M, R) — • Tor 2A(M, C) i s zero. Since B i s a balance d bi g Cohen-Macaula y algebr a fo r R an d ever y syste m o f parameters fo r A i s a syste m o f parameters fo r R, w e have tha t B i s a balance d bi g Cohen-Macaulay algebr a fo r A, an d henc e i s fla t ove r A, sinc e A i s regular. Thus , Torf (M,B) = 0 fo r al l i > 1 . Howeve r th e ma p Toif(M,R) — • Tor f (M,C ) factors throug h Tor^(M , B) an d s o i s zero . • The specia l cas e o f (9.7 ) i n whic h M i s th e residu e field o f A alread y ca n b e used t o prov e Theore m 9. 6 (Exercis e 9.6) . Th e specia l cas e in which 5 i s a field can be use d t o prov e th e direc t summan d conjectur e (Exercis e 9.7) . Th e statemen t o f the vanishin g theore m i n mixe d characteristi c i s a n importan t ope n problem . Exercises 9.1. I f h : M— • N i s pure ove r R an d I i s a n idea l o f R the n /i _1 (L/V) = IM. 9.2 a ) Le t R b e a ring , an d le t M , N, L b e i2-modules . I f M — > N i s pur e an d factors M —> L — > iV , then M — > L i s als o pure . b)Let R, M , N, L b e as above. I f M— > L an d L —> N ar e pure, then M — > N is als o pure . 9.3. Assum e tha t (R, m , k) i s Noetheria n local , M i s a finitely generate d fre e i?-module an d N i s a n i2-module . Prov e tha t a n injectio n M — > N i s pur e over R if f E (&R M— > E ®R N i s injective , wher e E i s a n injectiv e hul l o f k over R. 9.4. Assum e tha t (R,m,k) i s Noetherian local . Le t 5 b e a n arbitrar y R- algebra, and le t N b e a n arbitrar y 5-module . Suppos e tha t on e ha s a n .R-modul e homomorphism R — > N tha t i s pur e ove r R. Ther e exist s a maxima l idea l Q o f 5 suc h tha t R — > NQ i s pure ove r R. 9.5. Le t (R,m) b e a complet e loca l rin g an d suppos e tha t R — > M i s a pur e homomorphism o f i ? int o a n R- module M. Prov e tha t thi s ma p splits .

