25. COHOMOLOGICA L INVARIANT S O F Et
57
Let D b e suc h a n irreducibl e divisor . I f D ^ A , the n b y 24. 4 an d 24.6.1, th e
image o f a i n
H(KD,C)
belong s t o H{k{D)
1
C). Thu s a ha s residu e 0 at D. Fo r
D A, E^ n i s isomorphi c t o E' x E" wit h E' o f ran k 2 . A s show n above , thi s
implies tha t th e imag e o f a i n H[K/\^ C) i s 0. Henc e the residu e o f a wit h respec t
to th e discret e valuatio n correspondin g t o A i s 0.
Since a ha s residu e 0 wit h respec t t o al l discret e valuation s o f K/ko whic h
correspond t o irreducibl e hype r surf aces , a belong s t o H(ko,C) b y 10.1. That is , a
and th e constan t invarian t define d b y a agre e o n E
gen.
Henc e thes e invariant s ar e
equal sinc e E
gen
i s versa l (cf . 12.3). Sinc e a vanishe s fo r multiquadrati c algebras ,
a i s identically 0 .
REMARK 24.10. Le t H b e th e subgrou p o f S
n
generate d b y th e transposition s
(2j l,2j) , fo r j = 1, ... , m [n/2]. I t i s a n elementar y abelia n grou p o f typ e
(2,..., 2) an d ran k m. Theore m 24. 9 can b e restate d i n th e followin g form :
THEOREM
24.11 . The restriction map (cf . §13)
Resfn : Inv(Sn, C) - Inv(#, C)
is injective.
(Recall tha t Inv(S n, C) = Inv(Et n, C), se e 3.2. )
PROOF.
Indeed , i7-torsor s correspon d t o sequence s (^i,...,£'
m
) o f m qua -
dratic etal e algebras . Hence , i f an invarian t a is such tha t Res(a ) = 0 , one ha s
a(Ei x x Em) = 0 (i f n i s even, i.e. , n = 2m)
a(Ei x x E
m
x k) = 0 (i f n i s odd, i.e. , n = 2m + 1)
for al l possibl e choice s o f Ei, ..., E
m
ove r ever y k/ko. B y th e abov e theorem , thi s
implies a = 0 .
24.12. Recal l tha t a G Inv/
Co
(Etn, C) i s normalized i n th e sens e o f 4. 5 i f
a(Esplit) = 0 in H(k
0
, C) fo r £ s p l i t = k
0
x x fe0.
THEOREM.
If a e Inv(Et
n
,C) is normalized, then 2a = 0 .
PROOF.
B y Theorem 24.9, it is enough to prove that 2a(E) = 0 for E = E'xE"
where E' ha s ran k 1 or 2 .
Set E\ = k x E" o r k x k x E" a s neede d s o that i t ha s th e sam e ran k a s E.
The sam e constructio n a s at th e star t o f the proo f o f Theorem 24. 9 combined wit h
induction o n n show s that 2a{E\) = 0 .
There i s an extensio n k' o f k o f degre e a t mos t 2 such tha t E an d E\ becom e
isomorphic ove r k f. Henc e a(E) an d a{E\) hav e th e sam e imag e i n H(k f,C). B y
applying Corjjf w e get [k
f
: k]-a(E) = [k
f
: k]-a(Ei), henc e 2a(E) = 2a(E ±) = 0.
25. Cohomologica l invariant s o f Et
n
From no w on, unti l §33 , the characteristi c i s j^ 2.
In thi s section , w e fix C = Z/2 Z an d construc t invariant s
Etn-+H(k,Z/2Z) = H(k).
25.1. FIRS T CONSTRUCTIO N O F INVARIANTS : TH E GALOI S STIEFEL-WHITNE Y
CLASSES. Eac h u G H{Sn, Z/2Z ) define s a mo d 2 cohomological invarian t o f Et
n
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