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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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965).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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 $ …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar
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 …
Mikhail Borovoi's user avatar

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