# Questions tagged [galois-cohomology]

The galois-cohomology tag has no usage guidance.

165
questions

**2**

votes

**0**answers

148 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{...

**7**

votes

**0**answers

207 views

+100

### 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

74 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 ...

**1**

vote

**0**answers

64 views

### Reference for Selmer-Group coming from Galois representation associated with modular form

Is there any good reference (lecture notes) for the construction of Selmer-Groups associated with the Galois representation? In particular, i want to understand how they are using Deligne's and Mazur'...

**5**

votes

**1**answer

402 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 $\dim_F(D)=m^2$....

**5**

votes

**0**answers

181 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, \...

**3**

votes

**0**answers

133 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

109 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

248 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

112 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**

votes

**0**answers

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 $\...

**1**

vote

**0**answers

76 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 $n>1$? ...

**0**

votes

**0**answers

126 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

134 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

149 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

318 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

140 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**

votes

**0**answers

219 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

178 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 ...

**6**

votes

**0**answers

300 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

278 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

180 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

98 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

262 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

112 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

159 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

291 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

200 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**

votes

**0**answers

144 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**

votes

**0**answers

171 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 ...

**1**

vote

**0**answers

81 views

### The canonical map $\operatorname{Coind}_{H}^{G}(A)\to A$ is surjective?

This question is about a claim found in Serre's Galois Cohomology
Let $H$ be an open subgroup of a profinite group $G$ and $A$ be a discrete $H$-module.
We then can form the co-induced module $\...

**3**

votes

**1**answer

65 views

### Splitting variety for bicyclic algebras

Let $F$ be a field contains a primitive root of unity of order $p$, where $p$ is a prime number. Let $a,b \in F^\times$, then one can look at the cyclic algebra $(a,b)_p \in {_p}Br(F)$ where ${_p}Br(F)...

**3**

votes

**0**answers

173 views

### Why does the Galois twist of this cover specialize to a certain field extension?

I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-...

**5**

votes

**1**answer

236 views

### Brauer groups and field extensions

Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$...

**3**

votes

**0**answers

202 views

### Interaction between the Brauer group and abelian extensions

If $k$ is a field of characteristic zero and $k$ has no (non-trivial) abelian extensions (e.g. the composite of all solvable extensions of $\mathbb{Q}$), then $\text{Br(k)} = 0$ by the norm residue ...

**2**

votes

**0**answers

115 views

### complex numbers over algebraic numbers, continuous cohomology

This question is related to this one. Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from the algebraic closure of the field of rational numbers to the field of complex numbers. Is ...

**4**

votes

**2**answers

572 views

### Galois cohomology of finite fields

Let $k$ be a finite field, and let $G$ be the absolute Galois group of $k$, which is isomorphic to $\widehat{\mathbb{Z}}$. Let $\mathcal{C}$ be the category of $G$-modules. Then, we have the following:...

**5**

votes

**1**answer

260 views

### Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?

Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the ...

**1**

vote

**0**answers

130 views

### Levi decompositions of k-rational points of linear algebraic groups

Let $k$ be a field with characteristic zero and $G$ be a (connected or not) linear algebraic group defined over $k$. We know that $G$ has a Levi decomposition i.e., $G=R_u(G)\rtimes L$, where $R_u(G)$ ...

**1**

vote

**1**answer

236 views

### Spectral sequence of Galois cohomology over local fields

On page 530 in his paper,
Notes on etale cohomology of number fields, Ann. Scient. ENS (1973),
Mazur insisted that
$$
\text{Ext}^q_{G_K} (M,~\bar{K}^*) \...

**1**

vote

**0**answers

249 views

### Exercise in Galois cohomology

Let $K$ be a local field, and $\bar{K}$ its algebraic closure. Let $\mathcal{C}$ be the category of (continuous) $G=\text{Gal}(\bar{K}/K)$-modules.
Let $M$ be a finite $G$-module.
For any injective ...

**8**

votes

**3**answers

1k views

### Etale cohomology with coefficients in $\mathbb{Q}$

Let $X$ be a smooth variety of a field $k$. Then is
$$H_{et}^i(X, \mathbb{Q}) = 0$$
for all $i > 0$?
The result is true for $i=1$. This follows from the same argument given for $\mathbb{Z}$-...

**2**

votes

**0**answers

159 views

### Galois cohomology of cyclotomic extension

Let $K$ be a complete discrete valuation ring with algebraically closed residue field $F$ of characteristic $p > 0$. Suppose ${\Bbb Q}_p \subset K$ and the absolute ramification index v$_{\pi_K}(p) ...

**3**

votes

**0**answers

236 views

### Relation between Galois and etale cohomologies

Let $D$ be the ring of integers of a number field $F$.
Let $X=\mathrm{Spec} ~D$, and let $\pi$ be the etale fundamental group of $X$.
There are natural maps from $H^i(\pi, \mathbf{Z}/n)$ to $H^i_{...

**7**

votes

**0**answers

104 views

### defining Selmer groups using étale cohomology

Concerning http://swc.math.arizona.edu/aws/1999/99RubinES.pdf, especially section I.5:
Can one define the Selmer groups and the unramified cohomology groups as étale
cohomology groups of certain ...

**3**

votes

**1**answer

202 views

### induced isomorphism in continuous cohomology

Suppose that we have a morphism between profinite groups $f: G_{1}\rightarrow G_{2}$ such that $f^{\ast}:H_{cont}^{\ast}(G_{2},A)\rightarrow H_{cont}^{\ast}(G_{1},A) $ is an isomorphism for any finite ...

**4**

votes

**0**answers

138 views

### Quadrics contained in the (complex) Cayley plane

In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...

**26**

votes

**3**answers

879 views

### Consequences of Shafarevich conjecture

The Shafarevich conjecture states that the Galois group $\mathrm{Gal}({\overline{\mathbf{Q}}/\mathbf{Q}^{ab}})$ is a free profinite group, where $\mathbf{Q}^{ab}$ is the maximal abelian extension of $\...

**6**

votes

**1**answer

367 views

### The second Milnor $K$-theory of a field

Let $\mathbf{Q}^{\mathrm{ab}}$ be the maximal abelian extension of the field of rational numbers $\mathbf{Q}$. I'm interested in the following question:
Is it true that $K^{M}_{2}(\mathbf{Q}^{\mathrm{...

**1**

vote

**0**answers

235 views

### A cup product in etale cohomology of Elliptic curves

Another related question was posted here
A cup product in Galois cohomology of Elliptic curve
Prof. Silverman suggested me to post this in another thread so I post it here.
Suppose $E$ is an ...