All Questions
Tagged with class-field-theory group-cohomology
12 questions
4
votes
0
answers
166
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Problem understanding the cup-products for the modified cohomology in David Harari's book "Galois Cohomology and Class Field Theory"
I'm recently reading the very beginning of David Harari's book Galois Cohomology and Class Field Theory, and I met a problem with the definition of cup-products for the modified cohomology.
Before ...
- 525
3
votes
0
answers
141
views
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)...
- 41
18
votes
4
answers
2k
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What's the use of group cohomology for class field theory?
I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...
- 521
0
votes
1
answer
454
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Coboundary operators, 1-cocycles and computing cohomology
My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago on MathSE
and I set a bounty on it, but I didn't ...
6
votes
0
answers
496
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
$$...
- 14.2k
4
votes
0
answers
305
views
Explicit invariant map in local class field theory
Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...
- 953
3
votes
0
answers
187
views
Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
0
votes
0
answers
124
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When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
- 1,215
12
votes
0
answers
272
views
sequences in non-abelian group cohomology
In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
- 273
7
votes
4
answers
608
views
Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$?
I am reading Class field theory - Bonn Lectures by Neukirch.
Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules ...
user37663
21
votes
3
answers
1k
views
The Teichmüller's algebraic interpretation of $H^3$ in group cohomology
In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an ...
74
votes
10
answers
18k
views
Intuition for Group Cohomology
I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...
- 15.4k