1
vote
1answer
33 views

Existence of a suitable cover for $S^{2}$ and a given sheaf

I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all ...
6
votes
1answer
105 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
0
votes
0answers
13 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
7
votes
1answer
99 views

Strange case of Serre's duality

$\newcommand{\O}{\mathcal{O}}$ Let $X$ be a smooth projective curve and $D$ and effective divisor on it. The normal bundle of $D$ is defined as $$ \O_D(D)\; = \; \O_x(D)\;\otimes_{\O_X}\, \O_D$$ where ...
4
votes
0answers
59 views

Cohomology of $\mathcal O(k)$

I am reading a paper in which it is claimed that $H^1(\mathcal O(-k),\mathcal O)=0$, where $k\geqslant 1$. Moreover, the argument also requires that $H^2(\mathcal O(-k),\mathcal O)=0$. Here ...
2
votes
0answers
53 views

Grothendieck topology of sets and Cech cohomology

Base concepts: Grothendieck topology - http://en.wikipedia.org/wiki/Grothendieck_topology Cech cohomology - https://en.wikipedia.org/wiki/%C4%8Cech_cohomology Cohomology - What is the motivation ...
4
votes
0answers
69 views

Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
2
votes
0answers
101 views

Computing algebraic de-Rham cohomology via Čech cohomology

I have been reading this paper about de-Rham cohomology of hyperelliptic curves, and I have been trying to recompute some of what has been done in section 3. In particular, I am trying to see why ...
3
votes
1answer
73 views

Defining the coboundary map $\delta_*$ on the Sheaf Cech Cohomology groups

I'm having trouble understanding the definitions I've been reading, of what has been called an 'induced coboundary operator' or a 'connecting homomorphism' depending on what source you're reading. ...
17
votes
2answers
216 views

What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the ...
5
votes
0answers
75 views

Injective Resolutions in $\mathfrak{Ab}(X)$

Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
2
votes
0answers
95 views

Vanishing of local cohomology of constructible sheaves

Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$. Is there an analogous statement for ...
1
vote
1answer
69 views

Induced sequence of global sections

I'm reading Differential Analysis on Complex Manifolds by Raymond O. Wells. It states the following in the beginning of section 3 of chapter 2 on page 51: Consider a short exact sequence of sheaves: ...
6
votes
0answers
82 views

Example where Čech and derived functor cohomologies don't agree.

I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces. Is there a simple example of a topological space ...
7
votes
0answers
211 views

Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
4
votes
1answer
116 views

Cech cohomology of $\mathbb A^2_k\setminus\{0\}$

I'm trying to prove, via the Cech cohomology, that $S=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Consider the structure sheaf $\mathcal O_{\mathbb ...
4
votes
1answer
119 views

Existence of acyclic coverings for a given sheaf

Let $\mathcal{F}$ be a sheaf over $X$ and $\mathcal{U}=\{U_i\}_{i\in I}$ a covering of $X$. I say that $\mathcal{U}$ is acyclic for $\mathcal{F}$ if $H^k(U_{i_0 \ldots U_n}, \mathcal{F}|_{U_{i_0 ...
3
votes
2answers
244 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
3
votes
1answer
104 views

Pushing forward sheaves and the result on sheaf cohomology

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and let $\cal{F}$ be a sheaf on $X$. Is there an obvious map $H^\ast(Y,f_\ast \mathcal{F} ) \rightarrow H^\ast (X,\cal{F})$?