Questions tagged [galois-cohomology]

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

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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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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}\...
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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 ${\text{H}}^1({\mathrm{Gal}}(\overline{\Bbb Q}/K), {\Bbb Z}/n{\...
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$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) ...
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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 $\...
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1answer
69 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) \...
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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 \...
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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 ...
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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 \...
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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 $...
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45 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 ...
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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 $\...
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The Finiteness of H^n(H, Z/pZ) implies finiteness of H^n(U, Z/pZ) for any open subgroup U of H

On Page 175 of J.Neukirch et al.'s book Cohomology of Number Fields, it was remarked: If $H$ is a pro-$p$-group, then this is already true if $H^n(H, \mathbb{Z}/p\mathbb{Z})$ is finite ($n = cd_{p}H$)...
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exercise Galois Cohomology

enter image description here I'm completly lost on this. Any help?
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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{...
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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}(\...
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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 $k$. ...
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36 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{\...
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60 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 ...
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Understanding lemma regarding cup products.

I am trying to understand the following lemma on page 176 from appendix titled 'Computations of cup products' from Serre's book Local fields. Given $G$- modules $A,B$ and $a \in A^G $, let $f_a: \...
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1answer
70 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 \...
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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 ...
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What does it mean for a 1-cocycle to split?

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(\...
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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 $H^1(\text{Gal}(k^s/k),A)=H^1(k,A)$?
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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 ...
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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 $\chi\rho^*\kappa^n_{cyc}:G_{\mathbb{Q}}\rightarrow L^\times$, ...
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1answer
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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 ...
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A Theorem of Hirsch for Cohomology of Groups and for Hoschild Cohomology

The mathematician Morris W. Hirsch wrote on 1977 the paper "Flat Manifolds and the Cohomology of Groups". The Theorem C establishes the folloging : Let $G$ a nilpotent group acting linearly ...
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Open Problems in Homological Algebra

This time I want to ask about open problems in : 1.- Homological Algebra. 2.- Cohomology of Groups. 3.- Algebraic Topology. 4.- Galois Cohomology. Is there a list ? My motivation : I must to present a ...
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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 ...
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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 ...
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1answer
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On prime ideal and irreducible ideal in R[X].

For a noetherian domain $R$, an irreducible ideal $I$ implies $\sqrt{I}$ is a prime ideal. Irreducible implies primary, but not always vice versa. That said, I would like to ask whether the following ...
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Galois Cohomology and Loop Groups

I am trying to understand problem 8.5 in Kac's Infinite dimensional Lie algebras. It goes as follows. Let $G$ be a semisimple algebraic group, let $\alpha$ be an automorphism of $G$ of order $m$, and ...
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132 views

Exactness of Inflation restriction sequence, Galois Cohomology

I am trying to prove the following. Let $K/k$ be a finite Galois extension, $G= G(K/k)$, $k \subset F \subset K$ with $K/k$ normal and $H=G(K/F)$. Then: $ \rho : C^{2} (G,A) \rightarrow C^{2} (H,A) $ ...
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Why are central simple algebras classified by cohomology?

In their article on the Brauer group Wikipedia writes: Since all central simple algebras over a field $K$ become isomorphic to the matrix algebra over a separable closure of $K$, the set of ...
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Question about inertia groups and unramified extensions

Let $K$ be a number field, and $v$ a finite place. If $\bar{K}$ is a separable closure of $K$, then in $G_K=\text{Gal}(\bar{K}/K)$ we can find the decomposition group of (a place over) $v$, which is ...
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Relation between the selmer group and an etale cohomology group.

Let $E/\mathbb Q$ be an elliptic curve with good reduction away from a finite set of primes in $S$. Let $\mathscr E$ be a model for $E$ over $\mathbb Z[1/S]$. Then I know two ways to prove the weak ...
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Global Tate Duality Exercise in Neukirch

For $K$ a $\mathfrak p$-adic number field, local Tate duality yields a non-degenerate pairing $$H^1(K, \Bbb Z/n\Bbb Z) \times H^1(K, \mu_n) \longrightarrow \Bbb Z/n\Bbb Z,$$ where $\mu_n$ is the ...
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A question on finding an inverse image of $\varphi-1$ over some $(\varphi, \Gamma)$-module, from an article by Cherbonnier and Colmez.

