Questions tagged [cech-cohomology]
The cech-cohomology tag has no usage guidance.
52 questions
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Is it possible to use the Cech complex to compute coherent cohomology in practice?
Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute the coherent cohomology of $V$ with coefficients in a line ...
- 2,964
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141
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Computing the coherent cohomology of a quasiprojective variety
I have a quasiprojective variety given by some explicit quations. How do I compute its coherent cohomology (with coefficients in the structure sheaf)? Do I use the Cech complex for an open affine ...
- 2,964
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What do associated fibre bundles have in common?
Two fibre bundles are said associated if they have isomorphic associated principal bundles. I understand that this means they are defined by the same transition functions, but still is there some more ...
- 35
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The tautness property and the continuity property of cohomology theory
Let $H^{\ast }$ denote the Čech cohomology or Alexander-Spanier
cohomology.
Definition: (Tautness property of cohomology) Let $X$ be a
paracompact Hausdorff space and $A$ be a closed subspace of $X$. ...
- 1,367
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465
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Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
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162
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Multiplicative structure on Čech–Alexander complexes
I have the following basic question on Čech–Alexander complexes.
Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...
- 1,479
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1
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144
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About Čech cohomology in transformation groups
I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
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Determining a class in Dolbeault cohomology that defines a principal $\mathbb{C}$-bundle over a compact torus
This is a cross-post from MSE
Consider a standard complex torus $\mathbf{T}=\mathbb{C}/(\mathbb{Z}\oplus i\mathbb{Z}).$ It could be obtained in another way.
$\mathbf{T}$ is a quotient $(\mathbb{C}^\...
- 1,374
3
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0
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83
views
Cohomological interpretation of gluing conditions
This question comes from a problem of theoretical physics. Stated in its simplest form, there is a complex line bundle over $S^1$. For each $z \in S^1$, the fiber $F_z$ is the complex eigenspace of a ...
- 141
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Čech representatives for Chern classes in holomorphic Deligne cohomology
Let $X$ be a complex-analytic manifold with "nice" (e.g. Stein) cover $\mathcal{U}=\{U_\alpha\}$, and $E$ a holomorphic vector bundle on $X$ defined by transition functions $\{g_{\alpha\beta}...
- 1,109
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On Brylinski–McLaughlin's paper "Čech cocycles for characteristic classes"
In the paper "Čech cocycles for characteristic classes", the authors Brylinski and McLaughlin describe how to construct Čech cocycles with values in the Deligne smooth complex representing ...
- 609
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475
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Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
- 898
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77
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Reference Request: Cech cohomology of complexes on an arbitrary site
I am looking for a reference which is equivalent to this stacks project page [1], except formulated in the generality of an arbitrary site. I checked the "Cohomology on Sites" section of the ...
- 437
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3
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Are two different definitions for Čech cohomology equivalent?
In Spanier's book Algebraic Topology (Chapter 6 section 7) he defines Čech cohomology in terms of the nerves of open coverings.
I wish to know if this is equivalent, for a topological space A closed ...
3
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1
answer
175
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Example of locally contractible topological space where Čech cohomology does not coincide with singular cohomology
I believe that it is shown in EH Spanier's "Algebraic Topology" that if 𝑋 is paracompact and locally contractible, then singular cohomology and Čech cohomology of 𝑋 coincide, with ...
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When exactly does Čech cohomology coincide with singular cohomology?
I believe that it is shown in EH Spanier's "Algebraic Topology" that if 𝑋 is paracompact and locally contractible, then singular cohomology and Čech cohomology of 𝑋 coincide, with ...
1
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0
answers
98
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Cohomology with coefficient in sheaf of morphisms of an algebraic group
Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
- 1,177
4
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1
answer
602
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Relationship between Dolbeault and de Rham cohomology on Riemann surface
A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...
- 721
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When is Cech cohomology with compact support isomorphic to Singular cohomology with compact support?
This is a specific question regarding the understanding of a section of a paper. Having never posted here before I'm not sure that this is the right forum for this sort of question, but I hope someone ...
- 41
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478
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About an argument in Olsson's book
The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson.
I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end.
It seems that there might ...
- 1,906
3
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0
answers
137
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On the construction of principal $S^1$-bundles with prescribed characteristic form
I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-...
- 1,227
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446
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Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole
Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by the two affine ...
- 1,479
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1
answer
487
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Finding the right map between cohomology with local coefficients and Čech cohomology
Let $X$ be a space which is paracompact, Hausdorff, and sufficiently nice that it has a universal covering space (and map) $p:\tilde{X}\to X$. Also, let $\pi:=\pi_1(X)$ and $A$ some $\mathbb{Z}[\pi]$-...
