# On a-fibrations

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## On A*-fibrations

S .M . B h a tw a d e k a r 3 *, A m a r ty a K . D u tta b

■'School aj \latlwmatic.s. Tata Institute o f Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

^ Sun-M ath Lm i. Indian Statistical Institute, 203, B.T. Road, Calcutta 700 035, India

A bstract

In th is p a p e r w e i n v e s t i g a t e m in im a l su ffic ie n t fib r e c o n d itio n s fo r a fin ite ly g e n e r a te d fiat a lg e b ra o v e r a n o e th c ria n in te g r a l d o m a in to b e l o c a lly A* o r a t le a st a n A * -fib ra tio n . W e also d e s c r ib e th e s tru c tu re o f fin i te ly g e n e r a te d lo c a lly A * -a lg e b ra s .

1. Intro d u ctio n

In [3, 3.4], the fo llo w in g result has been proved.

T h eo rem . Let R be a noetherian normal domai n with quotient f i el d K a n d let A be a f i nit ely generat ed fai t hf ul l y f l a t R-algebra such that

( i) The generic f i br e K %-rA is a pol ynomi al ring in one variable over K .

( i i ) For each pri me ideal P o f R o f height one, t he fi bre ring k{ P) ®gA is geometrically integral over k ( P ) (where k ( P ) ~ Rp/PRp).

Then A is R-isomorphic to the Rees algebra R[ I T] o f an invertible ideal I o f R; in particular, A is an A ' - f b r a t i o n over R, i.e., the f i br e at every poi nt P o f Spec R is a p o l y n o mi a l ring in one vari able over k(P).

The result is som ew hat surprising as conditions on m erely the generic and codim en­

sion one fibres im ply th a t all fibres are A '. This phenom enon had also been observed

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e a rlie r in [2, 3 .1 0 a n d 3.1 2 ] f o r s u b a lg e b ra s o f p o ly n o m ia l a lg e b ra s . In th is p a p e r w e sh o w th a t a n a n a lo g o u s re s u lt h o ld s w h e n th e g e n e ric fibre is A' (i.e .. w h e n k i; A is a L a u re n t p o ly n o m ia l rin g K \ T , 7’- 1 ]). M o re p re c is e ly , w e p i o \ e .

T h e o r e m 3 .1 1 . L e t R be a n o e t h e r i a n n o r m a l d o m a i n wi t h g u o i i e n i h e l d k a i u l let A be a f i n i t e l y g e n e r a t e d f l a t R - a l g e b r a sueh t hat

( i ) Th e g e n e r i c fi bre K & R A is a L a u r e n t p o l y n o m i a l ri ng in one v a r i a b l e o v e r A . ( i i) F o r e ac h p r i m e i deal P o f R o f hei ght o n e . the f i b r e ri ng k ( P ) r A is g e o m e t ­

r i c a l l y i n t e g r a l b ut is n o t A 1 o v e r k ( P ) .

Then t h e r e e x i s t s an i nv e r t i bl e i d e a l I in R sueh t h a t A is a X- g r a d e d / { - a l g e b r a i s o mo r p h i c to t he R - s u b a l g e b r a R [ I T, / _ l T ~ l ] o f K [ T , r - 1 ]. In pa r t i c u l a r . A is l o a i l l y A* a n d h e nc e an A? - f ibrat ion o v e r R.

T h e c ru c ia l ste p in th e p r o o f is a p a tc h in g L e m m a 3 .1 . A s an a p p lic a tio n o t th e p a tc h in g le m m a w e sh all a ls o p r o v e th e fo llo w in g s tru c tu r e th e o re m fo r lo c a lly A ' a lg e b ra s o v e r n o e th e ria n d o m a in s .

T h e o re m 3 .4 . L e t A be a f i n i t e l y g e n e r a t e d a l g e b r a o v e r a n o e t h e r i a n d o m a i n R s uc h t hat f o r each m a x i m a l i de al M o f R, A M is a L a u r e n t p o l y n o m i a l ri ng R \ t [ T\ i . T u ' ].

Then t he r e e x i s t s an i nver t i bl e i d e a l I in R sueh t h a t A is a 7^-gradcd H- a h / e b r a i s o mo r p h i c to R \ I T , I ~ l T ~ 1].

T h e a b o v e re s u lt is an a n a lo g u e o f a re s u lt o f E a k in - H e i n z e r [4, 3.1] th at a ffin e d o ­ m a in s w h ic h a re lo c a lly A 1 a re th e sy m m e tric a lg e b ra s o f in v e rtib le id eals. (In fa c t, a little m o d ific a tio n o f o u r p r o o f w ill g iv e an a lte rn a tiv e p r o o f o f th e E ak in I le in z e r th e o ­ re m fo r n o e th e r ia n d o m a in s .) F in a lly , w e in v e s tig a te m in im a l su ffic ie n t c o n d itio n s fo r a fin itely g e n e ra te d flat a lg e b ra o v e r a n a rb itra ry n o e th e r ia n d o m a in to be an A '- f ib r a tio n a n d p ro v e th e fo llo w in g a n a lo g u e o f [3, 3.5].

T h e o re m 3 .1 3 . L e t R b e a n o e t h e r i a n d o m a i n wi t h q u o t i e n t f i e l d K a n d let A b e a f i n i t e l y g e n e r a t e d f l a t R - a l g e b r a s u e h t hat

( i ) T h e g e n e r i c f i b r e K <E>r A i s a L a u r e n t p o l y n o m i a l ri ng in one var i abl e o v e r K . ( i i) F o r e a c h p r i m e i deal P o f R o f he i g ht one , t he f i b r e r i ng k ( P ) '■ r A is g e o m e t ­

r i c a l l y i n t e g r a l b u t is n o t a n A 1- f o r m o v e r k ( P ) .

Then a l l t he f i b r e ri ngs a r e A t - f o r m s . In f a c t , t her e e x i s t s a f i ni t e bi r a t i o na / e x t e n s i o n R' o f R a n d an inverti ble i d e a l I o f R' such t h a t R' A is a Z - g r a d e d R ' - a l g e b r a i s o mo r p h i c to R ' [ I T , I ~ ] T ~ 1]. Fu r t h e r , i f R c o n t a i n s a f i e l d o f c h a r a c t e r i s t i c z e r o , a n d if a l l the f i b r e s have m o r e u n i t s than the r e s p e c t i v e r e s i d u e fields, then A is an A* -f i brat i on ov e r R.

W e a ls o g iv e ex a m p le s to s h o w t h a t th e c o n d itio n s in o u r th e o r e m s c a n n o t b e re la x e d .

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2. P r e lim in a r ie s

In th is se c tio n w e set u p th e n o ta tio n s , d e lin e th e te rm s used in th e p a p e r, state a tew e le m e n ta ry re s u lts a n d p ro v e a resu lt on A '- to r m s . T h ro u g h o u t o u r p a p e r w e w ill a s s u m e o u r rin g s to be c o m m u ta tiv e .

N o ta ti o n . I or a ring R. R' w ill d e n o te the m u ltip lic a tiv e gro u p o f u n its o f R. F o r a p rim e id eal I’ o f R. k ( P ) d e n o te s th e re sid u e field R p i P R p . T he n o ta tio n A = R ^ w ill m e a n th a t A is a p o ly n o m ia l rin g in n v a ria b le s o v e r R.

