All Questions
Tagged with sheaf-cohomology reference-request
22 questions
3
votes
0
answers
133
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Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
2
votes
0
answers
60
views
Relative Dolbeault cohomology using currents
I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...
- 2,054
4
votes
1
answer
408
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Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
- 518
6
votes
1
answer
443
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Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
6
votes
2
answers
789
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Reference request: Kleiman's proof of Snapper's Lemma
On page 4 of Nitin Nitsure's paper Construction of Hilbert and Quot Schemes, the author refers to the fact that Hilbert polynomials are indeed polynomials as
a special case of Snapper's Lemma, see &...
- 1,805
6
votes
1
answer
328
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Topology on cohomology of a sheaf of topological groups
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...
- 1,170
6
votes
1
answer
557
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Cohomology and base change without Noetherian assumption
In the "The Rising Sea" by Vakil one can find the base change theorem for proper morphisms over a locally Noetherian base (28.1.6). He later indicates (28.2.M) how one could exchange the ...
- 445
1
vote
1
answer
634
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Inception of modern view of Sheaf Cohomology in Mathematical Literature
From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
- 37
3
votes
0
answers
671
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Elementary reference for Borel-Moore/locally finite homology
There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic ...
- 2,533
9
votes
2
answers
4k
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Top cohomology detecting compactness
I am looking for a reference for the fact that the top cohomology $H^n(X;A)$ of an $n$-dimensional manifold $X$ is non-trivial precisely when $X$ is compact.
I tried to ask this question on Math....
- 1,211
1
vote
0
answers
160
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Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
- 1,479
6
votes
1
answer
334
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Naive question on local cohomology
Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims:
$$...
- 1,593
4
votes
2
answers
316
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Equivalence of different cohomology groups
Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups:
(1) singular cohomology $H_{sing}^*(X,A)$;...
- 21.8k
3
votes
2
answers
489
views
Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
- 49
3
votes
0
answers
285
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Reference for the Koszul--Malgrange Theorem
The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
- 31
9
votes
1
answer
1k
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Construction of generalized Eilenberg-MacLane spaces
The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing ...
- 99
5
votes
0
answers
377
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Push forward of the constant sheaf for a Serre's fibration
Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
- 21.8k
5
votes
0
answers
614
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Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?
It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e.
$$
H^i(X,\...
- 9,019
4
votes
0
answers
177
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Reference request: local cohomology in disjoint union
I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
- 41
7
votes
1
answer
1k
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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, ...
- 16.9k
5
votes
0
answers
151
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$A_\infty$ structure on sum of twists of structure sheaf
Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$.
I ...
- 10.7k
1
vote
0
answers
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