CHAPTER II I
Cohomological preliminaries : th e functio n field
case
9. Galoi s cohomolog y o f k(t)
Let Pi = P ^ b e the projective lin e over k. Th e function field of Pi i s K kit)
where t i s an indeterminate . A closed poin t o f Pi ca n b e identifie d wit h a discret e
valuation o n K whic h i s trivial o n k. Le t V b e the se t o f such valuations . Suppos e
that C i s a finite IVmodul e whos e orde r n i s not divisibl e b y th e characteristi c o f
k. A s above , fo r ever y v £ V w e cal l K
v
th e completio n o f K a t D ; it i s a loca l
field; it s residu e field k(v) i s a finite extensio n o f k. W e ma y appl y th e definition s
of 7.13; in particula r w e have a residue ma p
rv : H\K, C) -+
Hl-\k{v),
C(-l)) .
LEMMA 9.1. Let a be an element of H l(K,C). The set of v in V which are
poles of a (in the sense of 7.13) is finite.
PROOF.
Le t L/K b e a finite Galoi s extensio n o f K whic h i s larg e enoug h
so tha t a come s fro m a n elemen t o f H l(Gal(L/K),
CL),
wher e
CL
=
H°(TL,C).
Then r v(a) = 0 fo r al l v wher e L i s unramified . (I t woul d b e sufficien t tha t th e
ramification inde x o f L/K i s prime t o th e orde r o f C.) Thi s follow s fro m th e fac t
that th e kerne l o f the residu e ma p i s H
l{Gal{Kv^unr/Kv),
C).
This allow s u s to defin e a ma p
©r. : H\K, C) -+ e ^ y i f ^1 ^ ^ ) , C(-l) )
by a h- + (r
v
(a)). W e hav e th e followin g basi c result , whic h wil l b e prove d i n 9. 5
through 9.20 .
THEOREM 9.2 . The sequence
0 -+ H\k,C) -+ H
l(K,C)
^ 0 ^ -
1
( ^ ( ^ ) ^ ( -
1^
) ) H^fcCi-l)) - 0
v£V
is exact, where c is the direct sum of the corestriction maps
Cor: H^ikiv), C(-l) ) - ^ ff
i_1
(fc, ^(-1)) .
An elemen t o f H l(K, C) i s said t o b e constant i f it belong s t o H l(k, C).
The fac t tha t th e compositio n c o (0r
v
) i s zer o i s known a s th e "residu e for -
mula" ; it say s tha t th e su m o f th e corestriction s o f th e residue s o f a n elemen t o f
H\K,C) i s zero.
Note tha t th e poin t a t infinit y o f P i define s a place , whic h w e denot e b y oo ,
with residue field fc(oo) = k. Th e correspondin g corestrictio n i s the identity . Henc e
we may remov e H
l~1{k1C(
1)) fro m th e las t tw o group s i n th e exac t sequenc e i n
9.2. Thi s gives :
23
http://dx.doi.org/10.1090/ulect/028/05
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