D e fin itio n . An W -algebra A is d efin ed to be A* i f it is a L au ren t p o y n o m ia l rin g in o n e in d e te rm in a te o v e r R. i.e., i f th ere ex ists an e le m e n t T in A w h ic h is a lg e b ra ic a lly in d e p e n d e n t ox er R su c h th a t A = R[T. 7 '~ 1].

A n /^ -alg eb ra I is d e fin e d to be local ly A* \ f A m is A* o v er RM fo r e v e ry m ax im al ideal \ I o f R.

A fin itely g e n e ra te d flat /? -a lg e b ra A is d efin ed to b e an A*-fibration o v e r R if, at e a c h p o in t P o f Sp c c R . th e fibre rin g k ( P ) ■ !t A is A* o v e r k ( P) .

Let k be a field an d let k d e n o te the a lg e b ra ic c lo s u re o f k. A i - a lg e b r a B is said tii be g e o m e t r i c a l l y i n t e g r a l ( o v e r A ) if A7 ,-.k B is an in te g ra l dom ain. B is d e fin e d to he an A ' - f o r m over k i f k t B is A* o v e r k. A &-a lg e b ra C is said to b e a n At - f or m nvcr k if k t C -- k1

L e m m a 2 .1 . Let />’ I h e i n t e g r a l domai ns. S u p p o s e t hat there e x i s t s a non-zero e l e m e n t tt in B such t hat B[ 1 n] = A { \ / n \ a n d t he c anoni c al m a p B / n B —> A/ nA is injective. I hen B = A.

P roof. S in c e the m ap B nB — A/ n A is in jectiv e, it is e a sy to see th a t B

### n

n nA = n”B fo r all n - I. Let a £ A. T h e n a = bin" fo r so m e b € B an d n o n -n e g a tiv e in te g e r n.

T h e re fo re h --- n"a E n"B. H e n c e a £ B.

L e m m a 2 .2 . Let B he g e o m e t r i c a l l y i nt egr al o v e r t he f i e l d k. Then k is al g e b r a i c a l l y c l o s e d in B.

Proof. Let 1. be the a lg e b r a ic c lo s u re o f k in B. T h e n L B is an in te g ra l do m ain . S u p p o s e th a t L f k an d le t a € L \ k . L et / b e th e m in im a l p o ly n o m ia l o f a o v e r k an d let /_, = k ( a ) = k [ X ] l ( f ( X)). T h en L\ *>k £ ( - > L ® k B ) is an in teg ral d o m a in . O n the o th e r h a n d , L\ :■:* B { ^ B [ X ] K f ( X ) ) ) c a n n o t b e an in te g ra l d o m ain s in c e ( X - a) is a fa c to r o f f ( X ) in B [ X ] , T h e c o n tra d ic tio n sh o w s th a t L = k.

W e n o w sh o w th a t o v e r a p e r f e c t field k, a n y A * -fo rm h a v in g n o n -triv ia l u n its is A*.

Proposition 2 .3 . L e t k he a f i e l d a n d let B b e a k - a l g e b r a such t hat B* ^ k ■. Su p p o s e that t her e e x i s t s a s e p a r a b l e f i e l d ext e ns i on L o f k such t hat L B is A* over L.

Then B is A* ov e r k.

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Proof. L e t L(g>k B = L [ T , T ~ ' ] . W e id e n tify B w ith its im a g e in 1. k B. It is e a sy to see th a t B is fin ite ly g e n e ra te d o v e r k. H e n c e , th e re e x is ts a fin ite ly g c n c ia te d s e p a ia h le e x te n sio n L\ o f k su ch th a t L\ x\( B = L][T. T ~ y\. T h u s , re p la c in g I. by l.\ . w e m a \ a ssu m e L to b e fin ite ly g e n e ra te d o v e r k to start w ith.

W e first c o n s id e r th e c ase w h e n L is finite a lg e b r a ic o v e r A. R e p la c in g L b y its sp littin g field , w e m a y a ss u m e L to b e finite G a lo is o v e r k w ith G a lo is g ro u p ( / . s a \ . A n y a £ G c a n b e e x te n d e d to a 5 -a u to m o rp h is m o f L k B ( —L\T. T 1 ]) by d e fin in g o ( x ® b) = a ( x ) ® b fo r x £ L, b e B. L et

T = ClQ 0 1 + a 1 55 C\ + ' ' ■ + Or i &r-

w h ere l , e \ , . . . , e r fo rm p a rt o f a & -basis o f B a n d a, £ L. S in c e L is G a lo is , th e b ilin e a r m ap L x L —> k g iv e n b y (x, v ) —> T r a c e ( x y ) is n o n -d e g e n e r a te . H en ce, re p la c in g T b y a T ( a £ L) i f n e c e ss a ry , w e c a n a ss u m e th a t Tr(cij) ^ 0 fo r so m e / > 1. T h u s .

W = ^ a ( T ) = Tr ( a0) ® 1 + T r ( a\ ) x < ? ! + • • • + T r { a r ) e r is an e le m e n t o f B \ k; in p a r tic u la r , W 0.

W e n o w sh o w th a t B = k [ W , 1 j W \ L et / £ B* \ k * . S in c e k is a lg e b ra ic a lly c lo s e d in B b y ( 2 .2 ) , / is tr a n s c e n d e n ta l o v e r k a n d h e n c e o v e r L. T h e re fo re , f = aT'" for so m e a £ L* a n d so m e n o n -z e r o in te g e r m. R e p la c in g / b y 1 / i f n e c e ss a ry , w e m ay a ssu m e m > 0. S in ce B is in v a r ia n t u n d e r e v e ry a £ C , w e h a v e

a T m = f = G ( f ) - - = a { a ) ( o ( T ) ) m.

S ince a ( a ) £ L * , th e a b o v e r e la tio n sh o w s th a t ( (a ( T ) ) / T ) m £ L*\ an d h e n c e ( o ( T ) ) T ; L * . T h e re fo r e , a ( T ) = a„ T fo r s o m e a a G L*. H en ce, W = a T fo r so m e a £ L. S in c e W 7^ 0 , a £ L * . T h erefo re, L[ W, 1 j W ] = L[T\ = L B. N o w , L b e in g a fin ite e x ­ te n s io n o f k, L(8ik B is in teg ral o v e r B. H e n c e , B D ( L X/,- B) * = B * . T h e re fo re , 1 IV £ B.

N o w , as k [ W, l / W ] CB, b y fa ith fu l fla tn e ss o f L o v e r k, it fo llo w s th at B = k[l V. 1 Il'J.

W e n o w c o n s id e r th e c ase w h e n L h a s p o sitiv e tr a n s c e n d e n c e d e g re e o v e r k. N o w , sin ce L is a fin ite ly g e n e ra te d s e p a r a b le e x te n sio n o f k, th e r e e x is ts a p u re ly t r a n s c e n ­ d e n ta l e x te n s io n K = k ( X\ , . . . , X „ ) o f k such th a t L is a fin ite s e p a ra b le e x te n s io n o f K.

