# 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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### 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 ...
• 3,170
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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 ...
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### 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 ...
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### $H^1(G_{K_v}, M^D) \cong \widehat{H^1(G_{K_v}, M)}$, 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 ...
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### $0 \to E(K)/2E(K) \to H^1(G_K,E[2])\stackrel{\delta}{\to}H^1(G_{K_v},E)[2]\to 0$ 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 ...
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• 2,085
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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 ...
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• 1,558
1 vote
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### 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 ...
• 176
1 vote
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### 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 ...
• 501
1 vote
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• 1,407
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### 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(\...