9. A P P L I C A T I O N

S O F BI G C O H E N - M A C A U L A Y A L G E B R A S 6

5

(Hint: tenso r wit h a n injectiv e hul l o f the residu e field of R an d dualiz e int o it.) 9.6 Us e th e Vanishin g Theore m fo r map s o f To r t o prov e Theore m 9.6 . 9.7 Us e th e Vanishin g Theore m fo r map s o f To r t o prov e th e direc t summan d conjecture i n characteristi c p. 9.8 Le t R b e a locally excellen t Noetheria n domai n o f characteristic p. Suppos e that fo r ever y module-finit e extensio n domai n S o f R, th e inclusio n R — > S splits a s a n i^-modul e homomorphism . Prov e tha t R i s Cohen-Macaulay . 9.9 Le t (A,m) b e a complet e regula r loca l rin g o f positiv e characteristi c wit h A C S wher e S i s a module-finit e domai n extensio n o f A. Le t #i , ...,Xd b e a regula r syste m o f parameter s o f A. Prov e tha t th e inclusio n AC. R split s as a n A-modul e if f fo r al l positiv e integer s t , {x\ • • • Xd)1 £ (#i +1 > •••, x^^R. (Hint: Identif y a n injectiv e hul l o f A/m wit h liuiA/(x\ 1 ...,xtd) an d us e Exercises 9. 3 an d 9.5. ) 9.10 Le t (A,m) b e a complet e regula r loca l rin g o f positiv e characteristi c wit h ACS wher e S i s Noetheria n an d bigh t (mS) = ht(ra) . Prov e tha t th e inclusion o f A int o S splits . (Hint : Us e the monomia l theore m an d Exercis e 9.9.) 9.11 Le t (R,m) b e a complet e loca l domai n o f positiv e characteristic . Assum e that R C S wher e S i s a Noetheria n rin g an d bigh t (mS) = ht(m) . The n there i s a n .R-linea r ma p / : S— » R suc h tha t / ( l ) ^ 0 . I n particular , fo r every idea l / i n i? , IS D R C /* . (Hint : Choos e a regula r loca l rin g A ove r which R i s modul e finite an d us e exercis e 9.1 0 t o ge t a ma p fro m S t o th e canonical modul e o f R. The n imbe d th e canonica l modul e int o R. Fo r th e last statemen t se e Exercis e 1 .6. ) The nex t thre e exercise s analyz e th e tigh t closur e o f powers o f ideals generate d by parameter s whic h ar e tes t elements . Al l thre e o f thes e exercise s ar e impor tant point s i n th e application s o f tight closur e t o arithmeti c Macaulayfications , se e [HS]. A n arithmetic Macaulayfication o f R i s a Ree s algebr a R[It] whic h i s Cohen Macaulay. Th e existence of such arithmetic Macaulayfication s i s open, bu t ha s bee n proved i n severa l specia l cases , e.g . i f th e dimensio n o f th e no n Cohen-Macaula y locus i s eithe r 0 o r 1 . Se e [GY] , [Br2,4] , [Ku] , [Ab4] , an d [HS] . I t appear s tha t tight closur e method s ar e ver y usefu l i n thi s problem . 9.12 Le t (R,m) b e a complet e loca l Noetheria n domai n o f characteristi c p. Le t #i,..., Xd be a syste m o f parameters o f R generatin g a n idea l I. Assum e th e xi ar e tes t elements . Prov e tha t (J n )* n I n~l — In~lI*. (Hint : us e colo n capturing, Theore m 9.2. ) 9.13 Le t (R,m) b e a complet e loca l Noetheria n domai n o f characteristi c p. Le t #i,..., Xd be a syste m o f parameters o f R generatin g a n idea l / . Assum e th e xi ar e tes t elements . Prov e tha t (I 71 )* — In~lP. (Hint : us e Exercis e 9.1 2 and Theore m 9.2. ) 9.14 Le t (R,m) b e a complet e loca l Noetheria n domai n o f characteristi c p. Le t x\,...,Xd b e a syste m o f parameter s o f R generatin g a n idea l / . Assum e

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the Xi ar e tes t elements . Prov e tha t (xi,...,Xk)* C\ 1 < k < d.

I C (xi,...,x^ ) fo r al l

9.15 Le t (R,m) b e a complet e Noetheria n loca l equidimensiona l rin g o f charac teristic p wit h a n m-primar y tes t ideal . Le t / b e a n idea l generate d b y par t of a syste m o f parameter s havin g heigh t strictl y les s tha n th e dimensio n o f R. Prov e tha t th e tigh t closur e o f I i s exactly th e unmixe d par t o f i", which is the intersectio n o f al l th e minima l primar y component s o f / .

http://dx.doi.org/10.1090/cbms/088/12

CHAPTER 1 0

P h a n t o m Homolog y Many of the homological conjectures concer n free complexes having finite lengt h homology. Suc h complexe s satisf y th e standar d heigh t an d ran k conditions , an d i n actual application s man y o f the m com e fro m bas e chang e fro m a regula r rin g A . The flatness o f A+ ove r A the n give s a powerfu l tool . Le t (R, m) b e a complet e local Noetheria n domai n o f positive characteristi c p. Writ e R a s module-finit e ove r a complet e regula r loca l ring A, an d le t M b e any finitely generate d A-modul e wit h a finite fre e resolutio n F . Th e modul e M F e (M)), and thi s i s th e sam e a s th e R-sp&n i n F e ( M ) o f th e element s x q fo r x 6 N. N^ depends heavil y o n what M i s (or , mor e precisely , o n wha t N — > M is) : t o indicat e this dependenc e o n M i n th e notatio n writ e N)$ instead . Wit h thes e convention s MJ$ = F e ( M ) , an d I'l l sometime s writ e M ^ for F e ( M ) . If M = R an d N = I C R the n /I? ' i n thi s ne w notatio n i s identica l wit h 1 ^ .

