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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Project of understanding proof that the ampleness of pull back implies ampleness under some condition ( Gortz's Algebraic Geometry book, Vol.2. )

I am reading the Gortz's Algebraic Geometry, Vol.2, proof of Lemma 23.7 and stuck at some argument Lemma 23.7. Let $X$ be a quasi-compact ( quasi-separated ) scheme, and let $i: X' \to X$ be a closed ...
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Confusion in proof of $H^1(X,\mathcal{O})=0$ where $X$ is an open disk

In Otto Forster's Lecture on Riemann Surfaces, I have faced a small confusion in which I am certainly overlooking something in the proof of Theorem 13.4 that for $X:= \{z\in \mathbb{C}: |z|<R\}, 0&...
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Which Leray acyclicity theorems are true, and when?

I've been doing some reading around the subject and I've encountered three (arguably five) variations of the Leray acyclicity theorem given by different sources. I know three of them are true; I'm ...
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How to prove the de Rham complex of sheaves of modules.

My question: Let $X$ be a topological space. Let $\mathcal{A} \rightarrow \mathcal{B}$ be a homomorphism of sheaves of rings. Denote $d: \mathcal{B} \rightarrow \Omega_{\mathcal{B} / \mathcal{A}}$ ...
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F. Warner exercise 5.8 (foundations of Differentiable Manifolds and Lie groups)

Context In section 5.6 of Warner's Book the relationship between sheaves (the author calls sheaves to what is usually defined as Étale spaces) and presheaves is discussed. Given a sheaf S, we define ...
Juan MF's user avatar
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Significance of second (and higher) cohomology groups

I have been studying cohomology (sheaf cohomology and Čech cohomolgy) and I have an intuition of what the first cohomology group means. For example, the first derived functor measures to what extent ...
Juan MF's user avatar
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Long exact sequence of the exponential short exact sequence and elements of $H^1(X,\mathcal{O}^*_X)$

I am trying to understand how to think about the elements of $H^1(X,\mathcal{O}^*_X)$ I know one way is to think about it as the Picard group via the isomorphism. But when I try to "see" the ...
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Homology defining quasi-isomorphisms vs sheaf cohomology

I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after ...
Tom Gatward's user avatar
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Cohomology of $j_* L$

Let $j : \mathbb C^* \to \mathbb C$ be the inclusion. Write $U = \mathbb C^*$ and $X = \mathbb C$. I want to calculate the cohomology groups $H^*(X,j_* L)$ where $L$ is a local system on $U$. It is ...
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How is the sheafy De Rham cohomology functorial?

$\newcommand{\T}{\mathscr{T}}\newcommand{\C}{\mathscr{C}^\infty}$I've been enjoying Iversen's book on sheaf cohomology. He briefly mentions De Rham cohomology and a sheafy perspective on it but he ...
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Resolution and long exact sheaf sequence of cohomology groups

(I am reading Huybrechts's Complex Geometry) I also looked some classic sources, but I didn't find anything addressing the below. Proposition $B.0.35$ Let $$0\longrightarrow \mathcal{F}^0\...
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$R^2f_*\mathbb{Z}$ is isomorphic to the constant sheaf $H^2(X_0,\mathbb{Z})$

Let $f:X \rightarrow S$ be a smooth proper family of K3 surfaces and let $X_t=f^{-1}(t)$. Let's suppose $S$ is a disk in $\mathbb{C}^n$. I know that the Betti/Hodge numbers are constant, so the direct ...
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De rham cohomology over complex manifolds

I am studying sheaf cohomology of complex manifolds and, while reading some proof about Dolbeault cohomology, I realized that there is a $\bar{\partial}$-Poincaré Lemma which gives us the local ...
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How to prove Leray's theorem without using Sheaf cohomology

Let $X$ be a manifold. Leray's Theorem for Čech Cohomology states that if a covering $\mathfrak{U}$ is acyclic $(\check{H}^p(U_{i_0}\cap\ldots U_{i_k},F)=0$ for every finite intersections of elements ...
Juan MF's user avatar
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étale $\ell$-adic cohomology is a Weil cohomology theory

I was reading https://mathoverflow.net/questions/85078/ell-adic-weil-cohomology-theory and in the first paragraph it is said that $\ell$-adic cohomology is a Weil cohomology theory over separably ...
FreeFunctor's user avatar
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Holomorphic 1-forms as a subspace of the first de Rham cohomology group of a surface

If we fix a complex structure $c$ on a closed oriented surface $S$, then the space of holomorphic 1-forms $\Omega^1(S,c)$ has complex dimension equal to the genus $g$ of $S$ and - since they are all ...
Christian's user avatar
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Cohomology map induced by Frobenius morphism

Suppose $X$ is an elliptic curve over a field $k$ of characteristic $p>0$. Let $F\colon X\to X$ be the Frobenius morphism. Then $F$ induces a map $F^*\colon H^1(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}...
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Right Derived Functors from Acyclic Resolution ( Ref: Algebraic Geometry by Robin Hartshorne)

