P UBLICATIONS DU D ÉPARTEMENT DE MATHÉMATIQUES DE L YON
V ALENTINA B ARUCCI
On the Polynomial Ring over a Mori Domain
Publications du Département de Mathématiques de Lyon, 1988, fascicule 3B , p. 59-63
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ON THE POLYNOMIAL RING OVER A MORI DOMAIN
Valentina Barucci D i p a r t i m e n t o d i M a t e m a t i c a
I s t i t u t o 1G . C a s t e l n u o v o1 U n i v e r s i t à di R o m a fL a S a p i e n z a1
P i a z z a l e A l d o M o r o 2 00185 R o m a , Italia
O n e o f t h e o p e n p r o b l e m s a b o u t M o r i d o m a i n s i s t h e f o l l o w i n g : if A is a M o r i d o m a i n ( i . e . a d o m a i n w i t h t h e a s c e n d i n g c h a i n c o n d i t i o n on d i v i s o r i a l i d e a l s ) , is A [ X ] a l s o a M o r i d o m a i n ?
J . Q u e r r é h a s s h o w n t h i s is t r u e i f A i s i n t e g r a l l y c l o s e d (cf. [6, S e c t i o n 3, T h é o r è m e 5]) a n d M . R o i t m a n f o u n d t h e s a m e r e s u l t if A c o n t a i n s an u n c o u n t a b l e f i e l d ( c f . [ 8 ] ) . T h e r e are n o e x a m p l e w h e r e A is M o r i and A [ X ] is n o t .
I s h a l l g i v e a n e w c l a s s o f M o r i d o m a i n s for w h i c h t h e p o l y n o m i a l r i n g is a l s o a M o r i d o m a i n . A d o m a i n o f i t s m a y b e n o t i n t e g r a l l y c l o s e d or m a y n o t c o n t a i n an u n c o u n t a b l e f i e l d . So t h i s c l a s s c o n t a i n s n e w e x a m p l e s w i t h r e s p e c t t o t h e o n e s g i v e n by J.Querré or M . R o i t m a n .
S u p p o s e A is a M o r i d o m a i n a n d A * i s t h e c o m p l e t e i n t e g r a l c l o s u r e of A . S u p p o s e further t h a t (A:A*) f ( 0 ) .
I r e c a l l t h e a l g o r i t h m g i v e n in [2, S e c t i o n 1] t o g e t A * in t e r m s of p r i m e i d e a l s of A . Put A0 = A and
a) Ai + l = ( A i : ! ^ ) , w h e r e ¡R^ is t h e i n t e r s e c t i o n o f t h e s t r o n g m a x i m a l d i v i s o r i a l i d e a l s of A¿ (if A^ h a s s o m e s t r o n g m a x i m a l d i v i s o r i a l i d e a l ) ;
b ) Ai+1 = Aj_f if A¿ d o e s n o t h a v e a n y s t r o n g m a x i m a l d i v i s o r i a l i d e a l .
W e g e t in t h i s w a y a s e q u e n c e of M o r i o v e r r i n g s of A ,
that
is s t a t i o n a r y for some m a n d Am = A * ( c f . [ 2 , T h e o r e m 1.8] ) :
A = A0 C Ai C ... C Am = A * (*)
C o n s i d e r n o w t h e p a r t i c u l a r c a s e w h e r e for e a c h i, i = 0 , . . . , m - l , Tl^ is a r a d i c a l ideal of Ai + 1. I c a l l in t h i s case A
" s e m i n o r m a l in A * " ; t h i s d e f i n i t i o n is d u e t o t h e f o l l o w i n g p o i n t s :
i) if A is N o e t h e r i a n , t h e n A * = A (integral c l o s u r e o f A ) a n d o u r p a r t i c u l a r c o n d i t i o n h o l d s if a n d o n l y if A is s e m i n o r m a l (in A ) in t h e u s u a l s e n s e ;
i i ) t h e r e a r e m a n y s i m i l a r i t i e s b e t w e e n N o e t h e r i a n s e m i n o r m a l d o m a i n s a n d M o r i d o m a i n s " s e m i n o r m a l in t h e c o m p l e t e i n t e g r a l c l o s u r e " : in b o t h c a s e s , Aj_ is o b t a i n e d f r o m A ^+ 1 b y a g l u e i n g of p r i m e i d e a l s (cf. for d e t a i l s [10, T h e o r e m 2.1] a n d
[2, C o r o l l a r y 3 . 7 ] ) .
