Questions tagged [galois-cohomology]

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

Filter by
Sorted by
Tagged with
1 vote
1 answer
78 views

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 ...
user avatar
2 votes
0 answers
19 views

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 ...
user avatar
1 vote
1 answer
50 views

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 $G$...
user avatar
  • 11
1 vote
0 answers
19 views

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 ...
user avatar
  • 27
0 votes
0 answers
54 views

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 ...
user avatar
  • 167
1 vote
0 answers
56 views

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 ...
user avatar
  • 167
1 vote
0 answers
17 views

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 ...
user avatar
  • 1,965
0 votes
0 answers
62 views

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 ...
user avatar
2 votes
0 answers
58 views

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 ...
user avatar
  • 1,240
1 vote
0 answers
60 views

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 ...
user avatar
  • 225
4 votes
1 answer
92 views

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 ...
user avatar
2 votes
0 answers
53 views

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 ...
user avatar
  • 1,631
1 vote
0 answers
37 views

Cup product of cohomology.

Suppose $[K \colon {\Bbb Q}_p] < \infty$ and that $\mu_p \in K$. We shall consider the cup product $$ H^1(G, {\Bbb Z}/p) \times H^1(G, {\mu_p}) \overset{\cup}{\to} H^2(G, \mu_p) \cong {\Bbb Z}/p{\...
user avatar
1 vote
1 answer
59 views

Cochain from ${\Bbb Z}$ and $\widehat{\Bbb Z}$.

Suppose we shall prove $H^2({\Bbb Z}, {\Bbb Z}) = 0$ and $H^2(\widehat{\Bbb Z}, {\Bbb Z}) = {\Bbb Q}/{\Bbb Z}$. This is simply the matter of calculating $H^2(G, {\Bbb Z})$ or finding what the cochain $...
user avatar
4 votes
2 answers
126 views

Interpretation of vanishing of cohomology groups

Let $G$ be a group and $M$ a left $G-$module. It is well know that for some conditions all the cohomology groups $H^{i}(G,M)$, $i=0,1,2,.....$ vanish. The same can be do for the Hochschild cohomology ...
user avatar
0 votes
1 answer
64 views

On pairing and cup product in group cohomology

I'm reading the wiki page of local Tate duality. https://en.wikipedia.org/wiki/Local_Tate_duality#cite_note-2 Let $K$ be a non-archimedean local field, let $K^s$ denote a separable closure of $K$, and ...
user avatar
  • 909
0 votes
0 answers
45 views

Galois descent of Galois-invariant objects

Let $QProj_{k}$ stand for the category of quasi-projective varieties over a field $k$. Consider a Galois extension of fields $L\subset K$. From Galois descent theory we know that to give an object $X\...
user avatar
  • 167
2 votes
1 answer
67 views

Isomorphisms of local and global class field theory

Let $k$ be a number field and $K/k$ be a finite Galois extension of degree $d=[K:k]$ with Galois group $\Gamma=\Gamma_{K/k}$. Let $v$ be a place of $k$, and let $k_v$ denote the completion of $k$ at $...
user avatar
2 votes
0 answers
36 views

What is the image of a $\mathbb{Q}$-form of an algebraic group by a $\mathbb{Q}$-morphism?

Suppose we have two (affine) algebraic $\mathbb{Q}$-groups $G$ and $H$ and a $\mathbb{Q}$-morphism $\phi:G\rightarrow H$. Given a $\mathbb{Q}$-form ${}_{\xi}G$ of $G(\overline{\mathbb{Q}})$ is it true ...
user avatar
  • 187
1 vote
0 answers
44 views

What is the action on $H^1$ in Galois cohomology?

This is a question that I haven't been able to find answers for in the books I've looked at, so I wanted to ask it here to get some clarification. Given an abelian group $A$ and a Galois group $G$ ...
user avatar
  • 1,956
1 vote
1 answer
37 views

What is this path torsor appearing in this group cohomology formulation of the section conjecture?

Let $X$ be a (nice) scheme over $\mathbb{Q}$. Then there is an exact sequence $1\rightarrow \pi_1^{et}(\overline{X}, b)\rightarrow \pi_1^{et}(X, b)\rightarrow G_{\mathbb{Q}}\rightarrow 1$. This ...
user avatar
4 votes
0 answers
45 views

When does the Galois cohomology functor $H^1(Gal(K^{sep}/ -), G(K^{sep}))$ preserve pullback diagrams?

