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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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Group action on cohomology is trivial

Let $G$ be a group and $H$ be a normal subgroup. Let $M$ be $H-$module. I want to check $G/H$ acts on $H^1(H,M)$ by $(\bar{\sigma}*X)(g):=\sigma X({\sigma}^{-1}g\sigma)$ where $X $ is cocycle in $H^1(...
Pont's user avatar
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Is every extension of by an Abelian Group isomorphic to a central extension?

Suppose $A$ is an abelian group and $$0\to A\to H\to G\to 0$$ is exact. Does it follow that that this SES is isomorphic to one of the form $$ 0\to A\to H'\to G\to 0$$ such that $A$ is contained in the ...
mattematician 's user avatar
3 votes
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Proof of $\text{Cor} \circ \text{Res}=[n]$ for group cohomology

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index $n$. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology ...
Pont's user avatar
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$G$-module structure of augmentation ideal

I'm given the following: Suppose $G$ is a group and let $I$ be the kernel of the homomorphism $$\mathbb{Z}[G]\to\mathbb{Z}\quad\sum_{g\in G}a_gg\mapsto\sum_{g\in G}a_g$$ (i) Show that $I$ is a free $\...
Seth's user avatar
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Extensions of $G$-modules parametrized by $H^1$

Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess we can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
Conjecture's user avatar
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The definition of action of $G/H$ on group cohomology $H^1(H,M)$

Let $G$ be a group and $H$ be a normal subgroup of $G$. Let $M$ be $G$-module. Let $H^1(H,M)$ be first group cohomology. $G$ acts on $H^1(H,M)$ by $(\sigma*f)(g):=gf({\sigma}^{-1}g{\sigma})$・・・①. My ...
Pont's user avatar
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6 votes
2 answers
187 views

Kernel of restriction and cokernel of corestriction of group cohomology

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology defined ...
Pont's user avatar
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2 votes
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Cohomology Ring Calculation Query

I am trying to give the ring structure on $H^*(\mathbb{Z} \oplus \mathbb{Z},\mathbb{Z})$ using group cohomology techniques. So far, I have found that: $$H^n(\mathbb{Z} \oplus \mathbb{Z},\mathbb{Z}) = \...
Todd Burnett's user avatar
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$\text{Res}\circ \text{Cores}$ in the context of group cohomology

Let $G$ be an abelian group and $H$ be a finite index($=n$) subgroup . Let $A$ be a $G$ module. Let $H^1(G,A)$ denote group Galois cohomology. Restriction map $\text{Res}: H^1(G,A)\to H^1(H,A)$ is ...
Pont's user avatar
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Groups acyclic with group ring coefficients

I am looking for examples of finitely generated groups for which $H^i(G,\mathbb{R}G)=0$ in all degrees, so a sort of acyclicity with coefficients in the group ring. Are there such things?
user10439561's user avatar
16 votes
1 answer
160 views

What is the "higher cohomology" version of the Eudoxus reals?

The "Eudoxus reals" are one way to construct $\mathbb{R}$ directly from the integers. A full account is given by Arthan; here is the short version: A function $f: \mathbb{Z} \to \mathbb{Z}$ ...
user263190's user avatar
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Group Structures of a Group Extension

Let $A,G$ be groups with $A$ abelian, and $0 \to A \xrightarrow i E \xrightarrow \pi G \to 0$ a group extension. We denote the induced action of $G$ on $A$ as $g \cdot a$. A paper that I am reading ...
Mafematician's user avatar
1 vote
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80 views

If $G$ is a simply connected Lie group and $H^2 (\mathfrak{g};\mathbb{R})=0$, then $H^2 (G;\mathbb{R})=0$?

Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. My question is that: If $G$ is a simply connected and $H^2 (\mathfrak{g};\mathbb{R})=0$, then $H^2 (G;\mathbb{R})=0$? My question ...
Mahtab's user avatar
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2 answers
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Lifting map from finite cyclic group to integers

Suppose $\phi:G \to \mathbb Z/n\mathbb Z$ is a group quotient with finite cyclic image. Under what conditions can $\phi$ be lifted to a homomorphism $\tilde \phi:G \to \mathbb Z$? Specifically I'm ...
Ethan Dlugie's user avatar
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1 vote
1 answer
68 views

Skew-symmetric non-degenerate bicharacters over abelian groups

I'm struggling to solve a problem in group cohomology theory which could seem immediate to some more expert mathematicians here. Suppose to have a non-degenerate, skew-symmetric bicharacter $b\colon G\...
skewfield's user avatar
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1 vote
1 answer
75 views

Long exact sequence of the exponential short exact sequence and elements of $H^1(X,\mathcal{O}^*_X)$

I am trying to understand how to think about the elements of $H^1(X,\mathcal{O}^*_X)$ I know one way is to think about it as the Picard group via the isomorphism. But when I try to "see" the ...
領域展開's user avatar
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An isomorphim by J. Lewis

Suppuose that $A:\mathbb{Z}^{p}\rightarrow\text{GL}(q,\mathbb{Z})$ is an action. Define the acction of $\mathbb{Z}^{p}$ on $\mathbb{R}^{p}$ and $\mathbb{T}^{q}$ by: $$\ell\cdot x=A(\ell)x$$ $$\ell\...
José Luis  Camarillo Nava's user avatar
2 votes
1 answer
96 views

$H^1(G_{K_v}, M^D) \cong \widehat{H^1(G_{K_v}, M)}$, Tate dual and Pontryagin dual

Let $K$ be a number field and $G_K$ the absolute Galois group of $K$. Let $M$ be a finite $G_K$-module. The Tate dual of $M$ is defined as follows: $M^D = \text{Hom}(M, \mu(K))$, where $\mu(K)$ is the ...
Pont's user avatar
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2 votes
0 answers
30 views

$0 \to E(K)/2E(K) \to H^1(G_K,E[2])\stackrel{\delta}{\to}H^1(G_{K_v},E)[2]\to 0$ and Pontryagin dual

Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$. The short exact sequence $0 \to E[2] \to E \to E \to 0$ induces a famous exact ...
Pont's user avatar
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0 votes
1 answer
33 views

How to use dimension shifting in group cohomology

Let $M$ be a discrete $G$-module and $M'$ another $G$-module defined by the exact sequence \begin{equation} 0 \to M \to \text{Ind}_G(M)\to M'\to 0\end{equation} where $\text{Ind}_G(M):=\text{Maps}(G,M)...
Rocket_Rabbit77's user avatar
2 votes
0 answers
28 views

Spectral sequence of cohomology of $\mathbb{Z}$-extension

Consider an extension of groups $$1 \to N \to G \to Q \to 1$$ where $Q \simeq \mathbb{Z}$ is infinite cyclic. Such an extension necessarily splits as a semidirect product $G \simeq N \rtimes Q$. My ...
Henrique Augusto Souza's user avatar
13 votes
1 answer
335 views

ch. 8.3 Exercise 1 in Cohomology of Number Fields

Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\...
Snacc's user avatar
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3 votes
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Is galois cohomology invariant under inner forms and not just pure inner forms?

Let $G, G'$ be smooth algebraic groups over $k$ (absolute Galois group $\Gamma$) which are etale inner forms of each other, that is, there exists an isomorphism $G_{k_s} \cong G'_{k_s}$ and the ...
C.D.'s user avatar
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What is the definition of an affine $G$-module?

Let $G$ be a Lie group, $T:G\to GL(\mathbb{V})$ a representation of $G$ in a vector space $\mathbb{V}$. A $\mathbb{V}$-valued one-cocycle is a (smooth) map $S:G\to V$ satisfying the property $S(fg)=T(...
Mahtab's user avatar
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1 vote
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54 views

Triviality of $H^2(\mathbb{R},U(1))$

A $2$-cocycle of $\mathbb{R}$ (valued in the unitary group $U(1)$) is a map $c:\mathbb{R}\times \mathbb{R}\to U(1)$ such that $ \ \ \ \ c(x+y,z)c(x,y)=c(x,y+z)c(y,z)$ $ \ \ \ \ c(x,0)=c(0,x)=1$ I'm ...
Kandinskij's user avatar
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3 votes
1 answer
91 views

Confusion about projective resolution: group cohomology

I am totally confused about a detail in computing group cohomology. Let $G$ be a group, $A$ a $\mathbb{Z}[G]$-module. $H^i(G, -)$ is the $i$th right derived functor of $\operatorname{Hom}_{\mathbb{Z}[...
Emory Sun's user avatar
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1 vote
1 answer
85 views

