All Questions
Tagged with sheaves or sheaf-theory
979 questions
11
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1
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812
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Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory
By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
- 7,789
5
votes
0
answers
189
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Constructible sheaves on general stratified spaces
I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
- 21.8k
2
votes
0
answers
272
views
Double dual of ample sheaf
Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves).
Now I was ...
- 147
3
votes
0
answers
108
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When is a functorial coverage a sheaf, and what universal property does it have?
In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is
a function assigning to each object $A$ of $\...
- 8,659
10
votes
1
answer
529
views
Polylogarithm sheaves
In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...
- 1,428
2
votes
1
answer
163
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ample subsheaf contained in the tangent bundle of projective space
Let $\mathcal F$ be an ample subsheaf of $T_{\mathbb P^n}$. Is it actually locally free? If not, is there a counterexample?
- 147
6
votes
0
answers
141
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Expressing the stack of sheaves with 1-limits
Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent ...
- 10.5k
4
votes
0
answers
262
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Applications of cosheaf homology?
What are some applications of cosheaf homology within mathematics?
Some ones I've heard of Sheaves (not cosheaves) are computing global sections and the Picard Group with a sheaf on projective space.
- 913
2
votes
0
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1k
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Equivalent definitions for fine sheaves
There are some different definitions for fine sheaves.
Let X is a paracompact Hausdorff space, a sheaf F over X is a fine sheaf, if
a) Hom(F,F) is soft
b) For every two disjoint closed subsets A,B$\...
- 403
4
votes
2
answers
315
views
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
9
votes
0
answers
378
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Is there any notion of "smoothification" from $\mathbb{R}$-schemes to generalized smooth spaces?
I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
- 7,789
11
votes
2
answers
413
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Homotopy property of constructible sheaves on stratified spaces
Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
- 21.8k
0
votes
0
answers
191
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First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$
Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{...
user76513
9
votes
1
answer
804
views
Is the analytification functor part of a geometric morphism of topoi?
Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...
- 7,789
4
votes
1
answer
1k
views
On push-forward of the constant sheaf for fibrations
Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
- 21.8k
0
votes
1
answer
164
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Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf?
Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact.
Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $...
- 9,019
3
votes
1
answer
220
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Sheaf on a filtered topological space?
Is there any nice way of defining a sheaf of abelian groups on a filtered topological space?
Let $X$ equipped with filtration $X_0\subset X_1\subset X_2\subset ... \subset X_k=X$ be an object in the ...
- 913
8
votes
1
answer
529
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Topology on the space of constructible sheaves
Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
- 21.8k
5
votes
0
answers
377
views
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
7
votes
1
answer
310
views
On the ordered set of real numbers, does sheaf+cosheaf imply constant?
I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it.
Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers ...
- 4,949
2
votes
0
answers
160
views
Pre-cosheaf of connected components
Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...
- 229
2
votes
1
answer
293
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Co-stalk of co-presheaves and cosheaves
Consider the co-presheaf $\mathcal{F}$ of continous real-valued functions with relatively-compact support on a topological space $X$. Consider a point $x\in X$.
1) When $\mathcal{F}$ is considered a ...
- 229
11
votes
3
answers
935
views
Examples of calculating perverse sheaves on algebraic varieties with easy stratification
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...
- 123
5
votes
0
answers
310
views
Extension of ample vector bundles is ample
As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this ...
- 159
3
votes
1
answer
159
views
Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?
Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules:
$$
\cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
- 355
3
votes
0
answers
579
views
A question about the adjunction between pushforward and pullback of sheaves
I am reading this article: http://arxiv.org/pdf/1310.5978.pdf. In definition 2.6 on page 4 there is claim that is made and I don't see why it is true. I will recall it here:
Let $X$ be an integral ...
1
vote
0
answers
236
views
Canonicity of Čech cohomology
For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$.
For a sheaf $F$ on $X,$ the cohomology $H^...
- 781
2
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0
answers
102
views
Are there Coherent Cosheaves?
Is there a well-defined notion of coherent cosheaves in a similar sense to coherent sheaves? If so, what properties do they hold?
- 913
6
votes
2
answers
957
views
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...
- 7,789
8
votes
0
answers
338
views
Grothendieck - A group as a sheaf over simplicial complexes
In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
- 935
6
votes
0
answers
324
views
Many-sorted nominal sets as sheaves
The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
3
votes
0
answers
551
views
Defining Inertia Stack
Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
- 31
4
votes
1
answer
409
views
Does the nearby cycle functor commute with the Verdier duality?
I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
- 21.8k
6
votes
1
answer
2k
views
Information and intuition packed in the Chern character for coherent sheaves
even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...
- 155
6
votes
1
answer
752
views
Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user82886
3
votes
2
answers
488
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
7
votes
2
answers
1k
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Natural operators in differential geometry - why are they natural?
I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...
- 10.5k
8
votes
1
answer
459
views
why don't (can't?) we sheafify the structure presheaf of an adic space
In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One ...
- 359
2
votes
1
answer
714
views
If presheaf is zero on a covering is the sheaf zero?
Let $C$ be a site and $F$ an abelian presheaf on $C$. Suppose that for each object $U$ in $C$ there is a covering $\{ U_i\to U \}$ such that $F(U_i)=0$. Is it true that $F^{sh}=0$?
This should be ...
- 501
1
vote
2
answers
276
views
Sections of a sheaf of differentials on a weighted complete intersection
Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\...
user75933
3
votes
0
answers
334
views
Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
11
votes
1
answer
335
views
Reference request: sheaves on the site of d-manifolds
I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...
- 54.6k
13
votes
0
answers
481
views
Making the conceptual leap from locales to Grothendieck topologies?
I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
- 373
3
votes
1
answer
515
views
For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 9,019
9
votes
1
answer
174
views
W-types and inverse image functor
All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...
- 408
3
votes
0
answers
102
views
Can we extend a homotopy invertible chain morphisms between complexes of sheaves from a closed subspace to the whole space?
Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. ...
- 9,019
10
votes
1
answer
612
views
How do the direct and inverse image sheaf functors interact with homotopy?
This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$)...
- 10.5k
20
votes
3
answers
911
views
Brouwer's theorem for the Cauchy reals
Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
- 66.8k
3
votes
0
answers
155
views
Elementary examples on sheaf extension
Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition
$\mathit{...
- 3,187
8
votes
0
answers
355
views
Why do Kashiwara and Schapira require a base ring of finite global dimension?
In the book "Sheaves on Manifolds" by Kashiwara and Schapira, they work always with sheaves of $R$-modules, where $R$ is a ring of finite global dimension.
Why do they do this, what care ...
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