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0
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0answers
79 views

“Sheaf” on the nerve of a category

So I have the nerve of a category and want to communicate information about the sets at each level in the simplicial set (nerve). Is there a way to regard the nerve as a category itself? Then one ...
1
vote
0answers
39 views

Continuously variable *space*

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
4
votes
2answers
322 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? ...
56
votes
8answers
5k views

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
2
votes
0answers
94 views

Derived Categories provide a good Framework for Sheaf Cohomology?

I'm a bit new to this sheaf cohomology business. Can someone explain how derived categories provide a good setting for Sheaf Cohomology? I understand that sheaf coho arises as right derived functors, ...
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0answers
48 views

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 ...
23
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3answers
2k views

Sheaf Description of G-Bundles

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free O_X-modules of rank n and vector bundles of rank n. So, equivalently, principal ...
11
votes
2answers
225 views

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 ...
4
votes
2answers
194 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 ...
4
votes
1answer
159 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 ...
9
votes
0answers
148 views

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 ...
2
votes
1answer
538 views

Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf

Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane. Choose a closed point ...
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8answers
4k views

How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
3
votes
0answers
122 views

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, ...
5
votes
1answer
148 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 ...
0
votes
1answer
66 views

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. ...
6
votes
1answer
330 views

Needless axiom for Grothendieck topologies?

Hi, The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family. Why ...
5
votes
1answer
493 views

Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...
2
votes
0answers
82 views

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 ...
8
votes
1answer
298 views

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 ...
3
votes
1answer
130 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 ...
5
votes
0answers
130 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 ...
2
votes
0answers
149 views

Toward Axiomatic sheaf theory? (References)

It is known that one of Lawvere and Tierney's goals was to provide an axiomatic approach to sheaves. Their notion of elementary topos constituted a preliminary step in that respect. Question: Has ...
2
votes
0answers
61 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):= ...
1
vote
1answer
54 views

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 ...
7
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0answers
160 views

“Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations: $f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact. $f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...
8
votes
2answers
292 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 ...
5
votes
0answers
146 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 ...
5
votes
1answer
114 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 ...
3
votes
0answers
123 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 ...
2
votes
2answers
141 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 ...
7
votes
2answers
1k 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: ...
0
votes
0answers
119 views

Canonicity of Cech cohomology

For a topological space X, consider the Leray covering U_λ(i.e. ∩U_λ is sufficiently fine, e.g. affine for Zariski topology) of X. For a sheaf F on X, the cohomology H^i(X,F) is calculated via Cech ...
3
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0answers
87 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?
3
votes
2answers
357 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?
0
votes
1answer
170 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 ...
4
votes
1answer
197 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 ...
6
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0answers
184 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 ...
11
votes
1answer
627 views

Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...
3
votes
0answers
190 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 ...
4
votes
1answer
177 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 ...
6
votes
1answer
180 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 ...
4
votes
2answers
332 views

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 ...
0
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0answers
41 views

Hilbert Manifold structure on global sections of a sheaf

Let $<U_i,\phi_i>$ be an atlas on a Hilbert manifold $M$. If $F(M,U_i)$ is a Hilbert space for every $U_i$ then does this imply that $F(M,M)$ can be made into a Hilbert manifold also with ...
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votes
2answers
1k views

Inverse Image as the left adjoint to pushforward

This is a repost of a question on Math stackexchange. No one is biting at it there, so I guess it is harder than I thought. Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous ...
3
votes
0answers
107 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 ...
7
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0answers
212 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 ...
5
votes
0answers
160 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 ...
3
votes
1answer
141 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 ...
3
votes
0answers
85 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. ...