Questions tagged [galois-cohomology]

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Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. Is it true that ...
Andrei Jaikin's user avatar
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336 views

Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...
nxir's user avatar
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Twists of projective automorphisms

Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$ The twists of $X$ are classified by the Galois ...
Daniel Loughran's user avatar
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1 answer
321 views

Equality of Galois modules

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module. One can define two subgroups of the Galois cohomology group $H^i(k,M)$: the group of elements of $H^i(k,M)$ ...
Wanderer's user avatar
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Do algebraic tori have no $H^1$?

If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...
Evan O'Dorney's user avatar
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201 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 ...
youknowwho's user avatar
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265 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^{'})^* \...
Mugenen's user avatar
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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 $$...
Mikhail Borovoi's user avatar
4 votes
2 answers
1k 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:...
user1225's user avatar
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Biquadratic extension of global function fields with cyclic decomposition groups

Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$. Question. What would be an example of a ...
Mikhail Borovoi's user avatar
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2 answers
926 views

Galois cohomology H^1(Q_p, Z_p(2)) = 0?

For Tate twists Z_p(2), which is defined by the projective limit of \mu_{p^m}(2) over all m>0, I would like to calculate H^1(Q_p, Z_p(2)). I guess this is zero, but cannot prove it. Is it possible ...
pierre matsumi's user avatar
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1 answer
301 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 ...
Sam Streeter's user avatar
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2 answers
312 views

Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success. Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...
user33624's user avatar
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1 answer
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What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$ The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...
Lloyd Yu-West's user avatar
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1 answer
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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 $k$ ...
Marsault Chabat's user avatar
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1 answer
476 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 $p$...
Piotr Pstrągowski's user avatar
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1 answer
256 views

Local triviality of Galois cohomology classes over $\mathbb{Q}$

Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in $$\mathrm{H}^1(\mathbb{...
Daniel Loughran's user avatar
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1 answer
289 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 $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...
oleout's user avatar
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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 ...
Pol van Hoften's user avatar
4 votes
1 answer
614 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 ...
Nobody's user avatar
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1 answer
279 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] \...
oleout's user avatar
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0 answers
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Anisotropic semisimple groups with no real compact factor

Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...
jacob's user avatar
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A computation of nearby cycles

I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow. Now ...
Huang Chenxin's user avatar
4 votes
0 answers
59 views

Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?

$\newcommand{\wt}{\widetilde}$ Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$. There are finite subgroups $$ {\rm Sha}^1 \subset H^1,\quad ...
Mikhail Borovoi's user avatar
4 votes
1 answer
119 views

Real forms of the general linear Lie superalgebra

I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$. The real forms of the simple complex Lie superalgebras were classified by ...
Alistair Savage's user avatar
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0 answers
151 views

Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle of $G=\GL_{n,\R}\,$, that is, an invertible ...
Mikhail Borovoi's user avatar
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0 answers
185 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 ...
Adithya Chakravarthy's user avatar
4 votes
0 answers
107 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 ...
Mikhail Borovoi's user avatar
4 votes
0 answers
106 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, ...
Agustí Roig's user avatar
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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 ...
Asvin's user avatar
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4 votes
0 answers
247 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{...
debanjana's user avatar
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4 votes
0 answers
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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, \...
debanjana's user avatar
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4 votes
1 answer
222 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) $...
user avatar
4 votes
0 answers
114 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, \...
HASouza's user avatar
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0 answers
249 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 ...
Jiangwei Xue's user avatar
4 votes
0 answers
240 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-...
Somatic Custard's user avatar
4 votes
0 answers
157 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 ...
nxir's user avatar
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4 votes
0 answers
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Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
nxir's user avatar
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3 votes
1 answer
224 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 $G_{\overline{K}/K}$ ...
Neronoggshafareivh's user avatar
3 votes
2 answers
369 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 ...
Bradley04's user avatar
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3 votes
2 answers
394 views

Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?

For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...
unramified's user avatar
3 votes
1 answer
601 views

Galois cohomology of a non-abelian group over a function field

Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, ...
Sasha's user avatar
  • 5,492
3 votes
2 answers
618 views

Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
prochet's user avatar
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3 votes
1 answer
344 views

What's the minimum number of generators for the wild inertia?

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...
Kestutis Cesnavicius's user avatar
3 votes
1 answer
225 views

The second Tate-Shafarevich group of a permutation module is trivial

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $...
Tuvasbien's user avatar
  • 156
3 votes
1 answer
265 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 ...
Mikhail Borovoi's user avatar
3 votes
1 answer
223 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 ...
Muhammed Ali's user avatar
3 votes
1 answer
160 views

On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned: Is the map $g_3 \...
nxir's user avatar
  • 1,409
3 votes
1 answer
463 views

The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...
JJH's user avatar
  • 1,447
3 votes
2 answers
1k views

Restriction of scalars of simple algebraic groups

I'm trying to understand the following basic property of the restriction of scalars: Given an absolutely simple algebraic groups $G$ defined over a number field $k$, are there at most finitely (up-to ...
Menny's user avatar
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