8. FUNCTORIALIT Y O F TH E RESTRICTIO N AN D CORESTRICTIO N
21
PROPOSITION
8.6 . The diagram
0 Hl(k,C) Hl{K,C) W-^k.Ci-l))
e. Corj' Cor£ ' Cor* '
0 H l{k',C) U l{K',C) H'-^k^Ci-l)) 0 ,
commutes, where the upward arrow on the left is given by the product of the rami-
fication index e with Cor
fc
.
PROOF.
I t i s sufficien t t o prov e th e propositio n i n th e followin g tw o specia l
cases:
CASE I : k'/k i s purely inseparable .
CASE II : e = 1 and k'jk i s separable (th e "unramifie d case") .
Suppose that w e have proved the result i n both cases, and consider the general case.
There i s a uniqu e field k" wit h k C k" C k' suc h tha t k"/k i s separable an d k!Ik"
is purely inseparable . I t i s well-known tha t ther e i s a field K" wit h K C K" C K'
such tha t e(K"/K) = 1 and th e residu e field o f K" i s k h\ se e e.g . [Bou83 , App.] .
The extensio n K")K i s of the sor t occurrin g i n Cas e I I an d th e extensio n K'/K"
is a s i n Cas e I . Hence , i f th e propositio n hold s i n eac h o f thes e tw o specia l cases ,
then i t hold s i n general .
Consider first th e left-han d squar e
H\k,C) v H^K.C)
(8.7) e.Corj' J Cor£' J
Hl(k',C)
H
l(Kf,C)
of the diagra m i n 8.6 .
8.8.
DIAGRA M
(8.7 )
COMMUTE S I N CAS E
I . I n Cas e I, k' i s purely inseparabl e
over k an d w e hav e I V = T^ . Thu s th e restrictio n ma p H l{k,C)— H l{k',C) i s
surjective (eve n a n isomorphism) , an d i t i s enough t o prov e th e commutativit y o f
Hl(k,C)
H
l(K,C)
e.Cor^ l J c o r £
Hl{k',C) H\K',C)
Hl(k,C) H l(K,C).
(Here th e lowe r vertica l map s ar e th e restrictio n maps. ) Th e compositio n o f th e
vertical arrow s o n th e righ t i s multiplicatio n b y n = [K
f
: K\. Th e compositio n
of th e vertica l arrow s o n th e lef t i s e time s multiplicatio n b y / = [k 1 : k]. Sinc e
ef n, th e squar e obtaine d b y omittin g th e middl e ro w commutes . Sinc e th e
bottom squar e commute s an d th e vertica l ma p i n th e lowe r lef t i s a surjection, th e
top squar e which i s the squar e i n (8.7 ) commutes .
8.9.
DIAGRA M
(8.7 )
COMMUTE S I N CAS E
II . I n Cas e II , th e natura l ma p
TK
—* Tfc gives a bijection o f
TK/^K'
ont o I \ / I V. I n this situation, i t i s clear tha t
Diagram (8.7 ) commutes .
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