On Tate's duality theorems in Galois cohomology

Duality Theorems in Galois Cohomology over Number Fields

the direct limit taken over all finite Galois extensions K of k in which the integral closure Y of X is unramified over X, where GKJk denotes the Galois group of such an extension, and where GY denotes the group of points of G with coordinates in Y. For example, if X=k, our notation coincides with that of [10]. For any X, the group H(X,C) is the r-th cohomology group of the profinite group Gjr=...

Two Theorems on Galois Cohomology1

Notice that we have dropped the hypothesis that both k and K be Galois over the rationals. To see how Theorem 1 generalizes Yokoi's result, remember that if G has prime order p, then multiplication by p annihilates all the cohomology groups. Thus in this case the cohomology groups are determined up to isomorphism by their order. The technique used to prove Theorem 1 can be used to prove other r...

Galois Module Structure of Galois Cohomology

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . We determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined only for F a local field, a case se...

A Short Course on Galois Cohomology

Duality Based on Galois Connection. Part I

In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p. 179–183; 1.1–1.12]. The duality is based on the concept of the Galois connection. Let S, T be complete lattices. Note that there exists a connection between S and T which is Galois. One can prove the following proposition (1) Let S, T , S , T be non empty relati...