I am reading the article THEORIE D’IWASAWA DES REPRESENTATIONS p-ADIQUES D’UN CORPS LOCAL by Cherbonnier and Colmez https://webusers.imj-prg.fr/~pierre.colmez/CCjams.pdf Question: I have a question ...
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surjectivity of a map in Cohomology of Number Fields by Neukirch

This question is from Cohomology of Number Fields by Neukirch(page-62). Let $G$ be a profinite group and $H$ be an arbitrary closed subgroup of $G$. For every discrete $H$-module $A$, define $M= Ind^...
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If $A^\Gamma$ is finite why is $A/(\gamma-1)A$ trivial?

Let $A\cong (\mathbb{Q}_p/\mathbb{Z}_p)^r$ and let $\Gamma\cong \mathbb{Z}_p$ act continuously on $A$, where we take $A$ with the discrete topology. I am trying to work out an exercise which asks to ...
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Group action such that $H^1(G, L^{\times}) \neq \{1\}$.

I am looking for an example of a field $L$ together with a group morphism $G \to \mathrm{Aut_{field}}(L)$ (i.e. a group $G$ acts by field automorphisms on $L$), and such that the cohomology group $H^1(...
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Finite Galois Cohomology of Abelian Variety

Let $l/k$ be a finite galois extension let $A$ be an abelian variety over $k$. Then $A(l)$ is a $Gal(l/k)$-module. Hence it makes sense to study $H^1(Gal(l/k),A(l))$. I know that for $A=\mathbb{G}_m$, ...
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Prove $A^\times$ is an inner form of $GL_n$

Question Let $A$ be a central simple algebra over $F$ where $F$ is a field and $dim_FA=n^2$. $A^\times$ can be seen as an algebraic group over $F$ and let $G=A^\times$. Prove that $G$ is an ...
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$H^1(G,GL_n(K))$ is trivial.

Suppose $K/F$ is a Galois extension with group $G=\text{Gal}(K/F)$, how to prove that $H^1(G,GL_n(K))$ is trivial with Galois descent? Thanks.
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Brauer group of a field and norms

For $a,b \in K^\times$ the symbol $(a,b)$ denotes the element of the Brauer group of $K$ represented by the $2$-cocycle on the absolute Galois group $G_K$ of $K$ sending $(g_1,g_2)$ to $$ \sqrt{a}^{\...
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Explicit description connecting homomorphism in Galois cohomology

Let $K$ be a quadratic extension of $\mathbb{Q}$ and $G:=$Gal$(K/\mathbb{Q})\simeq \mathbb{Z}/2\mathbb{Z}$. Now we have a SES of the form $$ 1 \rightarrow \{ \pm 1\} \stackrel{i}{\rightarrow} K^\times ...
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Galois Cohomology (doubt in greenberg's paper)

I'm reading the paper "Iwasawa Theory for p-adic representation" in which I am not unable to follow one statement: Let $K \subset \overline {\mathbb Q} $ be a finite extension of $\mathbb Q$. Let $...
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1answer
165 views

Can you give me a walk through of proof for Fermat's last theorem? [closed]

Proof for Fermat's last theorem was presented in year '96 by British mathematician, it is said to be "a very long proof and involves latest advances in mathematics", can anyone please suggest what ...
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Vanishing of a Galois cohomology of a pro-$p$ group

Suppose $G$ is a pro-$p$ group and $M$ is a discrete $p$-primary module not necessarily finite. Suppose the $p$-cohomological dimension of $G$ is $n$. Can one conclude $$ H^n(G,M) \neq 0? $$ This is ...