ANDRZEJ PRÓSZYŃSKI WSP w Bydgoszczy
HIGHER-DEGREE FORMS AND m-APPLICATIONSX)
1
. G e n e r a l i t i e s . We r e c a l l some d e f i n i t i o n s o o n ta in e d i n [ 7] , [ l ] , and C2
] . A l l r i n g s and a lg e b r a s w i l l be com m u tative w it h1
.A p o ly n o m ia l la w on th e p a i r ( X , Y ) o f R-m odules i s a n a t u r a l t r a n s fo r m a t io n F = (F a ) : X ® * Y ® - , w here X ® - , Y ® : R - A lg — » S e t . I t i s o a l l e d a fo rm o f d e g r e e m i f f FA( x a ) = F A( x ) a m f o r any A, a e A and x € X ® A. Then we d e n o te FeJ^ ( X , y ) . I t i s p r o v e d i n £7] th a t
/
S’ o' H
8^ » R ) — R Гт ч i • • • t T ] • I n th e n a t u r a l way we o b t a in th e
i n Ш f u n c t o r 9 R-Mod° x R-Mod — R-Mod.Any fo rm F € SP r ( X , y ) has th e f o l l o w i n g sh ap e: F ( x ® a ♦ . . . ♦ xn ® an )= S I F и ( x , . . . , x ) в m + . . . +m_=m I ' n À П л
1
n QD a1
• • • an » w h ere F : Xй — * Y sure u n iq u e ly d e te r m in e d b y F . I nШ1
* • • •p a r t i c u l a r F s F B and F =PF i s m - li n e a r and s y m m e tric .
Ю К l l l t t f I /
(We w i l l assume t h a t m > o ) . PF can be o b t a in e d fro m F_ i n th e f o l l o w i n g w ay:
P F (x
1
, . . . , x n )= ( A mFR ) ( x1
, . . . , x m) : = X ( - l f " k FR( x * - +xi j1
kHence we h a ve th e n a t u r a l t r a n s fo r m a t io n : T“ : 9 r ( X , Y ) — + A p p lR( X, Y ) , T“ ( F ) = Fr ,
w h ere A p p lR( X , Y ) i s th e m odule o f a l l m - a p p lic a t io n s f : x — ^ Y, i . e . suoh m appings t h a t A mf i s m - lin e a r and f ( r x ) = r mf ( x ) f ° r any r € . R and X, I n th e f r e e c a s e T R g i v e s us th e f o l l o w i n g w e ll-k n o w n m apping:
36
T * l R [T l t . ; . , T n] B — » A p p l ^ R ^ R ) , ^ “ ( F ) ( X l , . . . , x n ) = F ( x 1, . . . , * n I t l e known t h a t l a ein Isom orp h ism i n th e f o l l o w i n g o a s e s s ( l ) i f m ^ 2 ( 2 ) i f ml € U (r) ( 3 ) i f X=R . I n th e g e n e r a l c a s e we w i l l i n v e s t i g a t e K e r (T ^ )= S P ^ f x , Y ) and I * ( l ^ ) = Hom“ ( X , Y ) « - : A p p l " ( X t Y ) . 2 . R e p r e a e n t a b l l i t y . O b s erve t h a t th e fu n o t o r * i n th e f o i l . n g e x a o t s e q u e n c e : , 0 > S>m( X f - ) С — > Jp M( X ) - ) — » A p p le ( X t - , a r e r e p r e s e n t a b le . ( a ) ( s e e Г
7
] ) P Ш( Х , - ) i s r e p r e s e n t e d b y th e m -th d i v i d e d p o v e r P m(x ) of X, and P ^ p n,(X ,Y ) corresponde to . fra ) Ÿé. HomfP m( x ) , Y ) Iff F lx , хп)»Ч’ ( х ^ и , х й Л ) . ( b ) ( s e e С1 Л )Арр1Ш( X , - ) I s r e p r e s e n t e d b y th e module Pe ( x ) d e f in e d b y th e g e n e r a t o r s ^"ш( х ) t o r x g. X and th e f o l lo w in g r e l a t i o n s : 1° ^ r x i s r ^ f x ) f o r any r € R , x £ X 2 ° i e m - l in e a r . The c o rr e s p o n d e n c e i s g iv e n b y th e f o l l o w i n g d ia g ra m : Y ( c ) ( s e e f 2 } ) t ” in d u c e s th e n a t u r a l homomorphism: h :Гш(х )
— » Г И( Х ) , h ( r<n( x ) ) = х ( ш ) . L e tP Ш(х )=
Im (h )= r [ x ^ ;x t x }
. S in o e Horn i s l e f t ^ x a c t i t f o l l o w s t h a t У ( X , - ) i s r e p r e s e n t e d b y P “ ( X ) = C o k e r (h )= Г В( Х ) / Г Ш( Х ) . ( d ) ( s e e Г2З )Th e f u n c t o r НотШ( Х , - ) i s r e p r e s e n t a b le e x a c t l y i n th e c a s e when th e e x a c t sequ en ce 0 — ^ Г И( х ) — ^ P Ш( x ) - —•) P Ш( X ) - - > 0 s p l i t s , and th en i t i s r e p r e s e n t e d b y P m( x ) . I n th e g e n e r a l c a s e we h a ve th e e x a c t s e q u e n c e : 0 * HomB( X , - ) -С----> Нот ( P “ ( X ) , - ) --- » E x t 1( r " ( x ) , - ) - - *—* E x t1( p B( x ) , - ) ( s O i f X i * p r o j e c t i v e ) .
