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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Sheaf cohomology of blowup - reference request
I am looking for a reference for the computation of the sheaf cohomology of a blowup where things are worked out in detail. I'd like to see at least sheaf cohomology of the structure sheaf of the ...
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Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Leray Spectral Sequence for Sheaf Cohomology?
Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$.
I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, ...
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Dimensions of cohomology of ideal sheaf
This question follows a previous question Sheaf morphism from closed subscheme is a closed immersion, it's just another part so I'll recall everything.
For $K=\bar{K}$ a field consider $X=\mathbb P^...
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Some questions about the functor of points perspective
If I were to define schemes as functor of points from rings to sets, what would be the morphisms? My guess is that it is not the set of all natural transformations between the functors, but some ...
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Questions about Gauss Manin connection.
Suppose $X,Y$ are two algebraic varieties over $\mathbb C$, and $f:X\to Y$ is a homomorphism. Then we can consider the relative de Rham complex $\mathcal H^i(X/Y):=\mathbf R^if_*(\Omega^{\bullet}_{X/\...
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How to construct a group-valued sheaf from set-valued sheaf
Suppose that I have a set-valued sheaf $S$ on a site $(\mathcal C, J)$.
Question 1.) Is there a canonical way to turn $S$ into a sheaf with Abelian group values?
I considered the following: for each ...
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vanishing cohomology for indiscrete Grothendieck topology
Let $\mathcal{C}$ be a category endowed with the indiscrete Grothendieck topology. Let $X$ be an object of $\mathcal{C}$ and $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Then I want to know how ...
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Compactly supported sections of a vector bundle form a cosheaf
I am trying to understand why the compactly supported sections of a vector bundle form a cosheaf. I have proven that they form a precosheaf: simply extending the compactly supported section by zero ...
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Exercise III.2.1(b) Hartshorne: Isn't the restriction of a constant sheaf on a connected space to a connected subspace again constant?
Consider Exercise III.2.1(b) in Hartshorne: Let $X=\mathbb{A}_k^n$ for $n\geq 2$ and an infinite field $k$, and let $Y\subseteq X$ be the union of $n+1$ hyperplanes in general position. Let $U=X\...
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Why are collections of all etale-maps sets?
Let $\mathbb{S} = (E,p,X)$ and $\mathbb{S'} = (E',p,X)$ be etale-sheaves over
a topological space $X$.
We write ${\rm Hom}_{et}(\mathbb{S},\mathbb{S'})$ for the set of all etale-maps between $\mathbb{...
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Finding a Leray Cover for $\mathcal{O}_X^*$
Let $X$ be a (projective) curve (or more generally a scheme).
I would like to "find" a open cover $\mathcal{U}$ of $X$ such that $H(\mathcal{U},\mathcal{O}_X^*) = H(X,\mathcal{O}_X^*)$, i.e. ...
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Induced Euler short exact sequence on wedge product
From any short exact sequence $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, we can construct the following induced exact sequence on wedge product (following apendix A):
$$ ...
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Cohomology of compact Riemann surface
Let $X$ be a compact Riemann surface. I will denote by $H^{.}(X, \mathbb{C})$ (respectively $H^{.}(X, \mathbb{Z})$) the Cesh cohomology associated to the constant sheaf $\mathbb{C}$ (respectively $\...
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Find $H^1(I_X(r))$ for all $r\geqslant0$ where $I_X$ is a four-point set's sheaf of ideals
Let $X\subset\mathbb P^n$ be a set of $4$ points and let $I_X\subset\mathcal O_{\mathbb P^n}$ be it's sheaf of ideals. I'd like to compute $H^1(I_X(r))$ for all $r\geqslant0.$
I've tried to consider a ...
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Describe $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by exact short sequence of sheaves
Let $i:X\subset\mathbb P^1$ be a set of $m$ points and let $I_X\subset\cal O_{\mathbb P^n}$ be its sheaf of ideals. Then the sequence
$$(1)0\to I_X(m-2)\to\mathcal O_{\mathbb P^1}(m-2)\to\mathcal i_*...
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Why is $()^{n}:\mathbf{G}_{m}\longrightarrow\mathbf{G}_{m}$ surjective in the $fppf$ topology?
Let $X$ be a scheme. Why is $()^{n}:\mathbb{G}_{m}\longrightarrow\mathbf{G}_{m}$ surjective in the category $X_{fppf}$?
(Here $\mathbf{G}_{m}(X)=X\otimes_{\mathbb{Z}}\operatorname{Spec}(\mathbb{Z}[T,X]...
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Adjunction formula over arbitrary fields
In Beuville's Complex Algebraic Surfaces we find the adjunction formula (or "genus formula", as he calls it) as follows:
Let $C$ be an irreducible curve on a surface $S$ (i.e. a smooth ...
