62
VII. COHOMOLOGICA L INVARIANT S O F ETAL E ALGEBRA S
E X A M P L E 26.2 . Le t M = Z/2 Z an d fix x,y G
iJ1
(G,Z/2Z) . I f on e write s X ,
y fo r th e image s o f x, y i n H 1 (k,Z/2Z), the n on e ha s X X = X - ( - l ) , y - y =
y •(—1), an d henc e
x-x-y + x-y-y = 2X-y-(-i) = o.
That is , x-y(x + ?/ ) i s negligible . I t i s nonzer o fo r well-chose n G , x, an d ?/ , fo r
example, whe n G = Z/2 Z x Z/2 Z an d {x, y} i s a basi s o f H l(G, Z/2Z) .
Note tha t x -x i s geometricall y negligibl e bu t i s no t negligibl e i n general .
T H E O R E M 26.3 . Let G Sn and assume that M is finite, with trivial action.
(1) An element x of H l(Sni M) is negligible if and only if its restriction to
every 2-elementary subgroup of S
n
generated by transpositions is 0 .
(2) For every x G Hl(Sn,M), i 0, the cohomology class 2x is negligible.
(3) An element x of H l{Sn, Z/2Z ) is negligible if and only if its restriction to
the subgroups of order 2 of S
n
is 0 .
P R O O F . Sinc e S
n
act s triviall y o n M , ever y elemen t o f H l(SniM) define s a
cohomological invarian t o f Et
n
a s i n 25.1. Thi s give s a natura l ma p
H\G, M) -+ I n v
Q
( 5
n
, M) = Inv
Q
(Et
n
, M) .
The kerne l o f thi s ma p clearl y contain s th e subgrou p H*i(S
n
,M) consistin g o f
the negligibl e classes . Conversely , an y elemen t o f th e kerne l i s negligibl e fo r ever y
field o f characteristi c 0 , henc e b y Propositio n 26. 1 i s negligible .
Property (1) i s then a consequenc e o f Theore m 24. 9 an d (2 ) follow s fro m 24.12.
Theorem 24. 9 combine d wit h th e nex t lemm a prove s (3) .
L E M MA 26.4 . [BS94 , Prop . 7.2.3 ] LetH be an elementary group of type ( 2 , . . . ,2) ,
and let a G Hl(H, Z/2Z ) be a cohomology class. The following are equivalent:
(i) a is negligible.
(ii) The restriction of a to every subgroup of order 2 of H is 0 .
(hi) If x\ is a basis of H l(H, Z/2Z)
7
a is in the ideal of the ring H(H, Z/2Z )
generated by the x\x^ + x\x 2^.
PROOF, (i ) = (ii ) becaus e 0 i s th e onl y negligibl e elemen t o f H(G, Z/2Z ) i f G
has orde r 2 sinc e suc h a G i s isomorphi c t o Gal(C/R) .
(ii) = (hi) : Th e rin g H(H, Z/2Z ) i s a polynomia l rin g ove r F 2 i n m variable s
x\ (wher e m rank H) , an d a ca n b e writte n a s a homogeneou s polynomia l P o f
degree i i n th e x\. Assumptio n (ii ) mean s tha t thi s polynomia l vanishe s wheneve r
the coordinate s ar e i n F2 . I t i s a simpl e exercis e t o sho w tha t thi s i s equivalen t t o
a bein g i n th e idea l generate d b y th e x\x^ + x\x 2.
(iii) = (i) : Th e elemen t x\x^ + xxx 2^ i s negligibl e a s explaine d i n Exampl e
26.2. D
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