The sheaf-theory tag has no usage guidance.

**22**

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### 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 ...

**9**

votes

**2**answers

213 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 ...

**3**

votes

**2**answers

180 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 ...

**-2**

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**0**answers

133 views

### Reference for algebraic manipulation of sheaves [closed]

I am currently playing with sheaves over families of algebraic varieties ($O_x$-modules) their torsion sub-sheaf, higher direct images and tensor products, I am looking for a good reference to learn ...

**4**

votes

**1**answer

154 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**

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146 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 ...

**0**

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**0**answers

81 views

### Reflexive sheaf and torsion free sheaf [closed]

Let $X$ be a smooth normal projective variety over $\mathbb{C}$. If $E$ is a torsion-free coherent sheaf on $X$, then the singularity set of $E$, that is the closed set where $E$ fails to be locally ...

**2**

votes

**1**answer

531 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 ...

**24**

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**8**answers

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**

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**0**answers

121 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**

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**1**answer

127 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

**1**answer

63 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

**1**answer

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**

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**1**answer

484 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**

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**0**answers

79 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**

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**1**answer

296 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**

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**1**answer

127 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**

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**0**answers

128 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**

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**0**answers

145 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**

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**0**answers

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):= ...

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**1**answer

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 ...

**6**

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158 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**

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**2**answers

288 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**

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**0**answers

145 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**

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**1**answer

112 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**

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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**

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**2**answers

137 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**

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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:
...

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116 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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86 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**

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355 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**

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**1**answer

167 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 ...

**2**

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248 views

### $\{$Affine schemes over $S$$\}$ $\cong$ $\{$$\mathcal{O}_S$ - algebras$\}$?

I've asked something very similar before on MSE but unfortunatly it hasn't recieved a lot of attention. I decided to ask again here.
Let $S$ be a fixed scheme. Is the following true?
"Theorem": ...

**4**

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**1**answer

191 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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183 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**

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**1**answer

621 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**

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183 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**

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**1**answer

176 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 ...

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**1**answer

174 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**

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**2**answers

328 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 ...

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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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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 ...

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104 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 ...

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210 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 ...

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154 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**

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140 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 ...

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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. ...

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111 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
...

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65 views

### Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions.
I am wondering whether hyperfunctions have any advantages over ...

**3**

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163 views

### Vanishing of sheaf cohomology with compact support

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.
1) Under what sufficient conditions on $F$ for any compact subset ...