8. FUNCTORIALIT Y O F THE RESTRICTION AN D CORESTRICTION 19
where r(a) , r(/3) , r(a-(3) ar e viewe d a s elements o f the cohomolog y o f K, vi a th e natura l
embedding H(k) » H(K). Th e cu p product s ar e taken relativel y to the natura l pairings ;
for instance , th e las t on e is relative to
A
(-l) x B(-l) x
Mn
- C7(-2) x
Mn
-+ C7(-l).
All th e term s o f the formul a thu s belon g to Fa + b - 1 ( i ^ , C ( - l ) ) .
[Hint: Decompos e a and / 3 as ao + (7r)
n
-ai an d /? o + (?r) n -/3i, as in Prop. 7.11, an d
use the identit y (7r)
n
-(—7r)n = 0.]
7.13. T H E N O N - C O M P L E T E CASE . W e no w dro p th e assumptio n tha t K i s
complete wit h respec t t o v. Choos e a n extension w o f v t o K
s
(al l such ar e TK-
conjugate), an d le t Decw b e the correspondin g decompositio n group , whic h consist s
of thos e g £TK suc h tha t gw = w.
Let K
S:W
b e the completion o f Ks relativ e t o w. Th e subfiel d K
s
-K
v
o f KSjW i s
the larges t algebrai c subextensio n o f KSjW/Kv; i t is separably close d b y Krasner' s
Lemma, henc e ma y b e identified wit h (K
v
)s. W e have
Dec™ = Gal(K
s
-Kv/Kv) = Gd((K
v
)8/Kv) = T
Kv
,
so tha t w e may appl y th e results abov e t o Dec^ .
More precisely , le t C b e a finite IV-modul e o f order no t divisible b y char (A:) .
Assume tha t C is unramified at v, i.e., tha t th e inertia grou p o f Dec^ act s triviall y
on C. Le t a G Hl{K, C) , an d le t av denot e it s image i n Hl{Kv, C) = ^ ( D e c ^ , C).
We defin e th e residue r
v
(a) of a at v to be the residue o f av i n Hl~1{k^ C(— 1)) a s
defined above . I f rv(a) ^ 0 , we say that a i s ramified a t v (o r has a pole a t v). I f
rv(a) = 0 , a i s said t o be unramified a t v. I n that case , a
v
ca n be identified wit h
an elemen t o f Hl(k, C) whic h w e denote b y a(v) an d call th e value of a at v.
We thu s hav e tw o canonical map s relatin g th e cohomology o f K wit h tha t o f
the residu e field k of v:
r^.H^K.C) - f P - ^ C t - l )) (residu e a t v)
ker(r
v
) —• H l(k,C) (valu e a t v).
When K i s complete, w e have see n tha t th e first ma p is surjective, an d that th e
second on e is an isomorphism. Thi s i s still tru e whe n K i s henselian, bu t not in
general.
8. Functorialit y o f the restrictio n an d corestrictio n
8.1. R E S T R I C T I O N . Recal l th e setting o f 7.1 an d 7.13: (K,v) i s a field wit h
a discret e valuatio n v an d residue field k\ w e do not assume i t i s complete . W e
write C for a finite T^-modul e whic h i s unramified a t v an d of order prim e t o the
characteristic o f k.
Let (K f, v') be an extension o f (X, v) with ramificatio n inde x e and residu e field
k'. W e have residu e map s
rv : H\K, C) - i T- ^ * ; , C{-1)) an d r
v
: H\K', C) -* Hl~l(k', C ( - l ) ) .
P R O P O S I T I O N 8.2 . (1) The diagram
H^K.C) —^— i T - ^ C X - l ) )
# * ( # ' , C ) ^ - W-^k^Ci-l))
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