All Questions
Tagged with sheaves or sheaf-theory
979 questions
10
votes
3
answers
2k
views
Representable Presheaf
I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
- 565
2
votes
1
answer
407
views
smooth manifold vs. exceptional inverse image
A well-known theorem in topology says that for a smooth manifold $M$ of dimension $n$ the map $f: M \rightarrow point$ satisfies
$$f^! \mathbf R = \mathbf R[n]$$
Here $\mathbf R$ is the constant sheaf....
- 2,040
9
votes
1
answer
471
views
When do sheaves which are constant along the fibers come from the base?
Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves (...
- 1,704
7
votes
1
answer
451
views
Coverage, itself considered as a presheaf
A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
- 291
12
votes
3
answers
845
views
Can one characterize those sheaves which have Hausdorff etale spaces?
Given a sheaf of sets $F$ on a space $X,$ under the equivalence of categories between etale spaces over $X$ and sheaves over $X,$ $F$ is associated to a local homeomorphism $$E\left(F\right) \to X$$ ...
- 15.5k
3
votes
1
answer
892
views
system of local coefficients on X, locally constant sheaves and orientation sheaves
Hi,
I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local ...
- 225
12
votes
1
answer
561
views
Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?
The category of finite sets is not a Grothendieck topos, but its Ind category
Ind(Finite-Sets) = Sets
is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete ...
- 226
2
votes
0
answers
175
views
Does mapping cylinder category have enough injectives?
Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \...
- 945
1
vote
0
answers
238
views
why is the local field Q_l not an etale sheaf over a scheme X?
I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?
- 11
2
votes
1
answer
258
views
Carving out subsheaves of local hom-sheaves of stacks of categories
Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack.
...
- 35.5k
1
vote
2
answers
2k
views
Sheafification of a presheaf through the etale space
I have some problems to show that the following contruction defines a sheafification:
Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, ...
- 11
19
votes
2
answers
3k
views
Explaining Mukai-Fourier transforms physically
A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...
- 10.5k
0
votes
1
answer
672
views
left adjoint to restriction functor
We know that for an immersion $j:U \to X$ the restriction functor $j^*:{\cal O}_X-mod \to {\cal O}_u-mod$ has a left adjoint $j!$.
I am looking for some condition to deduce that $j!$ takes its values ...
- 13
1
vote
1
answer
578
views
Descent of Morphisms of Sheaves
While reading Brylinski I am trying to understand the descent of morphisms of sheaves.
In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...
- 1,611
3
votes
1
answer
181
views
Which limits are preserved by prolongation of presheaves?
Let $F:C \to D$ be a full and faithful functor between small categories. Then we get a triple of adjoint functors $F_! \dashv F^* \dashv F_*$, with $$F_!:Set^{C^{op}} \to Set^{D^{op}}.$$
Notice that $...
- 15.5k
57
votes
3
answers
10k
views
What are the benefits of viewing a sheaf from the "espace étalé" perspective?
I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. ...
- 6,290
17
votes
3
answers
3k
views
Sheafification - Why does twice suffice?
Hi,
I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve.
It is known that ...
- 1,071
1
vote
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
5
votes
2
answers
1k
views
trying to understand the support of the sheaf of relative differentials
So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
- 10.7k
1
vote
1
answer
443
views
Is this Sequences of Complexes of Sheaves Exact?
So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ \underline{...
- 1,611
8
votes
2
answers
993
views
Example of a Sheaf (on the site of smooth manifolds) with Nontrivial Cohomology on $\mathbb{R}^n$?
Does such a sheaf of abelian groups exist? If not, is there a reference or a proof? Does such a sheaf of non-abelian groups exist?
I realized recently that while I've taken it for granted that ...
- 1,138
2
votes
1
answer
1k
views
Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
- 360
1
vote
2
answers
1k
views
Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
2
votes
1
answer
796
views
How does torsion behave under the direct image functor?
Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f_{*}$ induces an equivalence between the categoy of coherent $O_X$-modules ...
- 1,391
1
vote
2
answers
398
views
Cohomology of a cochain complex of acyclic sheaves
Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem:
Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of $F$....
- 360
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
23
votes
5
answers
1k
views
Sheafification via hypercovers
The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...
- 66.8k
11
votes
2
answers
4k
views
Is Sheafification Functor Exact?
I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.
My question is:
...
- 945
1
vote
1
answer
273
views
Do Categorical Quotients Preserve Covering Maps?
Before asking a question, please let me write down settings.
SETTINGS:
Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...
- 945
8
votes
2
answers
684
views
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 $\...
- 3,758
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
5
votes
0
answers
336
views
Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?
For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
- 16.9k
1
vote
0
answers
202
views
Soft sheaves on indiscrete paracompact spaces
Let $X$ be some space, I have basically 2 questions:
1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
- 417
22
votes
3
answers
3k
views
Necessity of hypercovers for sheaf condition for simplicial sheaves
I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.
For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
- 661
2
votes
0
answers
524
views
What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
0
votes
1
answer
328
views
Direct limit of sheaves over paracompact spaces
Let $\mathcal F_i, i\in I$, be a directed family of sheaves of abelian groups on a paracompact Hausdorff space $X$. Let $\mathcal F=\varinjlim F_i$ denote the direct limit sheaf. Is it true that $\...
- 741
0
votes
3
answers
400
views
Cohomology of complexes
I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*) $?
- 147
5
votes
1
answer
1k
views
Euler characteristic and inclusion-exclusion
Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C_1,...
- 303
4
votes
0
answers
457
views
Quantum sheaves
Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(...
- 1,368
1
vote
1
answer
146
views
Is the free R-module on a sheaf of sets still a sheaf?
Let $L$ be a sheaf of sets on some site $S$. Let $F$ be the presheaf obtained by composing $L$ with the free R-module functor, i.e. for any object $U$, we define $F(U)$ to be the free $R$-module on ...
- 5,637
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
- 363
6
votes
1
answer
2k
views
Is the stalk of the (co)limit of sheaves equal to the (co)limit of the stalks?
More precisely, if $\mathcal F_i$ is a system of sheaves, is it the case that
$$
(\lim \mathcal F_i)_p = \lim ((\mathcal F_i)_p)
$$
and similarly for colimits? I can see how to get a map
$$
(\lim ...
- 1,509
4
votes
1
answer
250
views
When do adjunctions preserve equivalence?
Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors $F_{\...
- 3,290
14
votes
2
answers
904
views
What's the easiest example of a morphism of topoi that is not from that of a site?
A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of ...
- 5,052
9
votes
1
answer
4k
views
surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
- 1,889
2
votes
1
answer
2k
views
Orientation Sheaf and Double Cover
The orientation sheaf of an $n$-manifold $M$ is $\mathcal{O}_n=Sheaf(U\mapsto H_n(M,M-U;\mathbb{Z}))$, with stalks given by $(\mathcal{O}_n)_x = lim H_n(M,M-U)=H_n(M,M-x)=\mathbb{Z}$ (the limit is ...
- 17.5k
5
votes
1
answer
468
views
Derived Equivalence of Sheaves and Homotopy
This question loosely elaborates on an earlier question. It is pretty silly, but I'd like to hear some authoritative answers.
Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of ...
- 2,684
11
votes
1
answer
1k
views
Etalé space construction for presheaves on a Grothendieck site
As it is described for example in [Mac Lane-Moerdijk, Sheaves in Geometry and Logic, II.6.], one can construct the sheafification functor very lucidly by associating to a presheaf a certain bundle (cf....
- 623
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
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