86 IX. TH E TRAC E FOR M I N DIMENSIO N 7
REMARK
32.31. I f q is as in 32.29, we write e(q) for the invariant (c) -(-1) '(-1)
in H
3(k)
a s i n 32.13. I f k doe s no t hav e propert y OQ, ther e exist s a n etal e algebr a
E o f rank 7 with trivia l Stiefel-Whitne y classe s an d e(qs) ^ 0 . Moreover , b y wha t
has bee n prove d i n §31, one may choos e for E a n algebr a o f the for m
E = Ei x E
2
x E4 , wit h rankl ^ = i ,
and wit h th e furthe r propert y tha t E4 i s give n b y a polynom e bicarr e a s i n th e
proof o f Th . 31.18.
33. Applicatio n o f trac e form s t o Noether' s proble m
Until 33.14, we place n o restrictio n o n th e characteristi c o f k.
33.1. VERSA L TORSOR S AN D NOETHER' S PROBLEM : DEFINITIONS . Le t G b e
a finite group . W e shall b e intereste d i n th e followin g propert y o f G :
Rat(G/fco) - There exist an extension K/ko and a G-torsorT over K such
that\
(1) T i s versal (as defined in 5.1);
(2) K i s fco-rational (i.e., is a purely transcendental ex-
tension of ko)
Many o f th e group s w e hav e considere d a t th e beginnin g o f thes e note s hav e tha t
property; fo r instanc e th e symmetri c grou p 5
n
, o r a n elementar y abelia n grou p of
type (2,... , 2).
A related propert y is :
Noe(G//co) - There exists an embedding p : G GLn(ko) such that, if
Kp is the subfield of ko(Xi, ..., X
n
) fixed by G, then K
p
is
ko-rational
Deciding whethe r Noe(G/&o ) i s tru e i s "Noether' s problem " fo r G an d &o - I f
Noe(G//co) i s false , w e sa y tha t Noether's problem has a negative solution for G
over ko.
REMARK
33.2 . Th e propert y Noe(G//co ) i s known t o b e equivalen t t o eac h of
the following :
(1) Ther e exist s a n embeddin g p fo r whic h K
p
/ko i s "stabl y rational " (i.e. ,
becomes rationa l afte r adjunctio n o f finitely man y indeterminates) .
(2) Fo r ever y embeddin g p , K p/ko i s stably rational .
THEOREM
33.3 . Noe(Gy/c
0
) implies Rat(G/fco) .
PROOF.
Indeed , i n th e situatio n o f Noe(G/&o), th e embeddin g G —» GL
n
(ko)
defines a versal torso r a s i n 5.4 .
REMARK
33.4 . Th e convers e implicatio n i s fals e i n general . Fo r instance , i f
G i s cycli c o f orde r 47 , the n Noe(G/Q ) i s no t tru e (Swa n [Sw69] , Voskresenski i
[Vo70]), but Rat(G/Q ) i s true [Vo77] . Se e e.g. Swans ' repor t [Sw83] .
33.5.
UNRAMIFIE D COHOMOLOG Y CLASSES .
W e nee d a generalizatio n o f th e
notion o f unramified cohomolog y clas s defined i n 9.4. Conside r a finitely generate d
extension K/ko] le t C b e a finite T^ Q-module.
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