Questions tagged [galois-cohomology]

For questions on Galois cohomology, the study of the group cohomology of Galois modules.

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Continuous galois cohomology with twisted profinite integer coefficients

Let $k$ be a field that is finitely-generated over its prime field and $\bar k$ a separable closure. I would like to compute $$ H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1)) $$ Since $\mathbb{Z}(1) = ...
Erich's user avatar
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Kummer map on the formal group of elliptic curve

In section 3(Page 50) of this paper, it is mentioned that the Kummer map, $\hat{E}(\mathfrak{m}_n)\rightarrow H^{1}(k_n,T)$ along with the Weil pairing induces a cup product of Galois cohomology ...
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The first cohomology group of automorphism group

I want to know $H^1(K(\mu_p)/K,\mu_p)$ where $K$ is a number field. Since $K(\mu_p)/K$ is cyclic, $H^1(K(\mu_p)/K,\mu_p)=H^{-1}(K(\mu_p)/K,\mu_p)$. Using calculation, $H^{-1}(K(\mu_p)/K,\mu_p) =0$. So,...
WHERE 234's user avatar
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Definition of Tate-Shafarevich group up to isomorphism of elliptic curves

Let $E/K$ be an elliptic curve. Tate-Shafarevich group $Sha(E/K)$ of $E/K$ is defined as $Sha(E/K)\stackrel{\mathrm{def}}{=} \text{ker}(H^1(G_K,E) \to \prod_{v\in M_K} {H^1(G_{K_v},E)})$. When $E/K\...
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How Galois group act on $H^1(G_L,E)$?

Let $E/K$ be an elliptic curve over a field $K$. Then, how $\sigma \in \text{Gal}(L/K)$ acts on first Galois cohomology group $H^1(G_L,E) $ where $G_L$ denotes absolute Galois group of $L$? $H^1(G_L,E)...
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Second group cohomology of cyclic groups

Let $M$ be a Abelian group and $G$ be cyclic group of order $2$. Let $M$ be a $G$-module. Suppose $G=\langle\sigma\rangle.$ Let $M^{G} =\{m\in M\mid \sigma m =m\}$. Define the norm map $N: M\...
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Galois fixed part of modules. Why does $|(1-\sigma)A||A^G|=|A|$ hold?

Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $G=Gal(L/K)$. Let $A$ be a finite $Gal(L/K)$ module. Then, Why does $|(1-\sigma)A||A^G|=|A|$ hold ? Here, $|A|$ ...
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Almost purity for perfectoid fields applied on a computation of Galois cohomology.

Let $K=\mathbf{Q}_p(\zeta_{p^{\infty}})$ and $C:=\widehat{\overline{\mathbf{Q}_P}}$. When I wrote an answer for a question a question answer, I have in my mind that "$\widehat{K}$ is a ...
Luc's user avatar
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$H^1(\text{Gal}(L/K), O_L)=0$ for local fields

Let $K$ be a localfield and $O_K$be its ring of integers, and $L/K$ be a quadratic extension. It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known ...
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Relation between Galois Cohomology $H^1(\text{Gal(L/K)}, L) = 0$ and $H^1(\text{Gal}(L/K), O_L)=0$

Let $K$ be a number field and $O_K$be its ring of integers, and $L/K$ be a quadratic extension. It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known ...
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Image of rational characters of $p$-adic groups and Galois cohomology

Let $F$ be a $p$-adic field with residue field order $q$ and the normalized absolute value $|\cdot|=|\cdot|_F$. Let $\mathbf{G}$ be a connected reductive group over $F$. Let me denote by $G=\mathbf{G}(...
youknowwho's user avatar
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Definition of coboundary of first Galois cohomology

Let $K$ be a field, $G=Gal(\overline{K}/K)$ be its absolute Galois group, and let $M$ be a discrete $G$-module. We define Galois cohomology as follows. $H^1(G,M)\stackrel{\mathrm{def}}{=}Z^1(G,M)/B^1(...
Pont's user avatar
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Why is coboundary $\sigma \to \sigma m-m$ automatically continuous?

