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 "Ks" denote a separable closure of "K", and let "GK" = Gal("Ks"/"K") be the absolute Galois group of "K".
Case of finite modules.
Denote by μ the Galois module of all roots of unity in "Ks". Given a finite "GK"-module "A" of order prime to the characteristic of "K", the Tate dual of "A" is defined as
(i.e. it is the Tate twist of the usual dual "A"∗). Let "Hi"("K", "A") denote the group cohomology of "GK" with coefficients in "A". The theorem states that the pairing
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 "GK" 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 "GK" (i.e. the Tate module of μ). A "p"-adic representation of "GK" is a continuous representation
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
(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 "GK" with coefficients in "V". Local Tate duality applied to "V" says that the cup product induces a pairing
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.