On the Complete Integral Closure of a Mori Domain

P UBLICATIONS DU D ÉPARTEMENT DE MATHÉMATIQUES DE L ^YON

M ^OSHE R ^OITMAN

On the Complete Integral Closure of a Mori Domain

Publications du Département de Mathématiques de Lyon, 1988, fascicule 3B , p. 25-29

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ON THE COMPLETE INTEGRAL CLOSURE OF A MORI DOMAIN

M o s h e R o i t m a n

D e p a r t m e n t of M a t h e m a t i c s and C o m p u t e r S c i e n c e U n i v e r s i t y of H a i f a , M o u n t C a r m e l , H a i f a 3 1 9 9 9 , Israel

( e - m a i l : RSMA 6 0 1 @ H A I F A U V M b i t n e t )

It is w e l l k n o w n that the c o m p l e t e integral c l o s u r e A of a d o m a i n A n e e d not be c o m p l e t e l y i n t e g r a l l y c l o s e d (R. G i l m e r and U. H e i n z e r ) . It t u r n s out that even if A is M o r i , t h e n A is not n e c e s s a r i l y c o m p l e t e l y i n t e g r a l l y c l o s e d or M o r i , thus a n s w e r i n g a q u e s t i o n of P r o f e s s o r V a l e n t i n a B a r u c c i ( U n i v e r s i t à di R o m a "La S a p i e n z a " ) . On t h e p o s i t i v e s i d e , for a n y Mori d o m a i n A , t h e d o m a i n A is c o m p l e t e l y i n t e g r a l l y c l o s e d (in short

*

c . l . c ) . If A is M o r i and r o o t - c l o s e d , t h e n A is c . i . c . U e

*

r e c a l l that b y a r e s u l t of V . B a r u c c i , if A is M o r i and (A:A )

* 0 , t h e n A is K r u l l .

In o r d e r to c o n s t r u c t a M o r i d o m a i n A s u c h that A * A , w e r e m a r k that for any d o m a i n A , w e h a v e : A =

U ( A^Q[ x , y ] n A ) , w h e r e AQ is t h e p r i m e r i n g c o n t a i n e d in

x , y ^y

in A

A . T h i s r e m a r k l e a d s us to c o n s i d e r c e r t a i n s u b r i n g s of D [ X , Y ] y for a d o m a i n D and to d e f i n e p o w e r f u n c t i o n s . F u r t h e r m o r e , as b e i n g c o m p l e t e i n t e g r a l c l o s e d and t h e M o r i p r o p e r t y are m u l t i p l i c a t i v e p r o p e r t i e s of a d o m a i n , w e d e a l h e r e not just w i t h r i n g p o w e r f u n c t i o n s but a l s o w i t h s e m i g r o u p p o w e r f u n c t i o n s .

Let x, y be e l e m e n t s of a c a n c e l l a t i v e s e m i g r o u p S w i t h u n i t . U e d e f i n e the f u n c t i o n *^Q : IN — • Wu{oo> as f o l l o w s :

*3 \ x, y

* (tn) = s u p ( n « N : ( x^m/ yⁿ) « S } ( H e r e x^m/ yⁿ b e l o n g n to

•3 » X , y ^ J

the l o c a l i z a t i o n S of S ) . A f u n c t i o n 4 : IN — • Nu{a>) w i l l V

be c a l l e d a s e m i g r o u p p o w e r f u n c t i o n if £ - *^c.__ for s o m e

n o n z e r o e l e m e n t s x^fy in a cancel lat ive s e m i g r o u p vS . A r i ng p o w e r f u n c t i o n is a f u n c t i o n of the form *⁰ , w h e r e S -

S ; x , y A \ { 0 ) for s o m e d o m a i n A .

Ue d e n o t e by Di the s e m i g r o u p ^ X ™ Yⁿ : nu=(N , n ^ Z j- , w h e r e

X,Y a r e ind et erminat es . Let M be a s u b s e m i g r o u p of IM c o n t a i n i n g X,Y . Ue d e n o t e by *M the f u n c t i o n *M . On t h e

$

o t h e r h a n d , for any f u n c t i o n 4 : IN — • K U { O D } , w e d e n o t e by A the set of all e l e m e n t s X.^/Y¹¹ in CM s u c h that n < $ ( m ) ; for a n y d o m a i n D , w e d e n o t e by D the d o m a i n D[A ]

It is easy to o b t a i n the f o l l o w i n g c h a r a c t e r i z a t i o n of s e m i g r o u p p o w e r f u n c t i o n s :

T H E O R E M I Let S : tN — » INU{oo} be a f u n c t i o n . T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

1) 4 is a ring p o w e r f u n c t i o n .

