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.

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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

Rubin, Karl (2000), Euler systems, Hermann Weyl Lectures, Annals of Mathematics Studies, 147, Princeton University Press, ISBN 978-0-691-05076-8, MR 1749177
Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR 1867431, translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964).

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[1] Serre 2002, Theorem II.5.2

[2] Serre 2002, §II.4.3

[3] Some authors use the term p-adic representation to refer to more general Galois modules.

[4] Rubin 2000, Theorem 1.4.1