All Questions
Tagged with sheaves or sheaf-theory
979 questions
7
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0
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815
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A question on a proof that fine sheaves are soft
Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof ...
- 5,019
5
votes
1
answer
723
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Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
CommunityBot
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17
votes
2
answers
1k
views
What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos)....
- 39.3k
4
votes
1
answer
639
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Morphisms between pure complexes of sheaves
I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
- 6,162
12
votes
0
answers
5k
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Wikipedia's definition of constant sheaf is wrong [closed]
According to wikipedia (constant sheaf) the constant sheaf $\underline{S}$ for an object $S$ is given by defining $\underline{S}(U)$ to be the set of functions $U \to S$, which are constant on each ...
- 9,366
8
votes
2
answers
4k
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Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
- 63.1k
5
votes
1
answer
3k
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Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves
I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
- 27k
6
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0
answers
2k
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group cohomology and cohomology of classifying space [closed]
Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $\tilde{M}$, which ...
- 1,457
9
votes
1
answer
3k
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Motivation for equivariant sheaves?
Hello everyone;
i'm looking for a motivation for equivariant sheaves (see http://ncatlab.org/nlab/show/equivariant+sheaf) ~ Why are we interested in them?
More explicitely: Can I think of G-...
- 5,232
8
votes
1
answer
562
views
Category of copresheaves over commutative monoids
Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets.
...
- 2,399
21
votes
3
answers
2k
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Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 12.3k
4
votes
1
answer
383
views
How to characterize Abelian sheaves that are quasi-coherent?
Let $X$ be a scheme. Suppose you are given a sheaf of Abelian groups $\mathcal{A}$ over $X$. How can you determine if $\mathcal{A}$ is the underlying Abelian sheaf of a sheaf of $O_X$-modules? In ...
- 23.8k
4
votes
1
answer
549
views
Does this condition reduce to the correct notion of irreducibility on schemes?
Consider the category of sheaves (of sets) on the affine étale site. It's a well known fact that a morphism of schemes is a Zariski-open immersion if and only if it is an étale monomorphism, so we ...
- 19.6k
5
votes
2
answers
830
views
Closed monoidal structure on the derived category of sheaves
Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should ...
- 52.6k
10
votes
3
answers
825
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Sheaves as full reflective subcategories
Hello everyone.
My question is concerned with the following statement.
"Having a grothendieck topology on a category C is equivalent to having a full reflective subcategory Sh(C) in the category PSh(...
- 19.8k
0
votes
1
answer
346
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O_X module with support Z \subset X vs O_S module?
Given a $O_X$ module $\cal F$ whose support is a closed subscheme $Z \subset X$. Under what conditions can we say that $ \cal F$ is an $O_S$ module ( how far off is $\cal F$ an $O_S$ module ? )
- 25.6k
36
votes
3
answers
4k
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What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 156k
6
votes
3
answers
1k
views
Definition of sheaves in wikipedia
In wikipedia, sheaves were first defined in the case of concrete categories (with usual identity and gluing axioms), then in the general case. (writing it as an "exact" sequence)
Do these two ...
- 19.6k
10
votes
1
answer
786
views
Sites which are stacks over themselves
A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
3
votes
3
answers
687
views
Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
- 2,132
7
votes
1
answer
1k
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Encoding fuzzy logic with the topos of set-valued sheaves
One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (...
- 1,292
24
votes
4
answers
6k
views
What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
CommunityBot
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6
votes
2
answers
790
views
Do quotients of representable sheaves represent quotients?
Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
- 11.3k
15
votes
2
answers
2k
views
Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
CommunityBot
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6
votes
2
answers
1k
views
Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 15.1k
7
votes
3
answers
562
views
Abstract Relation between Presehaves and Simplicial Sets
Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...
- 1,926
18
votes
9
answers
2k
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What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 2,992
11
votes
2
answers
2k
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Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 13.6k
2
votes
1
answer
406
views
Are there any criteria for a presheaf which is an etale sheaf to be a sheaf in the fppf topology?
I am happy to hear answers to variants too. For instance, my situation I actually have a sheaf in the smooth topology.
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