CHAPTER VII I
Witt invariant s
Recall tha t th e characteristi c i s assumed t o b e ^ 2 .
27. Propertie s o f th e Wit t rin g
27.1. Le t WGr(k) b e the Witt-Grothendieck rin g of k. Recal l that a n elemen t
of WGr(k) i s a formal differenc e q
qf
where q and q
f
ar e nondegenerate quadrati c
forms over k and the sum and product ar e those induced by the orthogonal sum an d
tensor produc t o f quadratic forms . Th e Wit t rin g W(k) i s the quotien t o f WGr(k)
by the idea l made up of integral multiples of hyp2 = (1,-1). Ther e is a short exac t
sequence
0 - I - W{k) - ^ Z/2 Z - 0
where / = I(k) i s the "augmentatio n ideal " o f W(k). Th e diagra m
WGr{k) - J ^ L , Z
i i
W{k) J^U Z/2 Z
is cartesian; i.e. , a n element o f WGr(k) ma y be viewed a s a pair (q, n), with n G Z,
q G W(fc), an d n = rank(g ) (mo d 2) .
Both W an d l^G r ar e functoria l i n k\ w e may tak e the m a s ou r "functo r H".
For mos t questions , W(k) i s mor e convenien t t o use . However , WGr(k) ha s th e
advantage tha t i t i s a X-ring i n the sens e of Grothendieck [SGA6 , Exp . V, §2] . I f q
is a quadrati c for m o n a /c-vecto r spac e V , le t b be th e uniqu e symmetri c bilinea r
form o n V suc h tha t q(x) = fr(x, x) fo r ever y x G V". Le t X pb be th e correspondin g
symmetric form on APV, cf . [Bo u 59, IX. 1.9]. Th e quadratic form on APV define d b y
x ^ (X
pb)(x,x)
i s denoted b y X
p(q).
I f q = (ai,.. . ,a
n
) , w e have X
pq
= J ]
7
(a/) ,
where / run s ove r th e subset s o f [1, n] with \I\ = p an d a / i s the produc t o f the ai
for i G i*. Th e powe r serie s
At(9) = X ) * * '
A
' ^ = ± + t-q +
t2-A2(g)
+ e WGr(fc)[[t] ]
is such tha t
Xt(q + q') = X t(q)-Xt(q').
Thanks to this multiplicativity property, X
t
(q) ma y be defined fo r every q G WGr(k),
see [SGA6 , loc.cit.] . W e thus hav e natura l exterio r powe r operation s
Xp: WGr(k)^ WGr(k).
63
http://dx.doi.org/10.1090/ulect/028/10
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