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 Ks denote a separable closure of K, and let GK = Gal(Ks/K) be the absolute Galois group of K.
Case of finite modules Denote by μ the Galois module of all roots of unity in Ks. Given a finite GK-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 GK 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. Since GK has cohomological dimension equal to two, the higher cohomology groups vanish.
Case of p-adic representations Let p be a prime number. Let Qp(1) denote the p-adic cyclotomic character of GK (i.e. the Tate module of μ). A p-adic representation of GK 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 Qp and GL(V) denotes the group of invertible linear maps from V to itself. 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, Qp)). In this case, Hi(K, V) denotes the continuous group cohomology of GK 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. Again, the higher cohomology groups vanish.
See also Tate duality, a global version (i.e. for global fields)
Notes
References Rubin, Karl (2000), Euler systems, Hermann Weyl Lectures, Annals of Mathematics Studies, vol. 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).
