29. W I T T INVARIANT S O F ETAL E ALGEBRA S 69
PROOF.
Le t a be a n invarian t Pfister
n
Quad^y . Composin g i t wit h th e ma p
Quad^— W give s a n invarian t b in Inv(Pfister n, W). B y 27.17, b may b e writte n
as
Kq) = fo + fi •( « " 1) i n W(k)
for som e form s /o , / i G Quad(/co), whic h on e ma y choos e t o b e anisotropic . Pu t
iV7 = rank(/o ) - f rank(/i) -(2 n 1). Sinc e a(q) an d b(q) hav e th e sam e ran k (mo d
2), w e have N' = N (mo d 2) . Ther e ar e tw o cases :
Case I: N f N. Pu t h = (N - N')/2. Defin e quadrati c form s cp
0
and y? i by :
A) = /o©ft'(l,-l) , ^ l = /i -
We hav e rank(^ o © £i *(# 1)) = N' + 2h = N. Sinc e a(# ) an d p o © y\ '(Q ~ 1)
have th e sam e ran k an d th e sam e imag e i n W(k), the y ar e equal . Thi s give s th e
formula w e wanted fo r a(q). It s uniquenes s i s clear .
Case II: N
f
N. Pu t h {N' N)/2. Th e sam e argumen t a s abov e show s
that, fo r ever y k/ko, an d ever y q G Pfistern(/c), th e quadrati c for m / o 0 f\ -(q 1)
contains h-(l, —1), hence is isotropic. Thi s contradict s Cor . 28.2 , applied t o fo an d
/i . Henc e thi s cas e i s impossible.
THEOREM
28.5 . Every invariant in Inv(Quadn, Quad^) can be written uniquely
as
q^Po + Vi-q-l y^Pm
'Amg,
where pi G Quad(/co) and cpi, ..., ^
m
are anisotropic.
(Note tha t (pi, ..., ^
m
ar e allowe d t o b e O-dimensional. )
PROOF.
Th e proo f i s analogous t o th e proo f o f Theorem 28.4 , except tha t on e
uses Cor . 28. 3 instead o f Cor . 28.2 .
29. Wit t invariant s o f etal e algebra s
THEOREM
29.1. A Witt invariant for Et
n
is 0 if it is 0 for all multiquadratic
algebras.
PROOF.
W e us e inductio n o n n. Le t a G Inv(Etn,VF) b e suc h tha t a(E) = 0
for al l multiquadrati c algebra s E. Usin g th e inductio n assumptio n on e shows , a s
in 24.9 , tha t a vanishe s o n ever y etal e algebr a whic h ha s a facto r o f ran k 1 or 2 .
Next on e observe s tha t th e invarian t a(E gen) o f th e versa l torso r E gen o f 24. 6 i s
unramified (i.e. , has residue 0) at al l the irreducible divisors of
Affn,
excep t perhap s
A. Thi s follows from 27.12. Furthermore , the versal torsor splits into the product of
a quadrati c algebr a an d somethin g els e "locall y a t A " (i.e. , over the loca l field K&,
as i n 24.6 ) Th e inductio n assumptio n the n show s tha t a(E
gen)
give s 0 in W(K&),
hence ha s residu e 0 at A a s well.
By 27.8,
a(Egen)
belong s to W{k 0). Sinc e
Egen
i s versal, a is constant b y 27.13.
Since a i s 0 on multiquadrati c algebras , i t i s identically 0 .
THEOREM
29.2 . Set m = [n/2]. Then Inv(Et
n
, W) is a free W(k
0
)-module
with basis the invariants
E ^
A2(g^)
for 0 i ra,
where qE is the trace form of E.
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