22
II. COHOMOLOGICA L PRELIMINARIES : TH E LOCA L CAS E
REMARK
8.10. Th e onl y propert y o f C use d i n 8. 8 an d 8. 9 wa s tha t i t i s a
T/c-module. Henc e Diagra m (8.7 ) commute s fo r C an y unramifie d module ; C nee d
not b e finite o f order prim e t o th e residu e characteristic .
To conclude th e proo f o f Prop. 8. 6 w e must sho w tha t th e right-han d squar e
W(K,C) - ^ - H
l-\k,C{-l))
( 8 - H ) Cor£' J j c o r j '
H\K',C) ^ U W-^k'^Ci-l))
commutes, wher e th e horizonta l arrow s ar e th e residu e maps .
Start wit h a n elemen t x' o f H l(K',C). W e wan t t o comput e th e residu e o f
y Cor^ [x'). B y Propositio n 7.11w e ma y writ e x' a s XQ + (n f) -x[ wher e TT'
is a uniformizin g elemen t o f K', x'
0
i s i n H l{k'',C), an d x[ i s i n H l~1 (kfJ C(— 1))
such tha t x[
VK'{X').
B y th e commutativit y o f (8.7) , th e corestrictio n Cor(xg )
belongs t o th e subgrou p H
l{k,C)
o f H
l(K,
C) , an d it s residu e i s 0. W e ma y the n
assume tha t x'
(nf)
-x[.
Let u s conside r separatel y th e case s I an d II .
8.12.
DIAGRA M
(8.11 )
COMMUTE S I N CAS E
I . I n Cas e I , th e restrictio n ma p
Hl~1(k1 C( 1))— » Hl~1{k'1 C( 1)) i s surjective . Henc e w e ma y writ e x[ a s a
restriction Res £ (xi) , wher e x\ i s an elemen t o f H
l~1(k,
C(— 1)). Ther e i s a "pro -
jection formula" whic h gives Cor(x-y) wher e x (o r y) i s Res(X) (o r Res(Y)), namel y
(8.13) Cor(x -y)=X- Cor(y) o r Cor(x ) Y,
see e.g . [Lang96 , p . 87 , Th . 3.1(2)] o r [NSW00 , p . 47 , Prop . 1.5.3.(iv)]. Henc e w e
have
y = Cor((7r
/)
-x[) = Cor((7r
/))
^ i-
It i s a standar d fac t tha t Cor((z)
n
) = {Nz)
n
, wher e N i s the norm . Henc e w e get
y = (iW)-xi , an d it s residu e i s equa l t o
V(NTT')'XI
= / x i , wher e / = [k' : &].
Since / x i i s the corestriction o f x\ wit h respect t o the purely inseparable extensio n
k'Ik, w e get th e formul a w e wanted .
8.14.
DIAGRA M
(8.11 )
COMMUTE S I N CAS E
II . I n Cas e II, we may take fo r n'
a uniformizin g elemen t TT o f K. B y th e projectio n formul a (8.13), we have:
Cor f (*' ) = (*) Corf (x[).
This show s that th e residu e o f Cor (a/) i s equal t o Cor(x
/
1), a s desired .
This complete s th e proo f o f Proposition 8.6 .
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