T h e p r o p e r t y i i ) i n d i c a t e s a l s o h o w t o c o n s t r u c t e x a m p l e s of d o m a i n s of this t y p e . W e get for e x a m p l e in t h i s w a y t h e d o m a i n s A = k + X Y k [ X , Y ] or B = k [ X ] + Y k [ X , Y , Z ] , N o t i c e t h a t , on t h e c o n t r a r y of t h e N o e t h e r i a n c a s e , in t h e M o r i c a s e , w e d o n o t h a v e a n y f i n i t e n e s s t y p e - r e s t r i c t i o n , so t h a t t w o lines can b e " g l u e d " in a p o i n t as in S p e c ( A ) or a p l a i n can b e
" g l u e d " in a l i n e as in S p e c ( B ) (cf. for o t h e r e x a m p l e s [2, E x a m p l e s 3 . 1 2 ] ) .
T H E O R E M . L e t A b e a M o r i d o m a i n s u c h t h a t (A:A*) ^ (0) . if. A is " s e m i n o r m a l in A * " , t h e n A [ X ] is a l s o a M o r i d o m a i n .
P r o o f . Let Aj_ = B C A ^+ 3 = C b e t h e g e n e r i c step of t h e s e q u e n c e (*) . T h u s C = ( B : R ) , w h e r e , if Pl f . . . , Pn a r e t h e s t r o n g m a x i m a l d i v i s o r i a l i d e a l s of B (cf. [2, P r o p o s i t i o n
1 . 5 ] ) , we h a v e H = P± n ...n Pn .
B y [2, P r o p o s i t i o n 2.7 and C o r o l l a r y 2 . 8 ] , we k n o w t h a t B = C n B1 n ... O Bn , w h e r e for e a c h j , j = l,...,n, Bj is t h e p u l l b a c k o f the d i a g r a m
- 1 /
Sj C > b^ C /
/ Sj P^
(where Sj = B - P j ) .
S i n c e A * is a K r u l l d o m a i n ( c f . [ lf C o r o l l a r y 1 8 ] ) , it is w e l l k n o w n that A * [ X ] a l s o is a M o r i (in fact K r u l l ) d o m a i n . T h u s to p r o v e the T h e o r e m it is e n o u g h to show (in t h e g e n e r i c s t e p of t h e s e q u e n c e (*), B C C) t h a t if C [ X ] is a M o r i d o m a i n , t h e n B [ X ] is also a M o r i d o m a i n .
I n d e e d B [ X ] - C [ X ] n B1[X] n ... n Bn[ X ] . S o , b y [7, T h e o r e m e 2 ] , it is e n o u g h to show that Bj[X] is a M o r i d o m a i n
(for j = 1, . . .,n) .
L e tfs fix an index j and let's d e n o t e , for s i m p l i c i t y , B j b y f i , Sj-iC b y C, Sj-!Pj b y X a n d k(Pj) by k. F r o m t h e p r e v i o u s p u l l b a c k d i a g r a m , w e g e t t h e f o l l o w i n g p u l l b a c k d i a g r a m (cf.[4, L e m m a 2 ] ) :
fi[X] > k [ X ]
C[x] > C/X [X] -C[X]/X[X]
If L is t h e q u o t i e n t f i e l d of k [ X ] , w e h a v e k [X] = C/T[X] n L, w h e r e t h e i n t e r s e c t i o n is m a d e in t h e t o t a l
q u o t i e n t ring of C/T[X]. T h u s , by a result of R o i t m a n (cf.[9, T h e o r e m 4 . 1 5 ] ) , fi [X] is a M o r i d o m a i n i f : i) C[X] is a M o r i d o m a i n , ii) T[X] is a M o r i i d e a l , iii) T[X] is a p r i m e i d e a l of B [ X ] . A c t u a l l y C[X] = Sj-iC[X] = S j " i ( C [ X ] ) is a M o r i d o m a i n , b e c a u s e C [ X ] is M o r i ( c f . [ 5 , C o r o l l a i r e 3 ] ) . M o r e o v e r T is a r a d i c a l i d e a l of C ( c f . [ 2 , P r o p o s i t i o n 3 . 3 , 2 ) ] ) a n d is a l s o a M o r i i d e a l , b e c a u s e it is a p r i m e (in fact m a x i m a l ) i d e a l of t h e M o r i d o m a i n fi ( c f . [ 8 , T h e o r e m 6 . 2 ] ) . T h u s , b y [9, P r o p o s i t i o n 4.9, ( b ) ] , 1 is a finite i n t e r s e c t i o n of p r i m e i d e a l s of C . W e e a s i l y d e d u c e t h a t T[X] is a l s o a f i n i t e i n t e r s e c t i o n of p r i m e ideals of the M o r i d o m a i n C [ X ] , T h u s , by
[8, T h e o r e m 6 . 2 ] , T[X] is a M o r i i d e a l . M o r e o v e r T[X] is p r i m e in fi [X] a n d so we c o n c l u d e that fi[X] is a M o r i d o m a i n .
