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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)

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Is my proof that $H^1(X,F) = 0$ for the skyscraper sheaf correct?

Let $X$ be compact connected. Let $F$ be the skyscraper sheaf over $p$ for a group $G$. Let $H^1(X,F)$ be the first Cech cohomology of the sheaf. We want to show $\ker(d_1) = im(d_0)$ is trivial. It ...
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Induced map from sheaves into cohomologies?

Suppose I have an short exact sequence of sheaves: $$0 \to \mathcal{E} \to \mathcal{F} \to \mathcal{G} \to 0$$ My book says that this induces a long exact sequence of cohomologies (by snake lemma): $$...
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Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand: In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ ...
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Hartshorne Exercise III 3.2: $X$ is affine iff every component is affine

I'm trying to solve the following exercise frome Hartshorne's Algebraic Geometry: Exercise III 3.2. Let $X$ be a reduced noetherian scheme. Show that $X$ is affine if and only if each irreducible ...
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are the global sections of a flat sheaf over a discrete valuation ring a free module?

Let $f:X\to \operatorname{Spec}\mathbf{Z}_p$ be a smooth proper $\mathbf{Z}_p$-scheme and $F$ a coherent sheaf on $X$ which is flat over $\mathbf{Z}_p$. Further suppose that $H^1(X_p, F_p)=0$ where $...
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Why are (Pre)sheaves more important than Co(pre)sheaves?

I'm learning Sheaf Theory, and this is an issue that's been bothering me. Fix a small category $\mathcal{C}$. A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor $F:\...
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Is $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\mathcal{O}(-p))$?

If $C$ is a complex curve, then any point is a hypersurface. To a point $p$ in $C$, suppose we have $w(p)=0$ for all $w\in\Omega_X$, then do we have $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\...
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Vector Bundle Transition Functions as Cech Cocycles

I am trying to understand the fact that vector bundles of rank $r$ over a space $X$ are classified by the Cech cohomology group $\check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$. I believe this ...
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Proof of upper semi-continuity of sheaf cohomology

$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\im}{im}$ Hello Math.Stackexchange.com-Community. Sorry for asking two questions at the same time, however they are part of one single step ...
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Cohomology theories and sheaf cohomology

Let $X$ be a paracompact Hausdorff topological space, $\mathcal U$ an open covering of $X$ and $\mathcal N(\mathcal U)$ the nerve of the covering (https://en.wikipedia.org/wiki/Nerve_of_a_covering). ...
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Confusion regarding sheaf cohomology and singular cohomolgy

Let $X$ be a smooth, projective curve (in particular irreducible) of genus $g$ at least $1$. We know that $H^1_{\mbox{sing}}(X,\mathbb{Z})=2g$. But, since $X$ is irreducible, the locally constant ...
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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 $...
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Cech cohomology of a quasi-coherent sheaf on an affine scheme and Leray acyclicity Theorem.

Let $X$ be an affine scheme, $\mathcal{F}$ a quasi-coherent sheaf on $X$. Let $\mathcal{U}=\{U_i\}_{i \in I}$ be an affine covering of $U$ (not necessarely made up of principal open subsets). Moreover,...
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Čech cohomology of a contractible space with integer coefficients

I am unable to see why the following statement is trivial: Since $\mathbb{C}^n$ is contractible, we see that $\check{H}^k(\mathbb{C}^n,\mathbb{Z}) = 0$ for $k>0$. Source: p. 46, "Principles of ...
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Question about Grothendieck's vanishing theorem

In the proof of Grothendieck's vanishing theorem, one of the steps says that if $X$ is irreducible of dimension o, then $H^{i}(X,\mathcal{F})=0$ for all $i>0$. This is because functor $\Gamma(X, .)$...
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A natural transformation between two functors

In Hartshorne's book, it states that there is a natural transformation between two $\delta-$functors $\text{lim}_{\rightarrow} H^{i}(X, .)$ and $H^{i}(X,\text{lim}_{\rightarrow} .)$. Here $\text{lim}_{...
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Suffices to compute Cech cohomology on an arbitrarily fine cover?

For a topological space $X$ and a sheaf $\mathscr{F}$, define the Cech cohomology groups in the usual way $$\check{H}^{q}(X, \mathscr{F}) = \varinjlim H^q(\mathfrak{U}, \mathscr{F})$$ where the limit ...
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Čech cohomology of a contractible space

I'm going through Čech cohomology at a gentle pace and as you know we need a colimit to obtain the sought-after sheaf cohomology. In practice, for calculating actual cohomology groups, I would invoke ...
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About $O_X$-modules?

