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 $\mathcal F(V)$, ...

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3
votes
1answer
40 views

Example of flasque but non-soft sheaves?

Does anyone have an interesting examples of a flasque but not soft $\mathscr{O}_X$-module over a ringed space? Of course with $X$ being non-paracompact.
0
votes
0answers
10 views

problem on sheaves

I want to show the following. Suppose $X$ is a smooth manifold and F,G are sheaves of $C^{\infty}_{X}$-modules, then the natural map $Hom(F,G)\to Hom(F(X),G(X))$ is injective. It's easy to see that ...
4
votes
3answers
70 views

What is the inverse image of a sheaf

Let $f : X \rightarrow Y$ be a continuous map of topological spaces and $\mathcal{G}$ a sheaf on $Y$. What exactly is $f^{-1}\mathcal{G}$? It seems like we should be able to describe the sections ...
0
votes
0answers
23 views

Why is the presheaf-“p-exterior power” of a sheaf separated?

In the first volume of the EGA ( http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1960__4_ ), p38, Grothendieck says that the presheaf-p-exterior power of a module in ringed space a separated ...
0
votes
1answer
18 views

Surjection Vs Surjective geometric morphism

Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is ...
5
votes
2answers
74 views

Is there a logical interpretation for equalizer and co-equalizer?

I know the logical equivalent to several universal constructions. For example product $\times$ is $\land$ and co-product $+$ is $\lor$. The associated arrows are projection and inclusion. The ...
0
votes
2answers
149 views

Non-cohomological proof that a quasi-coherent sheaf over an affine scheme is quasi-flasque

Let $\mathcal F$ be a quasi-coherent sheaf over an affine scheme $X$. Let $0 \rightarrow \mathcal F \rightarrow \mathcal G \rightarrow \mathcal H \rightarrow 0$ be an exact sequence of sheaves on $X$, ...
4
votes
1answer
68 views

Proof of a certain proposition on sheaves without using cohomology

Definition Let $\mathcal F$ be a sheaf of abelian groups on a topological space $X$. We say $\mathcal F$ is quasi-flasque if it satisfies the following condition. For every exact sequence of sheaves ...
1
vote
2answers
55 views

Forms on Riemann Surfaces

I want to show that the space of smooth $(1,0)$ forms on a compact Riemann surface $X$ has the natural splitting: $\mathcal{E}^{1,0}(X)=\Omega(X) \oplus \partial\mathcal{E}^{0}(X)$, where $\Omega(X)$ ...
4
votes
1answer
45 views

Do I correctly understand the constructions involved in definition of Cartier divisor?

Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any ...
1
vote
1answer
33 views

Existence of a suitable cover for $S^{2}$ and a given sheaf

I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all ...
1
vote
1answer
29 views

Proof that sheafification induces isomorphism on stalks using adjoints

Let $\mathcal{F}$ be a presheaf on some topological space $X$. It is not hard to prove directly that the map $\mathcal{F}\rightarrow \mathcal{F}^{sh}$ induces an isomorphism of stalks (Here ...
1
vote
0answers
35 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
4
votes
2answers
78 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
2
votes
1answer
72 views

How to prove the sheafification is a sheaf?

I know that this question might be too easy for you, but I have to study on my own, so please explain for me. In the page 64, Hartshone defined the sheafification of a presheaf $\mathcal{F}$ by ...
5
votes
2answers
94 views

Sheafification of the Presheaf of continuous and bounded functions

Let $X$ be a topological space. $U\subset X$ open. $\mathfrak{B}(U) = \{f:U\to \mathbb{R}| f \textrm{ continuous and bounded}\}$ is a presheaf. I would like to see the sheafification of this ...
1
vote
0answers
33 views

Example - Sheaf condition

In my tutorial the teacher wrote the following: Let $A$ be an integral domain. $f,g\in A$ such that $(f)+(g)=A$. Then the sheaf condition of $\mathscr{O}_X$ (with $X = Spec(A))$ gives ...
3
votes
1answer
45 views

Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$

One can describe a $\mathbb{CP}^{N-1}$ manifold with a Fubini-Study metric $g^{FS}$, and there is a connection one form $v$ on it. A is connection one form(gauge field) of a line ...
2
votes
0answers
26 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
3
votes
0answers
53 views

Understanding Limits and Colimits by Generalized Elements

We want to characterize the limit and colimit of a functor $D\colon J\to \mathcal C$ by generalized elements. The existence of limits theorem states that the limit of $D$ is the equalizer of ...
0
votes
1answer
30 views

$X=Spec(A)$. $X=\bigcup\limits_{i=1}^N D(f_i)\Rightarrow (f_1, …, f_N)=A$

Let $X=Spec(A)$ and note $D(f)\simeq Spec(A_f)$. $X=\bigcup\limits_{i=1}^N D(f_i)\Rightarrow (f_1, ..., f_N)=A$ We used this to proof a special case of $\mathscr{O}$ the sheaf of rings is a sheaf. I ...
1
vote
0answers
32 views

Existence of a long exact sequence for sheaf cohomology

Let $X$ be a normal variety over $\mathbb{C}$ , and let $U$ be a open subset of $X$, then there is an long exact sequence for singular or De Rham cohomology with compact support that relates the ...
0
votes
0answers
20 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
1
vote
0answers
40 views

Sheaves: pretopology versus comma pretopology

I'm reading about sheaves on sites and I have a question about a particular example in these notes: http://www.math.harvard.edu/~nasko/documents/stacks.pdf http://homepage.sns.it/vistoli/descent.pdf ...
2
votes
0answers
48 views

Example: Push-Forward Sheaf

Let $f: X\to Y$ be a continuous map, $\mathscr{F}$ a sheaf on $X$. $f_*\mathscr{F}$ is the sheaf on $Y$ defined by $f_*\mathscr{F}(U)=\mathscr{F}(f^{-1}(U))$ Uand $\rho_{VU}=\rho_{f^{-1}(V)f^{-1}(U)}: ...
3
votes
1answer
67 views

Sheafification - Construction of a Sheaf

I tried different books and lecture notes to understand sheafification, but for instance in Hartshore or Shafarevich's book, but I found it hard to understand. The following is the approach my ...
0
votes
1answer
37 views

Example of a Hausdorff sheaf of germs

Let $X$ be a topological space, and let $A$ be a presheaf on $X$. Let $\mathscr{A}$ be the sheaf of germs on $X$. We define a a topology on $\mathscr{A}$ as follows: Given an open set $U \subset X$, ...
2
votes
0answers
32 views

Is there a name for a sheaf with this property?

What is the name of the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is a connected open and $U$ is a nonempty open? In ...
4
votes
1answer
53 views

Presheaves and adjoint functors

I'm reading about Sheaf Theory from the point of view of categories and I have the following question: Suppose we have two small categories $\mathcal{C}_1, \mathcal{C}_2$ and $\alpha:\mathcal{C}_2\to ...
2
votes
0answers
28 views

Cech cohomology [duplicate]

There are 2 complexes computing Cech cohomology. The difference between them is that in the second one we require skew symmetry when you change the order of indices. How to show that they are ...
5
votes
1answer
79 views

When is the global section functor exact?

Given sheaves $\mathcal{F}_1,\mathcal{F}_2, \mathcal{F}_3$ on some scheme $X$, and an exact sequence $$ 0\rightarrow \mathcal{F}_1\rightarrow \mathcal{F}_2\rightarrow \mathcal{F}_3\rightarrow 0 ...
4
votes
2answers
104 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
0
votes
1answer
19 views

Abelian group of continuous sections

I'm trying to prove that the sheafification of a presheaf is indeed a sheaf. By definition, $\mathcal{F}^+(U)=\text{set of all continuous sections }s:U\rightarrow|\mathcal{F}|$. Part of the proof I'm ...
3
votes
1answer
47 views

