Questions tagged [galois-cohomology]

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

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70 views

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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23 views

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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1answer
52 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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28 views

Exact sequence for Galois cohomology groups

Let $K/k$ be a finite Galois extension and let $G=\mbox{Gal}(K/k)$. Let $F$ be a subfield of $K$ such that $k\subseteq F\subseteq K$ and $F/k$ is normal. Let $H=\mbox{Gal}(K/F)\trianglelefteq G$. ...
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2answers
91 views

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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15 views

Flat cohomology as a subspace of local Galois cohomology

Consider an elliptic curve $E$ over a local field $k$ with good reduction and with Neron model $\mathcal{E}/\mathcal{O}_k.$ The Galois group $G_k$ acts on $E[m]$, so we have a cohomology group $H = H^...
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1answer
41 views

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 ...
3
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0answers
77 views

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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1answer
28 views

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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0answers
26 views

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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18 views

Generalization of Hilbert 90 theorem for $H^1( \mathrm{Aut}(\mathbb C / \mathbb Q), \mathbb C^{\times} )$?

$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\...
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54 views

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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2answers
41 views

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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58 views

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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0answers
37 views

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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72 views

When does group cohomology $H^1(G,M)$ depend only on the image of $G$ in Aut($M$)?

To motivate the question (and narrow it down if the one I asked is too broad), I'm doing readings from Manin's cubic forms book. A while back I was asked to compute the Galois cohomology $H^1(G, Pic(...
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26 views

$H^1(G,GL_n(K))$ 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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1answer
57 views

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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1answer
34 views

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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27 views

Approaching the information of a topological group by that of a closed subgroup which may not be normal?

The title is vague and I explain more precisely on my question. Let $G$ be a topological group with $H\leq G$ a subgroup. I ask the question in the group cohomology point view, so actually the ...
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78 views

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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42 views

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 ...
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30 views

$p$ cohomological dimension of profinite groups

Suppose $p$ is an odd prime and $S$ is any finite set containing the primes above $p$ and the Archimedean primes. Does there exist any number field $K$ such that $\textrm{Gal}(K_S/ K_{cyc})$ has $p$-...
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1answer
124 views

Calculating the Galois cohomology of $U(n)$

I would like to know know what the Galois cohomology $H^1(Gal( \Bbb{C}/\Bbb{R}),U(n)(\Bbb{C}) ) $ is. The unitary group $U(n)$ can be defined as the real form of $Gl_{n,\Bbb{C}}$ relative to the ...
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1answer
49 views

Are the rank and the corank of the cohomologies of the given Tate module the same?

Let $G$ be a Galois group of two kinds: $G=Gal(K_S/K)$ where $K$ is a number field and $K_S$ is a maximal $S$-ramified extension. ($S$ is a finite set of primes containing primes above $p$ and $\...
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0answers
53 views

Finding an equation for a variety twisted by a cocycle.

I am trying to understand varieties $X’$ that are obtained by twisting a variety $X$ over a field $k$ by a cocycle $\gamma\in H^1(k^{sep}/k, Aut(X))$. To make this feel concrete, I would like to ...
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1answer
71 views

Does every simply-connected reductive group have trivial Galois cohomology?

Let $G$ be a linear algebraic group over a field $k$ with separable closure $k^s$ and absolute Galois group $\Gamma\!_k$. Consider the Galois cohomology group $$ H^1(\Gamma\!_k,G(k^s)). $$ This group ...
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1answer
40 views

Isomorphic elements in $H^i(k;\mu_2)$

An element $\alpha \in H^i(k;\mu_2)$ is called a pure symbol in case it has the form $\alpha = (a_1) \cup\dots\cup (a_i)$, $(a_i) \in H^1(k;\mu_2)$. It is known that for every pure symbol $\alpha $ ...
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1answer
125 views

Tensor product of Galois extension

Let $K/k$ be a finite Galois extension of fields with Galois group $G$. How to show that the (n+1)-fold tensor product $$K \otimes_k K \otimes_k K \cdots \otimes_k K$$ is isomorphic to $$\prod\...
6
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1answer
201 views

Nontrivial Twists of a Vector Bundle

Let $k$ be a number field, and let $X$ be a projective $k$-variety. Let $\mathcal{V}$ be a vector bundle on $X$ that is defined over $k$. A vector bundle $\mathcal{V}'$ on $X$ that is defined over $k$ ...
3
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1answer
101 views

Surjectivity of multiplication by $n$ on elliptic curves

Is there an abelian variety $A$ over a field $k$, such that $A(k^{\rm sep})$ is not a divisible group? The motivation of my question is the following : if $L$ is any algebraically closed field, then $...
4
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1answer
152 views

