All Questions
Tagged with sheaf-cohomology ag.algebraic-geometry
221 questions
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Global sections of twisting ideal sheaf of a smooth closed point on a projective space
Let $X = \mathbb{P}^n_k$ be a projective space over an algebraically closed field $k$ and $x$ be a closed point.
Given an integer $m$ and a positive integer $r$.
What are the global sections of $\...
- 1
13
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1
answer
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Etale homology via étale cosheaves
Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group ...
CommunityBot
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2
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2
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708
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Leray spectral sequence of the inclusion of an open subvariety
Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the ...
CommunityBot
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21
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2
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Cohomologically trivial stacks
The following theorem of Serre is well-known:
A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. (...
- 27k
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How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
2
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2
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645
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Vanishing of Ext group
Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$$
for some sheaf $...
- 6,920
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Cohomology of a sheaf with only one stalk
Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish).
I believe that ...
- 1,553
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answer
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Basic properties of Nisnevich cohomology; $l'$-topology?
I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...
- 1,553
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4
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Cohomology of line bundles
For sure answers to my questions are well known - but I never saw them anywhere.
Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...
- 16.2k
6
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2
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cohomology and $j_!$
I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,...
- 23.5k
1
vote
2
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315
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Cohomology groups interpreted as sheafs
Hi Folks,
I just came across a few lines where a (sheaf-)cohomology group of a scheme is treated as a sheaf. I've never seen this in Hartshorne.
Could you give any reference for this?
Thanks
Steven
4
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0
answers
362
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Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
CommunityBot
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7
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1
answer
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Are injective quasi-coherent modules acyclic?
Let $X$ be a scheme and $F$ be an injective object of $\mathrm{Qcoh}(X)$. Is it true that $F$ is acyclic with respect to the usual sheaf cohomology?
For noetherian schemes $X$ this is well-known; ...
- 63.1k
8
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2
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684
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Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?
Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\...
- 7,338
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2
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sheaves for which the derived (compact or not) pushforward is zero
Conventions: sheaf = complex of constructible sheaves (in the l-adic setup with etale tplg or in the complex coefficients setup with analytical tplg).
I would like to understand if there is an ...
- 35.2k
4
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2
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858
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If $f: X \to Y$ is a finite flat morphism of schemes, $g: Y \to Z$ is a proper morphism of relative dimension one, $Z$ is affine and $E$ is a vector bundle on $Y$ with $R^1g_*E=0$ then $H^1(X,f^*E)=0$?
Let $f: X \to Y$ and $g: Y \to Z$ be morphisms of schemes* such that f is flat and finite, g is proper and $R^{> 1}g_*E=0$ for all sheaves and Z is affine.
Let E be a vector bundle on Y such that $...
- 42.9k
2
votes
0
answers
151
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Intersections of components of 'simple' ('local") Zariski coverings
I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...
- 50.7k
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186
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Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?
Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...
- 16.9k
1
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249
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On inverse images with respect to Zariski-etale topology.
For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
- 16.9k
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The behavior of pure sheaves under functor Hom( F, -)
We know that a submodule A of B is pure if and only if the functor $Hom(M, -)$ is exact on the sequence
$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
for every finitely presented module ...
- 15.2k
5
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374
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Sheaf Cohomology on Zariski-Riemann Spaces
Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
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