Fix a field $K$ and a separable closure $K^{sep}$ of $K$. Consider the category of all separable field extensions $K \subset L \subset K^{sep}$ with field inclusions as morphisms. Consider the first ...
user avatar
2 votes
0 answers
28 views

Existence of weil group and cohomology of class fields

I currently reading an article by Tate in "Automorphic forms, representations and L-functions". The book consists of lectures given at Oregon state university in 1977. I am stuck at section (...
user avatar
  • 1,785
1 vote
0 answers
14 views

The direct limit of invertible functions on a variety

Let $X$ be a smooth geometrically integral variety over a number field $k$. We denote by $\bar{k}[X]^*$ the group of invertible functions on $\bar{X}$, and let $$G = \varinjlim_{n \in \mathbb{Z}_{>...
user avatar
0 votes
1 answer
60 views

$|H^0(\hat{\mathbb{Z}},A)|=|H^1(\hat{\mathbb{Z}},A)|$ 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 $A$ ...
user avatar
1 vote
1 answer
64 views

Why $\dim H^0(\hat{\mathbb{Z}},V)=\dim H^1(\hat{\mathbb{Z}},V)$ when $V$ is a finite $\hat{\mathbb{Z}}$-module?

In a lecture note of Joel Bellaiche, he mentioned that when $V$ is finite then we have $$\dim H^0(\hat{\mathbb{Z}},V)=\dim H^1(\hat{\mathbb{Z}},V)$$ where $$\hat{\mathbb{Z}}=\lim_{\longleftarrow}\...
user avatar
1 vote
0 answers
63 views

Canonical action of ${\mathrm{Gal}}(K/{\Bbb Q})$.

Let $K/{\Bbb Q}$ be a finite Galois extension, and ${\mathrm{Gal}}(K/{\Bbb Q})$ the galois group. The ${\mathrm{Gal}}(K/{\Bbb Q})$ module ${\text{H}}^1({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\...
user avatar
4 votes
1 answer
91 views

$H^1(\text{Gal}(\bar{k}/k),B(\bar{k}))$ finite for linear algebraic groups $B$ and fields $k$ with Serre's property $(F)$.

New poster, apologies if the formatting/conventions are incorrect. I am reading Serre's book Galois Cohomology. My question is about exactly how Serre's inductive principle (chapter 1 prop. 39 cor. 3) ...
user avatar
1 vote
0 answers
77 views

A question on Robba ring

With notations as in the question: some questions about the Robba ring. Moreover, we define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}$$ and define a Frobenius map $\...
user avatar
  • 103
1 vote
1 answer
81 views

Understanding a step in the proof of Silverman's Proposition 5.4 about twists of elliptic curves

On page 343 in the book Arithmetic of Elliptic Curves by Silverman, we have the following result: Proposition 5.4. Assume that $char(K)\neq 2,3$, and let $$ n = \begin{cases} 2 & \text{if }j(E) \...
user avatar
  • 947
4 votes
0 answers
98 views

Simple Calculation in Flat Cohomology

Question: Let $R$ be a PID and let $n \geq 2$ be even. What is $H^0_{\operatorname{fppf}}(R,\operatorname{SL}_n/\mu_2)$? Attempt at an Answer: From the short exact sequence of algebraic groups $1 \to \...
user avatar
5 votes
1 answer
113 views

On ${\Bbb Z}/m{\Bbb Z}$-torsors.

I would like to know the explicit construction of ${\Bbb Z}/m{\Bbb Z}$-torsor $Y$'s over a scheme $X$. It is explained that $X$ are classified by $H_{et}^1(X, {\Bbb Z}/m{\Bbb Z})$, which is far from ...
user avatar
7 votes
1 answer
353 views

some questions about the Robba ring

Notations and definitions Let $p$ be a prime integer, $k$ be a perfect field of characteristic $p$ and $W(k)$ its ring of Witt vectors. Definition 1 We put $$ \mathcal{R}_r=\bigg\{ \sum_{i\in \...
user avatar
  • 103
1 vote
0 answers
62 views

Galois quotient on character lattice of a torus

I am following up on a question asked here: Galois action on the character group of a torus Premise: Let $F$ be a (nonarchimedean local) field and let $T$ be an $F$-torus in a linear algebraic group $...
user avatar
3 votes
1 answer
55 views

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 ...
user avatar
4 votes
1 answer
91 views

On understanding $\mathrm{ad}^0 \overline{\rho}$ and $\mathrm{ad}^0 \overline{\rho}(1)$ in Taylor-Wiles method

I'm currently learning Taylor-Wiles method and modularity lifting and comming up with following difficulties, which I think is based on understanding how (global and local) Galois groups act on $\...
user avatar
  • 1,377
2 votes
0 answers
66 views

