66 VIII. W I T T INVARIANT S
COROLLARY 27.13. IfT is versal, then a(T) = 0 =^ a = 0 .
27.14. W I T T INVARIANT S O F ( 2 , . . . , 2 ) GROUPS .
Let G = Z/2 Z x x Z/2 Z (n copies) . A n arbitrar y elemen t o f H 1(k1G) i s give n
by a n n-tupl e ( a i , . . . , a
n
) G /c*//c* 2 x x /c*/&* 2- Fo r J a subse t o f [l,n] , w e
write (a/ ) wher e a / i s th e produc t o f a ^ fo r i G / a s i n 27.1.
Write ai fo r th e Wit t invarian t (cei,... , a
n
) i— (a/).
T H E O R E M 27.15. Inv/
Co
(G,W) z s a/re e W(ko)-module with basis (a/)/c[i,n] -
P R O O F . Suppos e firs t tha t n = 1, i.e. , G = Z/2Z . Le t k = k
0
(t), an d le t T
be th e k-G-torsor define d b y £ , a s i n th e proo f o f 16.2. I f a G Invfc
0
(G, W) , the n
a(T) i s a n elemen t o f W(k) whic h i s unramifie d outsid e {0 , oo} b y 27.11. B y 27.7 ,
a(T) i s o f th e for m q$ ® qi'(t), wit h qo, q\ G W(fco) . B y 27.13 thi s implie s tha t
a = qo ® gi -ai, wher e a i i s th e tautologica l invarian t (a) i— » (a). Thi s prove s th e
theorem fo r n = 1.
The genera l cas e follow s b y inductio n o n n , a s i n th e proo f o f Th . 16.4.
T H E O R E M 27.16 (Wit t invariant s o f Quad
n
). Inv(Quad
n
, W) is a free W(k
0
)-
module with basis the exterior powers {A 0, A1 ,... , A n }.
In othe r words , any invariant of a rank n quadratic form q can be written
uniquely as
n
q •- ^CiX^q), wit h a G W(k
0
).
2 = 0
PROOF O F T H. 27.16. Essentiall y th e sam e a s fo r Th . 17.3: Th e standar d
embedding G O
n
give s a restrictio n ma p (cf . §13)
Inv(Quad
n
, W) -* Inv(G , W)
which i s injective . B y Prop . 13.2, it s imag e i s containe d i n th e W(/co)-submodul e
of Inv(G , W) fixe d b y th e actio n o f S
n
\ thi s submodul e ha s a basi s mad e u p o f th e
A* = Yl\i\=i fl; for i = 0 , l , . . . , n . Th e resul t follows .
E X A M P L E 27.17 (Wit t invariant s o f Pfister
n
an d Oct) . Inv(Pfister
n
, W) i s a fre e
W(ko)-module wit h basi s 1 and th e identity . Th e cas e n 3 gives that Inv(Oct , W)
is a fre e W(ko) -module wit h basi s {1,93}, wher e q% i s th e nor m o f th e octonio n
algebra.
E X A M P L E 27.18 (Wit t invariant s o f Alb) . Similarly , Inv(Alb , W) i s a fre e
W(ko)-module wit h basi s {1,^3,^5}, wher e #3(.A) , q$(A) ar e th e 3-Pfiste r an d 5 -
Pfister form s associate d wit h th e Jorda n algebr a A a s i n 22.4 .
EXERCISE 27.19. Le t n b e eve n an d 4 ; se t m = n/2. Le t q b e a quadrati c for m
of ran k n , an d pic k a G k* represente d b y q. W e ma y writ e q a s q = (a) 0 q
a
, wit h
rank(ga) = n 1. Put :
£(a,q) = \ m(qa)
Then:
(1) I f d(q) = 1, prov e tha t £(a,q) doe s no t depen d o n th e choic e o f th e elemen t a
represented b y q. Henc e £(a,q) i s a n invarian t o f q, whic h w e denot e b y £(q).
One ha s 2£(q) = A m(g), s o that £(q) ma y b e see n a s a kin d o f "canonica l half "
oi\m(q).
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