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4 votes
Accepted

$H^2 (\mathbb{R}^3)$? (or in general $H^2 (\mathbb{R}^{2n+1})$)?

Santharoubane has proved in $1983$ in his article "Cohomology of Heisenberg Lie algebras" that $$ \dim H^i(\mathfrak{h}_n(\Bbb R))=\binom{2n}{i}-\binom{2n}{i-2} $$ for all $0\le i\le n$. For ...
Dietrich Burde's user avatar
4 votes
Accepted

Computing action on cohomology group induced by conjugation

So I have an indirect argument. Consider the exact sequence $$0\to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0.$$ Viewing all of them as a trivial $S_3$ module, we can take cohomology to ...
Snacc's user avatar
  • 2,240
4 votes
Accepted

Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?

This is a general phenomenon. It follows fron the existence, for any $G$-module $M$ (with $G$ a finite group) and any subgroup $H \leq G$, of a natural corestriction map $Cores: H^{\ast}(H,M) \...
Aphelli's user avatar
  • 34.7k
2 votes
Accepted

Long exact sequence and restriction

This is true. See proposition 1.5.2 of Cohomology of Number Fields by Neukirch, Schmidt, and Wingberg. The proof comes down to working on the level of cocycles and tracking through the definitions of ...
Snacc's user avatar
  • 2,240
2 votes
Accepted

Compute the cohomology ring of $G=\langle u,v\mid uvuv=vuvu\rangle$.

If I understand you correctly, the ring you are looking at is a free $\mathbb{Z}$-module with basis $e_0,e_1,e_2,e_3$ where $e_0$ is the identity, $e_3=\frac{1}{2}e_1\smile e_2$, $e_1^2=e_2^2=0$, and ...
Joshua Tilley's user avatar
1 vote

Cokernel of the ring map $k[t] \to k[x,y]$

Hope that this answer is correct. Two polynomials in $k[x,y]$ represent the same element if and only if their difference is in $k[x+y]$. Using the relations $x=(x+y)-y$, $x^2=(x+y)^2-2xy-y^2$, $x^3=(x+...
Desperado's user avatar
  • 2,303

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