Consider $F$ an $O_X$-module is true in general that $Hom_{O_X}(O_X,F) \equiv F(X)$ ? Morover I need to prove that if $F$ is a flasque sheaf then $H^n(X,F)=0$ for $n>0$. I think is beacuse the ...
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Is the genus of biholomorphic Riemann surfaces the same?

Is the statement above true for $X \cong_{bihol} Y$? I would say yes, since I can transform any holomorphic function on a open set in $X$ to one in $Y$ and vice versa.
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Exact sequence of sheafs

I am not quite sure whether my solution for this exercise is correct, more precisely the part $d: \mathcal{O} \rightarrow \Omega$. My solution: each holomorphic 1-form $\omega$ is locally exact (...
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Cech cohomology of coefficient in a presheaf and in its associated sheaf

I am reading Jean-Luc Brylinski's book on loop spaces. In his book, he claims that: Suppose that for any presheaf $F$ of Abelian group such that its associated sheaf $aF$ is $0$, the groups $\check{H}...
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Sheafs with $F(X)=\emptyset$

In the book "A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology" by Jean Gallier, the author states on page 214 that if F is a sheaf on a topological space X and for an open set U, $F(...
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Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a ...
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Cohomology of varieties defined solely by their embedding equation

In textbooks where sheaf cohomology is introduced, example computations are typically carried out on known spaces, e.g. $\mathbb{P}^n$. However, suppose we are handed a description of a variety ...
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Inducing Sheaf of Local Rings to Locally Closed Subspace

In this English translation of Serres FAC [working on p.38], for $X = K^r$ in the Zariski topology, $K$ algebraically closed, we define a locally closed subspace $Y$ as usual: the intersection of ...
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Cousin I problem in $\mathbb{C}$ and Mittag-Leffler theorem

In the wikipedia page about Cousin's problems (https://en.wikipedia.org/wiki/Cousin_problems) is stated that "the case of one variable is the Mittag-Leffler theorem"; I'm a bit in trouble with this ...
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Computing maps in Leray spectral sequence

Let $f:X_{s1} \to X_{s2}$ be morphism of sites ( Here $X$ is some scheme $X_{s1}$ refers to the site on $X$). Now using the Leray spectral sequence one gets the following exact sequence $0 \to H^1(...
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Importance of Vanishing Cohomology

As part of my masters project I have been working through Serre's FAC. Below are three closely related results I will be presenting as part of my defense. These results are from n$^{°}$ 52, page 63 ...
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Serre says open covers do not form a set, why? Directed sets and limits.

The following selection is from Serre's FAC (Chapter 1, §3, n°22, page 26). The relation `$\mathfrak{U}$ is finer than $\mathfrak{V}$' (which we denote hencforth by $\mathfrak{U} \prec \mathfrak{...
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Sheafification of constant real sheaf on smooth manifold and sheaf of smooth functions

I have a question that has been motivated by this one. Since $H^k_{dR}(M)\cong \hat{H}^k(M;\mathbb{R}_M)$ I was wondering if the constant sheaf $\mathbb{R}_M$ was isomorphic to the sheaf of $C^{\...
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References on de Rham and Poincaré via sheaves

I'm trying to understand the proof of de Rham's theorem and Poincaré duality via sheaf cohomology, as in the appendix to Hubbard's book ("Teichmuller spaces and applications...") and Conlon's ("...
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Dimension of an Open Covering - Serre's FAC

I am reading Serre's Faisceaux Algébriques Cohérents (Henceforth FAC) and he uses some terminology I have not seen. I have searched around a bit but can't get a clean and clear definition. Question:...
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Cech Cohomology and module structure

I just came across a question I had never thought about and that could be simple, but I can't answer it. If we consider a scheme $X$ and we pick a sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$, ...
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Not understanding a proof about coherent sheaves on projective schemes in Hartshorne

I have been stuck on the proof of the following statement for a while now. Let $S$ be a graded noetherian domain which is finitely generated by $S_{1}$ as an $A$-algebra where $A = S_{0}$ is a ...
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Reference for algebro-geometric definition of the $\cup$-product in (sheaf) cohomology.

