Questions tagged [galois-cohomology]

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2
votes
0answers
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
0answers
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
0answers
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
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0answers
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
1answer
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
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0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
3answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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 ...