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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1answer
45 views

Prove that $H^1(\mathcal{M}^*)=0$.

Let $X$ be a compact Riemann surface. For an open set $U$, let $\mathcal{M}^*(U)$ be the multiplicative group of nonzero meromorphic functions on $U$ ("nonzero" meaning "not identically zero"). This ...
3
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1answer
48 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
2
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1answer
46 views

Is this sequence of presheaves exact?

On p.298 of his Homological Algebra text, Rotman considers the sequence of presheaves on $X=\mathbb{C}-\{0\}$ : $0 \to \mathbb{Z}\to \mathcal{O}\to \mathcal{O}^\times \to 0$ where $\mathbb{Z}$ is the ...
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0answers
58 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
1
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1answer
49 views

Question on the definition of sheaves.

When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties. ...
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1answer
72 views

Only $f^\sharp_x$ makes the diagram commutative

By Algebraic Geometry I from Görtz, Wedhorn page 60 $f^\sharp_x$ is the unique ring homomorphism which makes the diagram $A\to B \to B_{p_x}$, $A\to A_{p_{f(x)}}\to B_{p_x}$ commutative. The first ...
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2answers
39 views

When are two morphisms of sheaves the same?

Suppose I have two morphism $\phi_1, \phi_2 : F \rightarrow G$, where $F$ and $G$ are sheaves of sets on $X$. Is it enough to show that $\phi_1(X) = \phi_2(X)$ (i.e. as maps from $F(X)$ to $G(X)$) to ...
1
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1answer
61 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
1
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1answer
51 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
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1answer
36 views

Why $F \leq F' \; \Leftarrow \; \forall x \in X \; F_x \subseteq F'_x$?

I cannot understand the proof of Proposition 3.11 in Sheaf Thoery by Tennison and Hitchin. If $F,F'$ are subsheaves of a sheaf $G$, then $$ F \leq F' \; \Leftrightarrow \; \forall x \in X \; ...
1
vote
1answer
27 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
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0answers
26 views

$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism [duplicate]

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
2
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1answer
42 views

$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
1
vote
1answer
51 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
3
votes
2answers
100 views

$f^+\mathscr{G}$ not a sheaf even if $\mathscr{G}$ is

I have read that there are continuous maps $f:X\to Y$ and sheaves $\mathscr{G}$ on $Y$ such that $f^+\mathscr{G}$ is not a sheaf. Do you have an easy example for me?
1
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1answer
65 views

Learning Fibre Bundle from “Geometry and Topology” by Bredon

Bredon defines bundle projection in the following way: Then he defines Fibre Bundle The he Remarks about the condition 3. He says the map $\theta :U \rightarrow K $ exists. The only important ...
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2answers
37 views

What is a natural morphism of stalks?

I am a bit confused with this question. Let $\pi: X \rightarrow Y$ be a continuous map and $F$ a sheaf of sets on $X$. If $\pi(p) =q$, then what is the natural morphism of stalks $(\pi_*F)_q ...
5
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0answers
56 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
0
votes
0answers
38 views

Direct limit of $ \ \ \mathcal{D} = ((\mathcal{F} (U))_{U \in \mathcal{V}} \, \ (r \ : \ \mathcal{F} (U) \to \mathcal{F} (V))_{V \subset U}) $. [duplicate]

Let $ X $ be a topological space. Let $ \mathcal{F} $ be a sheaf on $ X $. Let $ U $ be an open subset of $ X $. Let $ \mathcal{V} $ the set of open neighborhoods of $ U $, which is the filter for ...
2
votes
1answer
78 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
0
votes
1answer
56 views

Isomorphism of presheaves

I just want you to tell me if a morphism of presheaves $\varphi:\mathscr{F}\to\mathscr{G}$ is an isomorphism iff every map $\varphi_U$ is bijective. I think it is true. Here my proof for the ...
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0answers
30 views

Sheafification part 2: Uniqueness of $\tilde{\varphi}$ and a formal consequence

I'd like to go on discussing the proof which I started to discuss here. The book says sending $(s_x)_x\in\tilde{\mathscr{F}}(U)$ to $(\varphi_x(s_x))_x\in\tilde{\mathscr{G}}(U)$ defines a morphism ...
3
votes
1answer
71 views

Sheafification: Show that $\tilde{\mathscr{F}_x}=\mathscr{F}_x$.

My today's question is about a proof of this book. More precisely we are talking about the proof of Prop. 2.24 on page 52. The book says that we have $\tilde{\mathscr{F}_x}=\mathscr{F}_x$ for all ...
3
votes
1answer
56 views

Constructing the classifying space for a groupoid G

I am working with the classifying space BG of the groupoid G. One definition is as follows: $$ BG = \bigsqcup _n (G_n \times \Delta ^n) /(d_i(g),x) \sim (g, \delta _i (x)). $$ Where the $d_i$ are ...
5
votes
1answer
68 views

A question about Hartshorne III 12.2

In Hartshorne III 12.2, $X\to \text{Spec}\ A$ is a morphism, $\mathcal{F}$ is a coherent sheaf on $X$, flat over $\text{Spec}\ A$, $M$ any $A$ module, then we can construct the sheaf associated to the ...
1
vote
0answers
82 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
1
vote
2answers
31 views

Presheaves over a discrete space are necessarily sheaves?