Source for 5 term exact sequence in group cohomology

I know (see for example page 47 in Brown) that a short exact sequence of groups $$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$$ gives rise to a 5 term exact sequence in their homology ...
Chase's user avatar
  • 176
1 vote
0 answers
24 views

Group cohomology with coefficients in a representation

Let $G$ be a finite group, and $V$ a (say complex) finite dimensional representation of $G$. Let me view $V$ as a $G$-module in the obvious way. Is it true that $$H^n(G;V)=0$$ for $n\geq 1$? I suspect ...
JeCl's user avatar
  • 501
1 vote
1 answer
82 views

Projective dimension bounded by cohomological dimension

The cohomological dimension of a group $G$, denoted $\mathrm{cd}(G)$, is defined to be the projective dimension (i.e. length of shortest projective resolution) of the trivial module $\mathbb{Z}\in\...
bosshoggoutlaw's user avatar
2 votes
1 answer
59 views

Second Cohomology group with a non-trivial action

Let $G$ be a finite group and let $H^2(G,\mathbb{C}^*)$ be the second cohomology group which corresponds to the trivial $G$ action. Then there are known methods for computing $H^2(G,\mathbb{C}^*)$ for ...
user17292519's user avatar
3 votes
1 answer
112 views

Confusion about cohomology groups $H^i(S_3,\mathbb{Z})$

Let $S_3$ be the third symmetric group and let it act trivially on $\mathbb{Z}$. Consider the following theorem on page $154$ of Brown Group Cohomology (9.1) Theorem The following conditions are ...
Snacc's user avatar
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3 votes
0 answers
58 views

Action of base space on homology of homotopy fiber, when the fiber is an Eilenberg Maclane space.

I am trying to understand a proof I found in a paper by Wagoner about delooping of algebraic K-theory (proposition 1.2 for those interested). For this I have a fibration $BE\to BG\to BG^{ab}$, with $E=...
DevVorb's user avatar
  • 1,407
2 votes
0 answers
32 views

Modular symbols, Manin symbols, two and three term relation

Maybe someone could help me out. I consider a $SL_2(\mathbb{Z})$-module $\Omega$. We set \begin{align} S:=\left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right) \ \mathrm{and} \ U:=\left(\...
Running_mathematics's user avatar
3 votes
0 answers
49 views

Inflation-restriction exact sequence in nonabelian group cohomology

Let $G$ be a group with normal subgroup $H$ and let $M$ be an abelian group on which $G$ acts. The associated inflation-restriction exact sequence is $$1 \to H^1(G/H, M^H) \to H^1(G, M) \to H^1(H, M)^{...
gimothytowers's user avatar
1 vote
3 answers
102 views

Show that $S^2$ is not diffeomorphic to $T^2$ [duplicate]

I know this question have already been answered many times. However I want to prove this using the De Rham cohomology of them, showing that they are not isomorphic and thus $S^2$ and $T^2$ are not ...
NiJuice's user avatar
  • 41
1 vote
1 answer
61 views

The first cohomology group of automorphism group

I want to know $H^1(K(\mu_p)/K,\mu_p)$ where $K$ is a number field. Since $K(\mu_p)/K$ is cyclic, $H^1(K(\mu_p)/K,\mu_p)=H^{-1}(K(\mu_p)/K,\mu_p)$. Using calculation, $H^{-1}(K(\mu_p)/K,\mu_p) =0$. So,...
WHERE 234's user avatar
  • 115
2 votes
0 answers
69 views

Recover $G$-representation from restriction and quotient [closed]

Let $G$ be a finite group with subgroup $H$. Suppose there is an unknown $\mathbb{Z}[G]$-module $M$, where we know $M_1 = \mathrm{res}^G_H M$ and $M_2 = M\otimes_{\mathbb{Z}[H]}\mathbb{Z}$. Is there ...
kmera11's user avatar
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1 vote
0 answers
41 views

Can the inflation map be seen as a map between two $\delta$-functors?

