Questions tagged [galois-cohomology]

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2 votes
1 answer
93 views

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 views

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 ...
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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 views

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 $v$, ...
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3 votes
1 answer
127 views

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 ...
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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 \...
  • 23.1k
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 $M$, ...
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2 votes
0 answers
460 views

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^{'})^* \...
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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 ...
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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-...
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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\...
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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 ...
  • 439
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 ...
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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 ...
  • 443
1 vote
1 answer
205 views

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)}...
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3 votes
2 answers
330 views

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 ...
  • 487
7 votes
2 answers
1k views

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 ...
  • 487
6 votes
0 answers
367 views

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: ...
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4 votes
0 answers
96 views

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 answers
103 views

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 ...
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1 vote
1 answer
159 views

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 ...
  • 14.2k
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 views

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 views

$0$-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 ...
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2 votes
0 answers
90 views

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 ...

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