48
VI. COHOMOLOGICA L INVARIANT S O F O
n
, SO ,
21. Cohomologica l invariant s o f hermitia n form s
Let k\ ko(VS) b e a quadrati c extensio n o f fco- We will determin e th e grou p
Inv(Hermn,Z/2Z).
21.1.
RELATION S WIT H
Quad
n
AN D
Quad
2n
. Ther e i s a natura l ma p
Quadn— Herm
n
,
namely (ai,... , an)i— (a^,..., a n)H. Sinc e ever y hermitia n for m ma y b e diago -
nalized t o (ai,... , an)H wit h a ^ G &*, this ma p i s surjective.
Let V b e a fre e modul e o f ran k n ove r th e quadrati c algebr a k 0/c
o
/ci , and le t
ftbea nondegenerat e hermitia n for m o n V . The n V i s a 2n-dimensiona l k- vector
space endowe d wit h a quadrati c for m v ^ h(v,v); w e denot e thi s quadrati c for m
by Nh. Thi s gives a map N : Herm
n
» Quad
2n
whic h is well-known to be injective.
Hence we may view Hermn(&) a s a subset o f Quad2n(fc). (Whe n v ^ G &, this subse t
is {hyp} , wher e hy p i s hyperboli c o f ran k 2n ; i n th e genera l cas e i t i s th e se t o f
q G Quad2n(fc) whic h becom e hyperboli c ove r k(y/S).)
The ma p
(21.2) Quad n(/c) - Herm n(/c) ^ Quad 2n(/c)
is q h- (1, - %; i.e. , N(q H)= = (1, -J)•?.
21.3.
STIEFEL-WHITNE Y CLASSES .
Le t x G H(k
0
) b e a n elemen t whic h give s
0 i n H(ki), i.e. , whic h belong s t o th e idea l (S)-H(ko) generate d b y (5) [Ar75 ,
Cor. 4.6] . Le t i b e a n intege r 1. I f h G Hermn(fc), choos e q G Quadn(/c) whic h
gives h a s i n 21.1, i.e., suc h tha t qn = h.
LEMMA
21.4. The cup product x-Wi(q) G H(k) depends only on x and h (an d
not o n th e choic e o f q).
PROOF.
I t i s enoug h t o prov e th e lemm a whe n x = (S). Whe n 5 is a squar e
in fc, we hav e x = 0 an d th e statemen t i s obvious . W e ma y thu s assum e tha t 5 is
not a square , s o tha t th e field K = k(y/S) i s a quadrati c extensio n o f k. W e no w
consider thre e cases :
i) The case i = 1. I f q = (a
1 ?
..., a
n
), w e hav e wi(g ) = (ai---a
n
) . Th e
hermitian for m h q^ ha s determinan t d G k*/NK*, whic h i s th e imag e o f
ai an i n k*/NK* (se e [KMRT98 , p. 114]). Th e natura l embeddin g k*/NK* -
H2(k) transform s d int o (5)-(d ) = ((5 ) -wi(q). Thi s show s tha t ((5)-toi(g ) depend s
only o n h.
ii) The case n 2 . Sinc e Wi(q) = 0 if i n, w e ma y assum e tha t i n. B y
part i) , w e ma y als o assum e tha t i 1. Thi s onl y leave s th e cas e z = 2 , n = 2 .
The for m Nh G Quad4(fc) i s (a±, aiJ, »2, c^)- It s discriminan t i s 1, and it s 6 i
invariant (i n th e sens e o f 20.3 ) i s (a±) -{0.2) •(—«i5) = (ai ) -(0:2) '(^) = (^)"^2(^) -
This show s that (S) -W2(q) = bi(Nh) depend s onl y o n h = qn-
Hi) The case n 2 . Thi s follow s fro m th e cas e n = 2 b y Witt' s chai n
equivalence fo r hermitia n form s (cf . [W i 37, p . 36 ] not e tha t th e proo f fo r qua -
dratic form s i n [La m 73, 1.5.2] extend s t o th e hermitia n cas e wit h practicall y n o
change).
Thanks t o Lemm a 21.4, for ever y x G (6)-H(k) w e may defin e a cohomologica l
invariant i^
?x
G Inv(Herm n, Z/2Z) b y puttin g
Wi,x(h) = x-Wi(q)
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