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(10.0) DEFINITION . Le t N C M b e finitel y generate d module s ove r a rin g R of characteristi c p. Sa y tha t x G M i s i n th e tight closure N^ o fT V if ther e exis t c G R° an d a n intege r q' suc h tha t fo r al l q > q',cx q G Nj$. I f i V = 7V ^ w e sa y that N i s tightly closed (in M) . LEMMA 1 0. 1 (CONTRACTION S FRO M F I N I T E EXTENSIONS) . Let S CT be

a

module-finite extension of Noetherian domains of positive characteristic p. Suppose that N C M are finitely generated S-modules with M free such that w G M has the property that its image in T ® s M is in the tight closure of TN = Im(T ®s N —> T (8) 5 M). Then w is in the tight closure of N in M. P R O O F . (Compar e wit h (1 .7). ) W e ca n choos e a n 5-linea r ma p : T — > S such tha t 0(1 ) = d G S — {0} and w e can choos e c eT — {0} such tha t c( l ® w)q G (TW) M TMforallg. W e can choos e a nonzer o multipl e o f c in 5 , an d s o we may assume tha t c G S — {0} . I t follow s tha t fo r al l g , cw q i s a T-linea r combinatio n o f elements o f Njfy C T 0 5 M . Thinkin g o f M a s S T an d applyin g th e ma p (f) t o eac h coordinate, w e fin d tha t dcw q G N^ fo r al l G^- i i s in th e tight closure , withi n d, o f th e imag e # i o f Gi+ i— > G^ . I f thi s hold s fo r al H > 1 , we say tha t G . ha s phantom homology , or tha t G . i s phantom acyclic . I f F e (G # ) is phantom acycli c fo r al l e > 0 we day tha t G # i s stably phantom acyclic .

It i s not difficul t t o giv e example s (se e Exercis e 1 0.5 ) o f phanto m acycli c com plexes which ar e no t stabl y phanto m acyclic . I t i s the latte r whic h interes t u s mos t since w e ar e intereste d i n propertie s o f resolution s whic h ar e fairl y stabl e unde r Probenius. The main point i s that an y complex of finitely generate d fre e modules which sat isfies th e standar d condition s o n heigh t an d ran k wil l b e phanto m acyclic , an d wil l become exac t upo n tensorin g wit h i? + . Thi s follow s b y rewritin g th e Buchsbaum Eisenbud criterio n fo r exactnes s t o allo w tensorin g wit h possibl y no n finitely gen erated modules . Northcot t gives a versio n i n hi s boo k whic h work s i n th e non Noetherian cas e usin g th e concep t o f tru e grade . Barge r [Ba ] prove d tha t tru e grade an d Koszu l dept h ar e th e same . I f M i s a n R- module an d I = (xi , ...,x n ) i s an idea l suc h tha t IM ^ M , th e Koszul depth o f / o n M , denote d K-depth/(M) , i s n — j wher e Hj(xi, ..., x n; M) i s the highes t nonvanishin g Koszu l homolog y modul e on M. Th e generalize d Buchsbaum-Eisenbu d theore m the n state s (cf . [Abl]) : T H E O R E M (GENERALIZE D BUCHSBAUM-EISENBU D C R I T E R I O N ) . Let R be a

Noetherian ring. Let G m be a complex of finitely generated free modules 0 — • G n — > G n _ i — > • • • —> Gi — > • • - -> G\ — » Go —> 0 .

Let s t be the rank of the map G x ®R M — > Gi- ® # M for 1 < i < n and set sn+1 = 0 . Set U = I Si(di), where di : Gi — • G L-\ is the ith map in G # . Then G.®RM is acyclic iff 1. K-depthi t(M) > i for 1 < i < n, and 2. Si + s 2+ i = rank(G l) for 1 < i < n.

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The phanto m acyclicit y theore m show s mor e o r les s tha t finitely generate d free complexe s ar e phanto m acycli c if f the y satisf y th e standar d heigh t an d ran k conditions. However , thi s i s no t quit e true ; on e mus t worr y abou t nilpotents , an d iterates o f the comple x unde r Frobenius , neithe r o f which chang e th e height s o f th e ideals generate d b y minor s o f a give n siz e o f som e matrix . However , goin g modul o nilpotents ca n chang e th e rank s o f th e map s i n a give n complex , a s ca n applyin g the Frobeniu s morphism . I t the n make s sens e t o loo k a t th e reduced rank o f th e maps, tha t i s the ran k afte r killin g nilpotents . T H E O R E M 1 0. 3 ( P H A N T O M ACYCLICIT Y C R I T E R I O N ) . [HH4 , T h e o r e m 9.8 ] Let R be a Noetherian ring of characteristic p. Suppose that R is a homomorphic image of a Cohen-Macaulay ring and is locally equidimensional. Let G. be a complex of finitely generated free modules

0 — > Gn — > Gn-\ —••••—

• G i —>•••— > G i — > Go —• 0.

