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

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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