44
VI. COHOMOLOGICA L INVARIANT S O F O n, SO
n
, .. .
is 0. Sinc e 1 and i d ar e linearl y independen t i n Inv/
Co
(Z/2Z, Z/2Z ) b y 16.2, we hav e
Ai = 0 . Thi s prove s th e theore m fo r n = 2 .
18.4. O C T O N I O N S . A n octonio n algebr a O ha s a nor m polynomia l qo, whic h i s
a 3-Pfiste r (cf . 2.8) , an d th e invarian t o f thi s 3-Pfiste r i s a n elemen t e(O ) o f H 3(k).
What 18.1show s i s tha t Invfc
0
(Oct, Z/2Z ) is a free H(ko)-module with basis {1, e}.
In particular , ever y normalize d invarian t i s a multipl e o f th e e-invariant .
If O ha s basi s {1, ei, e2, •., e^} a s usual , wit h
e i e i = a i , e
2
e
2
= a
2
, e
3
e
3
= a
3
,
e4 = eie
2
= - e
2
e i , e
5
= e
2
e
3
= - e
3
e
2
,
^6 = e
3
e
4
= - e
4
e
3
, e
7
= e
4
e
5
= - e
5
e
4
,
then th e for m qo i s
go = (1, -on) 8 (1, - a
2
) ® (1, - a
3
,
and th e e-invarian t o f O i s th e cu p produc t (a±) -(a
2
) -(ce
3
) G H 3(k).
19. Cohomologica l invariant s o f Quad
n 5
( n odd )
Fix 5 G /CQ/A:Q 2, an d recal l tha t Quad
n 5
(fc) i s th e se t o f isomorphis m classe s o f
nondegenerate quadrati c form s o f ran k n an d discriminan t 5 ove r k a s i n 2.5 .
T H E O R E M 19.1 . For n odd, Inv
fco
(Quad
n s
, Z/2Z ) z s a /ree H(k
0
)-module with
basis {wo, iu
2
, w±,..., ii
n
-i}.
P R O O F . W e us e th e followin g tw o formulas , whic h ar e prove d b y direc t com -
putation: Fo r q a quadrati c form ,
(19.2)
Wi
(q 0 (a)) = {a) -Wi-^q) + w
t
{q)
and
I ( l ) - ^ _ i ( g) i t i i s even .
If q\ i s an y quadrati c for m o f ran k n 1, we ca n associat e wit h i t a ran k n for m
g wit h discriminan t S b y th e recipe :
(19.4) q = qi®(d{qi)S).
Hence ever y invarian t a o f Quad
n s
define s a n invarian t a\ o f Q u a d
n - 1
b y th e
formula a\{q{) a(q). Not e tha t a\ Res(a), wher e th e restrictio n ma p (§13) i s
relative t o th e natura l embeddin g O
n
_ i SO(Q^) , wher e Q$ = ( 1 , . . . , 1, S).
By Theore m 17.3, ther e ar e well-define d element s AQ , ... , A
n
_i o f H(ko) suc h
that
n - l
(19.5) ai{qi) = ^ A ^ ^ ( g i ) .
For q i n Quad
n s
, w e ma y writ e
q = ( a i , a
2
, - . , «n - i , ^ i « 2 - « n - i ) -
Set g o = (^i,... , an _
2
) an d q\ = q
0
0 ( a
n
_ i ) , s o tha t (19.4) holds . W e hav e
n - l
(19.6) a(g ) = a
1
(q1 ) = a^qo 0 ( a
n
- i » = ^ A * ' ^ 0 ( a™~i))-
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