25. COHOMOLOGICA L INVARIANT S O F Et
n
5 9
T H E O R E M 25.6 . (1) The map Res : Inv
fco
(Et
n
, Z/2Z ) - Inv
f c o
(#, Z / 2 Z )
5
- is
an isomorphism.
(2) The H(ko)-module Inv/
Co
(Etn, Z/2Z ) is free with basis 1, wf a , ... , wf% 1,
where m = [n/2].
(3) One has wf a = 0 for i m .
PROOF. Th e argumen t i n the proo f o f Theorem 17.3 shows that Inv(iJ , Z / 2 Z ) 5 m
is a fre e H(ko,Z/2Z)-xnodu\e wit h basi s consistin g o f th e elementar y symmetri c
functions ]T ^ ...i.(xii) *' ' ( xij) r 0 3 m - Bu t suc h a symmetri c functio n i s
the imag e o f w^ a b y Exampl e 25.4 . Thi s show s tha t th e ma p
Inv(Et
n
, Z/2Z ) - Inv(ff , Z/2Z) S™
is surjective . I t i s injectiv e b y Theore m 24.11 , s o i t i s a n isomorphism . Th e wf a
are 0 fo r i m sinc e thi s i s tru e fo r multiquadrati c algebra s b y 25.4 . Th e theore m
follows.
25.7. SECON D CONSTRUCTIO N O F INVARIANTS : S T I E F E L - W H I T N E Y CLASSE S
OF TH E TRAC E FORM . I f E G Etn(fc), it s trace form q
E
i s th e quadrati c for m o n E
defined b y q
E
(x) = Tr
E
/k(x2). Thi s give s a morphis m Et
n
» Quadn . Th e Stiefel -
Whitney invariant s o f Quad
n
(a s i n 17.1 ) thu s defin e cohomologica l invariant s W{
on Et
n
b y
(25.8) w
l
{E) = w
l
{qE).
E X A M P L E 25.9 . Le t E b e th e multiquadrati c algebr a fro m Exampl e 25.4 . Th e
trace for m o f E i s q
E
= q
El
0 0 q
Em
, an d q
Ei
= (2 , 2x{).
T H E O R E M 25.10. [Kahn84 ] For E e Et
n
(fc),
s^(E\
=
\
W
ME)
ifi is odd
\^I(QE) + (2)'Wi_i(q
E
) ifi is even
in H(k, Z/2Z).
P R O O F . B y Theore m 24. 9 i t i s sufficien t t o prov e th e formul a whe n E i s mul -
tiquadratic. Le t u s assum e tha t n i s even . (Th e cas e wher e n i s od d i s analogous. )
By Exampl e 25.9 ,
to
= ( 2 , . . . , 2 ) 0 ( 2 z i , . . ., 2z
m
) .
v
v '
m time s
Hence q
E
= (2)q
E
wher e q
E
= ( 1 , . . . , 1) 0 (x\, ..., x
m
) . I t i s clea r tha t Wi(q
E
) =
wfa (E). Th e formul a follow s fro m Lemm a 25.12 below , whic h relate s Wi(q
E
) wit h
Wi(qE). D
EXERCISE 25.11Sho . w tha t Th . 25.10 ca n b e restate d i n term s o f tota l Stiefel -
Whitney classe s a s
w^(E)=w(qE)-(l + (2)id)),
where d is the discriminan t o f E, s o that (d) w\{qE) = wf a (E). Sho w that thi s formul a
is equivalent to :
w{qE) = w^(E)il + (2 ) -(d)).
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