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Local duality combined with Tate's local Euler characteristic formula provide a versatile set of tools for computing the Galois cohomology of local fields.
Statement [ edit ]
Let K be a non-archimedean local field, let K s denote a separable closure of K , and let G K = Gal(K s /K ) be the absolute Galois group of K .
Case of finite modules [ edit ]
Denote by μ the Galois module of all roots of unity in K s . Given a finite G K -module A of order prime to the characteristic of K , the Tate dual of A is defined as
A
′
=
H
o
m
(
A
,
μ
)
{\displaystyle A^{\prime }=\mathrm {Hom} (A,\mu )}
(i.e. it is the Tate twist of the usual dual A ∗ ). Let H i (K , A ) denote the group cohomology of G K with coefficients in A . The theorem states that the pairing
H
i
(
K
,
A
)
×
H
2
−
i
(
K
,
A
′
)
→
H
2
(
K
,
μ
)
=
Q
/
Z
{\displaystyle H^{i}(K,A)\times H^{2-i}(K,A^{\prime })\rightarrow H^{2}(K,\mu )=\mathbf {Q} /\mathbf {Z} }
given by the cup product sets up a duality between H i (K , A ) and H 2−i (K , A ′ ) for i = 0, 1, 2.[1] Since G K has cohomological dimension equal to two, the higher cohomology groups vanish.[2]
Case of p -adic representations [ edit ]
Let p be a prime number . Let Q p (1 ) denote the p -adic cyclotomic characterof G K (i.e. the Tate module of μ). A p -adic representationof G K is a continuous representation
ρ
:
G
K
→
G
L
(
V
)
{\displaystyle \rho :G_{K}\rightarrow \mathrm {GL} (V )}
where V is a finite-dimensional vector space over the p-adic numbers Q p and GL(V ) denotes the group of invertible linear maps from V to itself.[3] The Tate dual of V is defined as
V
′
=
H
o
m
(
V
,
Q
p
(
1
)
)
{\displaystyle V^{\prime }=\mathrm {Hom} (V,\mathbf {Q} _{p}(1 ))}
(i.e. it is the Tate twist of the usual dual V ∗ = Hom(V , Q p )). In this case, H i (K , V ) denotes the continuous group cohomology of G K with coefficients in V . Local Tate duality applied to V says that the cup product induces a pairing
H
i
(
K
,
V
)
×
H
2
−
i
(
K
,
V
′
)
→
H
2
(
K
,
Q
p
(
1
)
)
=
Q
p
{\displaystyle H^{i}(K,V)\times H^{2-i}(K,V^{\prime })\rightarrow H^{2}(K,\mathbf {Q} _{p}(1 ))=\mathbf {Q} _{p}}
which is a duality between H i (K , V ) and H 2−i (K , V ′) for i = 0, 1, 2.[4] Again, the higher cohomology groups vanish.
See also [ edit ]
^ Some authors use the term p -adic representation to refer to more general Galois modules.
^ Rubin 2000 , Theorem 1.4.1
References [ edit ]
Rubin, Karl (2000), Euler systems , Hermann Weyl Lectures, Annals of Mathematics Studies, vol. 147, Princeton University Press , ISBN 978-0-691-05076-8 , MR 1749177
Serre, Jean-Pierre (2002), Galois cohomology , Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag , ISBN 978-3-540-42192-4 , MR 1867431 , translation of Cohomologie Galoisienne , Springer-Verlag Lecture Notes 5 (1964).
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Local_Tate_duality&oldid=1045235325 "
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● T h i s p a g e w a s l a s t e d i t e d o n 1 9 S e p t e m b e r 2 0 2 1 , a t 1 5 : 0 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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