2 ) $ is a s e m i g r o u p p o w e r f u n c t i o n .

FQr al 1 m , n in IN it h o l d s : <&(m+n) > $ (w)+$ ( n ) . 4 ) For a n y d o m a i n D , it h o l d s : $ = #

D ;X,Y

$ $

5 ) For a n y d o m a i n D , it h o l d s : A = DinD

E X A M P L E S of p o w e r f u n c t i o n s : Let c>0 in IR . U e d e n o t e by

the f u n c t i o n [cn] . For c>0 , we d e f i n e ty^(n) as the g r e a t e s t i n t e g e r w h i c h is < cn for n>0 and set 0 ^ ( 0 ) = 0 . Ue a l s o d e f i n e ty^(n) = 0 for all n > 0 ^f thus c/^ = T ^ . U e d e n o t e by

the f u n c t i o n I ( n ) - [c(n - l o g ( n + l ) ) ] (n € IN) . By T h e o r e m I, c

<y , T and / are p o w e r f u n c t i o n s , c c c

G i v e n a c l a s s £ of c a n c e l l a t i v e s e m i g r o u p s w i t h u n i t , a

£ - p o w e r f u n c t i o n is a p o w e r f u n c t i o n of the form , w h e r e

o ; x , y

S is a s e m i g r o u p in £ . For e x a m p l e , w e will deal h e r e w i t h Mori s e m i g r o u p p o w e r f u n c t i o n s , e t c . (The M o r i p r o p e r t y for c a n c e l i a t i v e s e m i g r o u p s is d e f i n e d s i m i l a r l y to the Mori ring p r o p e r t y ) . U e u s e a s i m i l a r t e r m i n o l o g y for r i n g p o w e r f u n c t i o n s . A n y M o r i s e m i g r o u p p o w e r f u n c t i o n w h i c h is not i d e n t i c a l l y oo is n e c e s s a r i l y f i n i t e .

For a n y f u n c t i o n $ : IN — • R f w e d e n o t e by AS : IN — • R t h e f u n c t i o n A $ ( n ) = $ ( n + l ) - $ ( n ) and by 6 $ : INxlN — * R the f u n c t i o n 6 $ ( m , n ) ~ $ ( m + n ) - $ ( m ) - $ ( n ) . For e x a m p l e , c o n d i t i o n 3 ) of the T h e o r e m 1 for the c a s e that $ is f i n i t e , can be s t a t e d in the

form: 6 ( $ ) > 0 .

U e n o w c h a r a c t e r i z e M o r e s e m i g r o u p p o w e r f u n c t i o n s :

T H E O R E M 2 Let $ be a f i n i t e p o w e r f u n c t i o n . T h e f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t :

(i) $ is s e m i g r o u p M o r i .

( i i ) T h e f u n c t i o n $ h a s the f o l l o w i n g two p r o p e r t i e s : ( 1 ) A3> is b o u n d e d .

( 2 ) A n y i n f i n i t e set I G IN h a s a f i n i t e subset F such that for all m € IN it h o l d s :

m i n 6 $ ( m , k ) : k <= I ^ = m i n ^ 6 $ ( m , k ) : k € F ^ .

( i i i ) T h e s e m i g r o u p A is M o r i .

C o n d i t i o n ( i i ) (2) of the p r e c e d i n g t h e o r e m is e q u i v a l e n t to the C C^X in the s e m i g r o u p M / M Y^r for every r .

For any f i n i t e p o w e r f u n c t i o n S , we d e n o t e sup S ( n ) / n by n

c„ . C l e a r l y for S = / , & or T w e h a v e : c = c- . It is

$ c c c * easy to s h o w that lim fc(n)/n = c^ and c^$ sup A * for any

n — » 0 0

p o w e r ftinction 5 . T h u s , if A * is f i n i t e , in p a r t i c u l a r if $ is a f i n i t e s e m i g r o u p M o r i p o w e r f u n c t i o n , t h e n c_ is f i n i t e .

