Local Tate Duality

In Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absolute Galois group of a non-archimedean local field. It is named after John Tate who first proved it. It shows that the dual of such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual.

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

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

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

:$A^\prime=\mathrm(A,\mu)$
(i.e. it is the Tate twist of the usual dual ''A''^∗). Let ''Hⁱ''(''K'', ''A'') denote the group cohomology of ''G_K'' with coefficients in ''A''. The theorem states that the pairing

:$H^i(K,A)\times H^(K,A^\prime)\rightarrow H^2(K,\mu)=\mathbf/\mathbf$

given by the cup product sets up a duality between ''Hⁱ''(''K'', ''A'') and ''H''^2−''i''(''K'', ''A''^′) for ''i'' = 0, 1, 2. Since ''G_K'' has cohomological dimension equal to two, the higher cohomology groups vanish.

Case of ''p''-adic representations

Let ''p'' be a prime number. Let Q_''p''(1) denote the ''p''-adic cyclotomic character of ''G_K'' (i.e. the Tate module of μ). A ''p''-adic representation of ''G_K'' is a continuous representation
:$\rho:G_K\rightarrow\mathrm(V)$
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. The Tate dual of ''V'' is defined as
:$V^\prime=\mathrm(V,\mathbf_p(1))$
(i.e. it is the Tate twist of the usual dual ''V''^∗ = Hom(''V'', Q_''p'')). In this case, ''Hⁱ''(''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

:$H^i(K,V)\times H^(K,V^\prime)\rightarrow H^2(K,\mathbf_p(1))=\mathbf_p$

which is a duality between ''Hⁱ''(''K'', ''V'') and ''H''^2−''i''(''K'', ''V'' ′) for ''i'' = 0, 1, 2. Again, the higher cohomology groups vanish.

See also

*Tate duality, a global version (i.e. for global fields)

Notes

References

*
* {{Citation , last1=Serre , first1=Jean-Pierre , author1-link= Jean-Pierre Serre , title=Galois cohomology , publisher=Springer-Verlag , location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-42192-4 , mr=1867431 , year=2002, translation of ''Cohomologie Galoisienne'', Springer-Verlag Lecture Notes 5 (1964).

Theorems in algebraic number theory
Galois theory
Duality theories