The sheaf-cohomology tag has no usage guidance.

**0**

votes

**0**answers

145 views

### Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...

**4**

votes

**1**answer

204 views

### Surjectivity of certain cohomology groups on hypersurfaces of high degree

I had been reading an article by Spencer Bloch. There is a remark in this text which states the surjectivity of a particular map between cohomology groups without explaining further. I had been trying ...

**2**

votes

**0**answers

68 views

### Cohomology and quotients for the canonical topology

Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example ...

**4**

votes

**1**answer

458 views

### The Gauss-Bonnet theorem for Sheaves

Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem
Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage ...

**0**

votes

**0**answers

126 views

### Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that:
$m^*(L) \simeq L \boxtimes L $.
Then why is $L$ an equivariant sheaf on $G$ with the action the ...

**1**

vote

**3**answers

368 views

### Higher cohomology of sheaves on a projective space

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf
...

**6**

votes

**1**answer

356 views

### infinite grassmannian in algebraic geometry

Geometric realization of $B{\mathbb G}_{\mathfrak m}({\mathbb C})$ is ${\mathbb C}{\mathbb P}^\infty=\varinjlim_n~ {\mathbb C}{\mathbb P}^n_k$; what if one considers a separable field $k\neq ...

**3**

votes

**1**answer

309 views

### Cohomology and proper base change

Let $\pi:\mathcal{X} \to B$ be a flat, projective surjective morphism over $\mathbb{C}$. Assume that $B$ is a smooth quasi-projective curve. Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$, ...

**4**

votes

**2**answers

738 views

### Why Cech cohomology does not compute sheaf cohomology on an open annulus

Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets:
$U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...

**2**

votes

**0**answers

237 views

### Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...

**3**

votes

**4**answers

704 views

### Sheaves with no cohomology

Let $X$ be a smooth projective variety of dimension $d$ over a field $k$. Suppose $\mathcal F$ is a coherent sheaf on $X$ such that $H^i(X,\mathcal F) = 0$, for all $i$. What can one say about ...

**2**

votes

**1**answer

240 views

### A functorial property of higher right derived functors

Let $f:X \to Y$ be a projective morphism of complex Noetherian schemes. Assume $Y$ is smooth and for all $y \in Y$, $f^{-1}(y)$ is of pure dimension $1$. Let $\mathcal{F}_1, \mathcal{F}_2$ and ...

**1**

vote

**0**answers

205 views

### Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...

**1**

vote

**2**answers

319 views

### Cohomology of sheaf extended by zero

Let $X$ be a projective scheme of pure dimension $1$. Let $U$ be a open subscheme and $j:U \to X$ the open immersion. Let $\mathcal{F}$ be a coherent sheaf on $U$.
Denote by $j_!(\mathcal{F})$ the ...

**1**

vote

**1**answer

131 views

### Restriction of sheaves on curves

Let $C$ be a scheme of pure dimension $1$. Let $C_1$ be a closed subscheme of $C$ of pure dimension $1$. Denote by $i:C_1 \hookrightarrow C$ a closed immersion. Given a sheaf $\mathcal{F}$ on $C$, ...

**0**

votes

**1**answer

119 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 ...

**0**

votes

**1**answer

307 views

### Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve

By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?
...

**2**

votes

**1**answer

415 views

### Compare global sections of restriction and pullback of sheaves

Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) ...

**4**

votes

**1**answer

210 views

### Relating deformations of a scheme to deformations of its singular locus

Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...

**3**

votes

**0**answers

62 views

### Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push ...

**3**

votes

**1**answer

306 views

### Explicit examples presheaves associated to higher direct images which fail to be sheaves

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and ...

**2**

votes

**1**answer

563 views

### Is the higher direct image sheaf of a locally free sheaf over $\mathbb{P}^1$ locally free?

Let $f:X \to \mathbb{P}^1$ be a projective flat morphism, $X$ is a projective scheme. Let $\mathcal{F}$ be a locally free sheaf on $X$. Are the higher direct image sheaves $R^if_*\mathcal{F}$ locally ...

