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).
1,114 questions
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Relationship between étale cohomology and group cohomology of the étale fundamental group
I am wondering to what extent and under which conditions there exists a generalization of the well-known relationship between the cohomology of the separable Galois group of a field and the étale ...
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Action on the second group of cohomology
Let $G$ be a finite group and let $M$ be a G-module. So we are given an action
$$G\times M\to M$$ by $(g,m)\mapsto g.m$.
Let $H^2(G,M)$ be the second cohomolgy group. Define the following action of $G$...
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Second cohomology group $H^2(G,A)$ and normalized $2$-cocycles
Let $G$ be a group and $A$ be a $G$-module.
A $1$-coboundary $f_2^B\in B^2(G,A)$ satisfies $f_2^B(g,h)=d_1(f_1^C)(g,h)=g\cdot f_1^C(h)-f_1^C(gh)+f_1^C(g)$ for some $1$-cochain $f_1^C\in C^1(G,A)$.
A $...
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Finite group with $H^3(G,U(1))$ trivial
Let $ G $ be a finite group. Today I learned that, for a certain category made from $ G $, the different associator structures making it a tensor category correspond to 3-cocycles, and the number of ...
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On the kernel of the maximal pro-$p$ quotient of a profinite group
Let $p$ be a prime, let $G$ be a profinite group and $G(p)$ its maximal pro-$p$ quotient. Let $H:=\ker(G\to G(p))$.
Is it true that $H$ must be a pro-$p'$ group (i.e. the order of any finite quotient ...
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Image of the transgression map and Hochschild-Serre sequence
Let $1\longrightarrow M\longrightarrow H\longrightarrow G\longrightarrow 1$ be a short exact sequence of finite groups, where $M$ is cyclic normal subgroup of $H$. Let $\mathbb{F}$ be a field of ...
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On the zero-th cohomology group of finite Galois module
Let $k$ be a field of characteristic zero. Let $G_{k}=Gal(\overline{k}/k)$ denote the absolute Galois group of $k$ where $\overline{k}$ denotes the algebraic closure of $k$, and let $M=\mathbb{Z}/p^{i}...
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$\mathbb{Z}G\otimes_{\mathbb{Z}H}A=\bigoplus_{j=1}^mx_j\otimes A$ where $G=\bigsqcup_{j=1}^mx_jH$
Let $H$ be a finite index subgroup of $G$ with coset decomposition $G=\bigsqcup_{j=1}^mx_jH$. Let $A$ be a left $\mathbb{Z}H$-module. Then since $\mathbb{Z}G$ is a $(\mathbb{Z}G,\mathbb{Z}H)$-bimodule,...
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Vanishing of First Cohomology for Twisted Tensors
I have been attempting (so far without success) to prove the vanishing of the following cohomology group:
$\
H^{1}(\mathbb{P}^{4}, \Omega_{\mathbb{P}^{4}}^{1} \otimes \Omega_{\mathbb{P}^{4}}^{2}(3 - d)...
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Cohomological description of homomorphisms of group extensions' groups
Let $B$ be a finite abelian group, and $G$ be a finite group. Let $G_1, G_2$ be two central group extensions of $G$ by $B$ with associated cohomology classes $\lambda_1,\lambda_2\in H^2(G,B)$.
We can ...
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Cohomology of a cyclic group: dependence on a generator
Let $G$ be a finite cyclic group of order $n$.
Choose a generator $\sigma$ of $G$.
Let $A$ be a $G$-module.
We have an isomorphism
$$\phi_\sigma\colon H^2(G,A)\overset\sim\longrightarrow A^G/N(A),$$
...
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Shapiro's lemma for a finite cyclic group: the inverse map
Let $G$ be a finite cyclic group, and let $W$ be a left $G$-set on which $G$ acts transitively.
Let $M$ be a left $G$-module (an abelian group on which $G$ acts).
We consider the $G$-module
$$ M[W]=M\...
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How does one calculate rational cohomology of a group using proper actions?
Let $G$ be a group such that $G$ acts cocompactly on a contractible CW-complex $X$ such that the stabilizer of each cell is finite. I would like to know how/if one can use the chain complex $C_n(X)$ ...
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Example of $\mathbb{Z}[S_3]$-module with "highly" non-trivial first cohomology
I've got a question concerning the cohomology of $S_3$, the symmetric group on three letters. According to my calculations, its first cohomology group with coefficients in arbitrary $\mathbb{Z}[S_3]$-...
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Does the spectral sequence stabilize on second page if vertical differentials in a double complex are zero?
In Parker: Higher extensions between modules for $SL_2$ (link), the author constructs some version of LHS spectral sequence. He has an interesting corollary.
Corollary 2.3: Suppose the $d_0$ are all ...
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Understanding Nakaoka's Theorem for Group Cohomology
I have been trying to understand the statement and proof of Nakaoka's Theorem on cohomology of wreath products of groups. I believe I should understand the statement before posting the question ...
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Verify my proof of Exactness of $\operatorname{Ind}_H^G$
In case it matters, I'm studying this for group cohomology reasons, though obviously it's also just representation theory. $\newcommand{\Ind}{\operatorname{Ind}} \newcommand{\Mod}{\operatorname{Mod}} \...
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Cohomology classes in $ H^2(G, K^\ast)$
Let $G = \mathbb{Z}_p \times \mathbb{Z}_p $, where p is an odd prime. Consider a field $K$ with the multiplicative group $K^\ast$ and let $ r \in K^\ast $ be a primitive root of unity in $K^\ast $. ...
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2-cocycle with no trivial Galois action and roots of unity
Let $n$ be a positive integer and $\zeta_n$ a $n$-th primitive root of unity.
$\Gamma=Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ acts on $\mathbb{Q}(\zeta_n)$ via Galois automorphisms. Observe that $|\Gamma|...
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2-cocycle into Abelian group trivialized by homomorphism into complex numbers
Let $G$ be a group and $A$ an Abelian group. Let $[\omega] \in H^2(G,A)$ be a cohomology class represented by a 2-cocycle $\omega:G \times G \to A$. Suppose that we are given that for any homomorphism ...
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Does group cohomology with respect to different groups commute?
Let $G,K$ be two groups, and $A$ be a $G\times K$-module. There is an obvious isomorphism
$$H^0(G,H^0(K,A))\cong H^0(K,H^0(G,A)).$$
My question is whetherwhen this extends to higher degrees. I.e. when ...
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Top degree transfer homomorphism is surjective for finite index subgroups [closed]
Let $H$ be a finite index subgroup of a group $G$ whose cohomological dimension is an integer $m$, i.e., $\text{cd}(G)=m$. Let $M$ be an $\mathbb{Z}G$-module. How do I show that the transfer ...
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Why does Schur Multiplier equal to $H^3(G,\mathbb Z)$
By definition, the Schur Multiplier of a group $G$ is the second cohomology $H^2(G,\mathbb C^\times)$ with trivial action. But here it says this is equal to $H^3(G,\mathbb Z)$. Where does this ...
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Group cohomology with 2-adic coefficients
I am trying to compute the cohomology groups $H^i(C_2; \mathbb{Z}_2^\times)$, where the 2-adic units $\mathbb{Z}_2^\times$ have trivial $C_2$-action. I believe that my answer is correct, but I am ...
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Problem Constructing the Weil Group
I have been trying to construct the Weil Group for an arbitrary class formation $(G,C)$ where $G$ is profinite. We assume that $C$ has trivial universal norms (ie. for any open subgroup $U \subset G$ ...
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