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Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to
We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
- 5,998
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283
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Triviality of holomorphic vector bundles over $\mathbb{C}$
Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle.
I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
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1
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343
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Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?
Given a topological space $X$, and a cover $\mathcal{U} :=\cup_{\alpha \in I}U_{\alpha}$ of $X$, one can define a groupoid called Čech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\...
- 1,215
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Computing cohomology using bounded hypercovers
Let $G$ be a Lie group (paracompact, not necessarily compact), and $A$ an abelian Lie group. I want to write down cocycles in $\mathrm{H^n}(\mathbf{B}G,A)$, the cohomology in the cohesive $\infty$-...
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Can we recover $\pi_2(S^2)$ from this simplicial set?
Let $S^3 \rightarrow S^2$ be the Hopf fibration. Can we recover $\pi_2(S^2)$ of $S^2$ from the simplicial set $X : \Delta^{op} \rightarrow \text{Set}$,
$$ X(n) = \pi_0 (S^3 \times_{S^2} \cdots \times_{...
user30211
1
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1
answer
521
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Connecting homomorphism in Cech cohomology
Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves
$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...
- 19
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160
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Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
- 1,479
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2
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Different definitions for integral de Rham cohomology classes
Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$.
Now, one ...
- 553
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116
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Compute Cech cohomology with two open sets
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
- 1,479
2
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1
answer
274
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Čech cocycles and monodromy
It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
- 553
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1
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The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
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464
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Why the third stage of Cech nerve a Lie 2-groupoid?
In the page https://ncatlab.org/nlab/show/Lie+2-groupoid the Lie 2-groupoid is defined as the 2 truncated $\infty$-Lie groupoid.
I am not much comfortable with the language of higher category theory ...
- 1,215
2
votes
1
answer
291
views
Čech complex of rigid $K$-space - Closed image of boundary maps
Let $(X,\mathcal{O}_X)$ be a rigid $K$-space with a finite affinoid covering $(U_i)_{i\in I}$ such that any intersection of the $U_i$ is affinoid too.
Equipping the direct products with the maximum ...
- 473
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301
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Naive question in Cech cohomology
Let $X$ be a smooth, projective variety and $F$ a coherent sheaf on $X$. Let $\{U_i\}_{i \in I}$ be an open affine covering of $X$ and $\{f_{ij}\}_{i<j}$ with $f_{ij} \in \Gamma(U_{ij},F)$ ...
- 2,022
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3
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Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space
If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group.
I hear that this is a classical result but I fail ...
- 6,023
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0
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142
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Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
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1
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Étale hypercohomology of complexes
Let $K^{\bullet}$ be a bounded complex of abelian étale sheaves on a quasi-compact and quasi-separated scheme $X$.
For any étale cover $\mathcal{U} :=\{ U_i\to X\}_{i\in I}$, can we find a refinement ...
user120812
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2
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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 ...
- 1,761
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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....
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How does Cech cohomology get around computing a delooping?
I'm trying to understand how the nlab's definition of cohomology works concretely. There, it is (very convincingly) claimed that every incarnation of "cohomology" in mathematics is a special case of ...
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2
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Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,367
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0
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236
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Canonicity of Čech cohomology
For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$.
For a sheaf $F$ on $X,$ the cohomology $H^...
- 781
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115
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Characterization of global sections (which are not products) of a sheaf which is locally a product
In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
- 397
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0
answers
563
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Principal bundles and Čech cohomology with non-good open covers
I'm trying to compute characteristic classes of principal bundles by defining transition functions and computing them in Čech cohomology. However, it seems all the constructions are defined in terms ...
- 61
2
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1
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412
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Vanishing Cech cohomology
Let $X$ be a manifold such that $dim(X)=n$. It is well-know that if $\mathcal{F}$ is a coherent sheaf $H^m(X,\mathcal{F})=0$ for all $m >n$ (where I denote with $H(-)$ Cech cohomology). But is ...
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The Abelian Group of Equivalence Classes of Gerbes
Are there any good references/explanations for understanding the "contracted product" (as Brylinski calls it) of a gerbe with abelian band? I am finding it difficult, using Brylisnki's definition. to ...
- 1,466
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2
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496
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Cech cohomology as a colimit over maps to a CW complex
Given a topological space $X$, we consider the following category $\mathsf{CW}_{X\to}$. The objects are finite CW complexes $Y$ equipped with a continuous map $X\to Y$. The morphisms are continuous ...
- 18.7k