All Questions
Tagged with class-field-theory galois-cohomology
15 questions
4
votes
0
answers
166
views
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
3
votes
1
answer
369
views
Local Tate duality for F-vector space
A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...
- 14.2k
3
votes
0
answers
117
views
Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients
I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...
- 12.9k
5
votes
0
answers
213
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 ...
- 709
47
votes
1
answer
3k
views
A three-line proof of global class field theory?
There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern ...
- 1,347
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
$$...
- 14.2k
7
votes
1
answer
343
views
Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...
- 14.2k
4
votes
1
answer
334
views
A Kummer exact sequence involving $\mu_\infty$
Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...
- 16.5k
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 ...
- 18.4k
1
vote
1
answer
138
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 ...
- 31
3
votes
0
answers
107
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 $\...
- 1,283
6
votes
1
answer
407
views
Unramified non-abelian extension and Galois cohomology
Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class ...
- 13.6k
7
votes
0
answers
470
views
Explicit $H^2(K, \mu) = Q/Z$?
In the development of local class field theory, a very fundamental theorem is that, for every local field $K$ of characteristic zero,
$H^2(K, \mu) \cong \mathbb{Q}/\mathbb{Z}$. $(*)$
Neukirch et al. ...
- 980
3
votes
2
answers
651
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?
- 620