Coherent sheaves of finite length over $\mathbb{P}^n_k$

Let $k$ be an algebraically closed field. Are there any nonzero coherent sheaves on the projective space $\mathbb{P}^n_k$ that are supported at (only) finitely many closed points? If they don't exist, ...
2
votes
1answer
48 views

Every Presheaf Is Colimit of Representables

I am working on the proof that each presheaf $F\in \mathbf{Set}^{\mathcal C^{op}}$ is the colimit of $\mathbf{y}\circ \pi\colon \int_{\mathcal{C}}F\to \mathbf{Set}^{\mathcal C^{op}}$ where ...
1
vote
0answers
40 views

$\operatorname{Spec} (\cdot)$ is functorial [duplicate]

If $A$ is a ring (with unity) I'm trying to prove that the assignement $A\mapsto\operatorname{Spec}A$ defines a contravariant functor from the category of rings to the category of affine schemes. If ...
5
votes
1answer
66 views

Functor of points $h_X$ is an fpqc sheaf on $\operatorname{Spec} \Bbb{Z}$

I want to show the following. Let $X$ be any scheme (say over the terminal object $\operatorname{Spec} \Bbb{Z}$ in $\textbf{Sch}$) and $A \to B$ a faithfully flat ring homomorphism. Then $$h_X(A) \to ...
1
vote
1answer
30 views

Section of pre-sheaf.

The notation is from O. Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981. Let $\mathcal{F}$ be a presheaf on the topological space $X$. Let $U\subset X$. Let $s:U\rightarrow ...
5
votes
1answer
69 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
4
votes
1answer
63 views

Using the cocycle condition to glue sheaves

Given a cover $\{U_i\}$ of a space $X$ and for each $U_i$ a sheaf $\mathcal{F}_i$ and isomorphisms $\phi_{ij}:\mathcal{F_j}|_{U_i \cap U_j} \rightarrow \mathcal{F_i}|_{U_i \cap U_j}$ satisfying the ...
1
vote
1answer
42 views

Linear systems and rational maps

I'm following Beauville's book on Complex Algebraic Surfaces. If $D$ is a divisor on a surface $S$, we write $|D|$ for the set of all effective divisors linear equivalent to $D$ and we call it a ...
4
votes
2answers
36 views

Stalk of the quotient presheaf

Consider a sheaf of abelian groupS $\mathscr F$ over a topological space $X$. If $\mathscr F'$ is a subsheaf of $\mathscr F$ (over $X$), then we can construct the quotient presheaf $\mathscr ...
1
vote
1answer
27 views

About the sheafification

Look at the following definition of the shefification (the source is the Stacks Project): I don't understand what is the projection $\prod_{u\in U}\mathcal F_u\longrightarrow\prod_{v\in ...
2
votes
1answer
54 views

Example of epimorphisms such that the product is not an epimorphism in the category of sheaves

I've heard that in the category of sheaves over a topological space $X$, products of epimorphisms are not epimorphisms. I think that it's equivalent to saying that $\mathbf{Sh}(X)$ does not satisfy ...
6
votes
1answer
105 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
4
votes
1answer
65 views

Universal property of quotient sheaves

Recently, I was doing exercise 2.3 (b) in chapter 2 of Hartshorne's book celebrated book on algebraic geometry. The exercise is as follows: Let $(X,\mathcal{O}_X)$ be a scheme. Define a presheaf by ...
7
votes
0answers
68 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
4
votes
1answer
57 views

Restriction of a sheaf to a fibre

I have come across the notion of the restriction of a sheaf to a fibre, but I haven't been able to find a proper definition, could anyone perhaps supply one? Suppose that $f: X \to Y$ is a morphism ...
3
votes
1answer
50 views

Flasque Constant Sheaf

It is easy to show that if $X$ is an irreducible topological space, then the constant sheaf $\mathbb{Z}$ is flasque. Is the converse true?
4
votes
0answers
73 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...