S ince L ® K( K ^ kB ) = L [ T , 7’- 1 ], b y th e p re v io u s ca se , it fo llo w s th a t K J i K \ U . I II 1 fo r so m e W £ K B. S in c e B is fin ite ly g e n e ra te d o v e r k, it is e a sy to see th a t th e re ex ists a p o ly n o m ia l F ( X U . . . , X „ ) £ k [ X u . . . , X „ \ s u c h th a t

B \ X U .. ,X„, 1 / F ( X { , . . . , X n) } = k [ X u . . . ,X„, l / F ( X i , . . . , X n), IV, 1 ) W \ ( * ) I f k is a n in fin ite field, th e n w e ca n ch o o se e le m e n ts £ k s u c h th a t F ( c \ , . . . , c n) / 0. L e t N b e th e m a x im a l ideal o f k [ X \ , . . . , X „ , \ / F ( X \ , . . . , X n)] g e n e r ­ ated b y X\ — c i , . , . , X n — c„, \ / F ( X \ , . . , , X „ ) — l / F ( c i, . . . , c„). F ro m E q. (* ) , it fo llo w s , b y ta k in g q u o tie n t m o d u lo th e id e a l N , th a t B is A* o v e r k.

I f k is a fin ite field, le t N b e a n y m a x im a l id e a l o f k [ X u . . . , X „ , \ / F ( X U . . . , X n )]

an d le t k' = k [ X u . . , , X„, 1 / F { X x , X „ ) ] / N . T h e n k' is a fin ite v e c to r sp a c e o v e r k b y H ilb e r t’s N u lls te lle n sa tz an d s e p a r a b le o v e r k. S in ce k' ® k B is A* o v e r k' b y E q . ( * ) , it fo llo w s, b y th e p rev io u s c a s e , th a t B is A* o v e r k.

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R e m a r k 2 .4 . T h e a s s u m p tio n th a t B* ^ k* is e s s e n tia l in th e a b o v e resu lt. F o r in stan ce, c o n s id e r th e c o -o r d in a te r i n g o f th e re a l c irc le , i.e ., B = R [X , Y ] / ( X 2 + Y 2 — 1 ). T h e n C r B is A ' o v e r C , th o u g h B is n o t A* o v e r R.

3. M a i n th e o r e m s

In th is s e c tio n w e sh a ll p r o v e o u r m a in resu lts. W e first p ro v e a p a tc h in g L e m m a 3.1 an d d e d u c e a stru c tu re T h e o re m 3 .4 fo r lo c a lly A * -a lg e b ra s. N ex t w e p ro v e o u r resu lt (3 .1 1 ) on A “ -Ite ra tio n o v e r K ru ll d o m a in s an d fin a lly w e in v e stig a te th e g e n e ra l case (3 .1 3 ).

L e m m a 3 .1 . L e t R be an i n t e g r a l d o ma i n wit h q u o t i e n t f i e l d K a n d let A b e a fl at R-al gehr a. S u p p o s e t h a t t h e r e e x i s t s non- z er o e l e m e n t s x , y in R such t h a t

( i ) .r a n d y ei t her f o r m a n R- s e q u e n c e or ar e c o m a x i m a l in R.

( i i) A[ 1 .y] is A* over /?[l/.v ], (iii) A[ 1 v] is A* o v e r / ? [ l / v ] .

Then t her e e x i s t s an i n v e r t i b l e i d e a l I in R such t h a t A = ( CK [ T , T '] ) us a Z- g r a d e d R-al gebra.

P ro o f . L et

^ .v = 0 i ( t r a n d A r = @ R v W".

WE'/. hGZ

T h en 4v.v =

n^/ n£Z

T h e re fo re , it is easy to see th a t W is e ith e r /. T o r / . T ~ ] fo r so m e / £ R*y . R e p la c in g T b y T - ' i f n e c e ss a ry , w e a s s u m e th a t W = /.T. L e t ?. = a / x my m w h e re a G R a n d m is a n o n -n e g a tiv e integer. A g a in , re p la c in g IV b y y"' W a n d T b y T/xm, w e a s s u m e th a t

W = a T fo r so m e a & R

### n

R*v.

Since A is /f-flat an d (Rx T " ) y = ( R y W n)x, u sin g c o n d itio n ( i) , it is e a sy to se e th a t A = Ay PI Ay = A„,

fl(EZ w h ere

A„ = Rx T n n R y W ” = ( R x n a nR y ) T n = { R

### n

a nR y ) T n. ( * ) T h u s A„ is /?-flat fo r e v e ry n. N o te th at, b y c o n d itio n ( i ) , Aq( ~ Rx D Ry ) = R, sh o w in g th a t A is a Z -g ra d e d 7?-algebra. N o w let

I = R P i a R r a n d J = R H a R x.

T h erefo re,

IX = RX, l v = a R y , J x = aRx an d J y = R y .

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W e n o w s h o w th a t I J -= aR. S in c e Rx n R y — R. c le a rly I f'\ J — a R so th a t U ~ aR.

L et M d e n o te th e m o d u le aR/ I J . R e c a ll th a t A\ = I T is /?-flat s o th a t I is flat o \ e i R.

H en ce I/ IJ is flat o v e r R/ J. F ro m th e c o n stru c tio n o f J it fo llo w s e a s ily th at .v is a n o n -z e ro d iv is o r in R / J . H e n c e , b y fla tn e ss , x re m a in s a n o n -z e r o d iv is o r in / /./ a n d h e n c e in th e s u b m o d u le M . B u t M x = a RxjIxJx = 0. H e n c e M = 0. i.e., IJ — ciR. T hus / is an in v e rtib le id e a l o f R.

L e t 5 = ® „ e z F T " . S in ce F C R C \ a nR y fo r all «, b y ( * ) , B Q . 4 . I b e in g in v e rtib le . B is 7?-flat. H e n c e , fro m c o n d itio n ( i ) , it fo llo w s th a t B = Bx n B v. N o w , sin c e a is a u n it in Rxy a n d JX= R X, it fo llo w s f r o m ( * ) th a t ( A„) x = RXT" = I" RXT", so th a t i . />’ . S im ila rly ( A n) v = a ”R v T" = I nR v T ”, s o th a t A v = B v. T h e re fo r e . A = A, O .1, = B y B y = B = ® n e z r r ’: □

E x a m p le 3 .2 . T h e a ssu m p tio n o f fla tn e s s is e ss e n tia l in L e m m a 3 .1. F o r in s ta n c e , let R — C [ X , Y , Z , W y { X Y — Z W ) a n d le t x , y , z an d w b e th e im a g e s in R o f A’. Y . Z a n d W, re s p e c tiv e ly . L e t I = ( x , z ) R a n d le t A ~ R [ 1 T , I ~ ' ' T - 1 ]. T h e n c le a rly A is n o t /? -lla t alth o u g h A x a n d A v are A* o v e r R x a n d R y, re s p e c tiv e ly .

W e s h a ll n o w a p p ly th e p a tc h in g L e m m a (3 .1 ) to p ro v e a s tru c tu r e th e o re m fo r locally A* a lg e b ra s. F o r c o n v e n ie n c e , w e first p ro v e th e s tru c tu r e th e o re m o v e r s e m i-lo c a l n o e th e ria n d o m a in s.

L e m m a 3 .3 . L e t R b e a s e m i - l o c a l i n t e g r a l d o m a i n a n d l e t A he an R - a l g e h r a wh i c h is local l y A* o v e r R. Then A is A* o v e r R.

P ro o f . C le a rly , A is fin itely g e n e ra te d a n d flat o v e r R. L e t b e th e m a x im a l ideals o f R. W e p ro v e th e re s u lt b y in d u c tio n on n, th e n u m b e r o f m a x im a l id e a ls o f R.

