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.
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
(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
Case of p-adic representations
(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
which is a duality between Hi(K, V) and H2−i(K, V ′) for i = 0, 1, 2. Again, the higher cohomology groups vanish.
- 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).