A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, ...

learn more… | top users | synonyms

0
votes
1answer
21 views

Show that a smooth manifold modulo diffeomorphism group is a smooth manifold

Would like help in starting this exercise: Suppose $\Gamma$ is a group of diffeomorphisms of a manifold $\left( {X,C_X^\infty } \right)$. Suppose that the action of $\Gamma$ is ...
1
vote
0answers
15 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 ...
7
votes
1answer
114 views

Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of ...
3
votes
0answers
30 views

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 ...
2
votes
1answer
42 views

morphism of sheaves on $\mathbb{R}/\mathbb{Z}$

Let $\mathscr{Z}$ be an arbitrary sheaf on $\mathbb{R}/\mathbb{Z}=X$ (with the quotient topology). Let $\mathscr{F}$ and $\mathscr{G}$ denote the sheaves of continuous functions on $X$ with values in ...
0
votes
0answers
34 views

Cartier divisor and map to associated line bundle

Given a Cartier divisor $D$ on an integral, separated scheme $X$ of finite type over an algebrically closed field $k$. Does such a divisor always induce a map $\mathcal{O}_X \to \mathcal O_X(D)$? I ...
0
votes
1answer
29 views

Local argument for proving that $\operatorname{Ext}^1$ vanishes for two sheaves

I am trying to compute the vanishing of $\operatorname{Ext}^1$ for two sheaves of $\mathcal{O}_X$-Modules and I was wondering if it was possible to use some local argument to reduce the problem to ...
0
votes
1answer
61 views

About alternative ways of computing $H^1(X,\mathcal{O}_{\mathbb{P}^n}(m))$.

This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
9
votes
0answers
91 views
+50

Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
3
votes
0answers
33 views

When does the graph of a morphism of schemes induce the usual pushforward on quasi-coherent sheaves?

Let $f: X \to Y$ be a morphism of $S$-schemes s.t. the graph $\Gamma_f : X \to X \times_S Y $ is qcqs with ideal sheaf $\ker\Gamma_f^{\sharp} = \mathcal{I}_f$ (and s.t. the projection $\pi_2$ is ...
2
votes
0answers
29 views

Should “diagonal” always mean “scheme theoretic image of the diagonal”? (Jet sequence of a non-seperated scheme).

There's a subtle point that always bothers me whenever the diagonal is considered. Let $X \to Y$ be a non-seperated morphism. As I understand so far this is equivalent to the fact that the comorphism ...
4
votes
0answers
51 views

Elegant formalism for gluing spaces over open subsets (Vakil 17.2.B.)

This question is about exercise 17.2.B. from Vakil's algebraic geometry notes. Let $X$ be a scheme. The following data is given: For each affine open set $U \subset X$ a scheme $\pi_U :Z_U \to U$. ...
3
votes
1answer
75 views

Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
5
votes
1answer
56 views

Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and ...
-1
votes
0answers
32 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 ...
1
vote
1answer
37 views

Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
1
vote
1answer
29 views

Is assuming the gluing axiom for any pair of sections equivalent with gluing arbitrary collections of sections?

In Griffiths and Harris the gluing axiom for sheaves is given as For any pair of open set $U,V$ and sections $\sigma\in\mathcal{F}(U)$, $\tau\in\mathcal{F}(V)$ s.t. $\tau|_{U\cap V}=\sigma|_{U\cap ...
1
vote
1answer
23 views

What are the stalks of $\Gamma_Z(F)$ for a locally closed subset $Z$ and a flabby sheaf $F$

My references for the following notations are Iversen & Hotta,Takeuchi, Tanisaki. Let $Z \subset X$ be a locally closed set, and $F \in Sh(X)$. Let $Z \subset W \subset X$ be an open subset of ...
2
votes
0answers
77 views

Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb ...
2
votes
0answers
21 views

Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter ...
-2
votes
1answer
54 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
0
votes
0answers
32 views

Recommend guide book of algebraic geometry [duplicate]

I have a little knowledge about geometry and algebraic topology . I want to learn some basic conception and thought of algebraic geometry. Besides , I want to know main of theory of sheaves. What book ...
0
votes
0answers
34 views

Finite type assumption necessary for this property of very ample sheaves?

The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement: ...
0
votes
1answer
53 views

Terminal object in the category of sheaves?

Let $\mathsf{C}$ denote one of the categories $\mathsf{Set}$, $\mathsf{Group}$, $\mathsf{Ring}$, or $\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their ...
4
votes
1answer
38 views

Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
0
votes
1answer
26 views

Quotient Sheaves

Let $X$ be a ringed space, and $J$ be a sheaf of ideals of the structure sheaf. Define, $Y = \{x\in X ~ | ~ J_x \not = \mathcal{O}_x\}$, this is a closed set. We have an inclusion $i:Y\to X$. Is there ...
1
vote
0answers
31 views

When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
0
votes
1answer
25 views

Representation of an effective Cartier divisor: is there an imprecision in Liu's book?

