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Local Tate duality

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Duality for Galois modules for the absolute Galois group of a non-archimedean local field

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 denote a separable closure of K, and let GK = Gal(K/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. 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 H(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 H(K, A) and H(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, H(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 H(K, V) and H(K, V ′) for i = 0, 1, 2. Again, the higher cohomology groups vanish.

See also

Notes

  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

References

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