# Questions tagged [galois-cohomology]

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185
questions

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votes

**1**answer

124 views

### Galois cohomology of $\mathbf{Z}_\ell(m)$ over finite fields

My question is surely a classical one in the algebraic number theory, but I'm not working in it and I do not know the results and references. Please excuse me.
Let $k = \mathbf{F}_q$ be a finite field ...

**2**

votes

**0**answers

82 views

### What are the Tits algebras of $\mathrm{SO}(A, \sigma)$ if $A$ is split?

Given a connected linear algebraic group $G$ over a field of characteristic zero, there are several constructions of the so called Tits algebras (see Sechin and Semenov - Applications of the Morava K-...

**6**

votes

**1**answer

169 views

### Real forms of complex reductive groups

I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in ...

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85 views

### Amitsur cohomology and limits

Let $R \longrightarrow S$ be a faithfully flat ring extension and $\{ G_\alpha\}$ an inverse system of affine group schemes. I would like to know if there is some hope for a bijection like
$$
H^1(S/R, ...

**3**

votes

**1**answer

104 views

### Globalising tori and weak approximation

Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a ...

**6**

votes

**1**answer

193 views

### Which cases of Beilinson-Bloch-Kato for elliptic motives are known?

Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that
$$
\operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\...

**5**

votes

**2**answers

318 views

### Non-abelian Ext functor and non-abelian $H^2$

Let $G$ be a group and
$$0\rightarrow K\rightarrow M\rightarrow N\rightarrow 0$$
a short exact sequence of groups. Now these are abelian groups, if I want to show that $\text{Hom}(G,M)\rightarrow \...

**2**

votes

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58 views

### Bloch–Kato–Selmer group of a 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_{\text{cyc}}:G_{\mathbb{Q}}\rightarrow L^\...

**3**

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114 views

### Central simple algebras via cohomology

I am following the book Central Simple Algebras and Galois Cohomology, by Gille and Szamuely (I am using the second edition). In section 2.4, the authors remark that the tensor product induces a ...

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78 views

### Can the cohomology group $H^1(H, E[p])$ be trivial for all subgroups $H$ of $Gal(K(E[p])/K) \simeq GL_2(\mathbb Z/p)$?

Let $p$ be a fixed odd prime, $K$ be a number field and $E$ be an elliptic curve defined over $K$. Set $L =K(E[p])$.
We know that $G=Gal(L/K)$ is a subgroup of $GL_2(\mathbb Z/p)$.
Suppose $G =GL_2(\...

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104 views

### Field extensions that preserve given cohomology classes

Let $G/\mathbb{Q}$ be a connected reductive group, let $C \subset H^1(\mathbb{Q}, G)$ be a finite set of points and let $n \ge 2$ be an integer. Is it always possible to find a finite extension $F/\...

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**0**answers

50 views

### Commutativity in local class field theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $L,M$ be finite Galois extensions of $K$ such that $L\subset M$. Let $\phi_L\in Z^2(\text{Gal}(L/K),L^\times)$ such that $[\phi_L]$ is a generator ...

**7**

votes

**1**answer

203 views

### Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?

Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...

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95 views

### The profinite topology on the Mordell Weil group

In this lecture of Serre on his open image theorem, around 6 minutes, Serre mentions the following theorem of Tate:
Let $A/k$ be an abelian variety over a number field and consider the Mordell-Weil ...

**3**

votes

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155 views

### Is the Cassels “$x - \theta$” map algebraic in some sense?

Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(...

**3**

votes

**1**answer

153 views

### Quotienting $G(\mathbb{Q})_{+}$ by $G^{\text{sc}}(\mathbb{Q})$ and inner forms

Let $G/\mathbb{Q}$ be a connected reductive group, let $G^{\text{ad}}$ be the adjoint group, let $G^{\text{der}}$ be the derived group and let $\rho\colon G^{\text{sc}} \to G^{\text{der}}$ be the ...

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vote

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108 views

### Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...

**6**

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163 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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**1**answer

2k views

### Proof of Prop 1.1 in Wiles' “Modular elliptic curves and Fermat's last theorem”

On p. 459 of "Modular elliptic curves and Fermat's last theorem", proof of Prop. 1.1, where it says "Since $H^2(G,\mu_{p^r}) \rightarrow H^2(G,\mu_{p^s})$ is injective for $r \leq s$...", is there any ...

**7**

votes

**1**answer

714 views

### What is known about the cohomological dimension of algebraic number fields?

What is the cohomological dimension of algebraic number fields like $\Bbb{Q}$, $\Bbb{Q}[i]$, $\Bbb{Q}[\sqrt{3}]$ or similar? I'm interested in computing the cohomological dimension of $\Bbb{A}^1_k$ ...

**7**

votes

**1**answer

303 views

### Imperfect Tate (cup product) pairing in Galois cohomology?

Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$
and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$.
Let $M$ be a finite $\Gamma$-module, that is, a finite abelian ...

**3**

votes

**0**answers

189 views

### p cohomological dimension of a profinite group

I would like to know what is the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$. Here $S$ is a finite set of primes containing $p$ and the Archimedean primes and $\mathbb{...

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233 views

### A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence

I've been reading the book 'Cohomology of Number Fields' for years.
But I couldn't check the commutativity of the diagram
on page 126 until now. So I ask for help.
The diagram is induced by taking ...

