bilinear form defined over k. Special cases of this norm theorem have been used in a crucial way by Arason and Pfister in [1] and by Elman and L a m i n [5].
ACTA
ARITHMETICA
XXIV
ndes,
(1973)
Math.
Specialization of quadratic and symmetric bilinear forms, and a norm theorem
University,
by MANFBED
KNEBUSCH Dedicated
(338)
(Saarbrücken)
to Carl Ludwig
Siegel on hü 75 birthday
Introduction. I n t h e first p a r t of t h i s paper ( § l - § 3 ) we s t u d y t h e specialization of a s y m m e t r i c b i l i n e a r o r q u a d r a t i c f o r m over a f i e l d K w i t h respect t o a place X: K->Lvoo, p r o v i d e d t h e f o r m has " g o o d reduct i o n " . W e have t o d i s t i n g u i s h between s y m m e t r i c b i l i n e a r a n d q u a d r a t i c forms since we d o not exclude fields of characteristic 2 . A t y p i c a l result o b t a i n e d b y t h i s theory i s the f o l l o w i n g : W e denote a s y m m e t r i c b i l i n e a r f o r m b y the corresponding s y m m e t r i c m a t r i x of i t s coefficients. L e t k(t) be the f i e l d of r a t i o n a l functions i n independent v a r i a b l e s t , t over a f i e l d k. Consider s y m m e t r i c b i l i n e a r forms (/#(*))> over k(t) whose x
r
coefficients f {t) g (t) are p o l y n o m i a l s . A s s u m e t h a t t h e f o r m (g i(t)) i s represented b y (fq(t)). A s s u m e f u r t h e r t h a t c i s a n r - t u p e l i n Jf such t h a t the f o r m (fij(c)) over k is n o n singular. I f charfc # 2 t h e f o l l o w i n g holds t r u e : ij
1
kl
k
(i) I f also (g i(c)) is n o n singular, t h e n t h i s f o r m i s represented b y k
[faie))
over k (see § 2 ) .
(ii) I f (g i(t)) is a diagonal m a t r i x w i t h m r o w s a n d c o l u m n s a n d if c i s a n o n singular zero of each p o l y n o m i a l g^t), t h e n the f o r m (/#(£)) has W i t t i n d e x > m/2 i f m is even a n d > ( m + l ) / 2 i f m i s o d d (see§3). k
T h e assertion (i) m a y be considered as a generalization of the p r i n c i p l e of s u b s t i t u t i o n of Cassels a n d P f i s t e r ([15], p . 3 6 5 ; [10], p . 20). A t t h e e n d of S e c t i o n 3 ( P r o p o s i t i o n 3.6) w e s h a l l also generalize t h e s u b f o r m t h e o r e m of Cassels a n d P f i s t e r ([15], p . 3 6 6 ; [10], p . 20). U s i n g t h e result quoted a b o v e a n d a s i m i l a r r e s u l t f o r charfc = 2 we p r o v e i n the last section § 4 a t h e o r e m a b o u t t h e p o l y n o m i a l s i n k[t] w h i c h c a n occur as norms of s i m i l a r i t y over Jc(t) f o r a f i x e d s y m m e t r i c b i l i n e a r f o r m defined over k. S p e c i a l cases of t h i s n o r m t h e o r e m have b e e n used i n a c r u c i a l w a y b y A r a s o n a n d P f i s t e r i n [1] a n d b y E l m a n a n d L a m i n [5].
280
Manfred
Knebusch
I n general our results about q u a d r a t i c forms are m u c h less complete t h a n those about b i l i n e a r forms. A l t h o u g h t h e language of forms is quite n a t u r a l t o describe t h e m a i n results of this paper, we use i n the b o d y of the paper the geometric language of q u a d r a t i c a n d b i l i n e a r spaces, since t h e geometric language seems t o b e more suitable t o u n d e r s t a n d t h e proofs. T h e t h e o r y developed here w i l l be a p p l i e d i n a subsequent paper about t h e b e h a v i o r of q u a d r a t i c forms i n transcendental field extensions [9]. § 1. Preliminaries about bilinear and quadratic spaces. W e r e c a l l some s t a n d a r d notations a n d w e l l - k n o w n facts about s y m m e t r i c b i l i n e a r a n d q u a d r a t i c forms over a (not necessarily noetherian) l o c a l r i n g A. F o r proofs of statements g i v e n here w i t h o u t further reference a n d moreover for t h e basic t h e o r y over a r b i t r a r y c o m m u t a t i v e rings t h e reader m a y consult C h a p t e r V of [3], [13], [7] a n d § 1 of [8]. I n t h e present p a p e r essentially o n l y t h e case t h a t A is a f i e l d or a v a l u a t i o n r i n g w i l l p l a y a role. A free (symmetric) bilinear module (EjB) over A is a f i n i t e l y generated free ^.-module E equipped w i t h a s y m m e t r i c b i l i n e a r f o r m B: ExE->A. W e often denote (E, B) b y a s y m m e t r i c m a t r i x (a ) w i t h a = B(x , Xj) f o r some basis x ..., x of E over A. W e say t h a t (E, B) — or B — is non singulary or t h a t (E, B) is a bilinear space, i f det(a^) lies i n t h e u n i t group A* of A i.e. i f x\-^B{—, x) is a bijection f r o m E t o the d u a l m o d u l e H o m ^ ( _ E , A). A free quadratic module (E, q) over A is a f i n i t e l y generated free ^.-module E e q u i p p e d w i t h a q u a d r a t i c f o r m g, i.e. w i t h a m a p p i n g q: E-+A such t h a t q(cx) = c q(x) a n d B(x,y): — q(oc + y) — q(x) — q(y) is b i l i n e a r i n x a n d f o r c i n JL, x a n d y i n E. W e s a y t h a t {E q) — o r q — is non singular or t h a t (E, q) is a quadratic space i f t h e associated b i l i n e a r f o r m B is n o n singular. A q u a d r a t i c m o d u l e (E, q) w i l l often be denoted b y a s y m m e t r i c m a t r i x [a#] i n square b r a c k e t w i t h a = q(Xi)j a = B(x Xj) if i # j , f o r some basis x , x of E. If 2 i s a u n i t i n A there i s n o essential difference between q u a d r a t i c a n d b i l i n e a r modules, since t h e n a n y b i l i n e a r f o r m B corresponds t o a u n i q u e q u a d r a t i c f o r m q(x) = \B(x, x). {j
{
19
{j
Let 0
(1.0) for quae thermor the n u n ; determr write t case the s t i l l uni< space o: t = ind.