I f n = 1, th e re is n o th in g to p ro v e . So let n > 2 a n d a s s u m e th e re s u lt w h e n th e n u m b e r o f m a x im a l id eals is < n - 1. L et Si = R \ (P\ U • • • U j ) an d S 2 = R \ P „ . B y in d u c tio n h y p o th e s is , S ] ]A a n d are A* o v e r S ^ l R a n d S / f ' R , re s p e c tiv e ly . S in c e A is fin ite ly g e n e ra te d o v er R, it fo llo w s e a sily th a t th e r e e x is ts a p a ir o f e le m e n ts x £ S \ , y € S2 su c h th a t A[ 1/x] a n d A [ \ / y ] are A* o v e r R [ \ / x ] a n d R [ l / y ] , re s p e c tiv e ly . C learly , x a n d y are c o m a x im a l so th a t fro m (3 .1 ) it fo llo w s th a t A is A* o v e r R. □ W e n o w p ro v e th e stru c tu re th e o r e m fo r lo c a lly A* a lg e b ra s .

T h e o re m 3 .4 . L e t R be an i n t e g r a l d o m a i n which is e i t h e r n oe t he r i an o r a K r u l l domai n. L e t A b e a f i ni t e l y g e n e r a t e d R- a l g e b r a whi ch is l o c a l l y A" ov e r R. Then there e x i s t s an invertible i de al I in R such t hat A is i s o m o r p h i c to R [ I T , I ~ l T ~ ' ] a s a Z - g r a d e d R- s ub a l g e b r a o f K [ T , 7 " 1], whe r e K is t he q u o t i e n t f i e l d o f R.

P r o o f . S in c e A is fin itely g e n e ra te d , fro m th e g iv e n c o n d itio n , it is e a s y to s e e th a t th ere e x is ts x 6 R su ch th a t ^ 4 [l/x ] is A* o v e r R11 x j. I f x c R ’ . w e are th r o u g h . I f not, th e n sin c e R is eith er n o e th e r ia n o r a K ru ll d o m a in , x R h a s fin ite ly m a n v c r im e

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d iv iso rs. Let P \...P„ b e th e p rim e d iv iso rs o f xR. L e t S — R \ ( P \ Li ■■■ Li P n). T hen S lR b e in g a se m i-lo c a l in te g ra l d o m a in , b y (3 .3 ) , is A* o v e r .S'- 1 ./?. H e n c e there ex ists v c. S such th a t .([1 v] is A* o v e r ^ [ 1 / y ] - B y c o n s tru c tio n , x an d y e ith e r fo n n an W -sequence o r are c o m a x im a l. H en ce, A b e in g fiat, th e re s u lt fo llo w s fro m (3 .1 ).

## □

By a re s u lt o f A s a n u m a [1, 3 .4 ], an A '-fib ra tio n o v e r a n o e th e ria n rin g R is n e c e s­

sarily an fl-s u b a lg e b ra of a p o ly n o m ia l a lg e b ra o v e r R\ in p a rtic u la r, th e re is a re tra c t tro m A to R (i.e ., an /^ -a lg e b ra h o m o m o rp h is m fro m A to R). B y c o n tra st, th e fo llo w ­ ing c o ro lla ry sh o w s th a t e v e n w h e n A is lo c a lly A* o v e r a n o e th e ria n d o m a in R, th ere w o u ld b e a re tra c t fro m A to R i f a n d o n ly i f A is it s e l f A* o v e r R.

C o r o ll a r y 3 .5 . Le t A he a f i ni t e l y g e n e r a t e d l o c a l l y A* al ge br a over a noet her i an d o m a i n R. S u p p o s e t h a t t h e r e e x i s t s a r e t r a c t f r o m A t o R. Then A is A* o v e r R.

P r o o f . B y (3 .4 ) . A = © „ eZ I " T ” fo r so m e in v e rtib le id e a l / o f R. L e t </> b e a re tra c t fro m A to R. L et = (j)(IT) a n d J 2 = <p(l ' T ~ {). W e sh o w th a t th e id e a ls J\ a n d J 2 o f R are a c tu a lly th e u n it id e a l. L e t a t , . . . , a n e l a n d b\ , . . . , b„

### e

/ “ ' b e su c h th at

1 = £ cfbj = '}2(ai T ) ( h j T~ 1). T h e re fo re , 1 = </>( 1) = ] T (j> (a ,T )< P (b J -] )

### e

J \J 2

sh o w in g th a t J \ J 2 = R a n d h e n c e J\ = J 2 = R. T h u s th e re is an /?-su rjectio n fro m 1 to R sh o w in g th a t / is p rin c ip a l. T h e re fo re A = R[T, T ~ 1].

E x a m p le 3 .6 . T h e a s s u m p tio n o f finite g e n e ra tio n is e ss e n tia l in T h e o re m 3.4. F or in s ta n c e , c o n sid e r R = Z a n d A = Z [ X / 2 ,2 / X , X / 3 , 3 / X , . . . , X / p , p / X ,. . . ] w h e re p varies o v e r th e set o f p rim e in te g e rs. T h e n Qt i z A = Q [ X , \ / X \ a n d Z ( P)!8)zA = Z ( p ) [ X / p , p / X ] fo r e a c h p rim e in te g e r p. T h u s A is lo c a lly A* o v e r R. B u t A is n o t fin ite ly g e n e ra te d o v e r R.

W e n o w in v e stig a te m in im a l su fficien t c o n d itio n s fo r a fin itely g e n e ra te d o v e rd o m a in o f a d is c re te v a lu a tio n rin g to b e A*.

P r o p o s itio n 3 .7 . L e t R b e a di s c r e t e val uat i on ri ng wi t h uniformi si ng p a r a m e t e r n a n d res i due f i e l d k. L e t A b e a f i ni t e l y g e n e r a t e d o v e r d o m a i n o f R such t h a t

( i ) The generi c f i b r e A [ \ / n ] is A* o v e r R [ \ / n \

( ii) The c l o s e d f i b r e A / n A is g e o m e t r i c a l l y i n t e g r a l o v e r k.

Then t here are p r e c i s e l y t w o p o s s i b i l i t i e s : ( a ) I f ( A / n A )* ^ k * , t hen A is A* o ver R.

( b ) I f ( A/ nA) * = k *, t hen A = R [ X , Y ] / ( n mX Y + olX + [ 1 Y + y ) f o r s o m e a, [3 G R * , y & R a n d p o s i t i v e i nt eger m. In p ar t i c ul ar , A / n A = k^K

Proof. L e t A = R [ t ] , . . . , t p ]. S in c e n is a p rim e e le m e n t in A a n d A [ \ / n \ is fa c to ria l, it fo llo w s th a t A is fa c to ria l. F r o m th e fa c to ria lity o f A, it is e a sy to see th a t th e r e ex ists

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an e le m e n t T £ A su ch th a t T ~ l G A a n d A [ \ / n \ = / ? [ l / 7t ] [ 7’, T 1 ]. L et A tt = R\ I . I |.

[ f Ao = A , th e n A is A* o v er R.

