Questions tagged [group-cohomology]
a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).
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questions
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transfer of rigidity between subgroups
Assume $R$ is a commutative ring, $G$ an arbitrary group and $G'$ an arbitrary subgroup of $G$. Let also $M$ be a rigid module over the group ring $RG$, where by rigid I mean $Ext^i_{RG}(M,M)=0$, for $...
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0answers
12 views
Conjugation action on group cohomology trivial if subgroup in the center
If we have $H$ a normal subgroup of $G$ it is known that the conjugation action induces an action of $G$ on the cohomology $H^*(H;A)$ for any $G$-module $A$. This action is considered for the ...
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Cohomology group $H^{i}(G;\mathbb{Z})$ where $G= (\mathbb{Z}_{p_1})^{n_1} \times \cdots \times (\mathbb{Z}_{p_k})^{n_k} $. [closed]
How to calculate the cohomology group
$H^{i}(G;\mathbb{Z})$
where $G= (\mathbb{Z}_{p_1})^{n_1} \times \cdots \times (\mathbb{Z}_{p_k})^{n_k} $, and $p_1, \cdots , p_k$ are prime numbers and $n_1, \...
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25 views
The Finiteness of H^n(H, Z/pZ) implies finiteness of H^n(U, Z/pZ) for any open subgroup U of H
On Page 175 of J.Neukirch et al.'s book Cohomology of Number Fields, it was remarked:
If $H$ is a pro-$p$-group, then this is already true if $H^n(H, \mathbb{Z}/p\mathbb{Z})$ is finite ($n = cd_{p}H$)...
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27 views
Trivial group cohomology implies trivial profinite cohomology
Let G be a profinite group, M - discrete G-module. Suppose $H^1(G,M)=0$ as abstract group cohomology. I think it is always the case that $H_p^1(G,M)=0$ where $H_p^1(G,M)=\varinjlim H^1(G/U,M^U)$ is ...
2
votes
1answer
44 views
Induced module and surjective morphism
I am trying to solve the following question:
Let $G$ be a finite group and $H$ a subgroup of $G$. Let $A$ be a $G$-module. Show that $\pi: I^{H}_{G}(A)\to A$ defined by $\pi(f)=\sum_{g\in G/H}g\cdot f(...
4
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1answer
59 views
On the definition of cohomological dimension
Let $G$ be a group and $R$ a commutative unital ring. We define the $R$-cohomological dimension of $G$ to be
$$cd_R(G) := \sup \{ n : H^n(G, M) \neq 0 \text{ for some } R[G]\text{-module } M \}.$$
I ...
3
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1answer
42 views
Cohomological dimension of a group via infinite direct product
Let $R$ be a ring and $G$ an infinite group. The $R$-cohomological dimension of $G$ can be defined as $cd_R(G) = \min \{ n \geq 0 : H^{n+1}(G, R[G]) = 0\}$, which implies that $H^i(G, M) = 0$ for all $...
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Group action on cohomology over a normal subgroup
I feel a slight difficulty to conceive the following definition:
Setting: $G$ any group and $N \unlhd G$ a normal subgroup, $A$ is a $G$-module. The maps $c_g : N \rightarrow N$ and $m_g : A \...
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76 views
Group cohomology of infinite cyclic group
I am trying to compute the group homologies $H^n(G,M)$ on a $G$-module $M$ where $G$ is the infinite cyclic group. I think I managed to get the case $n = 0$ and $n > 1$, but I'm unsure about the ...
5
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1answer
82 views
A question on exact sequences and group cohomology
Question:
If $0\longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0,$ is an exact sequence of $G$-modules, then show that $$0 \longrightarrow A^G \longrightarrow B^G \longrightarrow ...
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29 views
Groups with isomorphic group rings have the same homology
I am trying to do exercise 9.19 in Rotman's Intro to Homolgical Algebra. Admittedly, I have skipped much of the content leading up to it (which is probably causing my issues). Here's where I'm at:
Let ...