In problem number 5.42 on p.302 of Homological Algebra text by Rotman it is asked to prove that every presheaf of abelian groups over a discrete space is a sheaf. However it looks to me that I have a ...
0
votes
1answer
69 views

Hartshorne Exersice 1.17 Skyscraper sheaf Chapter II Schemes

I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf. Observation- If $P\notin U$, then $U\cap ...
1
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0answers
16 views

Example to show that presheaf maps agreeing on stalks are not necessarily equal

We know that if two maps $\phi,\psi : \mathcal{F}\to \mathcal{G}$, where $\mathcal{F} $ is a presheaf and $ \mathcal{G}$ is a sheaf, agree on stalks then they are equal. Can we find an example to show ...
5
votes
1answer
74 views

Is there a coherent sheaf which is not a quotient of locally free sheaf?

Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf? (Hartshorne II Cor 5.18 showed that on every projective variety, ...
0
votes
1answer
22 views

Cokernel of a sheaf morphism not being a sheaf

Does anyone know a good example where the cokernel of a sheaf morphism is not a sheaf?
9
votes
0answers
117 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
5
votes
2answers
87 views

Abstract proof that internal hom presheaf is a sheaf

Let me recall some definitions first. Let $D$ be a small category and let $J$ be a Grothendieck topology on $D$. A presheaf $F$ on $D$ is called a sheaf when for every covering sieve $\psi \in J(d)$ ...
1
vote
2answers
94 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
0
votes
1answer
52 views

What is a Presheaf (intuitively) and help with the technical machinery.

I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if ...
0
votes
1answer
64 views

Prove: $U \mapsto \mathrm{Hom}(U, Y)$

Rewording this problem via what Zhen Lin's notion of the original question is. For $X$ and $Y$ ringed spaces Prove: For each open $U \subset X$ the Presheaf $U \mapsto \mathrm{Hom}(U, Y)$ is a ...
1
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0answers
18 views

Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
3
votes
3answers
118 views

Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
4
votes
1answer
83 views

Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
3
votes
1answer
24 views

Isomorphism of sheaves between $\mathcal{O}$ and $\Omega$

I want to show that the sheaves $\mathcal{O}$ (holomorphic functions) and $\Omega$ (holomorphic $1$-forms) are isomorphic. Because every $\omega \in \Omega$ has the form $\omega=f \text{d} z$ for ...
4
votes
1answer
66 views

Cover of a sheaf

We define $U_1:=\mathbb{P}^1\setminus\left\lbrace \infty\right\rbrace$ and $U_2:=\mathbb{P}^1\setminus\left\lbrace 0\right\rbrace$. $\Omega$ is the sheaf of holomorphic $1$-forms. How can I ...
2
votes
0answers
54 views

How to prove that $\mathcal{Y}_X$ is a sheaf by using epimorphic families

I'm trying to prove this: Let $(\mathcal{C}, J)$ be a site and suppose that for every $X\in ob(\mathcal{C})$ and $R\in J(X)$ the family $\{ \bar{f}_Y: \mathcal{Y}_Y\rightarrow \mathcal{Y}_X \}_{Y\in ...
2
votes
1answer
57 views

An equivalence of categories of presheaves.

Let $C$ and $D$ be two small categories. Consider the corresponding categories of presheaves $PSh(C)$ and $PSh(D)$. Suppose we have an equivalence of categories $F: PSh(C) \to PSh(D)$. Asking for an ...
0
votes
0answers
14 views

Local systems on algebraic groups

Two questions about local systems: Given an algebraic group G over a perfect field $k$ and given a colection of vectorial spaces $(F_y)_{y\in G}$ why there is an unique local system $F$ such that the ...
1
vote
1answer
40 views

Stalk is a local object of a sheaf

Let $X$ be a topological space. Let $\mathcal{F}$ be a sheaf on $X$. The stalk is the direct limit $$ \mathcal{F}_x = \lim_{\underset{V \ni x}{\longrightarrow}} \mathcal{F}(V) $$ Let $U \subset X$ be ...
1
vote
1answer
50 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
46 views

Sheaf of ideals [duplicate]

Let $(X, \mathcal{O}_X)$ be a scheme and and $f \in \mathcal{O}_X(X)$. We define the sheaf of ideals by the assignment $$ U \longmapsto f|_U \mathcal{O}_X(U) $$ and denote this sheaf by ...
3
votes
1answer
57 views

History of terminology: sheaves, presheaves, etc.

I've been looking at some old notes (1970s) on Riemann surfaces, trying to match up terminology with modern definitions (at least going by Wikipedia). The notes use the same terms as Gunning's ...
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0answers
51 views

Hom sheaf over a scheme in the case of quasi-coherent sheaf at first argument

Let $X$ be a scheme and $\mathcal{F},\mathcal{G}$ be two sheaves of $\mathcal{O}_X$-modules. I showed that the presheaf which assigns each open subset $U$ of $X$, $$ U \longmapsto ...
0
votes
0answers
8 views

Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that: $m^*(L) \simeq L \boxtimes L $. Then why is $L$ an equivariant sheaf on $G$ with the action the ...