Let $G$ be a group and $H$ be a normal subgroup. For each $r \geq 0$, the inflation map from $H^r(G/H, M^H) \to H^r(G,M)$ is defined by the composition of two maps $H^r(G/H, M^H) \to H^r(G, M^H) \to H^...
Kyaw Shin Thant's user avatar
0 votes
1 answer
139 views

Group action of residue group on group cohomology

Let $G$ be a group and $M$ be $G$-module. Let $H^1(G,M)$ be first group cohomology. I heard $G/H$ acts on $H^1(H,M)$ naturally. What is the standard action ? I came upon an action * by $(\sigma*f)(g)=\...
Pont's user avatar
  • 6,051
0 votes
1 answer
79 views

First group cohomology $H^1(G,A)$ of cyclic group of order $2$

Let $G=〈\sigma〉$ be a cyclic group of order $2$ generated by $\sigma$. Let $A$ be $G$-module. Let $H^1(G,A)$ be a first group cohomology. I want to prove $H^1(G,A)\cong \text{ker}(\sigma +1)/(\sigma -...
Pont's user avatar
  • 6,051
2 votes
2 answers
179 views

$H_2(G,\mathbb{Q})$ for finitely presented group $G$

I am learning about group homology and spectral sequences and I have read somewhere that we can use the Lyndon-Serre-Hochschild spectral sequence to prove that $H_2(G,\mathbb{Q})$ has finite rank when ...
QGM's user avatar
  • 613
0 votes
1 answer
49 views

How Galois group act on $H^1(G_L,E)$?

Let $E/K$ be an elliptic curve over a field $K$. Then, how $\sigma \in \text{Gal}(L/K)$ acts on first Galois cohomology group $H^1(G_L,E) $ where $G_L$ denotes absolute Galois group of $L$? $H^1(G_L,E)...
Pont's user avatar
  • 6,051
0 votes
1 answer
81 views

Second group cohomology of cyclic groups

Let $M$ be a Abelian group and $G$ be cyclic group of order $2$. Let $M$ be a $G$-module. Suppose $G=\langle\sigma\rangle.$ Let $M^{G} =\{m\in M\mid \sigma m =m\}$. Define the norm map $N: M\...
Pont's user avatar
  • 6,051
0 votes
2 answers
63 views

When cocycle $G_K\to \mu_2, \sigma \mapsto \alpha^{\sigma}/\alpha$ is unramified at $v$?

Let $K$ be a number field. Let $G_K$ be absolute Galois group of $K$. Let $M_K$ be a set of all places of $K$. Fix an element $a\in K^{\times}/{K^{\times}}^2$. When cocycle $f: G_K\to \mu_2, \sigma \...
Pont's user avatar
  • 6,051
0 votes
1 answer
34 views

Galois fixed part of modules. Why does $|(1-\sigma)A||A^G|=|A|$ hold?

Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $G=Gal(L/K)$. Let $A$ be a finite $Gal(L/K)$ module. Then, Why does $|(1-\sigma)A||A^G|=|A|$ hold ? Here, $|A|$ ...
Pont's user avatar
  • 6,051
0 votes
0 answers
63 views

Group cohomology of a compact connected orientable genus $g$ surface without boundary and removing two open disks

Let $V_g$ be a compact connected orientable genus $g$ surface without boundary. Further, let $X$ be $V_2$ with two disjoint open disks removed, and let $X'$ be $X$ with three distinct points ...
Anik Bhowmick's user avatar
2 votes
1 answer
87 views

Transfer for the group of coinvariants

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\...
Mikhail Borovoi's user avatar
0 votes
1 answer
101 views

Computing the homotopy groups $\pi_n(\mathbb RP^2 \times \mathbb S^3).$

How can I Compute the homotopy groups $\pi_n(\mathbb RP^2 \times \mathbb S^3)$ ? Which theorems will help me in this? Should I use Kunneth theorem? I have seen this post here Calculate $[\mathbb{S}^n,...
Brain's user avatar
  • 1,013
2 votes
0 answers
93 views

Proving bockstein is a stable cohomology operation

I am considering the bockstein $\beta$ for $p=2$, so for the exact sequence $$0\rightarrow \mathbb Z_2\rightarrow \mathbb Z_4\rightarrow \mathbb Z_2\rightarrow 0$$ I would like to prove it is a stable ...
Gentleman Bizaut's user avatar
0 votes
0 answers
84 views

Induced inverse limit sequence

This proposition states that given the functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ and the short exact sequence $0 \to A_\bullet \to B_\bullet \to C_\...
June in Juneau's user avatar

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