60 VII. COHOMOLOGICA L INVARIANT S O F ETAL E ALGEBRA S
LEMMA
25.12. If q is a quadratic form of rank n
7
then
w.n2\ )
=
\ WM) i fn
=
i ( mod 2
)
\wi(q) + {2)-Wi-i(q) if n ^ i (mo d 2) .
PROOF.
Firs t not e that w e have (2 ) -(2) = 0 in H(k, Z/2Z ) sinc e the quadrati c
form (1,-2,-2 ) represent s 0 .
Write q ( a i , . . . , a
n
) . Recal l tha t Wi(q) i s th e i-t h elementar y symmetri c
polynomial i n the (a^)'s . W e may expan d Wi((2)q) usin g (2aj) = (2 ) + {ptj)- Sinc e
(2) -(2) = 0 , most o f the terms occurring i n the expansion ar e 0 and w e are left wit h
wi({2)q)=wi(q)+li2)-wi-1{(i).
where £ is the intege r n i + 1.
Theorem 25.10 combine d wit h Theore m 25. 6 gives:
THEOREM
25.13. Set m = [n/2]. Then Inv
fco
(Etn, Z/2Z ) is a free H(k
0
)-
module with basis 1, w\, ... , w
m
. We have Wi 0 for i m + 1, and w
m
+i is
equal to 0 if m is even and is equal to (2 ) -Wm if m is odd.
REMARKS
25.14. (1) Fro m a practical poin t o f view, the u^(g# ) ar e ver y eas y
to compute . If , fo r instance , E = k[x]/P(x), on e can write qE in terms of
the coefficient s o f P , diagonaliz e it , an d ge t th e Wi. Th e sam e woul d no t
be tru e fo r th e wf a (E).
Indeed, th e cas e i 2 [Se84 ] ha s bee n use d ver y ofte n t o comput e
wf\E), se e e.g. [Vi85] , [BLV86] , [Vi88] , [Cr89], [Cr90] , [Bo90] , [BF91],
[Ha 92], [Mes95] , [Mes98] . Thi s invarian t ha s a Galois-theoreti c inter -
pretation: B y 25.3 , i t i s th e obstruction t o liftin g th e homomorphis m
(fE' T/ c S
n
a s i n 3. 2 t o a homomorphis m F^ S n.
(2) Not e that , i f n i s even , th e cohomologica l invariant s are the same for n
and n + 1. Thi s fit s wel l with Theore m 31.23.
25.15.
GENERALIZATIO N T O WEY L GROUPS .
Her e w e assum e fo r simplicit y
that ko ha s characteristi c 0 . Le t W b e a Wey l group , i.e. , a finite Coxete r grou p
which stabilize s a lattice i n R n (i.e. , a product o f Weyl groups o f type A, ... , E$).
The standar d representatio n o f W i s defined ove r Q , henc e ove r ko. On e ca n thu s
make W ac t o n Aff n, wit h quotien t isomorphi c t o Aff n. Usin g this actio n (instea d
of the actio n o f S n), on e ca n extend t o W mos t o f what ha s bee n prove d above . I n
particular:
THEOREM.
An element of lnv(W
1
C) is 0 if its restriction to every abelian
subgroup ofW generated by reflections is 0.
This implie s tha t an y normalize d invarian t i s killed b y 2 .
25.16.
INVARIANT S MO D
2
m
. No w let C b e a finite T^-module whos e order i s
a powe r o f 2 . Fo r E a n etal e algebra , Wi(E) G
Hl(k,
Z/2Z ) i s liftable i n the sens e
of §23 . Fo r Xi G H(ko, C(—i))2- w e write Wi xi fo r th e invarian t
E ^ Wi(E) x
%
i n Inv(Et n, C),
where i s as define d i n §23 . B y th e sam e argument s a s fo r Inv(Et
n
, Z/2Z) above ,
we have:
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