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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100 views

fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one (so living in standard commutative algebra setting) and let $\phi: R \to S$ a faithfully flat. Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ ...
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Zariski cohomology of an etale sheaf vs etale cohomology.

Let $\mathcal{F}_{Zar}$ be a Zariski sheaf (on the big site of schemes or $S$-schemes) and $\mathcal{F}_{et}$ be its etale sheafification on the big site. Let $X$ be a scheme. If we know that $H_{Zar}^...
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Relation between first sheaf cohomology of scheme X and the global section of sheaf on diagonal.

Let $X$ be a scheme and $\mathfrak{U} = \{U_i\}$ be an open cover of $X$. Let $\mathscr{F}$ be a sheaf on $X$. Denote $C^1$ for a first component of Cech complex. What I observe is : $C^1 = \Pi_{i &...
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Construction of Quot schemes and construction of certain closed subschemes

I am still studying Nitsure's Construction of Hilbert and Quot schemes, and I am yet again stuck, this time on page 25. I understand what lemma 5.4 says, but I do not get its proof. Let me sketch the ...
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Injective map between sheaf cohomologies

I am currently working through Forster's Lectures on Riemann Surfaces and I may be on the right track on this exercise, but I'd like to make sure. The exercise in question is 12.3: Let $X$ be a ...
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Semi-continuity theorem and construction of Hilbert and Quot schemes

I am studying Nitsure's wonderful essay Construction of Hilbert and Quot schemes, and I am stuck on page 21. Given a Noetherian schemes $S$ and a coherent sheaf $\mathcal{F}$ over $\mathbb{P}^{n}_{S}$,...
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Cohomology of external tensor product of sheaves

Let $\mathcal F$ and $\mathcal G$ be sheaves on topological spaces $X$ and $Y$ respectively. The external tensor product of $\mathcal F$ and $\mathcal G$ is the sheaf on $X\times Y$ defined as $\...
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Ravi Vakil's book on Algebraic Geometry , Exercice 19.2.A

In what follows, $C$ will be a projective, geometrically regular, geometrically integral curve over a field $k$, and $\mathcal{L}$ is an invertible sheaf on $C$. $19.2.A.$ EXERCISE. Suppose $\mathcal{...
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Contradiction with “cohomology group of coherent sheaf of module on projective scheme over field is finite dimensional”

While working on an exercise, I came up with something that seems like a contradiction and I don't seem to find out why one of my two reasoning is wrong. Consider $X = \mathbb{P}^1_k$ and $Z = P_1\cup\...
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Castelnuovo-Mumford regularity and a short exact sequence

Let $k$ be a field and $\mathcal{F}$ coherent over $\mathbb{P}^{n}_{k}$, $H\subset\mathbb{P}^{n}_{k}$ a hyperplane. In Nitsure's paper Construction of Hilbert and Quot schemes, page 9 right at the ...
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Relationship between sheaf cohomology and $H^n(F)$

If $F$ is a complex of sheaves in the derived category of sheaves, what is the relation (if there is any) between $H^n(X,F)=H^n(R\Gamma(X,F))$ and $H^n(F)$?
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Why are line bundles called principal $\mathbb G_m$-bundle?

We see the standard statement that $H^1(X,F)$ claasifies principal $F$-sheaves (say $F$ is a sheaf of groups) on $X$. By definition, these are sheaves $G$ of sets with an $F$-action such that they ...
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Simple proofs of $H^1(X, \mathcal{O}^*)=0$ when $X$ is an open Riemann Surface

I am trying to get proof that $H^1(X, \mathcal{O}^*)=0$ when $X$ is an Open Riemann surface. Looking at some books I have seen the following two approaches: Use Mittag-Leffer distributions, Runge ...
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1answer
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Global sections and Fiber products $X \times_S \operatorname{Spec} k$

Let $X $ a $S$-scheme ($S$ another scheme). Suppose $k$ is a field and $S$ has a $k$-valued point, that is a map $p: \operatorname{Spec} k \to S$. If $X \times_S \operatorname{Spec} k$ denotes the ...
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Question about the construction of Godement Resolution and flabby sheaves being acyclic

I am reading an introductory book to Sheaf theory, a kind of gentle introduction, which is nice for the most part, comparing it with other more technical books on the subject, which I am not yet ...
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If $D$ is nef, then $h^1(D)=h^2(D)=0$

