24 III . COHOMOLOGICA L PRELIMINARIES : TH E FUNCTIO N FIEL D CAS E
THEOREM 9.3 . The sequence
0 -+ iF(fc, C) -• iT(K , C) -• 0 i T
- 1
^ ) , C(-l) ) - 0
v G V -
{OO }
25 exact.
9.4.
UNRAMIFIE D ELEMENTS .
A n element a G H l(K,C) i s called unramified
(resp. unramified outside 0 , oo ) if it ha s zero residu e fo r all v G V" (resp. D G 7 -
{0, oo}). Becaus e of the residue formula, a is unramified i f it has zero residue for all
v G y-{oo}, i.e. , on the affine line . Writ e Hlunr{K, C) (resp. Hlunr
outsid e 0?oo
(K, C))
for th e group of such classes . Theore m 9. 3 shows that an y unramified a belong s to
H'faC), i.e. , Hlm{K,C) = H l(k,C).
If a G K
nroutside0oo
(K,C) ha s residu e
ai
G ^ - ^ ^ ^ ( - l )) a t 0 , the n
& (t)n-ai i s unramifie d b y 7.11. Henc e suc h a n a ca n be writte n uniquel y a s
^o + (t)n'&i, w ^ h a o G Hl(k,C) an d ai G Hl~1(k, C{— 1)). B y comparing wit h
Proposition 7.11 , this show s tha t th e natural map
^unr . outsid e 0,oo(^C ) H l(KVQ,C)
is an isomorphism, wher e K
Vo
= &((£) ) is the completion o f K a t the place 0.
9.5.
PROO F O F THEORE M
9.2 . W e assume first tha t k i s perfect (fo r the
general case, see 9.18), so that k s(t) = k(t). Le t N = Gal(if s//c(£)). I t is well-known
(cf. [Se65 , II.3]) tha t th e cohomological dimensio n o f k(t) i s 1, so Hl(N, C) 0 for
all i 2 .
For v G V choos e an extension w to Ks an d let Iw an d Dec^ denote the inertia
and decompositio n group s o f w, which ar e subgroups o f TK- Th e diagram
1 I
w
D e c^ r
f c
(
v )
1
(9.6) «„ J
{ {
1 N T
K
T
k
1
commutes, where the vertical maps are the obvious ones. I f one chooses v such tha t
k(v) = k (e.g. , v oo), by 7.6 and 7.13 the upper sequenc e splits , henc e th e lower
one doe s as well.
We hav e checke d al l of the hypotheses o f Theorem 6. 1 for the exact sequenc e
1— N -^ TK ~^ ^k 1- We thus hav e a n exact sequenc e
0 - iF(fc, C) -• H\K, C) -^ iT
_1(/c,
Hom(7V, C)) -* 0.
It remain s t o show tha t
(9.7) i T - ^ H o m ^ C ) ) ^ ker(eF
2
-
1
(A:(^),C(-l))-^^-
1
(A:,C
f
(-l))),
and
(9.8) the v-component of r is the residue map r
v
.
We do both by looking more closely at the structure of Hom(7V, /xn) and Hom(iV, C).
9.9.
STRUCTUR E O F
Hom(iV, /xn). Le t
F = Pi(fc ) = {discret e valuation s o f &(£)},
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