# Questions tagged [galois-cohomology]

The galois-cohomology tag has no usage guidance.

234
questions

2
votes

1
answer

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### Vanishing of the degree 2 cohomology of a p-adic field with coefficients Q/Z and action of the Frobenius and the Pontryagin dual of the inertia

Let $K$ be a $p$-adic field with Galois group $G$ and inertia subgroup $I\subset G$. Denote $(-)^\ast=\mathrm{Hom}_{cont}(-,\mathbb{Q}/\mathbb{Z})$. Using Tate local duality, we can compute $$H^2(G,\...

5
votes

1
answer

117
views

### Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and ...

4
votes

1
answer

241
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### Why descend a representation (of a finite group) over $K$ to a representation over $k$ with $k$ a subfield of $K$ is useful?

I heard that Schur was trying to answer the following question
Given a representation of a finite group $G \overset{\rho}{\rightarrow} \operatorname{GL}_{n}(K)$, how to find the smallest subfield $k$ ...

3
votes

1
answer

336
views

### Is this exact sequence known?

$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...

4
votes

1
answer

370
views

### Generalizations of global Euler characteristic formula

Let $ K $ be a number field, $ S $ a finite set of primes of $K $ including the archimedean primes and $ G_{K,S} $ be the Galois group of the maximal extension of $K$ unramified outside $ S $. Assume ...

2
votes

1
answer

96
views

### Image of Kummer map for CM Elliptic curves

Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...

5
votes

0
answers

113
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### Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory

I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...

11
votes

1
answer

215
views

### Galois cohomology class of a reductive group not coming from a torus

Let $G$ be a (connected) reductive group over a perfect field $k$, and let $\xi\in H^1(k,G)$ be a cohomology class.
By a theorem of Steinberg (Serre, Galois cohomology, Appendix 1 to Chapter III, ...

2
votes

0
answers

182
views

### Relation between the Tate-Shafarevich group of a number field and the Tate-Shafarevich group of an elliptic curve

Let $E$ be an elliptic curve over a number field $K$, and let $sha^1(K,E)$ be the Tate-Shafarevich group, defined by:
Let $v$ be a valuation on $K$, and denote by $K_v$ the completion of $K$ by $v$, ...

3
votes

1
answer

127
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### The torsion subgroup of the coinvariants for a $G$-module

Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...

2
votes

0
answers

91
views

### Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...

4
votes

0
answers

140
views

### Alternative formulation of the Ferrero-Washington Theorem

The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the cyclotomic $\mathbf{Z}_p$ extension $K^{\text{cyc}}/K$ has $\mu=0$.
In the paper "Iwasawa invariants ...

9
votes

0
answers

260
views

### What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?

Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...

0
votes

0
answers

101
views

### Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules.
See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$
and of the cup product pairings
\...

3
votes

1
answer

294
views

### Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action.
For the first Galois cohomology of $M$, ...

2
votes

0
answers

460
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### Confusion regarding Proposition 1.1 in Wiles's Fermat paper

This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...

3
votes

1
answer

214
views

### Proof of $V\cong \overline{K} \otimes_{K} V_K$ using $H^1(G_{\overline{K}/K},\operatorname{GL}_n(K))＝0$

This is from Silverman's book "The arithmetic of elliptic curves" (AEC), p.36, lemma 5.8.1.
Lemma 5.8.1 states
Let $V$ be a $\overline{K}$-vector space, and assume that
$G_{\overline{K}/K}$ ...

2
votes

1
answer

197
views

### Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$

Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. My question is: how do you compute the local cohomology group $H^1(K_v, \, E[p^{\infty}])$?
As to why I'm asking this, it ...

4
votes

0
answers

177
views

### Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari.
After some arguments, we get a exact sequence
$$
\mathbf{P}^1_S(k,M^{'})^* \...

5
votes

0
answers

268
views

### The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field,
and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$.
By local class field theory, there is a canonical isomorphism
$$...

5
votes

2
answers

631
views

### Embedding torsors of elliptic curves into projective space

Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...

1
vote

0
answers

112
views

### Inflation in degrees $0$, $-1$, and $-2$ for Tate cohomology of finite groups

Let $\pi\colon G'\to G$ be a surjective homomorphism of finite groups, and let $A$ be a $G$-module.
I need explicit formulas for the inflation maps
$${\rm Inf}^{r}\colon H^{r}(G,A)\to H^{r}(G',A)$$ ...

2
votes

0
answers

128
views

### Cup product in modified (Tate) group hypercohomology

Let $G$ be a finite group.
Let
$$ M^\bullet=\,(\dots\to M^{-1}\to M^0\to M^1\to\dots) $$
be a bounded complex of $G$-modules.
One can define Tate hypercohomology groups $H^n(G,M^\bullet)$ for all $n\...

3
votes

0
answers

176
views

### Cohomology of the ring of integers of a local or global field

Let $\mathcal{O}$ denote the ring of integers in the separable closure $K^{sep}$ of a local or global field $K$ with absolute Galois group $G_K$. What is known about the cohomology groups $H^i(G_K,\...

