Local Tate duality

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 ′ = 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 Hⁱ(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 Hⁱ(K, A) and H²⁻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

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

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, 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 ) × 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 Hⁱ(K, V) and H²⁻ⁱ(K, V ′) for i = 0, 1, 2.^[4] Again, the higher cohomology groups vanish.