Questions tagged [cech-cohomology]
The cech-cohomology tag has no usage guidance.
24
questions with no upvoted or accepted answers
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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....
9
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438
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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 ...
8
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181
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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}...
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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 ...
6
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551
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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 ...
5
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297
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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 ...
4
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326
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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 ...
4
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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 ...
4
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345
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cech cohomology in topos
Hi,
The following result seems to be well known, but I can't come up with a proof.
Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is
any abelian sheaf on $C$...
3
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139
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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,\...
3
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128
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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 $1$-...
3
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432
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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 ...
3
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193
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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_{...
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113
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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 ...
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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 ...
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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}^\...
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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 ...
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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 ...
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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? ...
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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 $\mathcal{F}'$ ...
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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 $\...
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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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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^...
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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 ...