Denote the map from Gi to Gi-\ by o^ . Let bi denote the rank of Gi, with the convention bi = 0 if i > n or i < 0 and let vi — T,™=i(—l)t~'lbt, 1 < i < n, Let aled be the result of tensoring with R red> Suppose that ran k c*[ ed = n, 1 < i < n (equivalently, bi — ran k a ^ + ran k a£ e d , 1 < i < n), and suppose that the height of the ideal U — I ri(oti) is at least i, 1 < i < n. Then Hi(F eGm) is phantom for all e G N and all i > 1 . Conversely, let R be an arbitrary Noetherian ring of characteristic p > 0, and let G m be a free complex over R with notation as above. If Hi(F eG9) is phantom for all e G N and all i > 1 then ran k a [ e d = r; , 1 < i < n, and the height of Ii — I ri[pii) is at least i, 1 < i < n. The origina l proo f o f Theore m 1 0. 3 wa s throug h a constructiv e proo f o f th e Buchsbaum-Eisenbud criterion , an d ha d t o d o with th e constructio n o f homotopie s after multiplyin g b y element s whic h kil l homology . However , on e ca n se e wh y this shoul d b e tru e b y passin g t o th e completion s o f R a t it s maxima l ideals , go ing modul o minima l prime s i n th e completio n (cal l suc h a rin g B), the n finally tensoring wit h th e plu s closure s o f thes e complet e domains . I n thi s cas e th e Gen eralized Buchsbaum-Eisenbu d criterio n give s exactness usin g tha t th e plu s closure s are Cohen-Macaulay . Passin g bac k t o th e ring s o f th e for m B, on e see s tha t th e cycles ar e the n force d t o b e i n th e tigh t closur e o f th e boundaries . Sinc e thi s i s true modul o ever y minima l prime , i t force s th e homolog y t o b e phanto m i n th e completions o f R a t th e localization s o f maxima l ideals , whic h finally force s th e theorem t o b e true . Th e othe r directio n use s differen t ideas . The theor y o f module s wit h phanto m resolution s ha s bee n full y develope d b y Aberbach. Sa y tha t a finitely generate d i?-modul e M ha s a phanto m resolutio n if ther e exist s a comple x G # o f finitely generate d projectiv e i?-module s suc h tha t Ho(G9) — M an d suc h tha t th e positiv e homolog y o f G # an d al l o f it s Frobeniu s iterates i s phantom . M i s sai d t o hav e finite phanto m projectiv e dimensio n i f i t has a finite phanto m resolution , an d w e writ e ppd#(M ) < oo . I n th e cas e R is loca l h e prove s a n analogu e o f th e Auslander-Buchsbau m formul a fo r phanto m projective dimension . I t turn s ou t tha t th e lengt h o f a minimal phanto m resolutio n is independen t o f th e minima l resolutio n chosen . Her e 'minimal ' mean s tha t th e images o f th e fre e module s ar e i n th e maxima l idea l time s th e targe t fre e module . Even more , th e rank s o f th e fre e module s i n th e phanto m resolutio n ar e uniquel y determined; hi s proof use s th e existenc e o f balance d bi g Cohen-Macaula y algebras .

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Theorem 1 0. 3 gives a powerfu l tool . Fo r example , wheneve r on e map s R t o a regular rin g S th e homolog y o f a phantom acycli c comple x disappear s afte r tensor ing wit h S; th e homolog y i s 'phantom' . Thi s follow s sinc e tigh t closur e persists , and al l submodule s ar e tightl y close d ove r regula r rings . Thi s approac h lead s t o powerful vanishin g theorem s o n map s o f Tor , simila r t o Theore m 9.7 . Th e metho d of proo f o f Theore m 1 0. 3 ca n als o b e applie d t o prov e a resul t whic h i s stronge r than th e improve d ne w intersectio n conjectur e whic h i s stated below : IMPROVED N E W INTERSECTIO N C O N J E C T U R E . Le t (R,m) b e a Noetheria n local eZ-dimensiona l ring . Suppos e tha t G* i s a comple x o f finitely generate d fre e modules