Let $K$ be a field. Let $G_K := Gal(\overline{K}/K)$ be absolute Galois group of $K$. Let $M$ be a $G_K$-module, that is, $G_K$ (with the Krull topology) acts continuously on $M$ with discrete ...
Pont's user avatar
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Galois cohomology of a finite cyclic group, 1st step

Let $G$ be a finite cyclic group of order $2$ whose generator is $\sigma$. Let $M$ be $G$ module. I want to prove Group cohomology $H^1(G,M)=Z^1(G,M)/B^1(G,M)$ is isomorphic to $ker(\sigma +1)/im(\...
Pont's user avatar
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Normal extension of given degree of a number field

Let $n>1$ be a natural number. Question. Let $K$ be a number field, and let $S\subset V_f(K)$ be a finite set of finite (non-archimedean) places of $K$. Does there exist a normal extension $L/K$ ...
Mikhail Borovoi's user avatar
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About the definition on cohomology of groups

I'm studying again cohomology of groups from Brown's book and other books and notes. I want to write some proof of well known statements but I have trouble with this one: Let be $G$ a group and $M$ a ...
José Luis  Camarillo Nava's user avatar
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Group cohomology $H^1(G,M)$ , where $G$ is finite and $M$ is finitely generated

Let $G$ be a finite group of order $n$. Let $M$ be a finitely generated abelian group generated by $m_1,m_2,・・・,m_k$. Let $H^1(G,M)$ be a group cohomology. Because $nH^1(G,M)=0$ from restriction and ...
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Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

Let $G$ be a simply connected absolutely simple group of one of the types $^1{\sf A}_{n-1}$ (inner) or $^2{\sf A}_{n-1}$ (outer) over a field $k$. All such groups are described on page 55 of Tits, ...
Mikhail Borovoi's user avatar
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What is the motivation to introduce Tate-cohomology groups?

What is the motivation to introduce Tate-cohomology groups ? Let $G$ be a Galois group and $M$ be a $G-$module. Let $H^n(G,M)$ be usual Galois cohomology. In group cohomology theory, we often ...
Pont's user avatar
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Etale cohomology of $Spec(𝔽ₚ^{\text{sep}}((t)))$

I am thinking about how norms $ν : L ⭢ ℤ$ on higher local fields could induce long exact sequences in different cohomologies. $𝔽ₚ^{\text{sep}}((t)),ℚₚ^{\text{sep}}$, and $ℂ$ are a local fields. What ...
Ronald J. Zallman's user avatar
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Finiteness of $n$th Galois cohomology$H^n(U, \mathbb{F}_p)$ for open subgroups $U$ of a pro-$p$-group $G$

I am reading a book named 'cohomology of number fields' by Neukirch, Schmidt, Wingberg. Let $G$ be a pro-$p$-group. Suppose the $p$-cohomological dimension $cd_p G=n<\infty$. Suppose $H^n(G, \...
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Proof of Proposition 2.27 in Diamond, Darmon, Taylor's "Fermat's Last Theorem"

See page 67 of the free online text for Proposition 2.27 below. All cohomology is Galois cohomology. Page 62 defines goodness, ordinaryness, and semistability. Page 65-66 defines $H^1_f$. The ...
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What is the group cohomology set $H^1(\mathbb{Z},GL_n(\mathbb{Z}_p^{ur}))$?

As in title, what is the group cohomology set $H^1(\mathbb{Z},GL_n(\mathbb{Z}_p^{ur}))$ ? Here the action of $\mathbb{Z}$ on $GL_n(\mathbb{Z}_p^{ur})$ is via Galois action: consider $\mathbb{Z}\...
Yuan Yang's user avatar
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Norm residue map maps uniformizing elements to Frobenius elements

I am reading Algebraic Number Theory by Cassel, Frohlich. I have a question about the proof that norm residue map maps uniformizing elements to Frobenius elements in unramified extension. This is in ...
Ja_1941's user avatar
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Cohomology of local fields in positive characteristic

It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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When is $f^*:H^1(G_K,N)\to H^1(G_K,M)$ injective?

et $K$ be a number field and $G_K$ be a absolute Galois group of $K$. Let $f:N \to M$ be an injetion of map between $G_K$modules. Then, induced map $f^*$ between Galois Cohomology, $f^*:H^1(G_K,N)\to ...
Pont's user avatar
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What is the maximum value of $\sharp H^1(\Bbb{Z}/2\Bbb{Z},\Bbb{Z}×\Bbb{Z}/2\Bbb{Z})$?