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1answer
44 views
Computing Homology group of three-dimensional manifolds [closed]
I'm preparing for an exam. I want to solve this past exam question:
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26 views
What does it mean that $ H^2(\mathbb{Z}_2 \times \mathbb{Z}_2, U(1) ) \simeq \mathbb{Z}_2 $?
I found in a physics paper this one, but I'm sure it's in other places (in fact, they point you here) that:
$$ H^2(\mathbb{Z}_2 \times \mathbb{Z}_2, U(1) ) \simeq \mathbb{Z}_2 $$
To avoid function $\...
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0answers
62 views
Short exact sequence to long exact sequence of cohomologies
Let $R, S$ be rings with unity and $F : {\mathcal{M}}_R \rightarrow {\mathcal{M}}_S$ be an additive functor. Let
$$
0 \rightarrow A \xrightarrow{\alpha} B \xrightarrow{\beta} C \rightarrow 0
$$
be a a ...
1
vote
1answer
46 views
group cohomology and singular cohomology for $K(G,1)$
For $K(G,1)$ space, I know that there is an isomorphism between group cohomology and singular cohomology. Is there an example for which this fact is useful (for example, it is hard to calculate group ...
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0answers
7 views
Edge morphism spectral sequence of a group split extension
I have the next question related with spectral sequences given the next group split extension
$1\rightarrow H\rightarrow G \rightarrow Q\rightarrow 1$
I have the next spectral sequence $E^{2}_{p, q}=...
1
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0answers
41 views
Pushout square of groups, and group cohomology
Suppose that we have (not necessarily injective) group homomorphisms $H \to G_1$ and $H \to G_2$, and we construct the pushout $G_1 \sqcup_H G_2$. Suppose we have a representation $V$ of the pushout ...
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0answers
37 views
Third homology group over and abelian group
I have this question about the third homology group $H_{3}(G,\mathbb{Z}
)$ when $G$ is an abelian group.
I know that there is an injective map of algebras (Kenneth Brown)
$\phi:\bigwedge^{3}(G)\...
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0answers
16 views
Coboundary relating different cocycles for the Weil representation
I am not sure if this is more appropriate for Stack Exchange or Math Overflow, but I think it has probably been known for a couple decades at least. Given a symplectic space $W$ defined over a local ...
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0answers
28 views
Co-invariants in homology of groups
Let $1 \mapsto L \mapsto G \mapsto \mathbb{Z}\mapsto 1$ be a short exact sequence such that $G$ is a group of type $FP_{\infty}(\mathbb{Q})$, $H_{1}(L;\mathbb{Q})$ is infinite dimensional and let $t\...
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0answers
46 views
Proof of Lyndon-Hochschild-Serre spectral
I am checking about the Lyndon-Hochschild-Serre spectral sequence, which given the next group extension
$1\rightarrow H\rightarrow G\rightarrow Q\rightarrow 1$ we have the next spectral sequence
$E_{p,...
1
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1answer
52 views
Boundary Map of Bar Resolution vs. Face Map of the Nerve of a Group
For a discrete group $G$ I have the following two definitions, which I think are correct:
The nerve of $G$ is $NG$, a simplicial set whose $n$-simplices are $G^n$ ($G^0$ being the trivial group $\{1\}...
2
votes
0answers
63 views
Group multiplications in $Spin(d)$ via $(a,q) \in (\mathbb{Z}/2, SO(d))$
We know that the short exact sequence $0 \to \mathbb{Z}/2\to Spin(d) \to SO(d) \to 1$.
Given the groups $A$ and $Q$, we require to have the additional data
the 2-cocycle $ f \in H^2(BQ,A)$ and the map ...
1
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0answers
30 views
Deriving the group cohomology of $D_{2n}$ from that of $D_2$
While looking for the group cohomology of dihedral groups, I came across this post, which gives the cohomology groups $H^{k}(D_{2n},\mathbb{Z})$. We can see that
\begin{align}
H^{k}(D_{2n},\mathbb{Z}) ...
