Local Tate duality

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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[edit]

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[edit]

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

(i.e. it is the Tate twist of the usual dual A). Let Hi(KA) denote the group cohomology of GK with coefficients in A. The theorem states that the pairing

given by the cup product sets up a duality between Hi(K, A) and H2−i(KA) for i = 0, 1, 2.[1] Since GK has cohomological dimension equal to two, the higher cohomology groups vanish.[2]

Case of p-adic representations[edit]

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

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.[3] The Tate dual of V is defined as

(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(KV) and H2−i(KV ′) for i = 0, 1, 2.[4] Again, the higher cohomology groups vanish.

See also[edit]

Notes[edit]

  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[edit]