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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6 views

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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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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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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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 ...
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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(...
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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 ...
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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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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 ...
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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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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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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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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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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 ...
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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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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}=...
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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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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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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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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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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,...
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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\}...
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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 ...
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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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$\mathbb Z$ 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 ...
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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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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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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 ...
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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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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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Functoriality of first homology of group

Hello I have the next doubt: Let $A$ be a commutative ring and $T:=T(A)=\left\{\left( \begin{array}{cc} u & 0 \\ 0 & u^{-1} \\ \end{array} \right) \;:\;u\in A^{\ast}\right\}$ ...
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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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$Z\pi \cong Z\oplus Z$ 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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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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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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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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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 ...
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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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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$_\...
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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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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 ...
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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]...
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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}(\...
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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 ...
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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 \...
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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 ...
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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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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 ...

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