20 II. COHOMOLOGICA L PRELIMINARIES : TH E LOCA L CAS E
is commutative, where the vertical arrow on the right is the product of e
and the natural map i T
"1
^ , C(-l)) - H
i~1,C(-1)).f(k
(2) The diagram
ker(rv) H l{k,C)
i i
ker(rv) iT(/c',C )
is commutative.
(For the definition o f the ma p ker(rv)— H
l(k,
C) , see 7.13.)
PROOF .
W e may assume tha t K an d K' ar e complete. Le t It(K) an d It(K')
be the corresponding tam e inerti a groups . B y 7.1, we have a natural isomorphis m
of It(K) wit h lim/x n. Thi s identificatio n use d a uniformizin g elemen t o f K, an d
changing K t o K' force s a ne w choice o f uniformizin g element . Th e result i s a
commutative diagra m
(8.3) I t(K) = limv
n
A A
e
I
t
( l f ' ) = l i m / i
n
,
where th e vertical ma p on the left i s the natural ma p and the vertical ma p on
the righ t i s multiplication b y the ramification inde x e . B y 7.9, we have C{ 1) =
Hom(it,C), wher e I
t
ma y be take n t o b e either inerti a group . Vi a these iden -
tifications, th e natural ma p Hom(/t(K), C) Hom(I
t
(K'),C) become s th e ma p
C( 1)— C( l) give n b y multiplication b y e. I t follow s fro m 6. 4 that on e has a
commutative diagra m
0 H l(k,C) H l(K,C) H'-^k.Ci-l)) 0
0 Hl{k\C) H^K'.C) H^ik^Ci-l)) 0,
where th e downward arro w o n the right i s given b y the product o f the ramifica -
tion inde x e with th e natural homomorphis m iiP
_1(/c,
C(—1))— H
l~1(k/1
C( 1)).
Prop. 8. 2 follows.
8.5.
CORESTRICTION .
Suppos e no w that K' i s finite over K, an d that K (and
hence als o K') i s complete. Then , on e has a corestriction map
Corf': H\K', C) -+ H\K, C)
defined t o be the product o f the usual Galoi s corestrictio n (correspondin g t o the
inclusion YK' YK) wit h th e degree o f inseparability [K' : K]i o f K' ove r K.
(This definition applie s to any finite field extension, for instance to the residue field
extension k'/k.)
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