Generalisation of Hilbert's 90 Theorem

Let $L/K$ be a finite Galois extension of fields with Galois group $G = Gal(L/K)$. According to the famous Hilbert's 90 we know that the first cohomology vanish: $$H^1(G, L^*)=\{1\}$$ My question is ...
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74 views

Certain Galois cohomology computation

Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
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63 views

Galois Action on Coherent Sheaves Exact Functor

Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined ...
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1answer
95 views

Brauer-Severi varieties as quotients of forms of $\text{GL}_2$

Let $L/F$ be a finite galois extension of fields, with galois group $\Gamma$. Let $X$ be a variety over $F$ such that $X_L \cong \mathbb{P}^1_L$ over $L$, corresponding to a cohomology class $\alpha \...
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1answer
56 views

Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
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2answers
127 views

Two variations of Hilbert's 90 theorem

Let $L/K$ be a finite Galois extension. I know that $H^1(Gal(L/K), L^{\times}) = 0$ and $H^r(Gal(L/K), (L,+)) = 0$ for all $r>0$. 1) Do we have $H^r(Gal(L/K), L^{\times}) = 0$ for all $r>1$ ? ...
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2answers
180 views

Hilbert's theorem 90 for $p$-adic topology

Let $L/K$ be algebraic extensions of $\Bbb Q_p$. Consider $L^{\times}$ with the $p$-adic topology. Is it true that the first continuous cohomology group $H^1_{cont}(\mathrm{Gal}(L/K), L^{\times})$ ...
2
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1answer
252 views

About the definition of l-adic Tate-twist

In the J. Tate's paper "Relations Between $K_2$ and Galois Cohomology" Let F any field $F^{\text{sep}}$ the separable closure of F $G_F=\text{Gal}(F^{\text{sep}}/F$) he defines the ($\mathbb{Z}_l,G_F$)...
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0answers
155 views

Vanishing of second Galois cohomology group

This most likely follows from a standard result but a lack of knowledge prevents me from seeing this. Let $K$ be a non Archimedean local field. Let $\Gamma$ be $\mathrm{Gal}(\bar{k}/k)$. Let $T$ be a ...
7
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1answer
287 views

Computing étale cohomology group $H^1( \text{Spec}(k), \mu_n)$ and $H^1( \text{Spec}(k), \underline{\Bbb{Z}/\mathord{n \Bbb{Z}}})$

I am starting to learn about étale cohomology and would like to compute a simple example. Let $k$ be a field with a fixed separable extension $k^s.$ I want to compute $H^1( \operatorname{Spec}(k), \...
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0answers
63 views

When does a number field have $p$-rank greater than $n$?

Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...
3
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1answer
386 views

What should one know before learning galois cohomology

I recently became interested in galois cohomology, but I don't know if I have enough math to learn about it or where to start learning it. What should I know before I start learning about Galois ...
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42 views

Galois extension of exponent $mp^r$ in characteristic $p$

Kummer theory treats Galois extensions of exponents that are not divisible by the characteristic. Artin-Schreier and Witt extend this theory for Galois extensions of exponents $p^r$ in characteristic $...
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2answers
170 views

Tate Cohomology of Squares

Let $L | \mathbb{Q}_p$ be a finite Galois extension with Galois group $G$. What is known about the group $H^{-1} (G, L^\times / (L^\times)^p)$? I'm particularly interested in the case of a non-...
3
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1answer
132 views

How can a point on an elliptic curve be considered a galois cohomology class?

For an elliptic curve $E$, I saw a paper once mention that you can interpret a rational point on $E$ as a degree $0$ Galois cohomology class. I'm familiar with group cohomology but not all that much ...
2
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1answer
193 views

Hilbert 90 and K-forms

Studying the proof of Hilbert's 90 theorem modern version, I went through this lemma:given a Galois finite extension $K \subset L$ and an $L$ algebra $A$,we define the $(A,K)$ forms as the $K$ ...
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1answer
65 views

Surjectivity of map of étale sheaves

Let $F\to G$ be a map of étale sheaves on a scheme $X$. Suppose that for any closed point $x\in X$, the map between stalks $$F_x\to G_x$$ is surjective. Note that I am not forming stalks at the ...
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2answers
108 views

First cohomology group of the $n$-torsion of an elliptic curve

Let $E$ be an elliptic curve defined over a number field $K$. Let $v$ be a finite place of $K$ not dividing $n$ such that $E$ has good reduction at $v$. This only excludes a finite set of places $\...
3
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1answer
86 views

$H^1(G, \mathbb{Z}/p \mathbb{Z})$ and linearly independent elements in open subgroups.

Let $G$ be a profinite group and $p$ a prime number, and consider the following condition on $G$: For every open normal subgroup $U$ of $G$ and any integer $N \geq 0,$ there are $N$ elements $$z_1, \...