CHAPTER V I
Cohomological invariant s o f O
n
, SO
n
, .. .
In thi s chapter , th e characteristi c i s assumed t o b e 7 ^ 2.
16. Cohomologica l invariant s o f (2,... , 2) group s
In thi s sectio n w e determin e Inv(G , Z/2Z) fo r G elementar y abelia n o f typ e
(2,2,...,2).
16.1.
COHOMOLOG Y MO D
2 . W e writ e H
l(k)
fo r W(k, Z/2Z ) an d H(k) fo r
the direct su m of the fP(fc). Thu s H°(k) = Z/2 Z an d H x(k) = k*/k* 2. Fo r a G k\
write (a) fo r th e correspondin g clas s i n H l(k), s o tha t (ab) = (a) + (b). Th e cu p
product, whic h w e denote b y x-y, give s H(k) a ring structure . Recal l tha t th e cu p
product (a) -(b) corresponds (vi a the usua l identificatio n o f H 2(k) wit h a subgrou p
of Br(fc) ) t o th e quaternio n algebr a (a , 6); i n particula r (a) -(b) 0 i f an d onl y i f
the quadrati c for m (1, —a, —b) represents 0 , i.e. , b is a nor m fro m th e extensio n
k(y/a)/k o r equivalentl y fro m th e quadrati c etal e algebr a k[x]/(x
2
a).
16.2.
INVARIANT S O F G =
Z/2Z . A n arbitrar y elemen t o f H l(k,G) i s give n
by a G /c*//c* 2. Writ e i d fo r th e invarian t (i.e. , elemen t o f Inv/
Co
(Z/2Z, Z/2Z) )
which send s a t o (a) G H x(k).
PROPOSITION.
Inv
fco
(Z/2Z,Z/2Z) is a free H(k
0
)-module with basis {1, id}.
PROOF. Exten d scalar s t o ko(t) an d conside r th e elemen t (t) G iif1(A:o(t)). I t
defines a G-torso r T ove r th e projectiv e lin e minu s {0,oo} , whic h fro m th e poin t
of vie w o f quadrati c form s correspond s t o (t). Le t a G Inv/c0(Z/2Z, Z/2Z) b e a n
invariant; w e hav e a(T) G H(ko(t)). B y 11.7 , a(T) i s unramifie d outsid e {0,oo} ,
hence b y 9. 4 ca n b e writte n uniquel y a s
(16.3) a(T) = A
0
+ (t) -\x i n H(k 0(t))
with Xi G H(k 0).
Now define a' t o b e th e invarian t A o -f Ai -id. Th e invariant s a and a' agre e o n
the torso r T , whic h i s obviously versal . Thu s th e tw o invariant s ar e equa l b y 12.3,
and 1 and i d spa n Inv(Z/2Z , Z/2Z) .
Note tha t th e elemen t A o occurring i n (16.3) i s a((l)) , th e valu e o f a fo r th e
trivial G-torso r ove r fco- Th e othe r coefficient , Ai , i s th e residu e a t t = 0 (o r a t
t = oo) of a((t)) eH(k
0
(t)).
Consequently, i f a is 0, the n A o = A i = 0 . Tha t is , the invariant s 1 and i d ar e
linearly independen t ove r H(ko).
The decompositio n o f a n invarian t a a s
a = a o + a\ -id
39
http://dx.doi.org/10.1090/ulect/028/08
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