1
vote
0answers
58 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
0
votes
0answers
42 views

First cohomology of a Galois group with finite base field

Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology. Then ...
0
votes
1answer
69 views

Can anyone check my proof that $H^1(\Sigma-\{p\})=0$ for a compact and orientable surface $\Sigma$?

I have the following problem: Let $\Sigma$ be a compact and orientable surface. Show that $H^1(\Sigma-\{p\})=0$ for every $p\in \Sigma$. Can anyone check my proof and give suggestions? Sketch of ...
0
votes
1answer
44 views

Prove that the $2$ form defines a symplectic structure

Prove that the $2$ form $$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$ defines a symplectic structure on $\mathbb{R}_x^4$. My definition of as ...
0
votes
1answer
70 views

Restriction map on a compact orientable manifold without a boundary.

I have the following problem: Let $M$ be and $n$-dimensional compact oriented manifold without boundary. Let $p\in M$ be and point and let $M_p=M\backslash\{p\}$. Let $j:S^{n-1}\to M_p$ be the ...
1
vote
0answers
38 views

Sending the Poincaré dual class of a point in a smooth manifold to 1

I have been given the following problem: For a compact oriented $n$-dimensional manifold, use a nowhere zero $n$-fold $\omega\in\Omega^n(M)$ to define a linear map \begin{equation} [M]: H^n(M)\to ...
1
vote
0answers
106 views

group action - compact complex torus with $H^2_{DR}(X,C) = 0$ (de Rham cohomology)

If $\mathbb{Z}$ acts on $\mathbb{C}^n \backslash \{0\}$ by $(m,z) \mapsto 2^m\,z$, I need to show that $H^2_{DR}(X=(\mathbb{C}^n \backslash \{0\})/\mathbb{Z},\mathbb{C})=0$. I start with showing it ...
3
votes
0answers
169 views

short exact sequence - split, as a semidirect product, with some cohomology

I've looked at several s.e.s. examples and I feel I am quite close but here is a question I am still a little confused on. Let $E$ be a group and $A$ an abelian normal subgroup s.t. have an exact ...
5
votes
2answers
310 views

About sheaf cohomology in algebraic geometry

In algebraic geometry, why is much more interesting to work with injective resolutions? Why is the main functor of global sections of a sheaf? I am reading Hartshorne's book Algebraic Geometry, and it ...
10
votes
1answer
297 views

Finite groups with periodic cohomology

I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...