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Questions tagged [galois-cohomology]

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

6
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1answer
147 views

Nontrivial Twists of a Vector Bundle

Let $k$ be a number field, and let $X$ be a projective $k$-variety. Let $\mathcal{V}$ be a vector bundle on $X$ that is defined over $k$. A vector bundle $\mathcal{V}'$ on $X$ that is defined over $k$ ...
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0answers
25 views

Surjectivity of multiplication by $n$ on elliptic curves

Is there an abelian variety $A$ over a field $k$, such that $A(k^{\rm sep})$ is not a divisible group? The motivation of my question is the following : if $L$ is any algebraically closed field, then $...
5
votes
1answer
59 views

Generalisation of Hilbert's 90 Theorem

Let $L/K$ be a finite Galois extension of fields with Galois group $G = Gal(L/K)$. According to the famous Hilbert's 90 we know that the first cohomology vanish: $$H^1(G, L^*)=\{1\}$$ My question is ...
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0answers
54 views

Certain Galois cohomology computation

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, \...
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0answers
47 views

Galois Action on Coherent Sheaves Exact Functor

Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined ...
4
votes
1answer
77 views

Brauer-Severi varieties as quotients of forms of $\text{GL}_2$

Let $L/F$ be a finite galois extension of fields, with galois group $\Gamma$. Let $X$ be a variety over $F$ such that $X_L \cong \mathbb{P}^1_L$ over $L$, corresponding to a cohomology class $\alpha \...
0
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1answer
27 views

Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
0
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2answers
59 views

Two variations of Hilbert's 90 theorem

Let $L/K$ be a finite Galois extension. I know that $H^1(Gal(L/K), L^{\times}) = 0$ and $H^r(Gal(L/K), (L,+)) = 0$ for all $r>0$. 1) Do we have $H^r(Gal(L/K), L^{\times}) = 0$ for all $r>1$ ? ...
2
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2answers
93 views

Hilbert's theorem 90 for $p$-adic topology

Let $L/K$ be algebraic extensions of $\Bbb Q_p$. Consider $L^{\times}$ with the $p$-adic topology. Is it true that the first continuous cohomology group $H^1_{cont}(\mathrm{Gal}(L/K), L^{\times})$ ...
1
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1answer
74 views

About the definition of l-adic Tate-twist

In the J. Tate's paper "Relations Between $K_2$ and Galois Cohomology" Let F any field $F^{\text{sep}}$ the separable closure of F $G_F=\text{Gal}(F^{\text{sep}}/F$) he defines the ($\mathbb{Z}_l,G_F$)...
2
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0answers
105 views

Vanishing of second Galois cohomology group

This most likely follows from a standard result but a lack of knowledge prevents me from seeing this. Let $K$ be a non Archimedean local field. Let $\Gamma$ be $\mathrm{Gal}(\bar{k}/k)$. Let $T$ be a ...
7
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1answer
167 views

Computing étale cohomology group $H^1( \text{Spec}(k), \mu_n)$ and $H^1( \text{Spec}(k), \underline{\Bbb{Z}/\mathord{n \Bbb{Z}}})$

I am starting to learn about étale cohomology and would like to compute a simple example. Let $k$ be a field with a fixed separable extension $k^s.$ I want to compute $H^1( \operatorname{Spec}(k), \...
3
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0answers
59 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 $\...
2
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1answer
168 views

What should one know before learning galois cohomology

I recently became interested in galois cohomology, but I don't know if I have enough math to learn about it or where to start learning it. What should I know before I start learning about Galois ...
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0answers
21 views

Galois extension of exponent $mp^r$ in characteristic $p$

Kummer theory treats Galois extensions of exponents that are not divisible by the characteristic. Artin-Schreier and Witt extend this theory for Galois extensions of exponents $p^r$ in characteristic $...
1
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2answers
145 views

Tate Cohomology of Squares

Let $L | \mathbb{Q}_p$ be a finite Galois extension with Galois group $G$. What is known about the group $H^{-1} (G, L^\times / (L^\times)^p)$? I'm particularly interested in the case of a non-...
3
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1answer
84 views

How can a point on an elliptic curve be considered a galois cohomology class?