**2**

votes

**1**answer

661 views

### Cohomology of tangent bundles

Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$
of $X$ along $Z$.
What is the relation between the cohomology of the ...

**2**

votes

**2**answers

305 views

### Topological information via cohomology of sheaves

On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The ...

**1**

vote

**1**answer

151 views

### global sections of higher direct images of étale sheaves

Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?

**7**

votes

**1**answer

836 views

### Sheaf cohomology with compact supports (and Verdier duality?)

Consider a manifold and a complex where cochains are sections of vector bundles and coboundary maps are differential operators, which are locally exact except in lowest degree (think de Rham complex). ...

**3**

votes

**1**answer

217 views

### Euler Characteristic of Coverings via Sheaf Theory

Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works),
$f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic.
By the ...

**1**

vote

**2**answers

162 views

### Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero

What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?

**0**

votes

**0**answers

421 views

### Global sections of twisting ideal sheaf of a smooth closed point on a projective space

Let $X = \mathbb{P}^n_k$ be a projective space over an algebraically closed field $k$ and $x$ be a closed point.
Given an integer $m$ and a positive integer $r$.
What are the global sections of ...

**8**

votes

**1**answer

611 views

### When does the sheaf cohomology of a topological space vanish?

The question is in the title. A more precise formulation is:
Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$?
The obvious example is a ...

**2**

votes

**2**answers

316 views

### Leray spectral sequence of the inclusion of an open subvariety

Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the ...

**9**

votes

**1**answer

586 views

### Etale homology via étale cosheaves

Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group ...

**11**

votes

**2**answers

1k views

### When does sheaf cohomology commute with arbitrary direct sums?

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map
...

**18**

votes

**2**answers

639 views

### Cohomologically trivial stacks

The following theorem of Serre is well-known:
A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. ...

**8**

votes

**0**answers

555 views

### Eilenberg-Steenrod axioms of sheaf cohomology

Cohomology of a space is often defined axiomatically: a cohomology theory is a functor
from pairs of spaces to abelian groups
satisfying the Eilenberg-Steenrod axioms. Is there a similar ...

**1**

vote

**0**answers

72 views

### How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?

For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to ...

**2**

votes

**2**answers

383 views

### Vanishing of Ext group

Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$$
for some sheaf ...

**4**

votes

**0**answers

355 views

### Cohomology of a sheaf with only one stalk

Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish).
I believe that ...

**7**

votes

**4**answers

1k views

### Cohomology of line bundles

For sure answers to my questions are well known - but I never saw them anywhere.
Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...

**5**

votes

**2**answers

306 views

### cohomology and $j_!$

I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between ...

**0**

votes

**2**answers

268 views

### Cohomology groups interpreted as sheafs

Hi Folks,
I just came across a few lines where a (sheaf-)cohomology group of a scheme is treated as a sheaf. I've never seen this in Hartshorne.
Could you give any reference for this?
Thanks
Steven
...

**2**

votes

**2**answers

744 views

### Top cohomology detecting compactness

I am looking for a reference for the fact that the top cohomology $H^n(X;A)$ of an $n$-dimensional manifold $X$ is non-trivial precisely when $X$ is compact.
I tried to ask this question on ...

**18**

votes

**2**answers

1k views

### How to Draw Complex Line Bundles

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples.
Background and Context
I am considering ...

**1**

vote

**0**answers

202 views

### why is the local field Q_l not an etale sheaf over a scheme X?

I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?

**1**

vote

**1**answer

308 views

### Is this Sequences of Complexes of Sheaves Exact?

So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ ...

**2**

votes

**0**answers

309 views

### Weil Kostant Integrality Result as Stated by Brylisnki

I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued ...

**1**

vote

**2**answers

277 views

### Cohomology of a cochain complex of acyclic sheaves

Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem:
Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of ...

**4**

votes

**1**answer

362 views

### Sheaf cohomology invariant of weak homotopy type?

Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a commutative ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps ...

**8**

votes

**0**answers

376 views

### Deducing properness from H^i(X,F) finitely generated over \Gamma(O_X)

Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is ...

**8**

votes

**2**answers

472 views

### Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?

Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on ...