Questions tagged [galois-cohomology]

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votes
1answer
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
0answers
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
1answer
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 ...
4
votes
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
votes
0answers
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 ...
2
votes
0answers
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(\...
5
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0answers
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/\...
0
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0answers
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
1answer
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 $\...
4
votes
0answers
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
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0answers
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
1answer
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 ...
1
vote
0answers
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
votes
0answers
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(...
9
votes
1answer
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
1answer
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
1answer
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
0answers
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{...
7
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0answers
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
0answers
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
1answer
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
votes
0answers
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, \...
3
votes
0answers
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
0answers
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
1answer
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
1answer
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
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
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 $n>1$? ...
0
votes
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
votes
0answers
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
1answer
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 ...
6
votes
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
votes
0answers
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
votes
0answers
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 ...