2
votes
1answer
52 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
3
votes
1answer
70 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
1
vote
1answer
32 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
0
votes
0answers
31 views

Uniqueness of the cohomological functor

This question is from the chapter 'Cohomology of Groups' by Atiyah and Wall in Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group. Theorem 1 on page 95 says that there is a ...
1
vote
1answer
29 views

A G-isomorphism of certain Hom groups

This question is from 'Cohomology of Groups' by Atiyah and Wall, p.95 of Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group and $A={\rm Hom}_{\mathbb Z}(\mathbb Z[G],X)$ where ...
5
votes
0answers
79 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
1
vote
2answers
54 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
4
votes
1answer
55 views

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n ...
5
votes
0answers
53 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
1
vote
1answer
58 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
7
votes
1answer
70 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
3
votes
1answer
78 views

Extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$

Given that $H^2(\mathbb{Z}_n,\mathbb{Z})=\mathbb{Z}_n$, it follows that up to equivalence there should be $n$ extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$, one for each cohomology class. I'd like to ...
7
votes
1answer
121 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
1
vote
0answers
53 views

A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
6
votes
5answers
179 views

(Elementary) applications of group (co-)homology

I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology. My background is, ...
4
votes
1answer
92 views

Order of the first cohomology group and subgroups

Let $M$ be a $G$-module and $H$ a subgroup of $G$. Is $\# H^{1}(H, M) < \# H^{1}(G, M)$?
3
votes
0answers
122 views

Cohomology of a tensor product

Let $k$ be a field of characteristic $p$ and $V$ be a $k^p$ vector space. Denote by $k_s$ the separable closure of $k$ and set $G_k := Gal(k_s|k)$. Prove that $$ H^0(G_k, V \otimes_{k^p} k_s^p) = V ...
1
vote
2answers
194 views

Conjugation action group cohomology

I have a question from Lang's Algebra (chapter twenty ex 6d). I think the vagueness confuses me as I am not even sure where to start If $H$ is a normal subgroup of $G$, we have the cohomology groups ...
2
votes
0answers
110 views

Question on $\operatorname{res}^G_U \circ \operatorname{cor}^U_G = N_{G/U}$

This is probably a very basic question, but I can't wrap my head around it. Given a normal open subgroup $U$ of a profinite group $G$ and a $G$-Module $A$ we have the following equation in group ...
4
votes
1answer
158 views

Isomorphism between group cohomology groups

Consider a profinite group $S$ acting trivially on $\mathbb{F}_p$. Choose $\chi \neq 0\in H^1(S, \mathbb{F}_p)$ and set $T = \ker(\chi)$. Let $X$ be the $S$-Module of all functions $S/T \rightarrow ...
3
votes
0answers
154 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
1
vote
2answers
165 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
2
votes
1answer
94 views

Other differentials for group cohomology other than the standard one.

In group cohomology, one defines $H^i(G;A)$ for $G$ a group and $A$ a $G$-module (an abelian group with a $G$-action) as the $i$-th right derived functor of the functor $$(-)^G: G-mod \rightarrow Ab, ...
6
votes
1answer
368 views

Group cohomology and Shapiro's lemma

This is a stupid question about group cohomology, but it confuses me a lot. Basically I think that the problem is the fact that I do not really understand Shapiro's lemma. Say we take a profinite ...
5
votes
2answers
115 views

$H^1(G,A)$ is killed by $|G|$ : Proof on the level of cocycles

Let $G$ be a finite group and $A$ a $G$-Module. It is well-known that $H^q(G,A)$ is killed by $|G|$ for all $q \geq 1$. This is usually proved using Restriction-Corestriction (applied with the trivial ...
2
votes
1answer
129 views

Group cohomology: $H^1$ of $\mathbb Z/4 \mathbb Z$

Consider the group $G = \mathbb{Z}/4\mathbb{Z}$ and its subgroup $H = 2\mathbb{Z}/4\mathbb{Z}$. Consider the obvious injection $\mathbb{Z} \to \mathbb{Z}[G]: 1 \mapsto N_G = \sum_{\sigma \in G} ...
6
votes
1answer
320 views

Torsion-free virtually-Z is Z

It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan... Is there a simple cohomological proof of the fact that a torsion-free group ...
7
votes
2answers
208 views

What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I've seen two "standard" complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis ...
9
votes
4answers
516 views

What do higher cohomologies mean concretely (in various cohomology theories)?

Superficially I think I understand the definitions of several cohomologies: (1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't ...
10
votes
1answer
298 views

Finite groups with periodic cohomology

I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
3
votes
0answers
192 views

group cohomology with coefficient in an induced module

We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second ...
15
votes
1answer
501 views

Group cohomology versus deRham cohomology with twisted coefficients

Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
2
votes
3answers
383 views

Acyclic vs Exact

I have a question about the words "acyclic" and "exact." Why does Brown use the term "acyclic" instead of "exact" in his book Cohomology of Groups? It seems to me that these two terms exactly ...