S u p p o se th a t Aq ^ A. L e t xq= T , y o = T~ 1 an d Fa(X. Y ) = X Y — 1. F o r an e le m e n t a A.

d en o te its im a g e in A / n A b y a. S in c e A \$ \ \ / n \ = A [ \ j n \ a n d Ao ^ A by h y p o th e s e s , th e c a n o n ic a l m a p A 0/ n A 0 —> A / n A c a n n o t b e in je c tiv e by ( 2 .1 ) . T h e re fo r e , d i m ( k|.v^. TTi | ) 0. S ince k is a lg e b ra ic a lly c lo s e d in A / n A b y ( 2 .2 ) a n d .% T o = 1, it fo llo w s th at .v(), yo • k*. H e n c e th e r e e x is t x \ , y \ € A a n d /-o,/io £ R* such th a t .\'o = 7t.V| + / . o and yn = 7r V| fh>.

L e t A \ = R [ x i , y \ ] , C le a rly A 0 C A f . S in c e /.0/70 = .v 0y 0 = 1, it fo llo w s th at - I --=ny, fo r som e e le m e n t 71 € R. N o w ,

F 0( n X + /o , n Y + //0) = ( n X + /.0 ) ( n Y + /<0 ) - 1

= n 2X Y + n j j g X + n/.0 Y + ny\ = n F \ ( X . Y).

w h e re F\ 6 Rp^. N o te th at, b y c o n s tru c tio n , F \ ( X , Y ) = n X Y + y. \X + ( i \ Y + 71. w h e re ai(= jU o ) € R * , ( i \ { = / .0 ) £ R* a n d 71 € R. T h e re fo re , F\ is ir re d u c ib le a n d h e n c e p rim e , a n d F \ ( x \ , y \ ) = Q (s in c e F(xo, y o ) = 0 ). H e n c e it fo llo w s th a t A\ = R[X. Y] { F \ ( X . >’ )).

I f ^ i = A , th e n w e are th ro u g h ( s in c e in th is c ase, ( A / n A ) * = ( k 1^)* = k* an d s ta te m e n t ( b ) is s a tisfie d ).

I f A] A, th e n w e sh o w th a t th e re e x is ts a fin ite in c r e a s in g c h a in o f rin g s Aq C A i C ■ ■ ■ C A „ c A „ + i C • ■ • C A „ , = A w ith A„ = R [ x„ , v„], a n d a se q u e n c e o f ir ­ re d u c ib le p o ly n o m ia ls F „ ( X , Y ) £ R[ X, K ](= /? [21), (1 < n < m ) , sa tis fy in g c o n d itio n s (I ) an d (11) b e lo w fo r 1 < n < m, a n d th e re c u r re n c e re la tio n s (I II) an d ( I V ) fo r

1 < n < m — 1.

( I ) F„(X, Y ) = n”X Y + anX + f j „Y + y„, w h e re y„ £ R a n d x„, /}„ £ R ’ .

( I I ) F „ ( x „ , y „ ) = 0 an d th e m a p R[ X, Y]/ F„(X, Y ) —> A„ d e fin e d b y X — .v„. Y — y„

is a n iso m o rp h ism .

(I II) x„ = Tixn+X + a „, y„ = n y n+\ + fo r so m e £ R.

(I V ) F„(x„+ i, y n+\ ) = t f „ + i(j:„ + i j „ + 1 ).

W e h a v e a lre a d y defin ed A\ a n d F\ s a tisfy in g c o n d itio n s ( I ) a n d ( I I ) fo r n = I . A ssu m e th a t w e h a v e d efin ed u p to A„ = R [ x „ , y „ ] a n d F„, fo r so m e in te g e r n > 1, su ch th a t c o n d itio n s ( I ) an d ( I I ) h o ld . W e sh o w th a t i f A„ ^ A, th e n it is p o s s ib le to co n stru c t A „ +l = R[ x „ + \ , y n+ \ \ a n d Fn+X u s in g re la tio n s ( I I I ) a n d ( I V ) , such th a t A„ is a p ro p e r s u b r in g o f A„+] an d c o n d itio n s ( I ) a n d ( I I ) a re s a tis fie d b y An + , an d F„ t ,.

S in ce A n\ \ / n \ = A [ \ / n \ a n d A„ ^ A, it fo llo w s, b y a r g u in g as in th e c a se n = 1, th a t x „ , y n g k. L e t G R b e s u c h th a t x„ = a n d ~y// = Jl//. H e n c e th e r e e x is t x t!+i , y „ + , e A su c h th a t re la tio n ( I I I ) h o ld s . L e t A„+ , = R [ x „ + i , y „ + ,]. C le a rly A„ C A„ , ,.

S ince b y in d u c tio n h y p o th e sis, c o n d itio n ( I ) is v a lid fo r F„, w e h a v e

F „ ( n X + a „, n Y + [i„) = % \ % X + ) ( n Y + fi„) + ot „( nX + ) + jJn( n Y + /;„) + y„

= n "+ 2 x Y + nX(cc„ + n ”n") + n Y ( / ] n + n "l „ ) + F„(A „, ).

N o w s in c e ( I I ) is v alid fo r F„, w e h a v e , 0 = F n( x n, y n) = F „ ( n x n+x + n y „ +t + w h ich s h o w s th a t F„( l „ , j i „ ) G nA D R = nR. L e t y„+l = F „ ( l „ , j . i „) / n( e R) . N o w b y th e

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p re v io u s e q u a tio n s w e c a n d e fin e

F„ . i ( .V. Y ) = F „ ( n X + n Y + / / „ ) /7i

= tt" 1 X Y + y n . | X + ji„ | i Y + y„ : i ,

w h ere y.n . \ ( = y.„ + n"fi„) G R*. ji,n , ( =/ !„ + n"/.„) G R* a n d y n ] \ are all e le m e n ts o f R.

T hus c o n s tru c tio n sh o w s th a t F„ , i is an irre d u c ib le ( a n d h e n c e a p rim e ) e le m e n t o f R[.\. } ] w h ic h satisfies c o n d itio n (1) an d the re c u rre n c e re la tio n (I V ). M o re o v e r,

F>, ■ i (-v„. i . y„ . i ) = F„(.y„, y„ ) / n = 0.

It fo llo w s th a t th ere is an ^ - is o m o r p h is m R [ X , Y ] / ( F „ ^ \ ( X , Y ) ) —> A„+ \ m a p p in g the im a g e s o f X and Y to .v „ , i a n d y „ - i , re sp e c tiv e ly . T h u s ( I I ) h o ld s fo r th e p a ir F„ ,i an d A „ .i .

W e n o w sh o w th a t A„ / A, n i . R e c a ll th a t A = R [ t j , . . tp\. L e t I „ b e th e le a s t in te g e r su c h th a t G A„ V /. 1 < j < p . S uch an in te g e r e x is ts since A „ [ \ / n \ = A [ l / n \ . M o re o v e r, f n > 0 sin c e A„ ^ A. H e n c e th ere e x ists 4>G such th a t

n " t j = (p( x„, y„)

= <p(nxnl l + z y n+] + /<„)

= <£(/.„,/<„) + n0 ( x „ M ,y „ + i)

fo r so m e 0 G S in ce / „ > 0, (/>(/.„,/;„) G nA n R = nR. T h u s if M X Y ) = n „ ) / n + 6 ( X Y ) ,

th en

7T tj lj/(xn^ | , V/,1 ] ) G A„+ \ sh o w in g th a t

0 < < / „ - \ <

T h is sh o w s th a t A„ ^ A n+\.