E xam ple. I t can b e p r o v e d t h a t P
22 )=Ж^ f ^ ( ł 2 )= » 2 } and h en ce
Hom^(2^ , —) i s n o t r e p r e s e n t a b l e t The e x a o t seq u en ce i s f o l l o w i n g :о Hernia2 ,-)— » ноиа (Г^(а2 ),-) a2 ® ---->• о .
3. The f u n c t o r P M. The fu n d a m en ta l p r o p e r t i e s o f t h i s fu n c t o r , g iv e n i n С
2
З апй^СзЗ* a r e f o l l o w i n g :( 1 ) P “ ( x ) ® A 2 P “ ( X ® A ) f o r A=US , A =R /l (b u t n o t i n th e g e n e r a l c a s e ! )
(
2
) Th eorem . L e t X b e a f i n i t e l y g e n e r a t e d R -m od u le. Then th e f o l l o w i n g c o n d it io n s a r e e q u i v a l e n t :( i ) Г ^ ( х ) = о
( i i ) r “ /3;( x / i x ) = 0 f o r any I & M ax(R )
( i i i ) f o r any l£ M a x (R )e it h e r d im ^ y ^ X / I X )^ o r m ^ IR /II * ( ( i ) ^ = ^ ( i i ) is the NakayamA Lemma together with ( l ) ,
( t t } < r X i U ) i s the c a s e of a fie ld ).
C o ro lla ry . T h e follow ing properties o re equivalent :
U ) P " =
0
( i i ) p “ (r2 ) =o( i l l ) * £ d ( R ) : = i n f [| R / I | ; I f c M a x ( R ) } .
( 3 ) Lemma . I f P Ê S p e c (R )- M a x (R )t h e n p “ ( x ) p=0 f o r any R-m odule X. (R / P i s i n f i n i t e and h en oe d (R p )= 00 . N e x t a p p ly ( l ) a n d (
2
) ) . C orolla ry. If dim ( R ) > 0 th en f*R ( x ) are torsion R-m odules.(И n o n -z e ro , they a re not f r e e ).
. •—О
It i s proved, in [_5 J t h a t P R ( x ) i s finite provided that R i s noetherian a n d X is finitely gen erated .
( k ) S t r u c t u r a l th e o re m s . L e t R b e a n o e t h e r ia n r i n g . Th en :
(A) Г R( x ) -
Pei5|x (R )
p R p( Xp)
( B ) P “ ( X ) ^ Г | ( Х ® R ) i f R i s a l o c a l r i n g .(c)
L e tX b e a f i n i t e l y g e n e r a t e d R -m od u le. Then
p “ ( x ) c © p " V (x/f^ (pK ) ‘ R P t Max( R ) R/Pk ( P ) U / F ^ X) f o r a l l s u f f i c i e n t l y l a r g e k ( P ) . I f * 6 5 ( o r « é 7 and2 6 u (R ) ) th en we can c h o o s e
k ( p)=1
and h e n c e : n e ( x ) а Ф V я , (X / P X ) ' R Y e M a x(R ) ’ R/P38
f o r any R-m odule X. (Com pare a ls o [ 5 ] » C o r o l l a r y 5 . Ю ) . The a b ove p r o p e r t i e s p e r m it us t o compute K e r fT ” ) in some o a s e s . The q u e s t io n o f C oker ( Т Ш) I s much more c o m p lic a t e d .
