23. COHOMOLOGICA L INVARIANT S MO D 2'' 53
Th e su m o f two liftabl e element s i s liftable .
Hence any sum o f cup products o f elements of degree 1 for example , an y Stiefel -
Whitney invarian t o f a quadratic for m q is liftable. (Milnor' s conjectur e implie s
that ever y z i s liftable. )
Let x G H(k, C(-i)) b e such that 2x = 0. Le t z G H^k, Z/2Z ) b e liftable. W e
define a cup produc t z x i n H(k, C). T o do so, choose m larg e enoug h s o that C
is killed b y 2 m _ 1 , an d choos e a lifting z
m
o f z , i.e. , a n elemen t o f H %(k, Z/2 mZ(i))
whose imag e i n H l(k,Z/2Z) i s z. Ther e i s an obviou s pairin g
Z/2mZ(i) x C(-i) - C ,
which allow s u s to defin e th e (usual ) cu p produc t z
m
-x in H(k, C).
LEMMA
23.1. z m-x does not depend on the choice of lifting z m, and is killed
by 2.
PROOF.
A different liftin g o f z in
Hl{k,
Z/2
mZ(i))
differ s fro m z
m
b y the imag e
of a cohomolog y clas s z f G Hl(k, Z/2 rn~1 Z(i)) b y th e natura l ma p "multiplicatio n
by T : Z/2 rn~1 Z(i) - * Z/2 mZ(i). Thi s modifie s z
m
-x b y 2z'-x, whic h i s 0 sinc e
2x = 0 by assumption .
We write z x fo r z
m
-x, an d cal l i t th e "cu p product " o f z an d x. I t belong s
to H(k,C)2, wher e th e subscrip t 2 means th e subgrou p o f elements kille d b y 2 .
23.2. INVARIANT S O F Z/2Z. Choos e x G #(/c0 ,C(-l))
2
. I f k i s an extensio n
of ko and z G
Hx{k,
Z/2Z) , th e elemen t z x o f iJ(fc, C) i s defined a s above . Th e
map z ^ z x i s an elemen t o f Inv/
Co
(Z/2Z, C) whic h w e denote b y i d x.
LEMMA.
Every normalized invariant in Inv/Co(Z/2Z, C) can be written uniquely
as id x with x G H(ko, C{— 1))-2
PROOF.
Le t a G Inv(Z/2Z , C) b e a normalize d invariant . A s i n 16.2, (t)
2
is a versa l /co(t)-Z/2Z-torsor . Th e elemen t a((t) 2) i s unramifie d outsid e {0 , oc}.
Its residu e a t t 0 i s a n elemen t x\ o f H(ko,C(—l)), an d on e see s a s i n 16.2
that th e invarian t i s the cu p produc t o f (t) 2m wit h x\. Sinc e (t)
2
= (t
3)2,
w e hav e
2(t)2m -x\ 0, hence 2x\ 0 by 7.11. Therefore , a((t) 2) = (t)
2
x\, a s desired.
23.3. INVARIANT S O F G = Z/2 Z x •• x Z/2Z . Recal l th e invarian t a
7
G
Inv(G,Z/2Z) fro m 16.4. Fo r z G H
l{k,G),
a 7(z) i s a n elemen t o f H
l(k,Z/2Z)
which i s obviousl y liftable . Fo r x G H(ko, C(— 1/|)),
2
w e writ e aj x fo r th e
invariant o f G given b y z ^ a/(^ ) x.
LEMMA.
Every normalized invariant of G = Z/2 Z x x Z/2 Z ( n times) can
be written uniquely as
y a / x / where xj G H(ko, C(— 1/|)).2
/C[l,n]
PROOF.
B y inductio n fro m 23.2 , as fo r Theore m 16.4.
23.4.
INVARIANT S
O F Quad
n
. Le t q b e a quadrati c form , an d le t x b e a n
element o f H(k,C(—i)) wit h 2x = 0 a s above . Sinc e wi(q) i s liftable , th e cu p
product Wi(q) x i s a well-defined elemen t o f H(k, C) 2.
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