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Results tagged with group-cohomology
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user 4149
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
19
votes
3
answers
2k
views
Second nonabelian group cohomology: cocycles vs. gerbes
In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title.
In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in G …
9
votes
1
answer
369
views
For which subgroups the transfer map kills a given element of a group?
$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\sub …
8
votes
Accepted
Group cohomology question, trivial Galois action on discrete Galois module means we can say ...
If the $G_K$-action on $M$ is trivial, then
$$H^1(K,M)=\mathrm{Hom}(G_K,M),$$
and by Chebotarev's density theorem
$$ F^1(K,M)=0.$$
For details see Lemma 1.1(i) of Sansuc, J.-J. Groupe de Brauer et ari …
7
votes
1
answer
440
views
Imperfect Tate (cup product) pairing in Galois cohomology?
Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$
and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$.
Let $M$ be a finite $\Gamma$-module, that is, a finite abelian g …
7
votes
0
answers
303
views
Albrecht Fröhlich's text `Groupoids, groupoid spaces and cohomology' (1965)
I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). In this text Fröhlich defines cohomology of a group with coefficients in a groupoid (this was …
6
votes
1
answer
143
views
Finite $\Gamma$-modules with trivial $H^2$, where $\Gamma$ is a group of order 2
Let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a finite $\Gamma$-module,
that is, a finite abelian group on which $\Gamma$ acts.
It is a hopeless problem to classify finite $\Gamma$-modul …
6
votes
Accepted
Non-abelian Ext functor and non-abelian $H^2$
EDITED, taking into account the comments of Donu Arapura.
As JLA wrote, a homomorphism $f\colon G\to N$ gives an extension
\begin{equation}\label{e:E}
1\to K\to E\to G\to 1.\tag{E}
\end{equation}
This …
6
votes
Accepted
The second Tate-Shafarevich group of a permutation module is trivial
We write $G_w={\rm Gal}(L_w/K_v)$.
Definition. For $n\ge 1$, we denote
$$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$
where $C$ runs over the cyclic subgroups of $G$.
Remark. $Ш^2(L …
6
votes
2
answers
264
views
Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any c …
6
votes
Hilbert's Satz 90 for real simply-connected groups?
In addition to the answer of Gro-Tsen: Let $G=\operatorname{SU}(3)$. Then $H^1({\mathbb R},G)$ classifies matrices of Hermitian forms in 3 variables with determinant 1. There are two equivalence class …
6
votes
0
answers
483
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
$ …
6
votes
1
answer
176
views
Restriction vs. multiplication by $n$ in Tate cohomology
$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and …
5
votes
1
answer
274
views
What is the meaning of this coboundary homomorphism for group hypercohomology?
$\require{AMScd}$
Let $\Gamma=\{1,\gamma\}$ be a group of order 2.
In my problem from Galois cohomology of real reductive groups I came to a commutative diagram of $\Gamma$-modules
(abelian groups wit …
5
votes
1
answer
613
views
Non-vanishing of the Tate-Shafarevich kernel in group cohomology
Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$).
We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional repr …
5
votes
Second nonabelian group cohomology: cocycles vs. gerbes
Nonabelian $H^2$ in Galois cohomology can be defined in terms of: (1) cocycles, (2) extensions, (3) gerbes. The relations between these three definitions are described in Section 2.2 of Le principe de …