complete ribe the eometric language it paper i exten-
L e t q>: A-+A' be a h o m o m o r p h i s m between (local) r i n g s (of course ^(1) = i ) . F o r a n y free b i l i n e a r or quadratic m o d u l e E we denote b y .(p*(E) the J Z - m o d u l e E^^A' equipped w i t h the b i l i n e a r resp. q u a d r a t i c f o r m w h i c h is deduced f r o m the f o r m o n E b y base extension ([4], § 1 no 4, § 3 no 4). W e often write E® A' or E®A! i n s t e a d of is considered. A
L e t E be a free q u a d r a t i c or bilinear module over A. W e c a l l a submodule V of the JL-module E a direct submodule i f E = V@W w i t h some other submodule W ( © means the module s u m , w i t h o u t r e g a r d i n g forms). If J . is a v a l u a t i o n r i n g t h e n for a n y submodule V of E the module V consisting of a l l x i n E such t h a t B(V, x) = 0 is a d i r e c t submodule, for E/Y is torsion free a n d f i n i t e l y generated a n d hence free. 1
"e r e c a l l bilinear r i n g A. iioreover ler mayit paper ill play ely gen'orm B: with a F. B) .%) lies om E to ~er A is f o r m q, B(x, y): I! i n E. uadratic module bracket ?7
1
W e c a l l the b i l i n e a r or q u a d r a t i c module E isotropic if E has a direct submodule V # 0 w h i c h is totally isotropic, i.e. q(V) = 0 i n the q u a d r a t i c case a n d B(VxV) = 0 i n the bilinear case. If E is n o t isotropic, we say E is anisotropic. N o t i c e t h a t if A is a field a n y a n i s o t r o p i c b i l i n e a r m o d u l e over A m u s t be a space, b u t t h a t i n case char J . = 2 there exist anisot r o p i c q u a d r a t i c modules w h i c h are n o t spaces. A quadratic space E over A is called hyperbolic, i f E contains a t o t a l l y isotropic direct submodule V such t h a t V = V. T h e n E is i s o m o r p h i c to the orthogonal s u m txH of t = .J-dimJE? copies of the hyperbolic plane L
E
=|J
Jj.-
S i m i l a r l y a b i l i n e a r space E is called
metabolic,
if J0 con-
tains a direct submodule V = T -. A metabolic b i l i n e a r space is i s o m o r p h i c rJ
w i t h some a i n
to the o r t h o g o n a l s u m of spaces
A.
E v e r y q u a d r a t i c resp. b i l i n e a r space E has a n o r t h o g o n a l position E •=
(*)
decom-
E ±2I 0
w i t h E anisotropic a n d 31 h y p e r b o l i c resp. m e t a b o l i c . N o w i n the q u a d r a t i c case W i t t ' s cancellation l a w is true, since J . is l o c a l [6], i.e. 0
E. itic and unique a
Of
E\ AG ^ I * ±G =>F g±
(1.0)
1
F
2
for q u a d r a t i c spaces F ,F ,G over A ("^" means " i s o m o r p h i c " ) . F u r thermore M ^ txH w i t h some > 0. T h u s i n the decomposition (*) the n u m b e r t = J-dimitf a n d u p to i s o m o r p h i s m the space E are u n i q u e l y determined b y E. W e c a l l t t h e index of E a n d i which is the case if and only if (a a^) = 6 = c a + d*a with c and d in A. 1
x
X
2
2
2
1
2
1
2
x
2
W e close t h i s section w i t h some r e m a r k s o n q u a d r a t i c modules. T h e f o l l o w i n g generalization of W i t t ' s c a n c e l l a t i o n t h e o r e m is a n i m m e d i a t e consequence of S a t z 0.1 i n [6]. P R O P O S I T I O N 1.2. Let M and N be free quadratic modules over a local ring A and let G be a quadratic space over A. If G±N represents G±M then N represents M.
over o s case eve If c a n d the Pre
->Wg(ii T h i s is ( I n [13 Prüfer : M ~ 2 case 21 1
Wi w i t h re of f u l l : By Len mined s depend even t l .
equivalent, spaces i f aces exist, ace E w i l l module E the f o r m :sp. h y p e r paces f o r m ad the i n oup W(A) 'es over A. implication A-modules Orms of E y we s h a l l h
