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In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules (abelian groups with $\Gamma$-action) \begin{equation*}%\label{e:cd} \begin{CD} 1 @>>>Q … For a group $\Gamma$ of order 2 (and also for any cyclic group $\Gamma$) the Tate cohomology and hypercohomology are periodic with period 2. …

If $k$ is a $p$-adic field, then by the Tate duality theorem (see Serre, "Galois cohomology", or Milne, "Arithmetic duality theorems", or Harari, "Cohomologie galoisienne et théorie du corps de classes … Question: What is an example of a field $k$ of characteristic 0, a finitely generated $\Gamma$-module $M$, and a cohomology class $x\in H^2(\Gamma, M^D)$ such that $x\neq 0$, but $x^0=0$? …

Alternatively, there exists a field $k$ of characteristic 0 such that ${\rm Br}(k)=0$, but $k$ is not of dimension $\le 1$; see Serre, Galois Cohomology, II.3.1, Exercise 1. …

From the short exact sequence $$1\to G\to R_{L/K}\mathbb{G}_{m,L}\to \mathbb{G}_{m,K}\to 1$$ and the induced Galois cohomology exact sequence $$ L^*\to K^* \to H^1(K,G)\to H^1(K,R_{L/K}\mathbb{G}_{m,L …

Consider the restriction and corestriction homomorphisms in Tate cohomology: \begin{align*} \Res\colon\, &H^{-1}(G,M) \to H^{-1}(H,M),\\ \Cor\colon\, &H^{-1}(H,M) \to H^{-1}(G,M). …