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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Why does the continuous homomorphism group equal the homomorphism group?
I have the following question on Silverman's book The Arithmetic of Elliptic Curves, 2nd edition.
First, let $K$ be a perfect field, $G_{\overline{K}/K}$ be the absolute Galois group of $K$, and $M$ ...
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Projective representations that are compatible with a group automorphism
We know that projective representations of a group $G$, $\rho(g) \rho(h) = e^{i\omega(g,h)} \rho(gh)$, is classified by the second group cohomology $H^2(G,U(1))$. Let us now specify a group ...
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Why the non-abelian 4-cocycle condition?
In a monoidal category it holds by definition (together with the identity coherence) an associativity coherence axiom, stating commutativity of the pentagon
Now, if we categorify vertically, we can ...
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Cohomology group for trivial group
Let $G$ be a group, $A, B$ be a $G$ - modulo. We can define the n-th Cohomology group of $G$ with coefficient in $A$.
$$H^n(G,A) =\text{Ext }_G^n(\mathbb{Z},A)$$
And the n-th Homology group of $G$ ...
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Understanding how group cohomology classifies extensions using the derived functor point of view
I am rereading some material about group extensions, in particular because I needed to recall the formula
$$H^2(G;A)\cong \mathcal{E}(G;A).$$
We have that $G$ is some group acting on an abelian group $...
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${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$.
For a field $k$, I am calculating ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$, where $\epsilon^2 = 0$. However, there seems no complete explanation as far as I checked. ...
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Contradiction in Computation of Homology Groups of the Mapping Class Group of a Surface?
One of the two main results of a paper by Nathalie Wahl on homological stability of the mapping class group of a surface is the following:
Theorem 1.2 The map $H_*(\delta_g) : H_*(\Gamma_{g,1};\mathbb{...
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Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?
'Galois cohomology of Algebraic number fields' written by K. Haberland reads the following lemma in page 66.
Let $H$ be a finite group. Let $p$ be a prime number and $H_p$ be a
fixed p Sylow subgroup ...
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Induced G-modules are a direct sum
In the book Class field theory by J. Milne he defines an induced $G$-module to be of the form $\operatorname{Hom}_{\textsf{Ab}}(\mathbb{Z}[G],A)=\operatorname{Hom}_{\textsf{Set}}(G,A)$ for some ...
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Cohomology class of automorphism group of Galois form
Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
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Long exact sequence and restriction
Suppose $G$ is a group and $0 \to A \to B \to C \to 0$ a short exact sequence of abelian groups considered as trivial $G$-modules. There is a corresponding long exact sequence in cohomology. I wonder ...
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$H^2 (\mathbb{R}^3)$? (or in general $H^2 (\mathbb{R}^{2n+1})$)?
We know that $\mathbb{R}^2$ is an Abelian Lie algebra and $H^2 (\mathbb{R}^2)\cong \mathbb{R}$. We can define the three-dimensional Heisenberg algebra $\mathbb{R}^3=\mathbb{R}^2 \oplus \mathbb{R}$ ...
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Cokernel of the ring map $k[t] \to k[x,y]$
Let $k$ be a field and $\phi : k[t] \to k[x,y]$ be a graded ring homomorphism generated by $\phi(1)=1$ and $\phi(t) = x+y$. Here $t, x, y$ having degree $1$. I am interested in finding a nice form of ...
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Unbounded Double Complex with Diagonal Boundary
I am new to spectral sequences and am not very familiar with convergence results once we lose boundedness.
Suppose we have a differential cochain complex $X^n$ of $R$-modules with differential $D$, ...
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Computing action on cohomology group induced by conjugation
Consider the extension $1 \to C_3 \to S_3 \to C_2 \to 1$. I am trying to see intuitively why the action induced by the conjugation action $C_2 \curvearrowright C_3$ on $H^{\bullet}(C_3, \mathbb{Z})$ ...
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