Can anyone give me a reference on how the $\cup$-product of sheaf cohomology is defined? I read somewhere that it has to do with the Yoneda pairing of Ext, but my naive approach did not work, because $...
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Is the total ring of fractions mod the regular functions flasque?

I want to show that $$0\to \mathcal O_{\mathbb{P}_k^1}\to \mathcal K\to \mathcal K/\mathcal O_{\mathbb{P}_k^1}\to 0$$ is a flasque resolution of $\mathcal O_{\mathbb{P}_k^1}$ with $k$ infinite, but ...
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On cohomology of flasque sheaves of abelian groups

$\underline {Background}$: Suppose we have a short exact sequence of sheaves of abelian groups on $X$ as follows, $0\to\mathcal {F}\to \mathcal {I}\to \mathcal {G}\to 0$ with $\mathcal F$ a flasque ...
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Chow groups with coefficients in a local system

$\newcommand{\CH}{\mathrm{CH}} \newcommand{\F}{\mathscr{F}} $ Let $X$ be a smooth projective variety over a field $k$. Let $\F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski ...
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Sheaf cohomology and singular cohomology

Let $X$ be a manifold, $\mathcal{F}$ be an abelian sheaf on $X$. We can consider the etale space $F$ of $\mathcal{F}$, which is the disjoint union of stalks, whose topology is generated by local ...
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Exact sequences of sheaves

I don't understand something : in the notes by Justin Campell "Some examples of the Riemann-Hilbert correspondence", it is stated that there is an exact sequence $$ 0 \to \delta_{\infty} \to j_!IC_{\...
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Cohomology of tensor product of pullback in $\mathbb P^1\times\mathbb P^1$.

Let $X=\mathbb P^1\times\mathbb P^1$, and let $\pi_1$ and $\pi_2$ be the projection maps. For each $a,b\in\mathbb Z$, we have a sheaf of $\mathcal O_X$-modules $\mathscr F_{a,b} = \pi_1^*\mathcal O(a)\...
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Perversity and minimal extension functor

Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary. Question 1 : Is it true ...
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How does isomorphism of schemes induce maps on cohomology groups

Let $f:X \to X$ be an isomorphism of schemes. Let $\mathcal{F}$ be sheaf of abelian groups on $X$. When does $f$ induces a map $H^i(X, \mathcal{F}) \to H^i(X, \mathcal{F})$? I've seen similar ...
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Exercise 18.2.F. Foundations of algebraic geometry

Suppose $\pi: X\to Y$ is any quasi-compact separated morphism, $\mathcal{F}$ is a quasi-coherent sheaf on $X$, and $Y$ is a quasi-compact separated $A$-scheme. Describe a natural morphism $H^i(Y,\pi_*...
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A question on Hartshorne Chapter III Proposition 2.6

When I read Hartshorne, I saw the Proposition 2.6 in Chapter III as follows: Let $(X,\mathcal{O}_X)$ be a ringed space. Then the derived functors of the functor $\Gamma(X,-)$ from the category of $\...
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The cokernel of $H^0(\hat{X},\sigma^*L^k)\to H^0(E,\mathcal{O}_E)\otimes L^k(x)$

Let $X$ be a compact manifold. $\sigma:\hat{X}\to X$ is the blow up of $X$ of $x\in X$. Denote $\sigma^{-1}(x)$ by $E$. $L^k\to X$ is a very ample line bundle. $L^k(x)$ is a fiber. And by $L^k\to X$ ...
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In what sense is a constant sheaf of abelian groups with stalks G isomorphic to G?

In the English translation of Serre's FAC (link to the PDF can be found in this mathoverflow discussion in the top answer) Serre gives his first example of a sheaf of abelian groups, the constant ...
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Application of Cartan's theorem

The question is as follows: Let $X$ be a Stein space. Show that any epimorphism $\mathcal{S}\rightarrow \mathcal{T}$ of coherent analytic sheaves induces an epimorphism $\mathcal{S}(X) \rightarrow \...
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Calculating cohomology of sheaves

I am trying to prove that the twisted cubic $C: (u,v)\rightarrow(u^3,u^2v,uv^2,v^3)$ has as resolution $$ 0\rightarrow \mathcal O(-1)^2\rightarrow \mathcal O^3\rightarrow \mathcal I_C(2)\rightarrow0, $...