For a n y f i n i t e p o w e r f u n c t i o n 5 , w e d e f i n e the f o l l o w i n g two f u n c t i o n s from IN to [Nu{oo> :

*

$ ( m ) := s u p inf [$(km + r ) / k ] , $ ( m ) := s u p [ $ ( k m ) / k ] .

r«?[N k<slN k«=£N B o t h $ and $ a r e p o w e r f u n c t i o n s . M o r e o v e r , $ is r o o t - c l o s e d (as a s e m i g r o u p or as a ring p o w e r f u n c t i o n ) if and

*

o n l y if $ = $ . S i m i l a r l y , $ is c . i . c . if and o n l y if $ ~ $ ( a g a i n t h i s h o l d s in b o t h s e n s e s : as a s e m i g r o u p or as a ring p o w e r f u n c t i o n ) .

U e s e e that we can t r a n s l a t e s e m i g r o u p or ring p r o p e r t i e s into p r o p e r t i e s of p o w e r f u n c t i o n s and c o n v e r s e l y . For e x a m p l e , w o can c h a r a c t e r i z e Mori r o o t - c l o s e d or c . i . c . p o w e r f u n c t i o n s . Indeed, let P : = ^ cr ^ : c>0 | and T ^:~ ^ ^{T c : c}- ° J- • ^{l t c a n} be s h o w n that is the set of al] r o o t - c l o s e d p o w e r f u n c t i o n s w i t h c. f i n i t e and T is the set of all c . i . c . p o w e r f u n c t i o n s w i t h c_ f i n i t e . A l s o , ^-.UJ"- is the set of f i n i t e r o o t - c l o s e d M o r i p o w e r f u n c t i o n s and T^ is the set of all f i n i t e c . i . c . M o r i p o w e r f u n c t i o n s (here J*^ is the set of all ct w i t h c r a t i o n a l and s i m i l a r l y for ) . As b e f o r e e v e r y t h i n g h e r e is in b o t h s e n s e s ) .

T h e set of all f a c t o r i a l finite ( s e m i g r o u p or r i n g ) p o w e r f u n c t i o n s e q u a l s .

For the M o r i p r o p e r t y w e h a v e a t w o - s i d e d t r a n s l a t i o n just for s e m i g r o u p p o w e r f u n c t i o n s by T h e o r e m 2 a b o v e and w e c o n j e c t that a s e m i g r o u p M o r i p o w e r f u n c t i o n is a l s o ring M o r i (the c o n v e r s e is c l e a r ) .

U e h a v e the f o l l o w i n g

T H E O R E M 3 Let S : IN — • be a f u n c t i o n w i t h the f o l l o w i n g p r o p e r t i e s :

( 1 ) all m , n in. tN it h o l d s : $ ( m + n ) > $ ( m ) + $ ( n ) . ( 2 ) T h e s e q u e n c e A $ ( n ) c o n v e r g e s .

( 3 ) 3. itn 6 $ ( m^tn ) = GO . m—>oo

n—•oo

$

Thi_eri K is Mori for a n y field K . In p a r t i c u l a r , [$] is a llori p o w e r f u n c t i o n .

T h e c o n d i t i o n s of the last t h e o r e m a r e f u l f i l l e d by the f u n c t i o n $ ( n ) := c ( n - l o g ( n + l ) ) for a n y c>0 , so all the f u n c t i o n s I for c>0 a r e r i n g M o r i .

c

T a k i n g into a c c o u n t T h e o r e m 3 and f u r t h e r p r o p e r t i e s as e.g.

*

Z - I = <y „ w e can o b t a i n our c o u n t e r e x a m p l e s : c c c

I

Let K be a f i e l d . Let c> 0 and let A : = K ^C , t h u s A is M o r i . U e h a v e :

*

( 1 ) For c r a t i o n a l , A = A is M o r i , but is not c.i.c.

( 2 ) For c i r r a t i o n a l , A = A is not M o r i , but is c . i . c .

( 3 ) For p o s i t i v e c o n s t a n t s a and b , w h e r e a is r a t i o n a l

f ^a l^b * and b is i r r a t i o n a l , t h e d o m a i n B := IK is M o r i , but B

I I.

13 n e i t h e r M o r i , n o r c . i . c . N o t i c e that B ~ K <s> K a b K

In p a r t i c u l a r , w e see that the i n t e g r a l c l o s u r e of a M o r i d o m a i n is not n e c e s s a r i l y M o r i , t h u s a n s w e r i n g a q u e s t i o n of P r o f e s s o r Evan G. H o u s t o n ( U n i v e r s i t y of N o r t h C a r o l i n a at C h a r l o t t e ) . U e r e c a l l that by a r e s u l t of V. B a r u c c i , the integral