18 II. COHOMOLOGICA L PRELIMINARIES : TH E LOCA L CAS E
P R O P O S I T I O N 7.7 . The sequence
0 - H\Y
k
, C) -+ H\TK, C) ^ H'- l{Tk, Hom(J , C)) -+ 0
is exact.
(For th e definition o f r, se e 6.1.)
This follow s fro m Theore m 6. 1 applied t o G = I V , N = I, an d T = Tfc.
7.8. T A T E TWISTS . Le t u s choos e a numbe r n no t divisibl e b y th e residu e
characteristic suc h tha t nC = 0 . I f d is an integer, w e define th e i-th Tate twist o f
C t o be
c(d) =
h*d®c [fd

[ }
[Hom(/if-
d,C)
i f d0.
This doe s no t depen d o n the particular choic e o f n, an d this definitio n work s ove r
any groun d field. (I n particular, th e ground field nee d no t be local. )
For example , C( 1) = Hom(/x
n
,C).
The Tat e twis t C(d) ca n als o b e define d whe n C i s no t assume d t o b e finite,
but merel y torsio n withou t an y elemen t o f orde r equa l t o th e characteristic . On e
writes C a s limC" , wher e th e C' ar e the finite submodule s o f C , an d on e define s
C(d) a s \im C(d). I n particular , C(— 1) = Hom(M , C), wher e M lim/x
n
an d
Horn denote s continuou s homomorphisms .
7.9. R E S I D U E S . Fo r K a s in 7.1, Hom(/,C) i s the same a s Hom ( J ^ C ), henc e
the sam e a s Hom(//
n
,C) . Th e exact sequenc e 7. 7 can be rewritten a s
(7.10) 0 - H\k, C) - H\K, C) - ^ H^ik, C ( - l ) ) -+ 0.
For a G Hl(K^ C) , r(a) i s called th e residue o f a.
7.11. Fo r x G K*, writ e (x)
n
fo r th e correspondin g elemen t i n K*/K* n =
P R O P O S I T I O N. Let IT be a uniformizing element of K. Every a G H l(K,C)
can be written uniquely as:
a = a
0
+ (7r)
n
-ai w#f t a
0
G ZT(ft, C) an d ax G Hl~1(k, C ( - l ) ) .
Moreover, we have r(a) = a i .
PROOF. Th e cu p produc t (7r)
n
-ai i s th e on e induce d b y th e natura l pairin g
/xn x C ( —1)— » C. Th e residue ma p
r:H\K,nn)-*H\k,Z/nZ)
sends (#)
n
to ^(x ) G Z / nZ by Remar k 6.2 . B y 6.5 , r((7rn )-ai) = a\. Thi s show s
the uniquenes s o f an y decompositio n o f a a s a = a$ - h (Tr)n'^i- Th e existenc e
follows fro m th e fact that , i f ai i s defined a s r(a), then th e residue o f a (7r)n -a±
is zero .
EXERCISE 7.12 (residu e o f a cu p product). Le t A, B, C b e finite IVmodule s wit h
a pairin g AxB-^C. Le t n 0 be prime t o char(/c ) an d such tha t nA 0, nB 0 ,
and nC = 0. Le t a G Ha(K,A), 0 G #b (X , J5) , and let a-0 denot e thei r cu p product i n
Ha+b(K, C). Prov e th e formula :
r(a-p) = r(a) '(3 + ( - l ) a a-r(f3) + r(a) T(/3) -(-l)
n
,
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