4
votes
0answers
43 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
1
vote
1answer
40 views

Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting ...
3
votes
1answer
54 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
1
vote
0answers
22 views

Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
4
votes
0answers
31 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
2
votes
0answers
30 views

To prove Pic$(X)$ is isomorphic to $H^1(\mathcal{U},O_X^*)$

We start with an invertible sheaf $\mathcal{L}$ and an open cover $\mathcal{U}=\{U_i\}_i$ for $X$, such that $\mathcal{L}|_{U_i}=O_X|_{U_i}$. So the line bundle is given by the information : ...
1
vote
1answer
42 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 ...
8
votes
1answer
257 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
23 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
113 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
61 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
59 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
71 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
122 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
102 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
249 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
90 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
103 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
73 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
1answer
117 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
247 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
131 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 ...
5
votes
1answer
152 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
270 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?
3
votes
1answer
109 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 $H^\ast(Y,f_\ast \mathcal{F} ) \rightarrow H^\ast (X,\cal{F})$?