54
VI. COHOMOLOGICA L INVARIANT S O F O
n
, SO
n
, .. .
This gives a method fo r constructing a n element o f Invfc0(Quadn, C): star t wit h
any X{ in H(ko, C(— i))2, an d tak e th e invarian t
q i- Wi(q) Xi.
We denot e thi s invarian t b y wi x^. Not e tha t i t take s value s i n th e subgrou p
H(k,C)2 o f elements kille d b y 2 .
THEOREM
23.5 . Any normalized invariant Quad
n
» H(*,C) can be written
uniquely as
n
2_,wi

x
i with Xi G H(ko, C{—i))2-
PROOF.
Th e resul t follow s fro m 23. 3 by the sam e metho d a s 17.3.
COROLLARY
23.6 . If q, q'
G
Quad
n
(/c) have the same Stiefel-Whitney invari-
ants, one has a(q) = a(q f) for every a G Inv(Quad
n
, C).
This "explains " why the Arason invariant es is not the restriction of an invariant
of Quad n, cf . [Ar75 , las t 3 lines]: Suppos e tha t 1 is a squar e i n ko and tak e tw o
3-Pfister form s q, q' ove r k/k^. On e check s tha t Wi(q) an d wi{q') vanis h fo r i 1.
If es wer e a linea r combinatio n o f th e w^, w e would hav e es(q) = ^ 3 (/'), whic h i s
not alway s true .
EXERCISE
23.7. Le t a G Inv(Quad n, C) be a normalized invariant. Sho w that 2a = 0.
[Hint: us e Th. 23. 5 or give a direct argumen t a s in the proof o f Th. 24.12 below.]
23.8.
RELATIO N WIT H MILNO R if-THEORY .
I n order to use Milnor's notation ,
we write F fo r th e groun d field instead o f k. Th e i-t h Milno r K-grou p o f F wil l b e
denoted by Ki(F). Le t C be a finite IV-module of order prime to the characteristic .
There exist s a natural pairin g
c: Ki(F) x H(F,C) - H(F,C(i))
characterized b y th e propert y that , i f x = (#i,.. . , xj i s a decomposabl e elemen t
of Ki(F) an d z belong s t o H(F, C) , the n c(x , z) i s the cu p produc t {x\) (xi) -z.
The fac t tha t c exists i s clea r whe n i = 1; th e crucia l cas e i = 2 is a theore m o f
Tate [Ta76 , Th . 3.1]; the cas e i 2 follows fro m th e cas e 2 = 2 and th e definitio n
of the Milno r if-groups .
It i s natura l t o se e c a s a kin d o f cu p produc t an d t o denot e i t simpl y b y
(x, z) i— x-z.
Consider no w th e Milno r /c-grou p ki(F) = K
l
(F)/2Ki(F). Cal l H(F,C)
2
th e
subgroup o f H(F, C) kille d b y 2 . I t i s clea r that , i f z belong s t o tha t group , the n
x-z depend s onl y o n th e imag e o f x i n ki(F). Henc e we get a pairin g
ki(F)xH(F,C)2^H(F,C{i))2.
Let no w q b e a quadrati c for m ove r F. B y [M i 70, §3] , th e Stiefel-Whitne y
class Wi(q) ma y b e define d i n ki(F). Henc e w e ma y us e i t t o ma p H(F,C)2 i n
H(F, C(i))2 , henc e als o to ma p H(F, C(—i))2 i n H(F, C)2 W e find the sam e ma p
as th e on e define d above .
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