**2**

votes

**0**answers

85 views

### Coboundary in Kummer theory

Let $K$ be a non archimedean local field whose residue field is of characteristic $p$. Denote by $G$ the absolute Galois group of $K$. Denote by $\mu_p$ the group of $p$-roots of unity and assume it ...

**6**

votes

**1**answer

494 views

### Example of a central simple algebra

Let $A$ be a finite dimensional central simple algebra over a field $F$ of characteristic $0$. So by Weddernburn's theorem, $A\cong M_n(D)$ for some division algebra $D$ over $F$. Let $\dim_F(D)=m^2$....

**4**

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196 views

### Calculating some Galois cohomology

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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142 views

### I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...

**2**

votes

**0**answers

115 views

### Fundamental Group of small Zariski open set

Let $Y$ be an integral affine variety over $\mathbb{C}$ and $K$ be its function field. How to find a sufficiently small Zariski open set of $Y$ such that it is isomorphic to $K(\pi,1)$? Here $\pi$ is ...

**6**

votes

**1**answer

317 views

### Integral Tate-Sen theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...

**4**

votes

**1**answer

154 views

### Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...

**3**

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101 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 $\...

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77 views

### Description of classes in $H^2(K,GL_n)$

Let $K$ be a field, we know elements in $H^2(K,\mathbb{G}_m)=\mathrm{Br}(K)$ can be represented by division algebras over $K$.
Do we have some description of elements in $H^2(K,GL_n)$ for $n>1$? ...

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votes

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129 views

### Orthogonal Grassmanians: cases where $\text{OG}( \mathbb{P}^1 , Q) \not \simeq \mathbb{P}^3$

Let $Q = \{ q(x_0, \dots, x_4) = 0 \}$ be a quadric-threefold over a field $k$. Are there cases where the orthogonal Grassmanian $\text{OG}( \mathbb{P}^1 , Q)$ is not a copy of $\mathbb{P}^3$?
Here'...

**5**

votes

**1**answer

135 views

### If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$.
Does there exist $M \in \...

**3**

votes

**1**answer

155 views

### Vanishing question for self-products in Galois cohomology

Suppose that $k$ is a number field. Let $G$ be the absolute Galois group of $k$ , let $M$ be a torsion $G$-module and $\alpha \in H^{1} (G, M)$. Is it true that
$$\alpha \cup \alpha \cup \ldots \cup \...

**5**

votes

**1**answer

366 views

### A question on the injectivity of a canonical map between galois cohomology groups

I'm currently reading the book "Galois theory of $p$-extensions" by Helmut Koch.
There, we calculate the cohomological dimension of the galois group $G(K/k)$ where $K$ is the maximal (normal) $p$-...

**3**

votes

**1**answer

172 views

### Real automorphisms of the “quaternionic” real group ${\rm SO}^*(4m)$

Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...

**8**

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**0**answers

247 views

### Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...

**2**

votes

**1**answer

198 views

### Semi-simple Galois actions on étale cohomology

Assume that semi-simplicity of the Galois action on $\ell$-adic cohomology of all smooth projective varieties over finite fields, were known.
Can one deduce that the Galois action on $\ell$-adic ...

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305 views

### Galois invariants in étale cohomology

Suppose $X$ is a smooth projective variety over a field $k$, with separable closure $\overline{k}$, Galois group $G$, and let $\overline{X}$ be $X_{\overline{k}}$.
Do we have
$$(H^j(\overline{X},\...

**6**

votes

**1**answer

317 views

### Unramified non-abelian extension and Galois cohomology

Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class ...

**4**

votes

**1**answer

196 views

### No lifts in an exact sequence of profinite groups?

In pg. 24 of his book on Galois cohomology, Serre gives the following exercise:
"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:
(i) $...

**4**

votes

**0**answers

99 views

### Relations between an projective variety and galois cohomology

Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...

**6**

votes

**2**answers

290 views

### The Tits classes of simply connected simple real groups

Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$:
...

**3**

votes

**1**answer

117 views

### Triviality of torsors after a field extension of bounded degree

Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for ...

**1**

vote

**1**answer

167 views

### Notation for the restriction map in Galois cohomology

My coauthors and I are writing a paper based on MO questions and answers:
Friedrich Knop's answer,
my answer 1
and
my answer 2.
For a linear algebraic group $G$ over a perfect field $k$, I consider a ...

**7**

votes

**0**answers

336 views

### Explicit $H^2(K, \mu) = Q/Z$?

In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero,
$H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$
Neukirch et al. ...

**2**

votes

**1**answer

213 views

### Rost Invariant of $E_7$

Let $E_7$ denote the split group of type $E_7$. Assume $G := \xi\overline{G}$ is a semisimple algebraic group over a field $k$ with characteristic zero for some $\xi \in H^1(k,E_7)$. Let $r(G)$ $\in$ $...

**2**

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**0**answers

163 views

### etale cohomology of tori

Let $k$ be an algebraically closed field.
Let $A$ be a strictly henselian local ring which is a $k$-algebra.
Let $T$ a torus over $k((t))$.
Can we compute $H^{1}(A((t)),T)$?

**4**

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200 views

### Galois cohomology of the Serre group in the proof of the fundamental theorem of CM

I am working through J.S. Milne's note on the fundamental theorem of complex multiplication over $\mathbb{Q}$. Let $E$ be a CM-field Galois over $\mathbb{Q}$, and $S^E$ the Serre group corresponding ...