46 VI. COHOMOLOGICA L INVARIANT S O F O
n
, S O
n
, . . .
20. Cohomologica l invariant s o f Quad
n s
(n even )
P R O P O S I T I O N 20.1. Assume that n is even, n 2 ; let X e H(k
0
) be such that
(S) -X = 0 in H(ko). Let q be an element of Quad
n
,$(&) Write q as (a\, ..., a
n
).
Then the element (ai ) -(a^ ) ( an - i ) * A 0 / #(& ) depends only on X and the iso-
morphism class of q. If we denote it by b\(q), the map q i— b\(q) is an invariant
Q u a d n,S ~ » #
[If 5 = 1, th e conditio n X-(5) = 0 i s empty , s o b\ i s define d fo r ever y A and i s
equal t o th e cu p produc t b\ -A.]
If q ( a i , . . ., a
n
) ha s discriminan t 5, w e writ e a fo r th e n-tupl e ( a i , . . . , a
n
)
and c(q,a ) fo r th e imag e o f (ai ) ( an _ i ) i n th e quotien t H n~1 (k)/(S) -H n~2(k).
Since multiplicatio n b y A defines a homomorphis m o f H n~l(k)/(8)'Hn~2(k) int o
H(k), th e followin g lemm a implie s Propositio n 20.1:
L E M M A 20.2 . c(q,a) depends only on q (bu t no t o n a) .
PROOF. Th e proo f i s i n tw o steps :
i) c(q
J
a) does not change when the ai are permuted.
It i s enoug h t o sho w thi s fo r a transpositio n (ij). I f bot h i an d j ar e n ,
the invarianc e o f c(g , a) i s clear . If , say , i = n an d j n , w e hav e t o sho w tha t
x-(aj) = x-(a
n
) i n H n~1 {k)/{5)-Hn"2(k), wher e x i s th e cu p produc t o f th e (a
r
)
for r 7 ^ j, n. B y assumption , w e hav e a\ an = 5 in /c*//c* 2, henc e (usin g th e fac t
that n i s even) :
(an) = (J ) + (ay) + 5^(-a
r
).
Since x-(—a
r
) = 0 fo r al l r ^ j , n, thi s give s b y multiplyin g wit h x:
x-(ctj) = x-(a
n
) mo d (5)'H n~2(k),
as desired .
ii) We have c(q, a) = c(q, /? ) z / g = ((3
U
. . . , (3
n
).
By Witt' s chai n equivalenc e theore m [La m 73, 1.5.2] i t i s enoug h t o prov e thi s
when Pi ^ fo r al l indice s i excep t tw o o f them , sa y j an d j ' . B y par t i ) (an d
the hypothesi s n 4 ) w e ma y assum e tha t j an d j ' ar e bot h n. Bu t i n tha t cas e
we hav e (Pj) -(Pjf) = (otj) -{ctj') sinc e th e quadrati c form s (Pj,Pj') an d (aj,aj) ar e
isomorphic. Henc e c(q,P) c(q,a).
Note tha t th e definitio n o f 6 A coul d hav e bee n give n differently : choos e a / 0
represented b y q, an d writ e q a s q'
a
0 (a) . The n 6 A is th e imag e o f w
n
-i(q'a) 'X i n
H(k). Lemm a 20. 2 say s tha t 6 A does no t depen d o n th e choic e o f a.
E X A M P L E 20.3 . Assum e n 4 , S = 1. Choos e a represente d b y q. The n
Q2 = {OL)1 i s a 2-Pfiste r form , an d qs = (1, —®)q2 is a 3-Pfiste r form . Th e for m # 2
is independen t o f th e choic e o f a , a s follow s fro m th e computatio n o f w2 o r fro m
Lemma 22. 2 below . W e hav e (i n th e Witt-Grothendiec k rin g o f fc, cf. 27.1):
an equatio n whic h determine s q^ and qs completel y (Lemm a 22.2) . Th e b\ invarian t
of q i s then :
h(q) = e
3
(q3).
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