It is not d i f f i c u l t t o s h o w that if A is an i n t e g r a l l y c l o s e d M o r i d o m a i n s u c h t h a t (A:A*) = ( 0 ) , t h e n A is
" s e m i n o r m a l in A * " and s o , b y the T h e o r e m , A [ X ] is M o r i . T h u s in t h i s c a s e t h e p o l y n o m i a l r i n g i n h e r i t s t h e M o r i p r o p e r t y a n d w e g e t a s a c o n s e q u e n c e of t h e T h e o r e m t h e m e n t i o n e d r e s u l t o f J.Querré in a p a r t i c u l a r c a s e :
C O R O L L A R Y . L e t A b e a M o r i d o m a i n s u c h t h a t (A:A*) ^ ( 0 ) . £ f A is i n t e g r a l l y c l o s e dr t h e n A [ X ] is a M o r i d o m a i n .
R E F E R E N C E S
[1] V . B A R U C C I . S t r o n g l y d i v i s o r i a l ideals a n d c o m p l e t e i n t e g r a l c l o s u r e of an i n t e g r a l d o m a i n . J.Algebra ( 1 9 8 6 ) , 1 3 2 - 1 4 2 .
[2] V . B A R U C C I . A L i p m a n ' s t y p e c o n s t r u c t i o n , g l u e i n g s a n d c o m p l e t e i n t e g r a l c l o s u r e . N a g o y a M a t h . J. (to a p p e a r ) .
[3] V . B A R U C C I - S . G A B E L L I . H o w far is a M o r i d o m a i n from b e i n g a K r u l l d o m a i n ? J.Pure A p p l . A l g e b r a 4 £ ( 1 9 8 7 ) , 1 0 1 - 1 1 2 .
[4] A . B O U V I E R - M . F O N T A N A . T h e c a t e n a r i a n p r o p e r t y o f t h e p o l y n o m i a l r i n g s o v e r a P r u f e r d o m a i n . S é m i n a i r e d ' A l g è b r e P . D u b r e i l et M . P . M a I l i a v i n . P r o c e e d i n g s , P a r i s 1 9 8 3 - 8 4 , 3 4 0 - 3 5 4 .
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[7] N . R A I L L A R D ( D E S S A G N E S ) . S u r l e s a n n e a u x d e M o r i . C . R . A c a d . S c . P a r i s 2 â û ( 1 9 7 5 ) , 1 5 7 1 - 1 5 7 3 .
[8] M . R O I T M A N . O n M o r i d o m a i n s a n d c o m m u t a t i v e r i n g s w i t h t h e c h a i n c o n d i t i o n on a n n i h i l a t o r s . J . P u r e A p p l . A l g e b r a (to a p p e a r ) .
[9] M . R O I T M A N . O n M o r i d o m a i n s and c o m m u t a t i v e r i n g s w i t h the c h a i n c o n d i t i o n on a n n i h i l a t o r s II ( p r e p r i n t ) .
[10] C . T R A V E R S O . S e m i n o r m a l i t y and P i c a r d g r o u p . A n n . S c . N o r m . S u p . P i s a 24 (4) ( 1 9 7 0 ) , 5 8 5 - 5 9 5 .