Cohomology of $G_m$

Suppose $\overline{k}$ be an algebraic closure of a field $k$. Usually $G_k \colon= {\mathrm{Gal}}(\overline{k}/k)$ is endowed with the Krull topology. When considering Galois cohomology $H^1({\mathrm{...
user avatar
2 votes
1 answer
144 views

Silverman *Arithmetic of Elliptic Curves* Problem 1.12 (a)

I am trying to understand Problem 1.12 (a) in Silverman's Arithmetic of Elliptic Curves. Here is the problem Let $K$ be a perfect field, $V/K$ be an affine variety, and let $G_K = \operatorname{Gal}(\...
user avatar
  • 1,208
3 votes
1 answer
236 views

Galois cohomology of projective linear group

I am currently trying to compute Galois cohomology $H^{1}(\overline{k}/k, PGL_2(\overline{k}))$. As far as I know these cocycles correspond to isomorphism classes of smooth genus-$0$ curves over $k$. ...
user avatar
  • 167
0 votes
0 answers
59 views

Skew-symmetricity of the Cup product

Let $f, g \in H^1(G, {\Bbb Z}/n{\Bbb Z})$ be two elements. Consider the cup-product \begin{equation} ∪ \colon H^1(G, {\Bbb Z}/n{\Bbb Z}) \times H^{1}(G, {\Bbb Z}/n{\Bbb Z}) \to H^2(G, {\Bbb Z}/n{\...
user avatar
1 vote
1 answer
71 views

Is the functor that takes an $R$-algebra to the group of finitely generated projective modules of rank one additive?

Let $P$ be the covariante functor from the category of commutative $R$-algebras to the category of abelian groups that takes an $R$-algebra $T$ to the group $P(T)$ of the finitely generated projective ...
user avatar
2 votes
1 answer
169 views

Discrete topological modules for profinite groups

The first step to define cohomology of a profinite group $G$ is to consider discrete $G$-modules. These are abelian groups with the discrete topology and a continuous action $\psi \colon G \times M \...
user avatar
  • 1,841
0 votes
1 answer
149 views

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 ...
user avatar
-1 votes
1 answer
58 views

What does it mean for a 1-cocycle to split? [closed]

Let $K$ be a finite Galois extension of a field $k$. Let $X$ be a variety defined over $k.$ In a paper I am reading, it mentions splitting of 1 -cocycles. For example, "a 1-cocycle $c \in Z^1(\...
user avatar
  • 690
2 votes
0 answers
37 views

Computing $H^1$ in Galois cohomology

I'm trying to learn the basics of Galois cohomology. If $k$ is a perfect field and $k^s$ is a separable closure, and $A$ is a finite group, how do you compute $H^1(\text{Gal}(k^s/k),A)=H^1(k,A)$?
user avatar
2 votes
0 answers
160 views

Understanding the proof on why the first cohomology group of a Galois extension is trivial

Currently, I am trying to understand the following theorem with its proof (taken from Milne's Fields and Galois Theory, page 70): As you can see, I am questioning the well-definedness of the sum for ...
user avatar
  • 947
2 votes
0 answers
91 views

Bloch-Kato-Selmer group of one-dimensional representation.

Let $L/\mathbb{Q}$ be a finite extension and let $V$ be a one-dimensional $L$-linear representation of $G_{\mathbb{Q}}$ which is given by $\chi\rho^*\kappa^n_{cyc}:G_{\mathbb{Q}}\rightarrow L^\times$, ...
user avatar
  • 63
2 votes
1 answer
55 views

Understanding lemma regarding fundamental class

I am reading Class Field Theory from Milne's notes. I do not understand a couple of things from the following lemma about fundamental class: https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.2.7 ...
user avatar
  • 2,585
1 vote
0 answers
145 views

What does it mean to say that $G$ has a 'canonical topological generator' $\text{Frob}_{L/K}$?

$G$ is $\text{Gal}(L/K)$ where $L$ is unramified extension (could be infinite) of a local field $K$. What does it mean to say that $G$ has a 'canonical topological generator' $\text{Frob}_{L/K}$? Does ...
user avatar
  • 2,585
3 votes
0 answers
111 views

Proving that uniquely divisible torsion group is trivial in case of $H^r _{cts}(G,\mathbb Q)$

$G$ be a profinite group. I want to see that the profinite cohomology group $H^2_{cts}(G, \mathbb Q ) $ is $0$. Using inflation maps we see profinite cohomology group as direct limit of cohomology ...
user avatar
  • 2,585