The sheaf-cohomology tag has no usage guidance.

**2**

votes

**0**answers

230 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

677 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

228 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

193 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

284 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

128 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

118 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

284 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

357 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

206 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

56 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

291 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

525 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 ...

**1**

vote

**1**answer

508 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

301 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

138 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

799 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

199 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

153 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

360 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

592 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

299 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

550 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 ...

**10**

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

620 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

517 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

70 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

371 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

350 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 ...

**6**

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

282 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

265 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

655 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

199 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

305 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

307 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

275 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 ...

**3**

votes

**1**answer

343 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

373 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 ...

**7**

votes

**2**answers

458 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 ...

**4**

votes

**1**answer

508 views

### Are injective quasi-coherent modules acyclic?

Let $X$ be a scheme and $F$ be an injective object of $\mathrm{Qcoh}(X)$. Is it true that $F$ is acyclic with respect to the usual sheaf cohomology?
For noetherian schemes $X$ this is well-known; ...

**2**

votes

**2**answers

378 views

### sheaves for which the derived (compact or not) pushforward is zero

Conventions: sheaf = complex of constructible sheaves (in the l-adic setup with etale tplg or in the complex coefficients setup with analytical tplg).
I would like to understand if there is an ...

**4**

votes

**2**answers

457 views

### If $f: X \to Y$ is a finite flat morphism of schemes, $g: Y \to Z$ is a proper morphism of relative dimension one, $Z$ is affine and $E$ is a vector bundle on $Y$ with $R^1g_*E=0$ then $H^1(X,f^*E)=0$?

Let $f: X \to Y$ and $g: Y \to Z$ be morphisms of schemes* such that f is flat and finite, g is proper and $R^{> 1}g_*E=0$ for all sheaves and Z is affine.
Let E be a vector bundle on Y such that ...

**2**

votes

**0**answers

132 views

### Intersections of components of 'simple' ('local") Zariski coverings

I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...

**4**

votes

**1**answer

335 views

### Derived Equivalence of Sheaves and Homotopy

This question loosely elaborates on an earlier question. It is pretty silly, but I'd like to hear some authoritative answers.
Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of ...

**6**

votes

**1**answer

513 views

### Basic properties of Nisnevich cohomology; $l'$-topology?

I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...

**0**

votes

**0**answers

166 views

### Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?

Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...

**1**

vote

**0**answers

207 views

### On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...

**1**

vote

**0**answers

211 views

### Non-realizable CR structures?

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$,
$v = ...