# Questions tagged [galois-cohomology]

For questions on Galois cohomology, the study of the group cohomology of Galois modules.

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### Hilbert's Theorem 90 for polynomial rings

I have seen here: Silverman *Arithmetic of Elliptic Curves* Problem 1.12 (a) that $H^1(G_{\overline{K} / K}, \overline{K}^+) = 0$ implies that $H^1(G_{\overline{K} / K}, I(V)) = 0$, where $I(V)$ is a ...
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### Proving that the first Galois cohomology group is direct limit of finite quotients

This question comes from Silverman's Arithmetic of Elliptic Curves, specifically the appendix on Galois cohomology. I am a cohomology beginner, interested (for now) in understanding just enough to get ...
1 vote
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### What's the group extension corresponding to the sum of 2 classes of $H^2(G,A)$

I'm learning Galois/Group Cohomology and i've just seen that the second cohomology group $H^2(G,A)$ (constructed as the factor systems quotient splitting factors) classifies the group extensions of $G$...
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### Understanding the action that comes with $H^1_{cont}(G_K,GL_n(\mathbb{C}_p))$?

I need to look through Sen's "Continuous Cohomology and p-Adic Galois Representations" 1990 paper, but I have confused myself. What is the $G_K$-action on $GL_n(\mathbb{C}_p)$, that makes it ...
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### Second Galois cohomology of elliptic curves over number fields

Let $E$ be an elliptic curve over a number field $K$. Do I understand correctly that the second Galois cohomology of $E$ is a product of $\mathbb{Z}/2\mathbb{Z}$ where each factor corresponds to a ...
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### Kernel of the local-global map for first galois cohomology of $m$-torsion group

Let $E$ be an elliptic curve over a number field $K$. Are there examples when the local-global map $$H^1(K,E[m])\to \prod_\nu H^1(K_\nu,E_{K_\nu}[m])$$ has a nontrivial kernel? I know that if we ...
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### Obstruction of existing a 2-cocycle in Galois cohomology with given local components

Let $K$ be a global field and $T$ be a torus which satisfy Hesse principle defined over $K$ and spilt over $L$. By tate-nakayama theorem we now that at every palace $l$,a cocharcter $\mu_l$ defined ...
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### If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character, why is $H^1(G,A)=0$?

If $\phi:G\longrightarrow A^{\times}$ is a nontrivial character and $M=A^1$, where $A$ is a ring, why is $H^1(G,M)=0$ if $G$ acts via $\phi$? This is essentially the claim that Wiles makes on p. 465 ...
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### Deriving functors on subcategories and profinite group cohomology

I am reading through Weibel's chapter on Galois Cohomology and there he defines profinite group cohomology as the right derived functors of the $G$-invariants functor, but restricting to $C_G$, a ...
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### Definition of Galois Cohomology

For Galois cohomology, one uses the cohomology constructed for profinite groups instead of the usual group cohomology. In other words, one also uses/takes into account the Krull topology we have on ...
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### Explicit Local Fundamental Class

Let $L/K$ be a Galois extension of local fields of degree $n:=[L:K]<\infty$ with Galois group $G:=\operatorname{Gal}(L/K)$. In short, my question is as follows. Is there an explicit computation of ...
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### Applications of Galois cohomology outside number theory

I'm giving a talk on Galois cohomology, but the majority if not all audience members do not work in number theory - the talk is part of a seminar on group cohomology generally. As such, I'm looking ...
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### $|H^0(\hat{\mathbb{Z}},A)|=|H^1(\hat{\mathbb{Z}},A)|$ when $A$ is finite

In the book Galois Cohomology by J-P. Serre, page no. 95, in the proof of proposition 18, it is written that $H^0(\hat{\mathbb{Z}},A)$ and $H^0(\hat{\mathbb{Z}},A)$ have the same cardinality when $A$ ...
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### Correspondences between nonabelian cohomology classes

Suppose $\Gamma$ is a finite group acting on another group $A$. If $A$ is abelian, then it follows from basic group theory that the cohomology classes $[c] \in H^1(\Gamma, A)$ are in one-to-one ...
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### On $K(\pi, 1)$ space.
As far as I know, $K(\pi,1)$ space is a manifold $M$ such that $\pi_n(M) = 0$ for $n > 1$ and $\pi_1(M) = \pi$. Q. Why is the singular cohomology $H_{\mathrm{sing}}^i(M, {\Bbb Z}/n)$ isomorphic to ...