For an elliptic curve $E$, I saw a paper once mention that you can interpret a rational point on $E$ as a degree $0$ Galois cohomology class. I'm familiar with group cohomology but not all that much ...
2
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1answer
97 views

Hilbert 90 and K-forms

Studying the proof of Hilbert's 90 theorem modern version, I went through this lemma:given a Galois finite extension $K \subset L$ and an $L$ algebra $A$,we define the $(A,K)$ forms as the $K$ ...
1
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1answer
59 views

Surjectivity of map of étale sheaves

Let $F\to G$ be a map of étale sheaves on a scheme $X$. Suppose that for any closed point $x\in X$, the map between stalks $$F_x\to G_x$$ is surjective. Note that I am not forming stalks at the ...
2
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2answers
66 views

First cohomology group of the $n$-torsion of an elliptic curve

Let $E$ be an elliptic curve defined over a number field $K$. Let $v$ be a finite place of $K$ not dividing $n$ such that $E$ has good reduction at $v$. This only excludes a finite set of places $\...
3
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1answer
70 views

$H^1(G, \mathbb{Z}/p \mathbb{Z})$ and linearly independent elements in open subgroups.

Let $G$ be a profinite group and $p$ a prime number, and consider the following condition on $G$: For every open normal subgroup $U$ of $G$ and any integer $N \geq 0,$ there are $N$ elements $$z_1, \...
0
votes
1answer
61 views

Surjectivity of the inv map in Global class field theory

Let $L/K$ be a Galois extension of degree $n$. On page 195, in the book Algebraic Number Theory, in the Chapter on Global class field theory by Tate, he writes that the inv map $$H^2(Gal(L/K),J_L)\...
2
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1answer
29 views

Square classes of a real closed field

The group of square classes $k^*/k^{*2}$ $\simeq$ $H^1(k,\mu_2)$ for $k$ the real numbers, is well known to be $\mu_2$ with the non trivial element represented by $(-1)$. Question: Assume $k$ is ...
3
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1answer
63 views

Galois cohomology elements becoming trivial over $k(\mathrm{SB}(A))$

Assume we work over a field $k$ of characteristic zero. Let $A$ be a central simple algebra over $k$ of index $\mathrm{ind}(A) = 2$ and let $[A]$ denote its class in Br$(k)$. Let $\alpha \in$ $H^3(k,\...
3
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1answer
147 views

Hilbert's Theorem 90 for infinite extensions

I have proven Hilbert's Theorem 90 for finite extensions, that is for a finite Galois extension of fields $L/K$ with Galois group $G$, $H^{1}(G,L^{\times})=1$. I'm unsure as to how to proceed to the ...
3
votes
1answer
349 views

What is the Weil group of a global field $K$?

The question is in the title. For context, I know some things about the local Weil group. I know that the abelianization is isomorphic to the multiplicative group $K^{\times}$, and I know that it is ...
3
votes
1answer
166 views

$p$-primary component of the $G$-module “commutes” with cohomology

This question is related to a claim found in J-P. Serre's Galois Cohomology. Let $G$ be a profinite group (or finite if you like) and $A$ be a torsion discrete $G$-module. Serre says one can easily ...
0
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1answer
83 views

Galois cohomology for quadratic extensions

Suppose that $L$ is a quadratic extension of $\mathbb{Q}$. Since $L/\mathbb{Q}$ is a Galois extension, a general form of Hilbert Theorem 90 implies that for every $m\geq 1$ $$ H^{1}(\operatorname{Gal}(...
1
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1answer
128 views

3rd Galois cohomology of $n$th root of unity vanishes

I'm reading Milne's class field theory, and there is a statement about a group cohomology which I cannot understand. Let $K$ be a field and $K^{sep}$ be a separable algebraic closure of $K$, and let $...
1
vote
1answer
136 views

What does the pushforward of an etale sheaf over a field correspond to in terms of Galois cohomology?

Let $k$ be a field with separable closure $\overline k$ and absolute Galois group $G$. We know that sheaves of abelian groups on the etale site of $k$ correspond to abelian groups with a $G$ action (...
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0answers
64 views

Unramified Galois cohomology of p-adic fields

I am reading Serre's Galois Cohomology and am stuck trying to understand a statement made on page 95, II.5.5 Proposition 18. Let $k$ be a $p$-adic field and $A$ be a finite unramified $G_k$-module. ...
1
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2answers
84 views

Galois cohomology group equal the ground field

In our lecture on Algebraic Number Theory, we were given the following task: Let $L/K$ be a Galois extension with Galois group $G$. Show that $$H^0 (G,L)=K.$$ and $$H^n(G,L)=0 ,$$ for $n>0$. I ...
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0answers
188 views

Why is this cubic polynomial generic for cyclic field extensions?

