Local Tate duality


In Galois cohomology, local Tate 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 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 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
. Let Hi 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 and H2−i 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 Qp denote the p-adic cyclotomic character of GK. A p-adic representation of GK is a continuous representation
where V is a finite-dimensional vector space over the p-adic numbers Qp and GL denotes the group of invertible linear maps from V to itself. The Tate dual of V is defined as
. In this case, Hi 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 and H2−i for i = 0, 1, 2. Again, the higher cohomology groups vanish.