Questions tagged [sheaf-cohomology]

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. (Def: http://en.m.wikipedia.org/wiki/Sheaf_cohomology)

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. Reference: Wikipedia.

This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.

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Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived functor of the global section functor. What I…
Mohan
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The idea behind the notion of dualizing sheaf

Well, studying sheaf cohomology, I've faced the notion of dualizing sheaf on a projective scheme over a field $k$. Recall that a dualizing sheaf on $X$ (according to Hartshorne) is a coherent sheaf $\omega_X^\circ$, such that the…
Alex
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Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of integers on $X$. In proving this, I have seen several…
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Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. What follows is an excerpt from Dieudonné's History…
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Relationship between Galois cohomology and etale cohomology.

Why is étale cohomology a natural generalization of Galois cohomology ? I would like to inform you that I have a few quite sufficient prerequisites Galois cohomology and its application to solve the $90$ - th problem of Hilbert . So I can…
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Cup/cap product: sheaf cohomology vs singular cohomology

Is anyone aware of a good resource which deals with how the cup/cap products of sheaf cohomology classes are a generalization of those in singular cohomology? I would say that I already understand the basics of both constructions. Specifically, I…
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Why only consider Dolbeault cohomology?

On a complex manifold we have the differential operators $$\partial:A^{p,q}\to A^{p+1,q}$$ $$\bar\partial:A^{p,q}\to A^{p,q+1}$$ which both square to zero. Hence one can define cohomology groups $$H^{p,q}_\partial=\frac{\ker\partial:A^{p,q}\to…
user2520938
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Thom–Gysin long exact sequence

I have read about the following exact sequence of cohomology: Let $V$ be an algebraic variety over $\mathbb{C}$. If $U\subset V$ is an open subvariety, then there is a long exact sequence for singular cohomology with compact support: $$…
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Exact sequence of sheaves with non exact sequence of global sections

Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms $$\mathcal{F}_1\longrightarrow \mathcal{F}_2\longrightarrow \mathcal{F}_3\longrightarrow... $$ is said to…
user74574
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Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to Y$. Is there an example of a concrete calculation…
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Understanding etale cohomology versus ordinary sheaves

I am a physicist trying to understand etale cohomology from Shafaverich, and I would like to check a misunderstanding, undoubtedly. When defining etale cohomology, it seems it is sheaf cohomology in the sense of right-derived functors, but with the…
JPhy
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Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space $X$, and $\mathcal{F}$ is a sheaf on $X$. Take…
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An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to think that probably he used this terminology to…
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Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault cohomology, just by putting $\mathcal{F}=\mathbb{R}$ or…
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Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have $H^i(X,\mathscr{F})=0$. Exercise III.2.1 in Hartshorne is…
Jared
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