I will use the notation of Liu's book which is also the standard one. If you don't feel comfortable with it please ask more information in the comments Let $X$ be a scheme, and consider a Cartier ...
1
vote
2answers
50 views

Sheaf associated to a Cartier divisor

This question is motivated by a construction, unclear for me, related to Cartier divisors. But, in the end it can be reduced to a question involving only sheaves on topological spaces. Let $X$ be a ...
0
votes
1answer
47 views

Twisting an exact sequence of sheaf gives an exact sequence

Reading through Hartshorne's Algebraic Geometry, and reading different notes about cohomology of sheaves, I have often seen the argument that if you have an exact sequence of coherent sheaves over a ...
1
vote
2answers
64 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this ...
5
votes
1answer
129 views

Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object ...
0
votes
0answers
25 views

Commutation of pullbacks with disjoint unions in sites?

From what I understand, in superextensive sites disjoint unions commute with pullbacks. I know this holds in every topos sense pullback has a right adjoint, and I have heard something along the lines ...
1
vote
0answers
33 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
3
votes
1answer
156 views

Alternative description of the sheafification

For me the sheafification of a given presheaf is this: Proposition-Definition: Given a presheaf $\mathscr{F}$, there is a sheaf $\mathscr{F}^+$ and a morphism $\theta \colon \mathscr{F} \to ...
1
vote
1answer
53 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self ...
1
vote
1answer
29 views

Prove that for $\mathcal{F}$ a sheaf on manifold $M$ we have $H^0(\{U\},\mathcal{F})=\mathcal{F}(M)$

I want to prove that that for $\mathcal{F}$ a sheaf on manifold $M$ we have $H^0(\{U\},\mathcal{F})=\mathcal{F}(M)$, where $\{U\}$ is any locally finite open cover of $M$. (Note that I think we really ...
0
votes
0answers
41 views

Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
0
votes
1answer
33 views

Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...
0
votes
1answer
53 views

Is Coherent sheaf acyclic?

I am no expert in sheaf theory so the following question may be trivial. Let $X$ a complex manifold, and let $\mathcal{F}$ a coherent sheaf on $X$. Is $\mathcal{F}$ acyclic? If not: can you give a ...
3
votes
0answers
66 views

For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $\mathcal{O}_X$-module $F$ has the structure of a $G$-representation

Reading Huybrechts & Lehn's "The geometry of moduli spaces of sheaves" I am stuck with a particular statement that they make in the chapter on GIT without explanation. Namely, they say that for an ...
1
vote
1answer
27 views

A confusion with the Definition of 'Skyscraper Sheaf' from 'Stack Project'

According to Hartshorne, (Chapter, Ex 1.17): Let $X$ be a topological space, let $P$ be a point, and let $A$ be an abelian group. Define a sheaf $i_p(A)$ as follows: $i_P(A)(U)=A$ if $P\in U$ and ...
3
votes
1answer
41 views

sheaf defined on the support of a sheaf

Let $X$ be a topological space and $\mathcal{F}$ a sheaf on $X$. Let $Y$ be the support of $\mathcal{F}$ (Hartshorne, exercise 1.14), i.e., $Y = \left\{ P \in X: \, F_P \neq 0\right\}$. Is it true ...
3
votes
1answer
42 views

Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
0
votes
1answer
26 views

$a\in X$. Show that the stalk $\mathscr{F}_a$ is isomorphic to the stalk of $\mathscr{F}|_U$ at $a$ on the topological space $U$.

This is exercise 3.24 from Gathmann's notes. Let $\mathscr{F}$ be a sheaf on a topological space $X$, and let $a\in X$. Show that the stalk $\mathscr{F}_a$ is a local object in the following ...
1
vote
1answer
39 views

Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...
1
vote
1answer
37 views

Stalks of ringed space

Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that ...
0
votes
0answers
12 views

Differences between locally free sheaves and local systems

Is every local system with fiber a vector space a locally free sheaf? What are the main differences between these two concepts? I was playing with the sheaf of sections of $Mo \to S^1$ ($Mo=$Möbius ...
3
votes
0answers
27 views

Computing the monodromy of a local system $\mathcal{L}$

I was trying to learn a little bit about local systems and their monodromy. In the notes I'm following they define the monodromy of a local system in the following way: Let $X$ be a topological ...
2
votes
1answer
66 views

Pullback of an invertible sheaf through an isomorphism

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback. Is it true ...