56 VII. COHOMOLOGICA L INVARIANT S O F ETAL E ALGEBRA S
of K wit h respec t t o vp. It s residue field i s the functio n field ko(D) o f D. Le t E^ n
denote th e imag e o f E gen unde r th e restrictio n ma p i7 1(K, S
n
) —- ^(KD, S
n
)y o r
equivalently unde r th e ma p Et
n
(K) E t
n
( i f o ) .
Let Disc r b e th e discriminan t o f the polynomia l t n + c\t n~x + + c
n
. Whe n
n = 1, w e hav e Disc r = 1. Whe n n 2 , Disc r i s a n irreducibl e polynomia l i n ci ,
..., c
n
, excep t whe n cha r (A:) = 2 , i n whic h cas e i t i s th e squar e o f a n irreducibl e
polynomial. I t define s a n irreducibl e diviso r A i n Aff n.
P R O P O S I T I O N. (1) If D ^ A, the K
D
-algebra E^ n is unramified (i n the sens e
of 24.3) .
(2) If D = A, then Ef^ n is the product of a quadratic algebra and an unram-
ified algebra of rank n 2.
P R O O F . (1): Fo r a basi s ( e i , . . . , e
n
) o f E gen, choos e (1,£,... , t n _ 1 ) . W e hav e
Tr (e{.ej) &o[ci,... , c
n
] an d the determinant o f the matrix M = (Trfa.ej)) i s equal
to Disc r ( c i , . . ., c
n
). Henc e M belong s t o GL
n
(i?£), wher e Ftp i s the valuation rin g
of VD. Criterio n (4 ) o f 24. 2 then show s tha t E^ n i s unramified .
(Note tha t th e "geometric " proo f o f (2 ) give n belo w als o provide s a proo f o f
(i)-)
(2): Le t Aff denot e Spec(fco[#i , ,xn]). Writ e Aff fo r th e affin e spac e wit h
coordinates c i , . . . , c
n
, whic h w e interpre t a s symmetri c function s i n x i , . . . , x
n
.
The grou p S
n
act s o n Aff ^ wit h quotien t Aff^ , an d i t act s freel y outsid e th e "ba d
locus" ( = ramificatio n locus ) consistin g o f the union o f the hyperplanes H^ define d
by th e equation s xi Xj fo r i ^ j . Th e ma p Aff ^ Aff ^ i s unramifie d outsid e
of th e imag e A o f th e ba d locus , an d completin g K t o obtai n K& correspond s t o
completingAff a t th e generi c poin t o f the ba d locus .
The algebr a E^
n
= E
gen
0K KD i s describe d b y a ma p TK
A
Sn an d th e
geometric descriptio n show s tha t th e imag e o f the inerti a grou p i s the subgrou p o f
Sn generate d b y a transpositio n (ij). Sinc e th e inerti a grou p i s norma l i n r ^
A
,
the algebr a E^ n i s isomorphi c t o th e produc t o f a ramifie d quadrati c K/\-a\gebTa,
corresponding t o {i,j} an d a n unramifie d i^A-algebr a o f ran k n 2 correspondin g
to th e complemen t o f {i , j] i n {1, 2 , . . ., n} .
EXERCISE 24.7 . I n case (2 ) of Prop. 24.6 , show that bot h factor s o f E^n ar e fields, if
n 3.
24.8. SPLITTIN G PRINCIPLE . W e cal l a n etal e algebr a multiquadratic i f i t i s
decomposable a s a produc t o f etal e algebra s o f ran k 1 or 2 .
T H E O R E M 24.9 . If a e Inv/
e o
(Et
n
,C) satisfies a(E) = 0 for every multi-
quadratic E G Etn(fc) [over every extension k of ko), then a = 0 .
P R O O F . B y inductio n o n n. Th e result i s clear whe n n = 1 or 2 , so we assum e
that n 3.
Suppose tha t E G Etn(/c) ca n b e writte n a s E = E' x E" wher e E' ha s ran k
i = 1 o r 2 . Defin e a n invarian t Et
n
_^/fe H b y F ^ a{E' x F) wher e F run s
through th e (n i)-etale algebra s ove r th e extension s o f k. Thi s invarian t vanishe s
for multiquadrati c extensions , s o by induction i t i s zero. Henc e i n this cas e w e hav e
a(E) = 0 .
Let g e n , A , etc . b e a s i n 24.6 . W e conside r th e residue s o f a = a(E gen)
corresponding t o irreducibl e divisor s o f the affin e spac e Aff n.
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