0—> G n— > G n-i — • • • •— > Gi — » • • • — > G\ —> Go —* 0 . If Hi(G %) ha s finite lengt h fo r i > 1 and HQ(G 9) ha s a nonzer o minima l generato r annihilated b y a powe r o f the maxima l ideal , the n n> d. The tigh t closur e versio n o f thi s theore m give s a stronge r conclusio n [HH8 , T h e o r e m 6.5] : T H E O R E M 1 0. 4 ( P H A N T O M INTERSECTIO complex of finitely generated projective modules

0 - > Gn - > >

N THEOREM) .

Let G. be a finite

G o -> 0

over a Noetherian ring R containing a field such that the standard conditions for rank and height hold for G r%ed. Suppose further that R is an excellent locally equidimensional homomorphic image of a Cohen-Macaulay ring. Let z G M — Ho(G9) be any element whose annihilator in R has height > n. Then z is in the tight closure of 0 in Ho(G 9). It i s eas y t o se e tha t n o minima l generato r ca n b e i n th e tigh t closur e o f 0 i n a non-zer o finitely generate d modul e ove r R (se e Exercis e 1 0.3) . I n particular , th e Phantom Intersectio n Theore m immediatel y implie s the Improved Ne w Intersectio n Conjecture. I will no t giv e a proo f o f Theore m 1 0. 4 bu t intuitivel y a n argumen t can b e mad e a s follow s i n th e cas e i n whic h R i s a domain . Tenso r th e comple x with i? + ; th e comple x become s exac t sinc e i t satisfie s th e standar d condition s o n height an d rank . B y standar d dept h argument s (whic h no w ca n b e mad e usin g Koszul depth) , th e Koszu l dept h o f th e zerot h homolog y modul e wil l b e a t leas t dim(.R) — n. Henc e th e elemen t z mus t ma p t o zero , whic h force s i t t o b e i n th e tight closur e o f 0 in Ho(G m). One o f th e famou s homologica l conjectures , solve d i n equicharacteristi c b y Evans an d Griffit h [EG ] i n th e earl y 80's , i s th e syzyg y theorem . I t admit s a tight closur e proof . T o clos e thi s section , I' d lik e t o poin t ou t tha t th e mai n poin t of th e syzyg y theore m follow s rapidl y fro m th e fac t tha t R + i s Cohen-Macaulay . The conjectur e itsel f say s tha t a /ct h syzyg y M o f a modul e o f finite projectiv e dimension suc h tha t th e ran k o f M i s less than k mus t b e free . However , th e hear t of th e proo f involve s th e heigh t o f orde r ideals , DEFINITION 1 0.5 . Le t M b e a n i?-module , an d le t x G M. Th e order ideal of x, denote d OM(X), i s the se t o f element s {/(# ) : / G Hom#(M , R)}.

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The orde r idea l i s actually a n ideal . Th e mai n point 1 t o th e syzyg y theore m i s (cf. [EG]) : T H E O R E M 1 0. 6 (HEIGHT S O F O R D E R IDEALS) . Let Rbe a d-dimensional complete local domain containing a field. Let M be a finitely generated kth syzygy of finite projective dimension, and let x be a minimal generator of M. Then height(OM(%)) > k. P R O O F . I'l l giv e th e proo f i n characteristi c p\ th e equicharacteristi c 0 cas e follows b y reduction t o finite characteristic . Assum e the theorem i s false an d choos e a prim e P containin g OM(X) suc h that heigh t (P) < k — 1. I n this case dim(R/P) > d — k + 1 , since R i s a catenar y domain . Le t G 9 b e th e fre e resolutio n o f a modul e N suc h tha t M i s the kth syzyg y o f N. Writ e

0— » G n — • G n-\—- > • • • — • Gi— > - • •— > G\ —- > Go —• 0 , and le t fi b e th e ma p fro m Gi — > G;_i . Thus , M = Im(//c) . Tenso r thi s comple x with (R/P)+ an d truncat e i t a t Gk' on e obtain s a finit e fre e comple x o f (R/P)+ modules o f length a t mos t pdn(N) — k