At the page 204 , Hartshorne defined right derived functors as $R^{i}F(A)$ = $h^i(F(I^{.}))$ , where $I^{.}$ is an injective resolution of $A$. Now on page 205 at Prop 1.2A he gives a natural ...
nullstellensatz92's user avatar
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Dimension of sheaf cohomology of divisor on $\mathbb{P}^1$ depends only on the degree of the divisor

I am struggling with the following exercise, which I have to do without Riemann-Roch: Show that the dimensions of $H^0(\mathbb{P}^1, D)$ and $H^1(\mathbb{P}^1, D)$ depend only on the degree of D. ...
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Complex of sheaves, Eilenberg-MacLane spectra and hypercohomology

This question is about the relation between the category of spectra and the category of chain complexes of abelian groups. Specifically, I am trying to understand the examples from Deligne cohomology. ...
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Cech cohomology stays the same restrcting to a subcover?

Let $X$ be a topological space with an open covering $\mathcal{U}=\{U_i\}_{i\in I}$. Let $\mathcal{V}$ be a subcovering of $\mathcal{U}$, i.e. there exists a subset $J\subset I$ s.t. $\mathcal{V}=\{...
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Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)

This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
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Computation of $\mathrm{Tor}^1$ for two structure sheaves of subvarieties

Let $X$ be a hyperplane defined by $(x_3=0)$ and let $p$ be a point $[1:0:0:0]$ in the projective 3-space $\mathbb{P}^3_{\mathbb{C}}$. Then I want to compute $\mathrm{Tor}^1_{\cal{O}_{\mathbb{P}^3}}({\...
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Deligne complex and sheaf of chain complex

Deligne complex $\mathbb{Z}(n)$ and some other complexes can be defined as follows, for example, in this paper, wikipedia and nlab, $$\mathbb{Z}(n):=(0\rightarrow \mathbb{Z}\rightarrow \Omega^0 \...
timaeus's user avatar
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Correspondence, cycle class map and Bloch's decomposition of the diagonal

I'm studying Weil cohomology theories, in particular étale $\ell$-adic cohomology, and I have found some problems related to the cycle class map. Let $X$ be a scheme of dimension $d$, my ingredients ...
FreeFunctor's user avatar
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Twists of morphisms and torsors

Let $S$ be a scheme and $\tau$ some topology on $\mathrm{Sch}/S$. (I'm primary interested in \etale topology). Let $\pi:Y\to X$ be a morphism of $S$-schemes. A twist of $\pi$ is a morphism $\pi':Y'\to ...
Ben's user avatar
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How to show naturality of the section when trying to prove gluing axiom for sheaf hom

The question is to show that the sheaf Hom$(U):=\{$morphisms of sheaves $\mathcal{F}|_{U}\to \mathcal{G}|_{U}\}$ is indeed a sheaf, where $\mathcal{F}$ and $\mathcal{G}$ are fixed sheaves on the same ...
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Cohomology of twisted tangent bundle of fake projective plane

Let $X$ be a fake projective plane, id est $X$ is a smooth projective surface with the same Betti numbers of $\mathbb{P}^2_{\mathbb{C}}$ but not isomorphic to $\mathbb{P}^2_{\mathbb{C}}$. These ...
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Proposition 2.7.2 from Kashiwara and Schapira, Sheaves on Manifolds

I am reading Kashiwara-Schapira's Sheaves on manifolds, and I am having trouble understanding their proof of Proposition 2.7.2 which says the following : Let $X$ be a Hausdorff space, $F \in D^+(\...
stratified's user avatar
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K3 surfaces: equivalent definitions

I am just starting to study complex K3 surfaces and I'm trying to understand why the different definitions I found are actually equivalent. Definition 1: a (complex) K3 surface is a compact, connected,...
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Is this sheaf cohomology group trivial?

Let $X$ be a smooth irreducible curve (i.e. quasi-projective algebraic variety over $k$ such that all its irreducible components have dimension $1$), and in fact let $X$ be projective. Let $D$ be an ...
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Restriction of flasque sheaf onto compact subset

On page 103 of "Theory of Stein Spaces" by Grauert and Remmert is the following theorem: Let $X$ be a paracompact space and $\mathcal{F}$ a sheaf of abelian groups on $X$. Let $$K_0 \...
Muhammad Haris Rao's user avatar
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Brouwer's fixed-point theorem

I'm trying to understand the cohomological proof of this theorem, but I'm stuck at a point, which brings me to ask a more general question: let $i:Z\rightarrow X$ be the inclusion of a closed set into ...
CarloReed's user avatar
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0 cohomology of Riemann sphere

Consider the Riemann sphere $S^2$, together with the sheaf $\Omega_{S^2}$ which is defined to be $\Omega_{S^2}:=\operatorname{Ker}\overline{\partial}$ with $\overline{\partial}\colon\mathcal A^{1,0}\...
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Lazarsfeld's proof of Mumford's regularity Theorem 1.8.3