7
votes

1
answer

226
views

### Explicit cocycles for the first Galois cohomology of a $p$-adic torus

Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...

3
votes

0
answers

115
views

### The Brauer group and the second Galois cohomology group

I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...

1
vote

1
answer

526
views

### Cohomology with coefficients in $\mu_\infty$

I'm encountering a lot of problems when dealing with the root of unity sheaf $\mu_\infty := \mathrm{colim}_n\mu_n$.
Let $X$ be a smooth geometrically integral variety over a number field $k$. Although ...

3
votes

0
answers

137
views

### Obtaining an exact sequence of Galois modules via derived functors

This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there.
Let ...

0
votes

1
answer

156
views

### The direct limit of invertible functions on a variety

(I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.)
Let $X$ be a smooth geometrically integral ...

4
votes

1
answer

191
views

### A Kummer exact sequence involving $\mu_\infty$

Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...

2
votes

1
answer

225
views

### Computing $H^1$ with coefficients in a torsion-free abelian group

Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that ...

4
votes

1
answer

249
views

### Distinguished triangle and hypercohomology

Let $X$ be a smooth geometrically integral variety over a number field $k$, we have an exact sequence of complexes of $\mathrm{Gal}(\bar{k}/k)$-modules
$$0 \rightarrow [\bar{k}^* \rightarrow 0] \...

0
votes

0
answers

208
views

### Proof of “Hochschild-Serre” spectral sequence

I'm looking for a detailed proof of the Hochschild-Serre spectral sequence for Galois Cohomology:
If we have $H$ a normal subgroup of $G$ (profinite group) and $T$ is a $G-$module, we get:
$$
0\...

10
votes

1
answer

691
views

### Third Galois cohomology group

It is well known that when $K$ is a local or global field the Galois cohomology group $H^{3}(K,K_{\text{sep}}^{\times})=0$ where $K_{\text{sep}}$ denotes the separable closure of $K$. Could someone ...

3
votes

1
answer

271
views

### Finiteness of cohomology group

Suppose $G$ is a finite Galois group, and $M$ is an infinite $G$-module. When can I say that $H^1(G, M)$ is finite?
I know this not true in general. Is it true under certain assumptions on $M$?
To be ...

0
votes

0
answers

69
views

### An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...

1
vote

1
answer

205
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### Taking quotient of a variety by the additive group

1. Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times _{\Bbb C} G\to X,\quad (x,...

3
votes

0
answers

263
views

### A question on the Robba ring

Notation is as in the question:
https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring.
We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...

3
votes

2
answers

330
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### Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \, \mathbb{Q}_{ \ell } (r) )^G $?

Let $ k $ be a field and let $ X $ be a smooth projective variety over $ k $ of dimension $ d $.
We denote by $ \overline{X} = X \times_k \overline{k} \ $ the base change of $ X $ to the algebraic ...

7
votes

2
answers

1k
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### The Mumford-Tate conjecture

The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...

6
votes

0
answers

367
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### Galois cohomology with coefficients in the unit group of a cyclotomic field

While understanding Fermat's last theorem's proof for regular primes, I bumped into a proposition (http://www2.biglobe.ne.jp/~optimist/algebra/fermat2_proof.html#proof10, written in Japanese) stating:
...

4
votes

0
answers

96
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### Neutral cohomology classes and restriction maps for $H^2$ in group cohomology

$\DeclareMathOperator\res{res}$
Let $G$ be a profinite group.
Let $A$ be a finite $G$-group (a finite group on which $G$ acts continuously), not necessarily abelian,
with center $Z=Z(A)$.
We say that ...

4
votes

1
answer

409
views

### Galois cohomology of separable closure

Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation.
In his paper on $p$...

3
votes

0
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103
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### Galois descent of a Hopf algebra

In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent.
As I ...

1
vote

1
answer

159
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### The groups $H^i(k,\mathbb{Z})$ for $i=1,2$

This question is related to my post Interpretation of some maps involving cohomology groups.
$C$ is a smooth geometrically integral affine curve over a number field $k$, and $C_1$ is its smooth ...

7
votes

0
answers

383
views

### Status of the conjectured vanishing of Bloch-Kato H^2

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...

9
votes

1
answer

495
views

### Forms of ${\rm SL}(2)$

I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$.
Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real ...

2
votes

0
answers

152
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### kernel and cokernel of corestriction map in cohomology of a profinite group

Let $G$ be a profinite group, $N$ a normal open subgroup and $A$ a discrete $G$-module. We have a corestriction map $cor: H^1(N, A)_{G/N} \to H^1(G, A)$. Are there any results on the kernel and ...

1
vote

1
answer

110
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### $0$-th Galois cohomology with topological Milnor K-groups coefficients

In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...

2
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0
answers

90
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### Iwasawa's results about relation between Galois cohomology and principal factorization

Let $K$ be a Galois number field with Galois group and units group $G$ and $U$, respectively. How we can relate the first cohomology group $H^1(G,U)$ to principal factorization in K?
I'd try to find ...