# Questions tagged [sheaf-cohomology]

The sheaf-cohomology tag has no usage guidance.

212
questions

**53**

votes

**8**answers

9k views

### Sheaf cohomology and injective resolutions

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...

**30**

votes

**3**answers

3k views

### How should a homotopy theorist think about sheaf cohomology?

As a student of homotopy theory or algebraic topology, I have a certain outlook as to how one ought to think of a cohomology theory. There are axioms that help us with rudimentary computations, there ...

**20**

votes

**3**answers

2k views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

**19**

votes

**1**answer

1k views

### Two points of view about Borel-moore homology

They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...

**18**

votes

**2**answers

824 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 $i>0$. (...

**17**

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 ...

**16**

votes

**2**answers

864 views

### Is there a complex which computes Cech cohomology?

Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open ...

**15**

votes

**1**answer

1k views

### Where am I suppose to actually learn how to compute hypercohomology?

I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with ...

**13**

votes

**1**answer

834 views

### Does every sheaf embed into a quasicoherent sheaf?

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...

**13**

votes

**2**answers

3k 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
$$...

**12**

votes

**1**answer

842 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 ...

**12**

votes

**1**answer

644 views

### Algebraic groups without torsors

If $G$ is an algebraic group such that $H^1(S, G) = 0$ for all schemes $S$, must $G$ be the trivial group?
My original motivation for the question is the rationale I always give students for studying ...

**12**

votes

**1**answer

479 views

### Hodge to de Rham spectal sequence with twisted coefficients

Let $M$ be a smooth compact Kahler manifold and let $\mathcal{F}$ be a local system on $M$.
Question 1: I assume that there exists a twisted Hodge to de Rham spectral sequence converging to $H^{p+q}(...

**10**

votes

**5**answers

3k views

### Non-cohomological proof that the pullback of an ample bundle by a finite morphism is ample

It is a standard fact that for any finite morphism of proper Noetherian $A$-schemes ($A$ being Noetherian), the pullback of an ample line bundle is ample. The usual proof of this fact is via Serre's ...

**10**

votes

**4**answers

2k 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 ...

**10**

votes

**1**answer

1k 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 ...

**9**

votes

**1**answer

620 views

### Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?

Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...

**9**

votes

**1**answer

479 views

### Deducing properness from $H^i(X, \mathcal{F})$ 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 ...

**9**

votes

**0**answers

137 views

### Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....

**9**

votes

**0**answers

397 views

### In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?

Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...

**9**

votes

**0**answers

958 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 ...

**8**

votes

**2**answers

569 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 $\...

**8**

votes

**4**answers

2k views

### Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...

**8**

votes

**1**answer

1k 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). ...

**8**

votes

**0**answers

346 views

### Leray-Hirsch theorem for Dolbeault cohomology

In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this:
Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...

**8**

votes

**0**answers

361 views

### Can we use sheaf cohomology to say anything interesting for vector bundles with non-flat connections?

Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...

**8**

votes

**0**answers

344 views

### Dualizing complex of the product of two locally compact spaces

Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...

**7**

votes

**1**answer

305 views

### Can one determine the trace map for a nonsingular projective variety explicitly?

I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the ...

**7**

votes

**1**answer

468 views

### Sections of the conormal bundle

Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...

**7**

votes

**1**answer

401 views

### What are the uses of coefficient systems for arithmetic cohomology theories?

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...

**7**

votes

**1**answer

522 views

### Local cohomology groups and linearity

I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of ...

**7**

votes

**1**answer

165 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

**7**

votes

**1**answer

785 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, ...

**7**

votes

**1**answer

481 views

### Relative version of de Rham cohomology with local coefficients

Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...

**7**

votes

**0**answers

310 views

### Defining the Euler class in different ways

Let $\pi: E\to M$ be a rank two real vector bundle over a manifold $M$. Bott and Tu defines the Euler class by:
giving $M$ a Riemannian structure,
taking a trivializing chart $U_\alpha$ of $M$,
...

**6**

votes

**1**answer

450 views

### Poincare duality on the level of complexes

The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes.
In particular, suppose $\mathcal{C}^*$ is a complex of $...

**6**

votes

**2**answers

350 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 $H^i(X,...

**6**

votes

**1**answer

600 views

### difference between the small and big étale/flat/… site

What is the difference between the small and the big étale (or flat or syntomic or ...) site? How does the cohomology vary? When should I use which one? Up to now, I have always used the small sites.

**6**

votes

**1**answer

838 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; ...

**6**

votes

**1**answer

319 views

### A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR

On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is ...

**6**

votes

**1**answer

445 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 {\...

**6**

votes

**1**answer

729 views

### Sheaves with isomorphic cohomology, but not quasi-isomorphic

Suppose I have two (constructible) sheaves of vector spaces $F$ and $G$ over the same base space that have isomorphic cohomology (degree by degree), but no sheaf map inducing this isomorphism (i.e. ...

**6**

votes

**1**answer

227 views

### Independence of embedding for higher sheaf cohomology of local cohomology on projective space

Suppose $Y$ is a projective variety over a field $k$. Fix an embedding $\iota: Y \hookrightarrow \mathbb{P}^n_k$ for some $n$, and consider the local cohomology sheaves $\mathcal{H}^j_Y(\omega_{\...

**6**

votes

**1**answer

231 views

### An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology

Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$.
Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as ...

**6**

votes

**1**answer

228 views

### Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is ...

**6**

votes

**1**answer

148 views

### When is derived category of ringed space perfectly generated?

Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...

**6**

votes

**0**answers

131 views

### Sheaves on Rectifiable Sets

Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...

**6**

votes

**0**answers

358 views

### What is the geometric intuition for the sheaf-theoretic terms “soft”, “fine”, and “flabby”?

The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
...

**6**

votes

**0**answers

311 views

### Generalities on sheaves - Where can I find the technology that can make this “proof” of Atiyah duality precise?

Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct).
Let $X$ ...

**5**

votes

**2**answers

449 views

### Log canonical counterexample to Kawamata-Viehweg vanishing

I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\...