I want to apologize first of all if my English is bad, I hope my message will still be understandable. I’m currently reading Lazarsfeld’s book "Positivity in Algebraic Geometry I" and I’m ...
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Hodge decomposition metric dependency

I’m currently studying Complex Geometry by Daniel Huybrechts and I can’t understand why it is crucial to prove that Hodge Decomposition does not depend on the chosen metric. I’m talking about ...
Matteo Rossi's user avatar
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Direct sum of two sheaves

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two sheaves on $X$ ,is $ H^{i}(X,\mathcal{F}_1 \oplus \mathcal{F}_2) \simeq H^{i}(X,\mathcal{F}_1) \oplus H^{i}(X,\mathcal{F}_2) $ ?
BouaichiAnas's user avatar
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Sheaf cohomology on quotient stacks

Suppose I have a scheme $X$ over $\mathbb C$, acted on by a finite group $G$. Let $\mathcal F$ be a $G$-equivariant coherent sheaf of $\mathcal O_X$-modules. Then I can form the stack quotient $[X/G]$,...
stacklearner's user avatar
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Subsheaf with supports of flasque sheaf is flasque

I'm trying to show that the sheaf $\mathcal{H}_Z^0(\mathcal{F})$ defined as $\mathcal{H}_Z^0(\mathcal{F}) = (U \mapsto \Gamma_{Z\cap U}(U,\mathcal{F}))$, which is a subsheaf of $\mathcal{F} \in \...
Anthony Lee's user avatar
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Construct cocyle from Leray-Serre spectral sequence

Let $F \xrightarrow{f} E \rightarrow B$ be a fiber bundle and $L$ is the locally constant sheaf on $E$. $B$ is NOT simply connected. We can apply Leray-Serre spectral sequence to compute the local ...
wsh's user avatar
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Lifting morphism of stalks to sheaf morphisms

I attempted to prove Hartshorne, Proposition 2.2, which is Proposition. Let $(X,\mathcal{O}_X)$ be a ringed space. Then the category of $\mathcal{O}_X$-modules has enough injectives. Pick some ...
Anthony Lee's user avatar
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A Problem With Simplicial DeRham Cohomology

I am getting a contradiction by calculating the simplicial DeRham complex of an arbitrary manifold and getting it to be trivial. I also get a similar contradiction using the Godement resolution for ...
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Why is $Ext_{\mathcal{S}}^{j}(H,G) = 0$ if $H$ is a free sheaf and $G$ a flabby sheaf?

So, I am reading Schapira's and Kashiwara's "Sheaves on Manifolds" and in the proof of proposition 2.6.3, it is stated that, for a free $\mathcal{S}$-module $H$ and a flabby $\mathcal{S}$-...
Duarte Costa's user avatar
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Čech Cohomology on Sheaf of maps for a Contractible space is 0

I have been pondering on this question for half a year and haven't gotten an ideal answer. Hope someone can help! Conjecture: Given a topological space $X$ and any dimension $i\in \mathbb{N}$, the ...
XiaoChen Xiao's user avatar
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Reference for result and proof that $R(g\circ f)\cong Rg\circ Rf$ for morphisms of ringed spaces $X\xrightarrow{f}Y\xrightarrow{g}Z$

I am trying to find a reference that states and proves the following Lemma: Lemma. Let $(X,\mathcal{O}_X)\xrightarrow{f}(Y,\mathcal{O}_Y)\xrightarrow{g}(Z,\mathcal{O}_Z)$ be morphisms of ringed ...
Elías Guisado Villalgordo's user avatar
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Sheaf cohomology free module

Let $X \rightarrow$ Spec$A$ be a scheme over a ring $A$. I know that if $X$ is a projective scheme over $A$, and $A$ is Noetherian. $H^p(X, \mathcal{O}_X)$ is afinitely generated A-module for all $p \...
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ADHM construction: why two bundles of the monad have the same ranks?

I'm reading The ADHM construction of Yang-Mills instantons by Simon Donaldson. A theorem in the section 4 says: Let $E$ be a rank $r$ holomorphic bundle over $\mathbb{CP}^3$ with $c_2 = k$. Suppose ...
Arith Geo's user avatar
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What is "torsion" in the context of cohomology, and why is it important?

I searched for some answers, but most answers discussed the meaning of torsion, instead of its definition. Not knowing how the torsion is defined (in cohomology) I couldn't understand those answers at ...
Youngsub Yoon's user avatar
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Question for exercise IV.3.6 b) Hartshorne

Consider $X$ smooth projective curve such that $X\subseteq \mathbb P^3$, $X$ is not contained in any hyperplane, and $g(X)=1$. In this exercise the idea is to consider the long exact sequence we get ...
raisinsec's user avatar
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Is left t-exactness of $Rf_\ast$ for $f : X\longrightarrow Y$ affine just Andreotti-Frenkel

Sorry I'm a little lost in those perverse sheaves. I think this left t-exactness property is called Artin-Grothendieck vanishing, which I take to be the relative version of $H^n(X,\mathcal{P})=0$ for $...
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