k . C oker (T * ° ). O b serve t h a t Hom“ ( X , - ) Ç Н о т (Г Ш( х \ - ) с А р р 1
11
^ Х , - ). I f Нот ( X , - ) I s n o t r e p r e s e n t a b le ( i t i s so v e r y o f t e n ) , th en th e f i r s t two f u n c t o r s a r e d i f f e r e n t and h en ce C o k e r (T m) i s n o n - z e r o . I t was c o n je c t u r e d b y M. F e r r e r o t h a t th e l a s t two fu n c t o r s a r e e q u a l, o r , more p r e c i s e l y , t h a t Г ш( х ) andp m (x)
a r e is o m o r p h ic b y h . T h is means t h a t HomB( x f Y )= = A p p lm( X , Y ) f o r any i n j e c t i v e Y ; i n p a r t i c u l a r , o v e r a f i e l d K, t h a t C o k e r(T ® ) i s z e r o f o r any К-m o d u le s . U n fo r t u n a t e ly , i t ' i s n o t t r u e i n g e n e r a l ,E xam ple. L e t K=F^, Х=К* , Y=K and f ( x
1
, х2
, х3
,х ^ )= х ^ х | х ^ х ^ . O b s erve t h a t f £ A p p lj^ (x , Y ) b e c a u s e :1°
f ( r x ) = r ® f ( x ) = r " * f ( x ) s in c e г**=г i n F^ ,2°
f = e2
w here and h en ce û5
f - ( û5
g ) 2-0s in c e ( ) ^ is additive in К and
g
is o f d e g re e 4.On th e o t h e r hand, f i s , , r e d u c e d ’ * ( a l l p ow ers i n f are l e s s than I К 1 ) and h en ce f ^ Hom^7(x, Y ) f o r m <
8
. ( s e e HtJ } .C oker(T*B) i s c o m p le t e ly d e s c r ib e d i n i n th e c a s e o f f i n i t e f i e l d s : any m - a p p lic a t io n i s a p o ly n o m ia l m apping (b u t n o t n e o e s s a r i l y o f d e g r e e m - as i n th e e x a m p le ), and we can f i n d th e s ta n d a r d b a s e s o f Homm, Appl™ and C o k e r (T m) f o r any К-m o d u le s . I n p a r t i c u l a r , we h a ve th e f o l l o w i n g
Th eorem . L e t К be an a l g e b r a i c e x t e n s io n o f 2^ ( f o r exam ple a f i n i t e f i e l d o f c h a r a c t e r i s t i c p ) . Then th e f o l l o w i n g c o n d i t i o n s a r e e q u iv a l e n t :
( 1 ) C o k e r (T ^ )= О ( f o r any K—m o d u le s ) (
2
) K=*p о г ш ~2
p.I n p a r t i c u l a r , C o k e r (T ^ )= 0 f o r m ^ k.
Remark 1. I f K^2 and C oker(TÎÜ )= 0 th en m ^ 2p < lК I and
' _ p л "
h en ce K e r (T ^ )= 0 .
Remark 2 . U s in g a n o t h e r m ethods we can p r o v e t h a t С о к е г (т ” )= 0 o v e r any f i e l d К f o r m=3» bu t th e c a s e o f m=i£ i s unknown.