[EDIT: There doesn't seem to be any interest in answering this question, so could anyone just provide me a reference for understanding (2), and if possible (1)? Hopefully that would be enough to help ...
2
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0answers
51 views

Galois extension over (commutative) rings

I was reading a paper by Childs, Garfinkel and Orzech and at some point they mentioned Galois extensions over commutative rings. I have never heard of this so I decided to look at one of the ...
1
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1answer
42 views

Realising finite abelian groups as Brauer groups of a field

Which finite abelian groups appear as Brauer groups of a field? Given a finite abelian group $G$, what are the (easiest) examples of fields with Brauer group equal to $G$?
2
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0answers
61 views

Field with trivial Brauer group that is not of dimension $\leq 1$

In Serre's book Galois cohomology he describes an example of a field with trivial Brauer group that is not of dimension $\leq 1$, as follows: Exercise II.3.1.1. Let $k_0$ be a field of ...
5
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0answers
58 views

Interpretation of the second continous group cohomology

In regular group cohomology theory, it is well known that $\mathrm{H^2(G,A)} \cong \mathrm{Ext(G,A)}$, where $\mathrm{Ext(G,A)}$ denotes the class of all groups $\mathrm{U}$ which make the following ...
9
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1answer
266 views

$\sqrt{2} \notin \mathbb{Q}$ and Galois Cohomology

If I look up infinite descent on Wikipedia we get the sample of proving that $\sqrt{2} \notin \mathbb{Q}$ -- it is irrational. $$\sqrt{2} = \frac{a}{b} \to 2 = \frac{a^2}{b^2} \to 2b^2 = a^2 \to 2b^2 ...
2
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1answer
83 views

What is the usual norm argument?

I'm currently reading this paper (PDF page 5/page 111) and the following WLOG statement has me stumped: Main Lemma: Let $K/\mathbb{Q}$ be a field extension of transcendence degree $1$ and $\ell\neq ...
3
votes
1answer
217 views

Unramified Galois representation and cohomology

I wish to understand a remark made by Rubin in his book "Euler Systems" (just after definition 5.1) about the first cohomology group of an unramified Galois representation. Let $K$ be a number field, ...
2
votes
1answer
46 views

Proof of $H^1(G,\operatorname{GL}_n(K))=0$

I have a confusion about the above proof. In the last paragraph, it says let $x_1 \ldots x_n$ such that $y_i=b(x_i)$ are linearly independent over K. How do we know such a $x_1 \ldots x_n$ exists? I ...
2
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0answers
190 views

Connecting homomorphism between cohomology groups

I am trying to prove that the connecting homomorphism between $\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ is the multiplication with $n$. From the exact sequence $0\rightarrow\...
3
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0answers
80 views

Cohomology groups of unramified field extensions

I have some problems concerning the chapter about "The Class Formation of Unramified Extensions" in Neukirch's "Class Field Theory". $K$ is a $\mathfrak{p}$-adic number field with finite residue ...
2
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0answers
115 views

Surjectivity of restriction map between $H^2$

Let $p$ be a prime and $G$ and $H$ be two profinite groups such that $H$ is open distinguished with $G/H$ cyclique of some order $m \equiv 0 \text{ (mod }p \text{)}$. Do we know some conditions on $G$ ...
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0answers
67 views

Edge morphisms coincide cup-products in the Tate spectral sequence

In the Tate spectral sequence, the edge morphism coincides with the cup product. The proof is written in Neukirch-Schmidt-Wingberg's book: Cohomology of Number Fields (Theorem 2.5.5,p125). https://www....
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0answers
106 views

Two basic conceptions of Galois cohomology.

$\quad$ I started learning $\textbf{Galois Cohomology}$ just 2 days ago and I am now confused with several conceptions. For example, there is a corollary written below. $\quad$ $\textbf{Corollary}$: ...
0
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1answer
74 views

multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
0
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1answer
121 views

Proof of finiteness of Selmer groups in Silverman's Arithmetic of Elliptic Curves

I'm trouble in understanding the proof of Lemma X.4.3 in Silverman's Arithmetic of Elliptic Curves (2nd edition), that claims $H^1(G_{\bar{K}/K}, M; S)$ is finite. In page 334, the book state the map $...
1
vote
1answer
48 views

Cyclic algebras of degree $4$ and period $2$

Recall that if a field $k$ has a primitive $n$-root of unity $\omega$, then the cyclic $k$-alegbras of degree $n$ (ie of dimension $n^2$) have the following familiar presentation : they are generated ...
1
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1answer
29 views

Problem defining morphism in Galois cohomology of algebraic group

Let $K$ be a field and $G$ an algebraic group defined over $K$. If $M\supseteq L$ are two finite Galois extensions of $K$, then the groups $\text{Gal}(M/K)$ and $\text{Gal}(L/K)$ act, respectively, on ...