Let $k$ be a number field and $S$ a smooth algebraic surface over $k$ which is birational to $\Bbb{P}^2_k$. I'm trying to understand why the following is true: If $D$ is nef divisor on $S$, then $h^1(...
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Vanishing of the Zariski cohomology $H^n(X,\mathbb{Z}[\mathbb{G}_m])$ for $n>1$

Let $X$ be a smooth and irreducible variety over a field. Does the Zariski cohomology $H^n(X,\mathbb{Z}[\mathbb{G}_m])$ vanish for $n>1$? Here $\mathbb{Z}[-]\colon Set\to Ab$ is (the Zariski ...
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sheaf and de Rham cohomology of projective lines glued to order $n$

Let $X = \mathbb{P}^1_k \cup_n \mathbb{P}^1_k$ be the union of two projective lines, glued together at a single point, where the gluing is of order $n$. I would like to compute the sheaf cohomology ...
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Meaning of certain cohomology classes and morphisms in Prop 3.3 of Deligne-Illusie's paper on mod $p^2$ liftings and decompos. of the de Rham complex

I am still struggling with Deligne and Illusie's paper (https://eudml.org/doc/143480). They say on page 261, in the course of the proof of theorem 3.3: The class $e(K)$ (which is associated to the ...
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Čech cohomology of sites: How does a “composite” morphism induce an element of $\operatorname{Ext}^{2}$?

This is yet another question concerning Deligne and Illusie's paper on $W_{2}(k)$ liftings and the degeneration of the Hodge-de Rham spectral sequence (https://eudml.org/doc/143480). In the proof of ...
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Étale cohomology vs flat cohomology

I'm recently reading the Milne's Etale Cohomology (1980 Princeton University Press). When I read the part of etale cohomology vs flat cohomology in page 115, I have trouble understanding the first ...
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How does a morphism in a distinguished triangle induce an element of $\operatorname{Ext}^{2}$

On p. 260 of Deligne and Illusie's paper (https://eudml.org/doc/143480) it says: Let $e(K)\in\operatorname{Ext}^{2}(H^{1}(K),H^{0}(K))$ be the class defined by the degree $1$ morphism in the ...
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Exactness of a pairing and the trace morphism

If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
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Why is the relative Frobenius morphism flat?

Let $f:X\longrightarrow S$ be any morphism of $\mathbb{F}_{p}$-schemes. Why is the relative Frobenius $F=F_{X/S}$ flat, as claimed here, and why does this imply $H^{\bullet}(\mathbf{Pullb}(F_{S},f),F_{...
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Cohomology(ies) of simplicial sheaves

Let $X$ be a topological space and denote by $Ab(X)$ the category of abelian sheaves on $X$. My question is on the category of simplicial abelian sheaves $[\Delta^{op},Ab(X)]$. A natural way to define ...
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Representable cohomology theories in motivic homotopy theory

I am reading Mazza, Voevodskys and Weibels book on Lecture Notes on Motivic Cohomology and have grown curious about the following question: Which cohomology theories on $Sm/k$ is representable, i.e. ...
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Hodge star operator and “Serre duality”

I am familiar with the Hodge star operator or Hodge duality in the theory of finite-dimensional differentiable manifolds, which gives an isomorphism $\star:\Omega^{i}(M)\longrightarrow\Omega^{n-i}(M)$ ...
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Liu Exercise 5.21

I'm trying to solve Liu's exercise 2.1 Chapter 5 which states: Here is what I'm thinking so far: a) This is vacuously true since J is a single element set so there is no $i<j$ and this is trivial. ...
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Proving that the Čech complex is actually a complex

All authors I have seen handwave the proof of the Čech complex actually being a complex as a calculation. But I tried to do it and I don't see how it works. Maybe there's a mistake in my calculation. ...
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45 views

Why are de Rham and Hodge cohomology groups coherent if the underlying morphism is proper?

I am reading a paper on Hodge cohomology in order to make myself more acquainted with the topic, and the authors take a few facts for granted that are unknown to me. In particular: If $X\...
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1answer
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Confusion about derived functors and inverse image sheaves

I am confused about the following (probably obvious) assertion. Let $X, Y$ be topological spaces and let $f: X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of abelian groups on $Y$ and let ...
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Computing cohomology of Cech-De Rahm Complex

Bott & Tu use what they call the "Cech-de Rahm complex" a lot, which is a double complex that uses the Cech differential horizontally and the de rahm differential vertically, with ...
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1answer
86 views

global sections of ideal sheaf quotient $\mathcal{I}/\mathcal{I^{2}}$

Let $X=\mathbb{P}_{k}^{2}$, $P$ a point in $X$ with its ideal sheaf $\mathcal{I}$. Then I would like to know what is $H^{0}(X, \mathcal{I}/\mathcal{I}^{2})$. My first thought was to consider the exact ...
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49 views

Projective bundle formula for sheaf cohomology

Given a projective bundle $p:P(E)\rightarrow X$ associated to a vector bundle $E$ and a line bundle on $P(E)$ of the form $\mathcal{O}_{P(E)}(1)\otimes p^*L$ where $L$ is a line bundle on $X$, is ...
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Is it possible for the pullback of an ample line bundle under projection to be big?

Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line ...
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Čech cohomology of quasi-coherent sheaves and Leray's acyclicity theorem

My professor told us that if $X$ is a separated scheme and $\mathcal{F}$ a quasi-coherent scheme over $X$, then for any affine cover $\mathcal{U}=\left\{U_{i}\mid i\in I\right\}$, $\bigcup_{i\in I}U_{...
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Hom, H$^n$ and tensor product

If $X$ is a scheme, $\mathcal{F}$ is a quasi-coherent sheaf and $L$ is an ample line bundle, why it holds $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?
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Semisimple perverse sheaf?

Let $\mathcal F$ be a complex of constructible sheaves on a stratified algebraic variety $X$ of dimension $d$. I read (*) that if $\mathcal F$ is perverse and $\mathscr H^j(\mathcal F) = 0$ for $j \...
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Calabi-Yau conditions for a threefold in the Grassmannian $Gr(2,7)$

I am trying to show a complete intersection $X$ in the grassmannian $G(2,7)$ is a Calabi-Yau in the strict sense. By that I mean $\omega_X\cong\mathcal{O}_X$ and $h^i(\mathcal{O}_X)=0$ for $i=1,2$. ...
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Sheaf cohomology of the structure sheaf of the grasmannian G(2,7)

How can I calculate $H^i(Gr(2,7),\mathcal{O}_{Gr(2,7)})$ over the base field $\mathbb{C}$? Or in general $H^i(Gr(k,n),\mathcal{O}_{Gr(k,n)})$? The first idea came to my mind is using Plücker embedding ...
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Sheaf Cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over arbitrary field

Let $k$ be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...
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Cohomology of doubly pinched torus via spectral sequences

Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
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How to verify that the map of Lemma 20.6.1 of the Stacks Project is a homomorphism

I have been going through the Stacks Project's treatment of sheaf cohomology. First it is shown that on a general topological space that $H^1(X,\mathscr{H})$ is in bijection with $\mathscr{H}$-torsors ...
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47 views

Spectral sequence from resolution

I am watching Scholze's masterclass on Condensed Mathematics and I don't understand or can find any references on something he said. You have a resolution $$ \dots \to \mathbb{Z}[\mathbb{R}^2] \to \...
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Exactness and obstruction in sheaf cohomology

Given a topological space $X$, sheaf cohomology 'measures' the lack of exactness of the global section functor $\Gamma(X, -) : \textbf{Sh}(X) \to \textbf{Ab}$. From another viewpoint, sheaf cohomology ...
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Description of Euler Characteristic Using $\operatorname{Ext}^i$'s

I am reading Nakajima's Lectures on Hilbert Schemes of Points on Surfaces, and I am confused about a detail at the top of page 13. He is sketching a proof that $X^{[n]}(=$ Hilbert scheme of $n$ points ...
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132 views

Is Hartshorne's Remark III.2.9.1 actually a valid argument?

In Hartshorne's book Algebraic Geometry, following Proposition III.2.9 which states that on a Noetherian topological space $X$ that cohomology of abelian sheaves commutes with direct limits he follows ...
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44 views

Superfluous assumptions on Vietoris-Begle mapping theorem in Iversen's Cohomology of Sheaves

I am trying to prove the Vietoris-Begle mapping theorem as indicated in Iversen's Cohomology of Sheaves on page 203. The statement is the following: Let $f: X\rightarrow Y$ be a proper map between ...
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1answer
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How are (pre)sheaves even well-defined on sites?

I am trying to learn a bit about etale cohomology from Milne's book and I have run into some early problems. I learned ordinary abelian sheaf cohomology from Hartshorne. There, sheaves are defined on ...
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75 views

Wrong solution in Hartshorne exercise III.2.7

I'm trying to solve Exercise III.2.7(a) in Hartshorne's Algebraic Geometry. And what I got is ${ H^1(S^1, \mathbb{Z}) = 0 }$ instead of expected ${ H^1(S^1, \mathbb{Z}) = \mathbb{Z} }$. My reasoning ...

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