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votes
1answer
36 views
$\mathbb Z$ dual of a projective resolution is exact
Suppose we have a projective resolution of $G$ modules (consider the standard resolution or example)
$$\cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_o \rightarrow \mathbb Z \rightarrow 0 $$
We ...
3
votes
1answer
80 views
Understanding cup products for Tate cohomology group
I am reading about 'Cup products' from the book 'Algebraic Number Theory' by Cassels and Frohlich. The following theorem appears on page 105.
Theorem. $G$ is a finite group then there exists one and ...
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27 views
Understanding lemma regarding cup products.
I am trying to understand the following lemma on page 176 from appendix titled 'Computations of cup products' from Serre's book Local fields.
Given $G$- modules $A,B$ and $a \in A^G $, let $f_a: \...
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votes
1answer
51 views
What is $\hat{H}^*(C_2 \times C_2, \mathbb{F}_2)$ as a ring?
I'm interested in computing the ring structure on the Tate cohomology $\hat{H}^*(C_2 \times C_2, \mathbb{F}_2)$. It's easy enough to compute the ring structure on nonnegative degrees, along with the ...
1
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1answer
35 views
Discrete topological modules for profinite groups
The first step to define cohomology of a profinite group $G$ is to consider discrete $G$-modules. These are abelian groups with the discrete topology and a continuous action $\psi \colon G \times M \...
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Equality involving cup products
I do not get the last line of the proof.
Specifically how $\delta (\overline \sigma) \smile (\delta ^{\vee})^{-1} (\chi)=f(\sigma -1)$.
I can see that $$\delta (\overline \sigma) \smile (\delta ^{\vee}...
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0answers
25 views
Commutativity of local artin/reciprocity map
I want to understand the following commutative diagram in class field theory (from https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.3.3):
Milne says this follows from the definition of Artin ...
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0answers
48 views
Functoriality of first homology of group
Hello I have the next doubt:
Let $A$ be a commutative ring and $T:=T(A)=\left\{\left(
\begin{array}{cc}
u & 0 \\
0 & u^{-1} \\
\end{array}
\right)
\;:\;u\in A^{\ast}\right\}$ ...
1
vote
1answer
41 views
Proving thm. 22.37 in “Modern Classical Homotopy theory” by Jeffery Strom
Here is the thm.
I want to prove it and I got a hint to use the universal coefficient theorem. I am confused about which statement of a universal coefficient theorem should I use and how. Here are ...
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votes
1answer
37 views
$Z\pi \cong Z\oplus Z$ and so $P \cong Z$ [closed]
If $\pi = Z$, then the augmentation ideal $P$ is projective and $0 \to P \to Z\pi \to Z \to 0$ is a projective resolution.
Here we know that $P$ has basis $\mathbb{Z}-$ ${0}$ Then how to show that it ...
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0answers
37 views
Understanding the map $F^\times \cap E^{\times n} \to H^1(G,\mu_n)$
Let $F$ be a field and $\zeta \in F$ be a primitive $n$-th root of unity. Also, let $E/F$ be a finite Galois extension with group $G$.
Now I would like to understand the map $f: F^\times \cap E^{\...
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0answers
34 views
Kummer Theory: Understanding the isomorphism $F^\times \cap E^{\times n}/F^{\times n} \to \operatorname{Hom}(G,\mu_n)$
Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$.
Now I am trying to understand the following section from Milne's Fields and Galois Theory (page 73):
The part "...
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votes
1answer
22 views
H$^1 (G,\mathbb{Z}) \cong$ Der$(G, \mathbb{Z})$/PDer $(G, \mathbb{Z})\cong$ Hom$(\mathbb{Z}, \mathbb{Z}) \neq {0}$.
$G \cong \mathbb{Z}$ is an infinite cyclic group.
Then H$^1 (G,\mathbb{Z}) \cong$ Der$(G, \mathbb{Z})$/PDer $(G, \mathbb{Z})\cong$ Hom$(\mathbb{Z}, \mathbb{Z}) \neq {0}$.
Is there something wrong with ...
