The sheaf-theory tag has no usage guidance.

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### 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 ...

**21**

votes

**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 ...

**2**

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85 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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177 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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100 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

**1**answer

110 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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80 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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105 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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54 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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141 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 ...

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**1**answer

313 views

### Grothendieck - sheaves as meter sticks

I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...

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**1**answer

443 views

### Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...

**3**

votes

**1**answer

99 views

### Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space

Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open ...

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74 views

### Surjectivity locus of a morphism of families of sheaves

Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$.
...

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**3**answers

2k 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 ...

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1k views

### Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...

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147 views

### Canonical (tautological) section of a family of sheaves

A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it ...

**14**

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**1**answer

786 views

### Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...

**1**

vote

**1**answer

122 views

### Is there a nonzero sheaf with all cohomologies vanish?

Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?

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198 views

### Constants sheaves on an open subset

Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}_U$, given by ...

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214 views

### A question on flasque sheaf

Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. ...

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67 views

### Isomorphism of sheaves

Given a smooth projective variety $X$ and a semiample and big $\mathbb{Q}$-divisor $D$.
We denote by $R:=\sum_{n\in \mathbb{Z}_{\geq 0}} H^0(X,\mathcal{O}_X(nD))$.
Denote by $\tilde R(n)$ the ...

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votes

**2**answers

861 views

### Non-zero sheaf cohomology

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is ...

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**1**answer

180 views

### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

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159 views

### Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...

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**1**answer

197 views

### Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...

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108 views

### What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?

It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor ...

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66 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...

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304 views

### Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...

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**1**answer

301 views

### Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...

**3**

votes

**2**answers

245 views

### Exact sequence of groups to exact sequence of sheaves

Disclaimer: This is a cross-listing of a math.stackexchange post. While not research level, after a week of no response, I figured I would ask it here.
For a topological group $G$ and a topological ...

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500 views

### Etale spaces using Kan extensions

Mac Lane - Moerdijk's "Sheaves" gives this cryptic hint in page 91 that the equivalence between etale spaces and sheaves on a space $X$ can be cooked up using formal methods.
More precisely, we are ...

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89 views

### Is a locally finitely generated sheaf of modules finitely generated on sections?

"Let $(X, \mathcal{O})$ be a ringed space. A sheaf of modules $\mathcal{F}$ on $X$ is finitely generated if for all $a \in X$ there exists a neighbourhood $U$ of $a$, an integer $n$ and a surjective ...

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135 views

### Coherence of $\mathcal O_X[T]$

Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module).
How to prove that also the ...

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1k views

### Grothendieck spectral sequence and Mayer-Vietoris sequence

Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence ...

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564 views

### global fibrations of simplicial sheaves

I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : ...

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374 views

### A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...

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**1**answer

415 views

### Simplifying the definition of a geometric context using sieves?

On Pages 1-3 of Cours 2 of Toën's Master Course on Stacks, he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over and above the ...

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195 views

### Sheaf cohomology on non paracompact topological spaces

I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.
My reference is Godement's book "Topologie algebrique et theorie dex ...

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50 views

### Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...

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**1**answer

252 views

### A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories
Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...

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559 views

### Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon ...

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**1**answer

780 views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...

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**1**answer

138 views

### Are the pullback functors of adjoint functors also adjoint?

Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by pre-composition, are these two also adjoint ...

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**1**answer

160 views

### Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R.
Is it possible to extend this function to a smooth function on R²?
...

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**1**answer

310 views

### Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...

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163 views

### Open subset of the moduli space of stable sheaves on a noetherian scheme

This is my question:
Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable ...

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**1**answer

486 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 ...

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209 views

### Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...

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134 views

### Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...