Local Tate duality

In Galois cohomology, local Tate 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.

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 (local) 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. Since G_K has cohomological dimension equal to two, the higher cohomology groups vanish.

See also

• Poitou–Tate duality, a global version (i.e. for global fields)
• Tate's Euler characteristic formula, Tate's formula for the Euler characteristic in the Galois cohomology of local fields

References

• Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR1867431, translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964).