33.APPLICATION OF TRACE FORMS TO NOETHER'S PROBLEM 87
DEFINITION
33.6 . A n element a G H(K, G ) i s said t o b e unramified over ko if ,
for ever y discret e valuatio n v o f K whic h i s trivial o n /co , the residu e o f a a t v (a s
defined i n 7.13) i s 0.
PROPOSITION
33.7 . If K/ko is rational, every unramified cohomology class in
H{K,C) is constant, i.e., belongs to H(ko,C).
This i s a consequence o f Th. 10.1.2.
REMARK
33.8 . Conversely , ever y constan t cohomolog y clas s i s obviousl y un -
ramified.
33.9.
UNRAMIFIE D COHOMOLOGICA L INVARIANTS .
Le t G b e a finite group ,
let C b e a s i n 33.5 , an d le t a b e a n elemen t o f Invjt
0
(G, G), i.e. , a cohomologica l
invariant o f the G-torsor s ove r Fields/
fco
wit h value s i n #(* , C).
We sa y tha t a i s unramified if , fo r ever y K/ko whic h i s finitely generate d an d
every G-torso r T ove r K, th e cohomolog y clas s a(T) G H(K, C) i s unramified ove r
ho in th e sens e define d i n 33.6 .
PROPOSITION
33.10. If a G Inv
fco
(G, C) is unramified overko, and z/Rat(G/fco )
is true, then a is constant.
PROOF. Choos e a versal G-torso r T ove r K a s in Rat(G//c
0
). B y 33.7 , a(T) i n
H(K, C) i s constant, i.e. , comes from a n element a$ G H(ho,C). B y replacing a by
a ao, we may assum e tha t a(T) = 0 . B y Th . 12.3, this implie s tha t a = 0 .
COROLLARY
33.11 . Suppose that Rat(G/fco ) is true and that a is normalized
(as define d i n 4.5 ) and unramified. Then a = 0 .
REMARKS
33.12. (1) I n wha t follows , w e shal l appl y 33.11i n th e followin g
way: fo r suitabl e G an d &o , we shal l construc t a non-zer o cohomologica l
invariant a which i s unramified an d normalized . B y 33.11, thi s wil l sho w
that Rat(G//co ) i s no t true , an d henc e (b y 33.3 ) tha t Noether's problem
has a negative solution for G over ko.
(2) Wha t w e ar e doin g her e fo r cohomologica l invariant s coul d als o b e don e
for Witt invariants.
REMARK
33.13. Th e strateg y w e ar e usin g i s essentiall y du e t o Sal t man an d
Bogomolov. However , ther e i s a difference : Bogomolo v an d Saltma n use d mainl y
cohomological invariant s comin g fro m element s o f H(G, C) (wit h trivia l actio n o f
G o n G ) a s i n 4.2 ; the invarian t w e shall us e will no t b e o f that for m (a t leas t no t
in a n obviou s way) ; rather , i t wil l com e fro m th e propertie s o f th e trac e for m i n
rank 6 or 7 proved i n §32.
EXAMPLE
33.14 ( A cohomological invariant fo r cyclic groups of order 8, 16, ...).
From no w unti l th e en d o f thi s section , th e characteristi c i s ^ 2 , an d G = Z/2Z ;
for k a n extensio n o f /co , we write a s usual H
l(k)
fo r H
l(k,
C).
Take G cyclic of order 2
m
wit h m 3. Ever y x G
Hl{k,
G) defines b y reductio n
(mod 2 ) a n elemen t d{x) o f H
l(k,
Z/2Z ) = H^k). Put :
b(x) = (2)-d(x) mH 2(k).
This define s a n invarian t b in Inv/
eo
(G, Z/2Z).
PROPOSITION
33.15. The invariant b defined above is normalized and unrami-
fied; if the ground field ko is Q, this invariant is not 0 .
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