Exam ple. L e t К be an I n f i n i t e a l g e b r a i c e x t e n s io n o f
Then К i s a sum o f f i n i t e f i e l d s and we d e f i n e f : K~’ —* К i n th e f o l l o w i n g w ay: i f x ^ x ^ x ^ x ^ € К and |к'| =2*:*. *ł
2
2
2
^2
^ then f ( x ^ i X ^ t Z ^ f Z ^ ) ^ XjXgX^ ^ • As i n th e p r e c e d in g e x a m p le , i t can be p r o v e d t h a t f i s a p r o p e r ly d e f in e d 5 - a p p l i c a t i o n - b u t f i s n o t a p o ly n o m ia l m apping. L e t us c o n s id e r th e homomorphism h : Р и - - •*>f m o v e r r i n g s . The p rob lem o f i n j e c t i v i t y o f h can b e re d u c e d t o th e l o c a l c a s e , b eca u se Г and P™ commute w it h l o c a l i z a t i o n s . L e tШ
(r,M ) be a l o c a l r i n g . Then we h ave th e f o l l o w i n g com m u tative d ia g ra m :
Рш(х )/м Ги(х ) --- ^
г
“ ( x ) / M f “ ( x )- ‘
I
Рю (х/м х)
»
Г “ (х/м х )
w here th e homomorphisme a r e d e f in e d i n th e n a t u r a l w ay. Th ey a r e a l l e p i , b u t n o t i s o i n g e n e r a l ( s e e [ 5 ] ) . Suppose t h a t R/M i s a p r o p e r a l g e b r a i o e x t e n s io n o f and 2 p < m ^ | R / M | . Then g i s i s o ( s i n c e Р ш = р Ш) , h* i s n o t i s o f o r some f r e e R-m odule X, and henoe h i s n o t i s o . U n fo r t u n a t e ly , th e d ia g ra m d o e s n 't g i v e p o s i t i v e ex am p les f o r o u r p ro b lem . I t i s known fro m f ć ] t h a t h i s i n j e c t i v e f o r m=3 ( o v e r R) i n th e f o l l o w i n g o a s e s : ( 1 ) f o r c y c l i c m odules ( 2 ) i f R i s von Neumanr r e g u l a r (
3
) i f no r e s id u e f i e l d o f R i s ( I ł ) i f R i s a DVR w it h th e p rim e e le m e n t 2 (5
) i f R i s a D ed ek in d domain o r R = Z fw j ( f o r f l a t R -m o d u le s ). Exam ple. L e t m=3. I t f o l l o w s fro m (3
) and (U ) t h a t h i s i n j e c t i v e f o r R e » . The , , g e n e r i c * * 3 - a p p l i c a t l o n s o v e r » a r e x^ , x y (x ± y )/ 2 and x y z . I f R = » C t 3 th en h i s i n j e o-ko
t l v e f o r f l a t , b u t n o t f o r a l l m od u les. I f R = s [x ,y 3 and (
3
) 1s n o t s a t i s f i e d th en h i s n o t i n j e c t i v e - e v e n f o r th e k e y c a s e o f R2 .REFERENCES
[ 1 ] FERRERO M ., MICALI A . , Sur l e s n - a p p l i c a t i o n s , B u l l . S o c . Math. F r a n c e , Memoire
5 9
, 1979, P . 33-53[ 2 ] PRÓSZYŃSKI A . , Some fu n c t o r s r e l a t e d t o p o ly n o m ia l t h e o r y , Fund.M ath. X C V I I I , 1978, p . 219-229
[
3
]
PRÓSZYŃSKI A . , Some f u n c t o r s r e l a t e d t o p o ly n o m ia l t h e o r y I I , B u ll.S o c ,M a t h .F r a n c e , Mém oire 59, 1979 , p . 125-129 [ 4 ] PRÓSZYŃSKI A . , m - a p p lic a t io n s o v e r f i n i t e f i e l d s , Fund.Math. C X II,
1 9 8 1
, p.205-21*t[
5
] PRÓSZYŃSKI A . , Forms and m ap pin gs. I : G e n e r a l i t i e s , Fund. Math. CXXII, 1984 p.2l9-235[6] PRÓSZYŃSKI A . , Forms and m ap p in gs. II : D eg ree 3 , to a p p e a r in Commentationes Mathematicae
[
7 ]
ROBY N . , L o is polynôm es e t l o i s f o r m e l l e s en t h é o r i e des m od u les, A n n .E c .Norm .Sup. 80,1 9 6 3, p . 2 1 3 -3 48
FORMY WYŻSZYCH STOPNI I m-ODWZOROWANIA
S t r e s z c z e n i e
P ra c a s ta n o w i t e k s t r e f e r a t u w y g ło s z o n e g o w M o n t p e l l i e r , p o ś w ię c o n a j e s t porów naniu dwóch d e f i n i c j i fo rm w y ż s z y c h s t o p n i . W s z c z e g ó l n o ś c i omówione z o s t a ł y fo rm y w y t w a r z a ją c e zero w e o d w zo ro w a n ia .
