# Tagged Questions

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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### The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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### Invariant cohomology for non-compact groups

Suppose I have a compact $G$-space $M$, and a differential form $\omega$ on $M$ with the property that $$\forall g\in G\quad g\omega = \omega + d\lambda_g, \quad(*)$$ i.e. $g\omega$ is cohomologous ...
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### Group cohomology classical exercise: Short exact sequence induces long exact sequence

This is probably so easy that I didn't find any other questions asking this exact question. Suppose that $1\to A\to B\to C\to 1$ is an exact sequence of $G$-modules. I can then easily prove that ...
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### Direct limit of a direct system looking like a cochain complex of objects.

I would like to ask you about a special kind of direct systems $(A_i, f_{i}^{j} )_{ i,j \in ( I , \leq ) }$ looking like a cochaîn complex $(A_i , f_{i}^{j} )_{ i,j \in ( \mathbb{N}^* , \leq ) }$ ...
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### Question on subgroup cohomology restricting proper, simplicial actions of an algebraic group

I have a question regarding an assertion made in p. 2 of these notes on Bruhat-Tits buildings. The question concerns the group $G_p=SL_n(\mathbb{Q}_p)$ and its subgroup $\mathbb{Z}^{n-1}$ (the ...
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### constructing a universal cover from K(G,1) space

Let G be a torsion-free group and G' a group of finite index. Suppose G' has finite cohomological dimension. Then it has a finite-dimensional K(G',1) complex. Its universal cover X' is also ...
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### If $G$ is cyclic of order $2$ show $|H^1(G,\mathbb{Z})|=2$.

Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has ...
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### Homomorphisms from a $p$-group to $\mathbb{F}_p$

I'm doing a problem on group cohomology and have reduced it to the following: if $P$ is a $p$-group then $\textrm{Hom}(P,\mathbb{F}_p) \simeq P/\Phi(P)$ where $\Phi(P)$ is the Frattini subgroup of ...
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### Doubt about the module of coinvariants

Suppose that $G$ is group, $R$ is a ring (commutative, associative with $1\neq0$) and $M$ is a left $RG-$módulo such that $M$ is free as a left $R-$module. Is it true that the module of coinvariants ...
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### Group cohomology for $\mathbb{Z}[G]$-modules versus $k[G]$-modules.

I am trying to get familiar with group (Tate) cohomology. I am for instance reading Brown's Cohomology of Groups. Now something seems unclear to me What information do we hope to attain from studying ...
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### Coefficients in Tate cohomology, mod-p Tate cohomology vs integral Tate cohomology

I am new to group cohomology and Tate cohomology. I have some questions in that regard. I have not yet understood exactly what information we hope to gain from the (Tate) cohomology modules. i) What ...
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### Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...
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### Does H2 depends only on abelian quotient?

Consider a finite group $G$ and an abelian group $N$. Let $G$ act trivially on $N$. Is $H^{2}(G,N)\cong H^{2}(G^{ab},N)$? ($G^{ab}=G/[G,G]$ the abelianization of $G$) I don't get group cohomology so ...
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### Group Homology with Coefficients in Chain Complex of trivial G-modules

I am reading through Kenneth Brown's Cohomology of Groups, and right now I am learning about spectral sequences. He seems use to some kind of "intuitive" idea of spectral sequences that I would very ...
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### Existence of class modules for finite groups

I'm reading Cohomology of Number Fields by Neukirch et al. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$: 1) $H^1(H,C)=0$ 2) $H^2(H,C)$ is cyclic of ...
The following proof of the inflation-restriction exact sequence is taken from Milne's notes on class field theory. My question is: why does $\phi':G/H \to M$ actually take values in $M^H$? In other ...