40 VI. COHOMOLOGICA L INVARIANT S O F On , S O
n
,
with ao , a\ G H(ko), i s the canonical decompositio n o f a as the sum (constant ) +
(normalized), cf . 4.5.
16.4. INVARIANT S O F G = Z/2 Z x •• x Z/2Z . A n arbitrar y elemen t o f
Hl(k,G) i s given b y an n-tuple ( a i , . . . , a
n
) G k*/k*2 x •• x /c*//c* 2. Fo r 7 a
subset o f [l,n] , w e write (a) / for the cup product n $ e j ( a i ) ^ H(k). W e have
(a)
0
= 1 G ff°(fe).
Write a / for the invarian t ( a i , . . . , a
n
) i— (a)/.
T H E O R E M . 7e £ G = Z/2 Z x x Z/2 Z ( n &raes). Tfte n Inv
fco
(G, Z/2Z ) z s a
/ree H(k^)-module with basis (aj)/c[i,n] -
P R O O F . B y induction o n n; the n = 1 case i s Proposition 16.2. Suppos e tha t
n 2 and writ e G ' for th e produc t o f n 1 copies of Z/2Z s o tha t G = Z/2 Z x G',
and on e write s ( a i , . . . , a
n
) a s (ai, a') wher e a ' i s an element o f Hl{k1 G').
Fix suc h a n element a' G 7T1(A:,G/). The n an y a G Inv&0(G, Z/2Z) define s
an invarian t a
a
/ G Invfc(Z/2Z , Z/2Z ) b y the formula a
a
/(ai ) = a ( a i , a ' ) . B y the
n = l case , a
a
/(ai ) = Ao,
a
' + (^i)Ai,a ' wit h A^
a
/ G H(k). Whe n a ' varies , the
maps a 1 i— Aja/ ar e als o invariants , i.e. , the y li e in Inv^0 (G / , Z/2Z) .
By induction , \j^ = Xl/ /cf2 nl( a ')i' / '\?7 / r s o m e ^j,i' £ H(ko). Conse -
quently,
a(ai,a 7 ) = (^(a')/'•*),/' ) + ( ^ ( a 7 ) / ' ' ( 0 : 1 ) ^ 1 , / ')
where th e sums ru n over al l subsets V o f [2,n]. Defin e A / as Ao,/ i f 1 0 7 and as
^i,/-{i} i f 1 G 7. Th e abov e formul a ca n the n b e rewritten as
/C[l,n]
which show s tha t th e a/' s span Inv(G , Z/2Z) .
Similarly t o what wa s don e i n the proo f o f 16.2, on e ma y recove r th e A/' s fro m
a. Fo r 7 = {z
1 ?
Z2,..., i
q
} a subse t o f [l,n] , defin e A; / to be the extension o f fco
generated b y indeterminates t ^ indexed b y 7. Defin e a n element a j o f i7 1(/c/,G)
by takin g ^ fo r the i-th coordinate i f i belong s t o 7, and 1 otherwise. Cal l bj th e
value o f a on a/, so 6/ is in H(ki). Tak e th e iterate d residu e ma p
r
7
: J7(*7 ) - #(fc/-{i
l }
) - •' - H(k
0
),
where al l residues ar e taken a t 0 (th e order i n whic h th e residues ar e taken i s
irrelevant). The n th e A / associated wit h th e invarian t a is:
A / = r
7
( 6 / ) .
Thus th e a/' s are linearl y independent .
EXERCISE 16.5 (invariant s of a product o f groups). I n this exercise, we write Inv/e
0
(G)
for Inv/c
0
(G, Z/pZ), wher e p is a fixed prime .
Let A, A' b e algebraic group s ove r ko. Fo r every extensio n k o f ko, w e identify
H^k.AxA') wit h th e produc t H 1 (k,A)xH1 (k,A'). I f a G Invfco (A) an d a' G Inv/e0(A'),
the cu p produc t a-a i s the elemen t o f lnvk0(A x A') defined by
(a-a')(T,Tf) = a{T)-a'{T') fo r ever y (T,T') e H^k.Ax A').
This define s a n H(ko)-linear ma p
CA,A' ' lnvko(A)g Invko(A') - Inv
feo
(A x A'),
where th e tenso r produc t i s taken ove r H(ko) = ©;7P(/c0, Z/pZ).
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