On Tate Local Duality

Let Fq be a finite field, and let g be its absolute Galois group. Hence g ∼= Ẑ, topologically generated by the Frobenius automorphism. Let A be a topological g−module, i.e. for each a ∈ A there is a positive integer n such that Frobn a = a (for us A will be either an elliptic curve or some torsion group). Thus the group A is the union of its subgroups Agn , the latter being g/gn-module, where gn = nẐ is an open subgroup of index n. The cohomology groups of g with values in A are defined by the formula H(g, A) = lim −→ H(g/gn, An)