$H^1(G,M)$ be group cohomology. This group depends on how $G$ acts on $M$. But I'm interested in the maximum valu of this group. What is the maximum value of $\sharp H^1(\Bbb{Z}/2\Bbb{Z},\Bbb{Z}×\...
Pont's user avatar
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The first Galois cohomology commutes with projective limits

I am reading Serre's paper "Sur les groupes de congruence des variétés abéliennes" (here is the link to this paper: https://www.mathnet.ru/links/016949238724700ec2209f00e507a40f/im3061.pdf). ...
Khainq's user avatar
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Tate Twists of Z/nZ

Let $k$ be a $p$-adic local field with absolute Galois group $G_k$. In Cohomology of Number Fields, the authors define the $n$-th Tate twist of a finite $G_k$-module $A$ as the $G_k$-module $A(n)$ ...
aspear's user avatar
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Is the subgroup of $WC(E/K)$ of curves with an $L$-rational point finite for $K$ local, $L/K$ finite Galois?

User BrauerManinobstruction's question "Weil–Châtelet group of a real elliptic curve is isomorphic to Z/2Z when Δ>0" relates to exercise 10.7 of Silverman's AEC, which asks the reader to ...
Keith Millar's user avatar
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Relationship between restriction and corestriction for Galois cohomology groups involving number fields and local fields

Let $L/K$ be an extension of number fields. Let $v \in M_K$ be a place of $K$ and $\nu \in M_L$ be a place of $L$ dividing $v$. Let $L_\nu/K_v$ be the extension of the corresponding local fields. Let $...
Dime's user avatar
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Let $E/L$ be an elliptic curve. Why is Galois cohomology $H^1(G,E(L))$ finite group?

Let $L$ be a finite extension of number field $K$. Let $G=Gal(L/K)$. Let $E/L$ be an elliptic curve defined over $L$. Let $E(L)$ be $L $ rational point of $E$. Then, why is Galois cohomology $H^1(G,E(...
Pont's user avatar
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Restriction map in semilocal Galois cohomology

Let $F/K$ be a finite extension of number fields, let $v$ be a prime of $K$ that is totally split in $F$. Call $w_1,\dots, w_n$ the primes of $F$ above $v$. Let $T$ be a $Gal(\bar{K}/K)$-module. Then ...
Fraz's user avatar
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Why discrete modules?

The profinite group cohomology of discrete modules can be defined by right derived functors. Its application includes Galois cohomology, Brauer groups etc. These facts demonstrate that defining ...
Ja_1941's user avatar
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1 vote
1 answer
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In definition of Galois cohomology, is continuity necessary?

Let $K$ be a number field. Galois cohomology $H^1(Gal(\overline{K}/K), E)$ is composed of representative of continuous maps $Gal(\overline{K}/K) \to E$ which satisfies cocycle relation. If we discard ...
Pont's user avatar
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Invariants of Iwasawa Modules

Let $\Lambda$ denote Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$(ring of $p$-adic integers) . The structure theorem of the Iwasawa module tells; If ...
math's user avatar
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Local invariant map in the case of closed points of a curve

In class field theory, we have the well-known local invariant map $\mathrm{inv}_v: \mathrm{Br}(k_v) \rightarrow \mathbb{Q}/\mathbb{Z}$, where $k$ is a field and $v$ is a place of $k$. Similarly, we ...
oleout's user avatar
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isomorphism of quaternion algebras