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Resolution of Castelnuovo-Mumford $m$-regular sheaves
Recall that a coherent sheaf $\mathcal F$ on $\mathbb P^n$ is $m$-regular if $H^i(\mathbb P^n,\mathcal F(m-i)) = 0$ for all $i\geq 1$. Here's an exercise from Jarod Alper's marvelous notes on Stacks ...
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Sheaf cohomology of Hopf surface.
Let $X$ be Hopf surface. (i.e. $X =\mathbb{C}^2\backslash\{(0,0)\}/∼$ , $(z_1,z_2)∼(2z_1,2z_2)$). $\mathcal{O}$ is the structure sheaf of $X$(i.e. $\mathcal{O}(X)=\{f\ |f$ is holomorphic on $X$$\}$). ...
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Example of sheaf cohomology
Suppose $X$ is a compact complex surface,
$$
X=\mathbb{C}^2\backslash\{(0,0)\}/\sim,
$$
where the relation is $(z_1,z_2)\sim(2z_1,2z_2)$.
Compute $H^0(X, \mathscr{O}), H^2(X, \mathscr{O}),H^1(X, \...
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turn a sheaf $ F $ into a cosimplicial sheaf via Godement construction
let $ X $ be a topological space and $ F $ a sheaf on $ X $. Let $ X_{\text{disc}} $ be the set $ X $ but now with the discrete topology.
One possible way to associate to $ F ...
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Functoriality of Sheaf Cohomology
Let $f:X\rightarrow Y$ be a continuous map and $\mathscr F$ be a sheaf of abelian groups on $X$ and $\mathscr G$ be a sheaf on $Y$. Then in general, how to define $H^i(Y,\mathscr G)\rightarrow H^i(X,f^...
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What are Picard groups for real projective spaces as smooth manifolds?
I am solving a list of problems related to smooth vector bundles over a smooth manifold considered as locally free modules over the sheaf of smooth functions, to Cech cohomologies with coefficients in ...
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Two definition of sheaves: thoughts?
This might be a naive question. However, as a absolute noob picking up cohomology theory I am deeply confused by these two definitions of sheaf that I have encountered. I definitely lack some ...
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Cohomology of a tensor product of sheaves of vector spaces
Let $k$ be a field and $I$ and $J$ two sheaves $\mathcal{C}^{op}\to \mathrm{kVec}$ of $k$ vector spaces on a reasonable site $\mathcal{C}$ with terminal object $X$ which are injective and acyclic ...
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Calculate $H^1(S^1, \mathbb{Z})$ [duplicate]
I'm learning Hartshorne section 3.2 and I'm wondering how to calculate $H^1(S^1, \mathbb{Z})$. I have refered the question Let $S^1$ be the circle (with its usual topology), and let $\mathbb{Z}$ be ...
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On the cohomology group of kernel of $\mathcal{F} \to \mathcal{F}(d)$
This is from Mumford-Oda's Algebraic Geometry 2, And here is pdf of chapter 7-8. https://www2.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf
My question is on page 243, I can't see why $H^{i+1}...
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Bott and Tu Exercise 10.7 (Cohomology of $S^1$ with twisted coefficients)
Bott and Tu exercise 10.7 is as follow (not an exact quote):
Let $\mathcal{F}$ be the presheaf on $S^1\}$ which associates to every open set the group $\mathbb{Z}$. They include a picture indicating ...
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Is $\chi=1$ for rational surfaces over an aribitrary field?
I've found in many textbooks that if $S$ is a smooth rational surface over an algebraically closed field, then $\chi(S,\mathcal{O}_S)=1$ (more precisely, $h^0=1$ and $h^1=h^2=0$).
I'm trying to find ...
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Concrete and simple definition of the cup product in sheaf cohomology
I would like a precise definition of the cup product in the simple form of
$$\mathcal{H}^{i}(X,F)\times\mathcal{H}^{j}(X,G)\to \mathcal{H}^{i+j}(X,F\otimes G)$$
Actually, I have seen the answer here ...
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The inclusion map is $i:X\to Y$ implies isomorphism of cohomology $ H^k(Y,i_*\mathscr{F})\cong H^k(X,\mathscr{F}). $
Suppose $X$ is a closed submanifold of $Y$ and the inclusion map is $i:X\to Y$. Let $\mathscr{F}$ be a sheaf of abelian groups over $X$. Prove that for any $k\geq 0$, we have
$$
H^k(Y,i_*\mathscr{F})\...
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Lower shriek pushforward of injectives.
Given an open subscheme $f: U\hookrightarrow X$ and an injective etale sheaf of abelian groups $\mathcal{I}$ on $U$, then is it necessarily true that $f_!(\mathcal{I})$ is also injective?
If not then ...
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Is this a correct way of constructing the Mayer–Vietoris sequence in sheaf cohomology?
I know that the Mayer–Vietoris sequence for sheaf cohomology can be derived from the spectral sequence relating Čech/presheaf cohomology to sheaf cohomology, but I am wondering if the following ...