S in c e th e ch ain o f in te g e rs 0 < ■ ■ < ( „ < ■ ■ ■ < A)

o b v io u s ly c a n n o t b e in fin ite , th e re e x is ts a p o s itiv e in te g e r m fo r w h ic h t m = 0 , i.e., A„, = A . In p a rtic u la r, A / n A = b y c o n stru c tio n o f A m.

W e c a n n o w d ed u ce c o n c lu s io n s ( a ) an d (b ).

( a ) I f ( A/ nA) * k * , th e n A = Aq (fo r o th e rw ise , b y o u r p re v io u s a rg u m e n ts , A —A m fo r a p o s itiv e in te g e r m a n d h e n c e ( A/ nA) * = ( & ^ ) * = k * , a c o n tra d ic tio n ). T h u s A is A* o v e r R.

(b ) I f ( A/ nA) * = k*, th e n o b v io u s ly A A 0 a n d h e n c e A = A m fo r so m e p o sitiv e in teg er m a n d th e re fo re b y c o n d itio n s ( I ) a n d ( I I), A = R[X, Y ] / ( n mX Y + a X + fi Y + y) fo r so m e a, ft G R* a n d y G R.

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R e m a r k 3 .8 . U n lik e the c ase o f A 1-fib ra tio n , th e c o n d itio n o f g e o m e tric in te g ra lity on th e c lo s e d fib re is n o t su fficien t to e n s u re th a t a fin itely g e n e ra te d o v e rd o m a m o f a d isc re te v a lu a tio n rin g , w h o se g e n e ric fibre is A*, is i t s e l f A ' . In la c t it is e asy to see th a t i f (R, n , k ) is a d is c re te v a lu a tio n rin g , th en fo r an y y.. ft e R \ y € R an d p o s itiv e in te g e r m, R[ X, Y ] / ( n mX Y + aX + f l Y + 7 ) is a fin itely g e n e ra te d fiat tf-a lg e b ra w h o s e g e n e ric fib re is A* a n d clo se d fib re is A1.

C o ro lla ry 3 .9 . L e t R be a P r i n c i p a l I d e a l D o m a i n wi t h q u o t i e n t f i e l d K a n d s u p p o s e t hat A is a f i n i t e l y g e n e r a t e d o v e r d o m a i n o f R such t hat

( i ) The g e n e r i c f i b r e K CEj rA is A* ov e r K .

( ii) Each c l o s e d f i b r e A/ PA is g e o m e t r i c a l l y i nt egral hut is no t A 1 o v e r R P.

Then A is A* o v e r R.

P r o o f . L e t P b e a m a x im a l id e a l o f R. B y the h y p o th e s e s . Rp is a d isc re te v a lu a tio n rin g an d A P is a fin itely g e n e ra te d fla t /?/>-algebra w h o se g e n e ric fibre is A”. an d w h o s e clo se d fib re k{P) <g)nA is g e o m e tric a lly in te g ra l, but k ( P) ZrA ^ k ( P)•'!. B ut th e n , by p a rt (b ) o f P ro p o s itio n 3.7, ( k ( P ) ' S R A) * ^ k ( P) * . T h e re fo re , b y p a rt ( a ) o f ( 3 .7 ) . A r is A* o v e r Rp.

T h u s, A is lo c a lly A* o v e r R. S in c e e v ery in v e rtib le id e a l o f a P ID is p rin c ip a l, b y T h e o re m 3 .4 , it fo llo w s th at A is A* o v e r R.

R e m a rk 3 .1 0 . S u p p o se th a t R is a P ID an d A is a fin ite ly g e n e ra te d flat tf -a lg e b ra su ch th a t th e g e n e ric fibre is A* a n d all the clo se d fib re s are g e o m e tric a lly in te g ra l.

T h en , b y ( 3 .7 ) , e a c h c lo se d fib re is e ith e r A* o r A 1. It is p o s s ib le th a t so m e a re A ' an d so m e A 1. F o r in stan ce, le t i ? b e a P ID w ith tw o m a x im a l id e a ls ( n\ ) an d ( n 2 ). Let A = R[X, Y ] / ( n2X Y + n tX + 7i2 7 + 1). T h en th e g e n e ric fib re o f A is A*, th e c lo s e d

fibre A/ t z]A is A* b u t th e c lo s e d fib re A / n j A is A1.

W e n o w p ro v e o u r m ain th e o r e m o v e r K ru ll d o m a in s.

T h e o re m 3 .1 1 . L e t R b e a K r u l l d o m a i n wit h quot i ent f i e l d K a n d let A he a f i n i t e l y g e n e r a t e d f l at R- al gebr a such t h a t

( i ) Th e g e ne r i c f i b r e K ® R A is A* o v e r K .

(ii) Fo r each p r i m e ideal P o f R o f hei ght one, the f i b r e r i ng k ( P) A is g e o m e t ­ r i cal l y i nt e g r a l but is n o t A 1 o v e r k ( P ) .

Then t h e r e e x i s t s an i nverti ble i d e a l I in R such t h a t A is i s o m o r p h i c to © „ e z T' T"

as a Z - g r a d e d R-al gebra. "

P ro o f . S in c e A is finitely g e n e ra te d o v e r R, b y c o n d itio n ( i ) , th e re e x is ts a n o n - z e r o e le m e n t x e R su c h th a t A [ \ / x } is A* o v e r R[ 1/x], I f x e R * , w e are th ro u g h . I f n o t, th en le t P x, . . . , P m be th e p rim e d iv is o rs o f x R an d le t 5 = R \ (/>, u . . . u pm). S in c e R is a K ru ll d o m a in , ht P t = 1 V/', 1 < i < m. T h e re fo re S ~ ]R is a se m i-lo c a l D e d e k in d d o m a in a n d h e n c e a PID . It fo llo w s , b y (3 .9 ), th a t S ~ ’A is A* o v e r S ~ lR. H e n c e th e r e

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e x is ts y t S such th a t A v is A ' o v e r R v. S in ce b y c o n stru c tio n x an d y e ith e r fo rm a s e q u e n c e o r are c o m a x im a l. th e re s u lt n o w fo llo w s fro m ( 3 . 1). □

R e m a r k 3 .1 2 . N ote th a t th e e x a m p le in (3 .1 0 ) sh o w s th a t, in th e s ta te m e n t o f T h e o re m 3 .1 1 . it is n e c e ssa ry to im p o s e th e co n d itio n th a t th e c o d im e n sio n o n e fib re s are not A 1. F ro m P ro p o sitio n 3 .7 , it fo llo w s (a s su m in g all o th e r h y p o th e s e s in ( 3 .1 1 ) ) th a t any c o d im e n s io n on e fibre w h ic h is n o t A 1, is a u to m a tic a lly A*. A lso n o te th a t, b y (3 .7 ), th e c o d im e n s io n o n e fib re s are A 1 i f an d o n ly i f th e y d o n o t h av e n o n -triv ia l units.

T h u s , c o n d itio n ( i i) in T h e o r e m 3.11 w ill be sa tisfie d , fo r in stan ce, u n d e r e ith e r o f the fo llo w in g h y p o th e s e s on th e fib res a t th e p rim e id e a ls P o f R o f h e ig h t o n e:

( i i ) ' i A ( / 'i R A y / l A i / ’ ) I . ( i i ) " k ( P ) - R A are A M o rm s .

W e n o w in v e s tig a te th e g e n e ra l case.

