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. Denote by μ the Galois module of all roots of unity in Ks.
The theorem states that the pairing given by the cup product sets up a duality between Hi(K, A) and H2−i(K, A′) for i = 0, 1, 2.
Let Qp(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 Qp and GL(V) denotes the group of invertible linear maps from V to itself.
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 H2−i(K, V ′) for i = 0, 1, 2.