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Cohomology groups (on curves) under closed immersions
Suppose $X$ is a separated, projective curve, and $Z = V(\mathcal{J})$ (where $\mathcal{J}$ is a nilpotent sheaf of ideals) is a closed subscheme of $X$ with closed immersion $i : Z \to X$.
Is it true ...
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Constructing a base change map $u^*R^if_*(F) \to R^ig_* v^* (F)$.
$\newcommand{\F}{\mathscr{F}}$ $\newcommand{\I}{\mathscr{I}}$ I am trying to understand the proof of Hartshorne's Proposition III.9.3, which states that if $f: X \to Y$ is separated and quasicompact, $...
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For divisor $D$, how to describe isomorphisms $H^0(X,\mathcal{O}_D(nD)) \xrightarrow{\sim} H^0(X,\mathcal{O}_D\left((n+1)D\right)$?
Let $X$ be a scheme, $D$ an effective divisor on $X$ with structure sheaf $\mathcal{O}_D$, and $U = X\setminus D$. I think I need $D$ either ample or affine. If necessary we can assume $\mathcal{O}_X$ ...
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Does quasi-isomorphic complexes of sheaves give the same cohomology groups?
Let $F^{\bullet}$ and $ I^{\bullet }$ be two bounded below complexes of sheaves of $O_X$-modules (on a scheme X) and let $F^{\bullet}\rightarrow I^{\bullet}$ be a quasi-isomorphism between complexes ...
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Nonconstant sheaf with microsupport contained in the zero section
Consider the inclusion $i:B^n\to \mathbb{R}^n$ of the open ball into Euclidean space. I want to understand the proper pushforward $F=i_!\mathbb{C}_{B^n}$ of the constant sheaf. Now since $B^n$ is open,...
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Vector bundles are first cohomology sets
$\textbf{Disclaimer:}$ As suggested in the comments, I have reposted this question on mathoverflow
Suppose $X$ is a variety, then we have the result that $\operatorname{H}^1(X, \mathcal{O}^*_X)=\...
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Quasi-isomorphism of complexes of sheaves
I am currently reading Robert Friedman's notes on sheaf cohomology and hypercohomology http://www.math.columbia.edu/~rf/cohomology.pdf
In it he defines quasi-isomorphism of complexes of sheaves by ...
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Is $Ext^1(\mathcal{O}, L)$ always $0$?
Consider an elliptic curve $X$.
The $\mathcal{O}_X$-module $\mathcal{O}_X$ itself is free, so in particular projective.
Thus, $Ext^1(\mathcal{O}_X, L)=0$ for all line bundles $L$ over $X$ by Exercise ...
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Euler characteristic when the base space admits a finite good cover
I am trying to solve Exercise 14.37 on the differential-form book by Bott and Tu:
Let $\pi:X\rightarrow Y$ be any map and $\{U\}$ a finite good cover of $Y$. Show that the Euler characteristics of $X$
...
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Sheaf cohomology of a space and cohomology of the espace etale
I have a very basic question about cohomology of sheaves. Suppose $\mathcal{F}$ is a sheaf of abelian groups over a topological space $X$. Then $\mathcal{F}$ itself is a topological space with a ...
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Functoriality of sheaf Cohomology [duplicate]
Let $f:Y\rightarrow X$ be a morphism of schemes. I would like to construct a natural map of homologies $$H^{q}(X,F)\rightarrow H^{q}(Y,f^{*}(F))$$ where $F$ is a sheaf on X. My idea is to take an ...
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anti-ample line bundles and global sections
Let $X$ be a variety say over $\mathbf{C}$, $L$ a very ample line bundle and $\mathcal{E}$ a coherent sheaf on $X$.
It is well known that $H^0(L^*)=0$. However, I'm curious what there can be said ...
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Hartshorne Theorem III.3.5 Proof
Let $X=\operatorname{Spec}A$ be the spectrum of a noetherian ring $A$. I am trying to understand the proof of the following fact from Hartshorne: For all quasi-coherent sheaves $F$ on $X$, and for all ...
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Induced group action on cohomology of vector bundle
Let $p:E\to X$ be a vector bundle on a smooth projective curve $X$. Let $\Gamma$ be a finite subgroup of $Aut(X)$ which has a lifted action on $E$.
How do we get an induced action of $\Gamma$ on $H^i(...
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Homology of the diagonal sequence of 3x3 commutative diagram of modules
Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules),
and suppose that we have a 3x3 commutative diagram, where ...
- 1,014
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Cohomology of pushforward and higher pushforward sheaves
I am reading through Peters-Steenbrink Mixed Hodge structures and I am having some trouble understanding what definitions of cohomology and hypercohomology are getting used. Let $U$ be a complex ...
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First higher direct image of a constant sheaf
Let $X$, $F$ be manifolds and $f:X\times F\to X$ be a natural projection.
Then is the first higher direct image of a constant sheaf $R^1f_*(\mathbb{Z}_{X\times F})$ also a constant sheaf on $X$ with ...
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