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0answers
73 views
Understanding the proof on why the first cohomology group of a Galois extension is trivial
Currently, I am trying to understand the following theorem with its proof (taken from Milne's Fields and Galois Theory, page 70):
As you can see, I am questioning the well-definedness of the sum for ...
0
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1answer
24 views
an isomorphism: $H^{q-1}(G,C) \xrightarrow{\delta} H^q(G,A)$.
This is from P$570$ of Rotman's 'Intro to Homological Algebra'.
$A^* = Hom_\mathbb{Z}(\mathbb{Z}G, A)$ where A is a $G$-module and $G$ is a group. There is an imbedding $i: A\to Hom_\mathbb{Z}(\mathbb{...
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0answers
36 views
each $u\in (\mathbb{Z}G\otimes_\mathbb{Z}A)^S$ has a unique expression $\Sigma_x{_\in}{_G}n_xx$
This is Corollary $9.80$ from Rotman's 'Introduction to Homological Algebra'.
$G$ finite group and $S$ normal subgroup of $G$, $A$ is a $G$-module. Then
Hom$_\mathbb{Z}(\mathbb{Z}G, A)^S \cong$ Hom$_\...
1
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1answer
31 views
The group of cohomology $H^3(G,\mathbb{Z})$ is finite when $G$ is finite.
The group of cohomology $H^3(G,\mathbb{Z})$ is finite when G is finite.
I am not sure how this is finite. We use the definite as follows:
$H^n(G,K) = Ext_\mathbb{Z}^n$$_G (\mathbb{Z}, K)$ and we use ...
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0answers
39 views
Pairing Cohomology with Homology on pg.510 in “Modern classical homotopy theory”.
Here is the paragraph in the book:
My questions are:
1- I do not understand why if $\alpha \in \widetilde{H}_{n}(X; B),$ then it should be represented by the given map below. Could anyone explain ...
0
votes
1answer
55 views
Chain complex and cohomology
I am trying to understant some ddetails from physical article.
In article there are two statements, which I wanna to understand.
We have complex $$
\mathcal{C}_* = \mathbb{Z}[1]\oplus \mathbb{Z}[2]...
1
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2answers
122 views
Why there is no suspension axiom for homology ? and why there is no excision axiom for cohomology theory?
Here are the axioms of reduced cohomology theory as given to me in the lecture:
1- $\tilde{H}^n(-;G): J_{*} \rightarrow Ab_{*}$ is a contravariant functor.
2- $\tilde{H}^n(X;G) \cong \tilde{H}^{n+1}(\...
2
votes
1answer
81 views
Fundamental Group of Klein Bottle acts on $\mathbb{R}$
It is well know that the Fundamental Group of the Klein Bottle can be defined (up to isomorphism) as the group with two generator and one relation
$$BS(1,-1)=\langle a,b: bab^{-1}=a^{-1}\rangle $$
In ...
2
votes
0answers
35 views
Proving isomorphism $H^r_T(G,M)\rightarrow H_T^{r+2}(G,M) $ for cyclic group using cup products
Suppose that we have cyclic group $G$ and a $G$-module $M$. Let $ \chi $ be isomorphism $G \rightarrow \mathbb Z /n \mathbb Z $ sending generator $\sigma \in G$ to $1+n \mathbb Z $. Consider $$0 \...
0
votes
0answers
21 views
A counterexample for a Morris Hirsch Theorem
Mathematician Morris Hirsch wrote on 1977 the paper "Flat Manifolds and The Cohomology of Groups". The Theorem C of his paper establishes the following statement:
Theorem C: Let $G$ be a ...
2
votes
1answer
49 views
Understanding lemma regarding fundamental class
I am reading Class Field Theory from Milne's notes. I do not understand a couple of things from the following lemma about fundamental class: https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.2.7 ...
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votes
0answers
14 views
A Theorem of Hirsch for Cohomology of Groups and for Hoschild Cohomology
The mathematician Morris W. Hirsch wrote on 1977 the paper "Flat Manifolds and the Cohomology of Groups". The Theorem C establishes the folloging : Let $G$ a nilpotent group acting linearly ...