T h e o r e m 3 .1 3 . L e t R b e a no et her i an do ma i n wi t h quot i ent f i e l d K a n d l et A be a fi ni t el y g e n e r a t e d f i a t R - a l g e b r a sueh t hat

( i ) The generi c f i b r e K .'.rA is A* over K.

( i i) For each p r i m e i d e a l P o f R o f hei ght one , t he f i b r e ring k ( P ) Hr A is g e o m e t ­ r i cal l y int egral but is n o t an A 1- f or m over k ( P ) .

Then the f o l l o w i n g r e s u l t s h o l d: ( a ) A l l the f i b r e ri ngs a r e A*-forms.

( b ) Ther e e xi s t s a f i n i t e bi r u t i o n a l ext ens i on R' o f R a n d an inverti ble i d e a l I o f R' sueh that R' v.rA is a Z - g r a d e d R- a l g e b r a i s o mo r p h i c to R ' [ I T , I ~ l J - 1 ].

( c ) I f R cont ai ns a f i e l d o f charact eri s t i c zer o, a n d al l the f i br e ri ngs ha v e mo r e uni t s than the r e s p e c t i v e resi due f i e l d s , then A is an A*-fibration o v e r R.

P r o o f . S u itab le m o d ific a tio n s in th e a rg u m en ts in [3, 3 .5] w o u ld g iv e a p r o o f o f (a ).

F o r th e c o n v e n ie n c e o f th e re a d e r w e sk etch th e p r o o f b elow .

Fix a p rim e ideal P o f R. R e p la c in g R b y Rp, w e a ss u m e th a t R is a lo c a l n o e th e ria n d o m a in w ith m a x im a l id e a l P. W e p ro v e (a ) b y in d u c tio n on ht P = d i m R. T h e case h t P — 0 is triv ial.

I f ht P ( = d i m R ) = 1, th e n , fro m the K r u ll- A k iz u k i th e o re m ([5 , 3 3 .2 ]) , it w o u ld fo llo w th a t th e n o rm a lis a tio n R o f R is a P ID a n d k { P ) are a lg eb raic e x te n s io n s o f k ( P ) fo r all m a x im a l id e a ls P o f R. T h e re fo re , fro m c o n d itio n (ii), it fo llo w s th a t for all PM a x R , k ( P ) ( R ®rA ) are g e o m e tric a lly in te g ra l b u t are n o t A 1 o v e r k ( P ) . M o re o v e r, b y co n d itio n ( i ) , th e g e n e ric fibre o f R<S>rA is A* o v er R. H e n c e , b y (3 .9 ), R %r A is A* o v er R. In p a rtic u la r, k ( P ) A is A* o v e r k ( P ) \ / P € M a x R . H ence k ( P ) Sir A is an A *-form o v e r k ( P ) .

I f ht P > 2 , th e n , b y in d u c tio n h y p o th e sis, w e a s s u m e th a t the fibre rin g s k(Q)<S)RA are A *-form s fo r all n o n -m a x im a l p rim e id eals Q o f R. L e t R d en o te th e c o m p le tio n o f R a n d le t A = R &rA. N o w R is a c o m p le te lo c a l rin g w ith m ax im al id e a l P su ch th at R / P = R / P an d A is a fin ite ly g e n e ra te d flat .R -algebra w h o se n o n -c lo se d fib res are all

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A *-form s. M o r e o v e r, i f Q is a m in im a l p rim e ideal o f R, th e n , s in c e R is /?-Hat. b y the “ g o in g -d o w n th e o r e m ” , Q c o n tr a c ts to ( 0 ) in R. H e n c e , it fo llo w s from c o n d itio n ( i ) th at, th e fib re s o f A at all m in im a l p rim e id eals o f / ? a re A ’ . Let be a m in im a l p rim e id e a l o f R su c h th a t d i m R = d i m ( R / Q a ). T h e n , re p la c in g /? by /? and A b y A / QqA, w e m a y a ss u m e R to b e a c o m p le te lo cal n o e th e r ia n d o m a in to start w ith , a n d a ssu m e A to b e a fin ite ly g e n e ra te d flat 7?-algebra such th a t th e g e n e ric fibre of . / is A ' a n d th e fib re s a t a ll n o n -m a x im a l p r im e id e a ls o f R a re A * -fo rm s. In ( |5 . 3 2 . 11). th e n o rm a lisa tio n R o f R is a finite /? -m o d u le a n d h e n c e a n o e th e r ia n n o rm a l local d o m a in . N o w , as b e fo r e , it w o u ld fo llo w th a t R Hr A is A* o v e r R s h o w in g th at k ( P ) r A is a n A *-form o v e r k ( P ) .

W e n o w p ro v e (b ) . L e t R d e n o te th e n o rm a lis a tio n o f R a n d let A = R « A. B y a th e o re m o f N a g a ta [5, 3 3 .1 0 ], R is a K ru ll d o m ain . C le a rly A is a fin itely g e n e ra te d flat a lg e b ra o v e r R an d its g e n e ric fib re is A*. M o re o v e r, s in c e th e re s id u e fields o f /?

are a lg e b ra ic o v e r th e re s id u e fie ld s o f R, b y re s u lt ( a ) , all fib re s o f A are A’ - f o r m s o v er th e ir re s p e c tiv e re sid u e field s. H e n c e , b y T h e o re m 3 .1 1 , th e re e x is ts an in v e r tib le ideal I in R s u c h th a t

R ®rA = R [ J T , r ' T - ' ] . ( * )

/ , b ein g in v e rtib le , is fin itely g e n e ra te d , say , / = ( a , , . . . , a„,)R. L et b \...b,„ e K be such th a t a \ b \ + • ■ • + a mb m = 1, so th a t I is g e n e ra te d b y b \ ,___b,„ as an ^ - m o d u le . S ince A is fin ite ly g e n e ra te d o v e r R, A = R [ t\ , . . . , t p ] fo r s o m e t\ . . . . , lp £ A. B y E q.

(* ), 1 ® tj = J 2 _ Si< i < ri (fji'T1 fo r s o m e c/7, 6 / ' , 1 < j < p . T h e c o e ffic ie n ts c/,, m a y b e e x p ressed as

9ji

T . cj i v ..ima \ ■ a!" fo r i > 0 ii H----\-im~i

y Cjir-i,„b '\ ■ - b m fo r i < 0 ,

w h ere c p i ...im £ R. A g a in , b y E q. ( * ) , w e h a v e a i T = Y L ui f ® vi/ a n d b f T ~ l = Y 2

!</ </ , fo r som e w,y,w,7 e R an d v u , z u e A.

N o w le t R' b e th e /?-su b a Ig e b ra o f R g e n e ra te d b y th e e le m e n ts a, , . . . ,a , c i j b / (w h e re 1 < / , / < m); Cjil. .,m (w h e r e / , + • • • + im = \i\, - Sj < i < r h 1 < j < p ) - U:/

(w h e re 1 < £ < q h 1 < i < m ) a n d w,y (w h e re 1 < £ < th 1 < i < m) . L e t I b e th e ideal ( a\ , . . . , a m) R ' . T hen R' is a fin ite b ira tio n a l e x te n s io n o f R a n d / is a n in v e r tib le ideal o f R ' .

