26. APPLICATION S T O NEGLIGIBL E COHOMOLOG Y
61
T H E O R E M . Every normalized invariant Et
n
H(*,C) can be written uniquely
as
[n/2]
y ^ Wi Xi with Xi G H(ko,C(—i))2-
The sam e statemen t hold s wit h Wi replace d wit h wf a .
The theore m show s tha t two etale algebras E and E
1
which have the same Wi's
have the same cohomological invariants for any C ; i n particular , thi s applie s whe n
QE = QE.
EXERCISE 25.17. Le t T b e a profinit e grou p an d le t X , X' b e tw o finite T-set s ( =
sets wit h a continuou s T-action) . W e sa y tha t X an d X' ar e weakly equivalent (cf . [Se71,
§13.1, Exere . 5] ) i f the correspondin g comple x linea r representation s o f Y ar e isomorphic ,
or, equivalently , i f every g G T fixes th e sam e numbe r o f points i n X an d i n X'.
When r = Tk an d X , X' ar e o f orde r n , th e action s o f T o n X an d X' giv e homo -
morphisms p, p : T » S
n
an d thu s defin e etal e algebra s E an d E' ove r k. Suppos e tha t
char(/c) ^ 2 and th e followin g holds :
If S is a 2-Sylow subgroup ofTk, then X and X' are weakly equivalent as S-sets.
Then:
(1) Sho w tha t qE an d qE hav e th e sam e Stiefel-Whitne y classes .
[Hint: Afte r replacin g /c by a tower of odd-degree extensions, we may assum e
that r i s a pro-2-group, i.e. , tha t T = S. Sho w that p an d p ar e then conjugat e
in O
n
(R), an d deduc e tha t the y giv e the sam e Wi\ use 25.16.]
(2) Giv e a n exampl e wher e qE an d q
E
ar e no t isomorphic .
[Hint: Tak e k = Q(u,v), wher e u, v ar e indeterminates ; writ e k
u
, k
v
, k
uv
for th e quadrati c algebra s define d b y u, v, uv. Tak e E = k
u
x k
v
x k
uv
an d
E' = k x k x (k
u
0 k
v
).]
(3) Assum e tha t k contain s a n algebrai c closur e o f th e prim e field. Sho w tha t qs
and qE' ar e isomorphic .
26. Application s t o negligibl e cohomolog y
Let G b e a finite grou p an d M a G-module . A n elemen t x G H(G, M) i s sai d
to b e negligible (cf . [BS94 ] o r [Se94] ) if , fo r ever y field k an d ever y (continuous )
homomorphism (p : Tk G , w e hav e cp*(x) = 0 i n H(Tk, M) = H(k, M).
(There i s als o a weake r notio n o f "geometricall y negligible " studie d b y Saltma n
[Sal 95], wher e on e ask s fo r th e sam e property , bu t onl y fo r fields k containin g Q. )
P R O P O S I T I ON 26.1. An element x e H(G,M) is negligible if p*(x) = 0 for
every field k of characteristic 0 and every (p:Tk G.
P R O O F . Le t x i n H %(G, M) b e negligibl e wit h respec t t o fields o f characteristi c
0. Conside r a field k o f characteristi c ^ 0 , an d a homomorphis m / : T^ G. W e
have t o sho w tha t f*(x) = 0 i n H l(Tk,M). W e ma y assum e tha t k i s perfect .
Consider th e rin g o f Wit t vector s ove r k an d it s field o f fraction s K. W e hav e a
natural ma p TK T& (se e 7.1), henc e / define s / o : ^K —* G. Sinc e char K = 0 ,
we hav e fj$(x) 0 i n H 1 (TK,M). Bu t TK » T^ split s b y Lemm a 7.6 , henc e th e
map H l(Tk, M) - if*(IV , M) i s injective , henc e f*(x) = 0 .
One ca n sho w tha t i t i s enoug h t o chec k th e conditio n (p*(x) = 0 whe n k i s o f
characteristic 0 , an d p correspond s t o a Q-versa l G-torsor .
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