The cech-cohomology tag has no usage guidance.

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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 $f_{ij} \in \Gamma(U_{ij},F)$ ...

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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 ...

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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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### 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 ...

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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 ...

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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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### 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,\...

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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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### 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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### 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 ...

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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 ...

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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 ...

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### An example where Čech and derived functor cohomologies don't agree. [duplicate]

Possible Duplicate:
Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?
Is there a simple example of a topological space $X$ with a sheaf $\mathcal F$ ...

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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 $C$...