S in ce A is flat o v e r R, R' <g>R A m a y b e id e n tifie d w ith its im a g e in R ®rA. T h e n it is e asy to se e th a t R' A = R ' [ I T J - ' T ~ ]],

P art ( c ) fo llo w s from (2 .3 ). □

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R e m a r k 3 .1 4 . T h e a b o v e p r o o f sh o w s th at in th e s ta te m e n t o f (3 .1 3 ), in c o n d itio n (i), it is e n o u g h to a ss u m e th a t th e g e n e ric fibre is an A * -form . (In the p r o o f ta k e R to be th e in te g ra l c lo s u re of R in L, w h e re L is a finite e x te n s io n o f K such th a t L-%R A is A' o v e r I.. )

S u p p o s e th at R is a o n e -d im e n s io n a l n o e th e ria n d o m a in a n d A is a fin ite ly g e n e ra te d Hat /^ -alg eb ra w h o se g e n e ric fibre is A" an d w h o se c lo s e d fibres are g e o m e tric a lly in te g ra l. W e h av e seen in ( 3 . 1 0 ) th at, in th is s itu a tio n , a clo sed fibre m ig h t b e A 1.

M o re o v e r, i f R is a P ID a n d a c lo s e d fibre is n o t A 1, th e n , b y (3 .7 ) , th a t c lo s e d fibre is n e c e s s a rily A". H o w e v e r, th e fo llo w in g e x a m p le s h o w s th a t if R is n o t n o rm a l, then, u n d e r th e a b o v e h y p o th e s e s , a c lo s e d fibre m ig h t b e a n o n -triv ia l A 1-fo rm . T h e re fo re w e d o n e e d the s tro n g e r h y p o th e s is in c o n d itio n ( i i ) o f T h e o re m 3.13 as c o m p a r e d to th e c o rre s p o n d in g c o n d itio n in T h e o re m 3.11.

E x a m p le 3 .1 5 . L et k b e a n o n -p e r fe c t field o f c h a ra c te ris tic p. L e t /i G k b e s u c h th at Z p — /) is irre d u c ib le in k \ Z \ . L et L = k [ Z \ / ( Z p — ft) = k(y.), w h e re y p = fi. N o w let R = k + ( U )Z.[[L']]. c o n s id e re d as a su b rin g o f £ .[[£ /]]. T h e n R is a o n e -d im e n s o n a l lo cal d o m a in w ith m a x im a l id e a l M = { U ) L [ [ U Y [ , q u o tie n t field K = L ( ( U ) ) a n d re sid u e field k. B e in g a finite m o d u le o v e r £ [ [ t / ] ] , R is n o e th e ria n .

L et X\ = X + y.Y an d Y\ = Y —X f . T h en it is e a s y to see th a t th at K [ X \ , Y\] = K [ X , Y ] an d U X }. Y, £ R[X. Y]. L e t F ( X , Y ) = U X I Y] + Yj + 1 a n d A = R[X, Y ] / ( F ( X , Y) ) . O ne can v e rify th a t A is i?-flat, th e g e n e ric fibre K ZrA is A* o v e r K a n d th e c lo s e d fibre k n A is a n o n -triv ia l A 1-fo rm o v e r k.

In [3, 3 .5 ], it w as s h o w n th a t i f R c o n ta in s th e field o f ratio n als, th en c o n d itio n s on g e n e ric a n d c o d im e n sio n o n e fib res are e n o u g h to c o n c lu d e th a t A is a n A '-fib ra tio n o v e r R. B u t b elo w w e g iv e an e x a m p le o f a fin ite ly g e n e ra te d flat a lg e b ra A o v e r a tw o -d im e n s io n a l n o e th e ria n lo c a l d o m a in R, w h o se fib re s a t all n o n -c lo s e d p o in ts o f S p c c R are A*, but w h o se c lo s e d fibre is a n o n -triv ia l A * -form . T h u s in th e n o n -n o rm a l situ a tio n , w e n eed a c o n d itio n o n a l l fibres (i.e ., th e e x is te n c e o f n o n -triv ia l u n its ) to c o n c lu d e th a t all fibres a re a c tu a lly A*.

E x a m p le 3 .1 6 . L et R a n d C d e n o te the field o f re a l n u m b e rs an d c o m p le x n u m b e rs, re s p e c tiv e ly . L et R = R + ( U , K )C [[(7 , V]] (c o n s id e re d a s a su b rin g o f C [ [ t / , V] ] ) . T h e n R is a tw o -d im e n sio n a l lo c a l d o m a in w ith m a x im a l id e a l M = ( U, V) C[ [ U, V ] \ , q u o tie n t field K = C( ( U, V) ) a n d re s id u e field R . B e in g a fin ite m o d u le o v e r R[[C7, V]], R is n o e th e ria n . L et A = R[X, Y ] / { X 2 + Y 2 - 1). T h e n A is a fin itely g e n e ra te d ^ - a lg e b r a an d b e in g a free m o d u le o v e r R[ X] , it is a lso flat o v e r R.

N o w le t R d en o te th e n o r m a lis a tio n o f R. T h e n ^ = C [[ti, V] ] a n d M is th e c o n d u c to r o f R in R. C le a rly R ® R A is A* o v e r R an d h e n c e k ( Q ) <g)R A is A* o v e r k ( Q ) for e v e ry p rim e id eal Q o f R.

S in ce M is th e c o n d u c to r o f R in R, fo r e v e ry n o n -m a x im a l p rim e id e a l P o f R, Rp = Rp so th a t k ( P ) % R A is A* o v e r k ( P ) . B u t k { M ) ® R A = R [X , Y \ f { X 2 + Y 2 - 1) is an A *-form o v e r & ( A / ) ( = R ) b u t is n o t A* o v e r k ( M ) .

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A c k n o w le d g e m e n ts

T he seco n d a u th o r th a n k s M . M iy a n is h i fo r ra isin g , in a p riv a te d is c u s s io n w ith him o n p a p e r [3], th e q u e s tio n o f A * -fib ratio n s. P a rt o f th e w o rk w a s d o n e d u rin g the v isit o f th e seco n d a u th o r to th e S ch o o l o f M a th e m a tic s , T IF R . H e is g ra te fu l to r the w arm h o sp ita lity d u rin g h is v isit.

R e fe re n c e s

[1] T. A sa n u m a , P o ly n o m ial fibre rin g s o f a lg e b ra s o v e r n o eth erian rin g s. In v en t. M ath. 87 (1 9 8 7 ) 1 0 1-127.

[2] S.M . B h a tw a d e k a r, A.K.. D u tta, O n A1-fib ra tio n s o f su b alg eb ras o f p o ly n o m ia l a lg e b ra s. C om p. M ath. 95 (3 ) (1 9 9 5 ) 2 6 3 -2 8 5 .

[3] A .K . D utta, O n A’ -b u n d le s o f affine m o rp h is m s , J. M ath. K y o to U niv. 35 ( 3 ) (1 9 9 5 ) 3 7 7 -3 8 5 . [4] P. E ak in , W . H e in ze r, A c an c e lla tio n p ro b le m fo r rings, C o n fe re n c e on C o m m u ta tiv e A lg e b ra .

L aw ren ce, K a n sa s, 1972, L e c tu re N o te s in M a th e m a tic s, vol. 3 1 1 . S p rin g er. B erlin . 1973.

pp. 6 1 -7 7 .

[5] M. N agata, L o c a l R ings, In terscien ce, N e w Y o rk , 1962.

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