I has a question reading 'Galois cohomology-Gille-Szamuely' I has a problem as follows. Let us define a k-algebra homomorphism $\varphi:(a,b) \to (u^2a,v^2b) $ which assigns $ui$ to $i$ and $vj$ to $...
Yong's user avatar
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1 answer
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Discrete G-module

Let $G_{\mathbb{Q}_\infty}$ be the absolute Galois group of $\mathbb{Q}_\infty$, where $\mathbb{Q}_\infty$ is the cyclotomic $\mathbb{Z}_p$- extension of $\mathbb{Q}$ and let $A$ be a discrete $G_{\...
math's user avatar
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1 answer
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Is $\mathbb{Q}$ an injective $\mathbb{Z}[G]$-module

Here $G$ is a finite group. I can show that $\mathbb{Q}$ (as a trivial $G$-module) is cohomologically trivial, since the inclusion $\mathbb{Q} \to \mathbb{Q}[G]$, $a\mapsto a(\sum_{g\in G} g)$ splits (...
J.Li's user avatar
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Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ Let $H$ be a subgroup of index $n$ of a finite group $G$, and let $M$ be a $G$-module, that is, an abelian group on which $G$ acts. ...
Mikhail Borovoi's user avatar
5 votes
2 answers
178 views

Real forms of $\mathbb{G}_m$ as a variety

Notation: Denote $k=\mathbb{R}$, $K=\mathbb{C}$ and let $G:=\text{Gal}(K/k)=\{\text{id},\sigma\}$, where $\sigma$ is complex conjugation. Background: In the Notices article A Gentle Introduction to ...
Thomas Preu's user avatar
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1 vote
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Splitting of exact sequence of G-modules

Suppose we have an exact sequence of $G$(finite)-modules; $$0\to A=B^G \to B \to C \to 0$$ where $A,B$ and $C$ are $p$-primary groups. If p does not divide the order of $G$, can I say that the above ...
math's user avatar
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1 answer
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Galois cohomology of product

By Hilbert 90 theorem we know that $H^1(Gal(L/K),L^\times)=\{1\}$ for any Galois extension $L/K$. Do we have a formula for $H^1(Gal(L/K),(L^\times)^n)$? Especially when the fields are extensions of $\...
Desunkid's user avatar
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Can a point on a curve always be mapped to another point with trivial residue map?

The following construction is taken from the book Central simple algebras and Galois cohomology by Gille and Szamuely in Chapter 6.4: Let $C$ be a smooth projective curve over a perfect field $k$ with ...
oleout's user avatar
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Composition series calculation for C2xC4-modules

Let $G=C_{2} \times C_{4}$ with $a=(1,0)$ and $b=(0,1)$. Let $1$ and $a$ denote the cosets modulo the normal subgroup generated by $b$. Similarly, let $1,b,b^{2},b^{3}$ be the cosets modulo the ...
Steppewolf's user avatar
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Multiplications on the group cohomology

Let ${\cal A}$ and ${\cal B}$ be abelian categories. Let $(F^n, \delta_F)$, $(G^n, \delta_G)$ be $\delta$-functors from ${\cal A}$ to ${\cal B}$. A morphism of $\delta$-functors from $F$ to $G$ is a ...
Pierre MATSUMI's user avatar
3 votes
1 answer
135 views

Equivalence of Hasse and Artin reciprocity laws

I learnt Hasse and Artin reciprocity laws when I was learning class field theory. Recently, I was looking for some facts about simple algebras written in Weil’s famous text Basic Number Theory. And I ...
youknowwho's user avatar
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2 votes
0 answers
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Applications of the Tate-Shafarevich group in class field theory

I'm studyng a course on class field theory and Galois cohomology this semester, and my lecturer said in one of the lectures that the Tata-Shafarevich group is indeed important is class field theory, ...
Or Shahar's user avatar
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1 vote
1 answer
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Künneth theorem for profinite cohomology.

Let $G,H$ be groups and $k$ a field, there is a well known formula for the group cohomology of the product: $$H^\ast(G\times H,k)\cong H^\ast(G,k)\otimes H^\ast(G,k)$$ I was wondering whether this is ...
Ben's user avatar
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