All Questions
Tagged with sheaf-cohomology ag.algebraic-geometry
221 questions
4
votes
1
answer
263
views
Relating deformations of a scheme to deformations of its singular locus
Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
- 8,998
7
votes
1
answer
1k
views
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, ...
- 16.9k
2
votes
2
answers
579
views
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 ...
- 887
2
votes
0
answers
287
views
$\delta$-functor and commutativity of pull-back with right derivation
Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...
- 833
1
vote
0
answers
445
views
Pull-back of globally generated sheaves
Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
- 1,981
0
votes
0
answers
239
views
Cohomology group vs sheaf of cohomology group
Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$...
- 401
3
votes
0
answers
130
views
Action of automorphisms on cohomology with supports
Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
- 525
2
votes
1
answer
293
views
global sections of higher direct images of étale sheaves
Is there a useful criterion for when $\Gamma(X, R^qf_*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?
user19475
1
vote
0
answers
132
views
Picard sequence for sujective morphisms
Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
- 101
2
votes
2
answers
645
views
Vanishing of Ext group
Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$$
for some sheaf $...
- 1,070
3
votes
0
answers
76
views
$H^{1}(C, N_{C/X}(-m)) = 0$, for $C$ a irreducible curve on $X$ through $m$ general points
I was reading A. de Jong and J. Starr's paper "Low degree complete intersections are rationally simply connected", which can be found at http://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf, and ...
- 283
1
vote
2
answers
315
views
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
- 13
0
votes
1
answer
156
views
The Existence of Pure Resolutions, Given a Degree Sequence?
I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
user24421
4
votes
0
answers
447
views
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
4
votes
0
answers
362
views
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-...
- 1,611
0
votes
1
answer
382
views
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 ...
- 55
5
votes
0
answers
374
views
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 ...
- 468
1
vote
0
answers
249
views
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
0
votes
0
answers
186
views
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
vote
0
answers
101
views
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
votes
0
answers
151
views
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 ...
- 16.9k