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{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)\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

\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^\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)\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.

Notes

↑ Serre 2002, Theorem II.5.2
↑ Serre 2002, §II.4.3
↑ Some authors use the term p-adic representation to refer to more general Galois modules.
↑ Rubin 2000, Theorem 1.4.1

References