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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72 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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0answers
19 views

How is a connection the same as Morphism between different pullbacks to the first infinitesimal neighbourhood

This question arose from the first pages of Delignes Equations Differentielles. There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data: For all ...
4
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0answers
48 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
3
votes
1answer
30 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
4
votes
1answer
58 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
2
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3answers
72 views

Presheaf that do not satisfy: If $\{U_i\}$ is an open cover of $U \subseteq X$ and $s \in \mathcal{F}(U)$, then $s=0$ iff $s|_{U_i}=0$ $\forall i$

Let $X$ be a topological space. Find an example of a presheaf $\mathcal{F}$ that do not satisfy: If $\{U_i\}_{i \in I}$ is an open cover of $U \subseteq X$ and $s \in \mathcal{F}(U)$, then ...
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1answer
25 views

What does the notation $\mathcal{O}_{\mathbb{P}^n}(1)$ mean?

I have tried looking at my sheaves notes but couldn't find anything.
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0answers
12 views

Sheafification and restriction to open subset

Let $X$ be a topological space and $\mathcal{F}$ be a presheaf on $X$. We denote by $\mathcal{F}^+$ the sheafification of $\mathcal{F}$. Let $U\subset X$ be an open subset. We denote by ...
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votes
0answers
10 views

Tensor product of sheaf

Let $M$ and $N$ be sheafs on a space $X$ then what is the relation of $\pi_1^* M\otimes \pi_2^* N$ and $\pi_1^* N\otimes \pi_2^* M$, where $\pi_i$ is the $i$ projection of $X\times X$
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vote
1answer
35 views

Definition of a sheaf: What is $s\rvert_{V_i}$ if $V_i\not\subseteq U$?

I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that: For all open sets $U$ and open coverings $\{V_i\}$ of ...
5
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1answer
97 views

Calculating global sections of sheaves

Consider the usual projective space $\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}$, and the Weil divisor $D = \{0\} \subset \mathbb{C} \subset \mathbb{P}^{1}$. Writing projective space as the union of ...
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votes
2answers
36 views

Where is sheafification in the definition of exact sequence of sheaves?

I am reading Andreas Gathmann's notes on Algebraic geometry,http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf Def 7.1.14(iv)says the following As usual, a sequence of sheaves ...
3
votes
0answers
62 views

sections of higher direct image sheaf

Let $f:X \rightarrow Y$, be a proper birational morphism of projective algebraic varieties with $X$ smooth. Denote by $R^if_* \mathcal{O}_X$, the higher direct image sheaves. Do exists a simple way ...
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0answers
63 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
3
votes
1answer
39 views

Step in the construction of the global spec of a sheaf of algebras

I'm working my way through the construction of the global spec of a sheaf of algebras. Here is the setup. Let $ Y $ be a scheme. Let $ \mathscr{A } $ be a quasi coherent sheaf of $ ...
2
votes
1answer
55 views

Sheafification as a Kan Extension of the Identity?

How can the sheafification functor be described in terms of a Kan extension of the identity on the category of $\mathsf{Set}$-valued sheaves (over some topological space)? How about general $\mathsf ...
6
votes
0answers
80 views

Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to ...
3
votes
1answer
48 views

Subsheaf generated by one section is coherent

I'm working on exercise II.5.15 in Hartshorne's book. I need to prove the following bit. Let $ X $ be a noetherian scheme. Let $\mathscr {F }$ be a quasi coherent sheaf on $ X$. Then the subsheaf ...
1
vote
1answer
30 views

Decomposing Sheaves into a direct sum

I have two questions about sheaves inspired by representation theory: Is there a theorem that tells me when a sheaf of vector spaces can be decomposed into the direct sum of two "smaller" sheaves? ...
2
votes
0answers
32 views

“Étalé space almost looks like a sheaf”

In Daping Weng's A Categorical Introduction to Sheaves, in the paragraph preceding section 3.2 (page 8), the author says A sheaf space [= Étalé space] almost looks like a sheaf... I would very ...
4
votes
1answer
88 views

What is the definition for a fine sheaf/ a partition of the unity on a sheaf?

From Griffiths and Harris, p. 42: A sheaf $\mathcal F$ on $M$ is called fine, if for $U=\bigcup_i U_i\subseteq M$ there is a partition of the unity subordinate to the cover $(U_i\mid i\in I)$. By this ...
4
votes
0answers
46 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
0
votes
1answer
25 views

Canonical group structure of germs

Given a topological space $X$ and a presheaf of abelian groups on $X$, $A$, we can construct the set of germs at a point $x\in X$ by taking $\mathscr{A}_x=\lim\limits_{\rightarrow} A(U)$ where $x\in ...
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vote
1answer
36 views

Characterisation of closed subschemes of projective spaces

I'm trying to understand a step in the proof of corollary 5.16 in Hartshorne. The statement Let $ A $ be a ring. if $ Y$ is a closed subscheme of $\Bbb{P}_{A}^{r} $ then there is a homogeneous ...
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vote
1answer
36 views

Pushforward and sheaf-hom

If $S$ is a surface over the complex numbers $\mathbb{C}$, and $C$ is a curve in $S$, and $i:C\longrightarrow S$ is the inclusion morphism. If $A$ is a line bundle over $C$, then is it true that ...
0
votes
1answer
80 views

Is this sheaf simple?

Let $S$ be a surface and $C$ be an effective divisor in $S$. That is $C$ is a curve in $S$ and $i:C\longrightarrow S$ is the inclusion morphism. Let $E$ be line bundle over $C$, so $i_*E$ is a ...
0
votes
1answer
31 views

Is any reflector from a presheaf category $PSh(K)$ to a topos $C$ necessarily left exact?

Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$. Is $T$ left exact? $D$ ...
0
votes
0answers
30 views

Compute some sheaf cohomology supported at a point

This question is from the Kashiwara's book: Sheaves on Manifolds Ex3.1 Let $X=\mathbb{R}$,$Z=\left\{\frac{1}{n}; n\in\mathbb{N}\setminus\{0\}\right\}\cup\{0\}$ and $Y=X\setminus Z$. Prove that ...
0
votes
0answers
48 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme ...
2
votes
1answer
87 views

Why are sections of sheaves called sections?

Generally, a section is a right inverse. On the other hand, if $F$ is a ($\mathsf{Set}$-valued) sheaf, then the elements of $FU$ are usually also called sections. Why is this terminology justified ...
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votes
0answers
27 views

Subfunctor of a sheaf which is not a sheaf

The following is proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1. Proposition 1. If $F$ is a sheaf on $X$, then a subfunctor $S\subset F$ is a subsheaf ...
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0answers
21 views

Theory around the Cellular Sheaf

I have lately stumbled upon cellular (co)sheaves, which look very interesting. To understand them better, I would like references that systematically develop the theory behind them (preferably in ...
1
vote
1answer
59 views

Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ ...
3
votes
2answers
66 views

An example of a coproduct of sheaves in the category of presheaves that is not a sheaf

For a site $C$ there is an adjunction $a\colon PSh(C)\leftrightarrows Sh(C)\colon \iota$ with sheafification $a$ and inclusion $i$. $a$ preserves finite limits. The inclusion $\iota$ does not preserve ...
5
votes
2answers
75 views

De Rham-Weil theorem

I am having trouble understanding a couple of points with regard to the De Rham-Weil theorem and was hoping that someone might be able to shed some light. Let $X$ be a smooth manifold and ...
2
votes
2answers
70 views

Sheafification, stalks and quotient

I gave a problem that I can't finish by myself. Any help would be appreciated. Consider a sheaf $\mathcal{F}$ of abelian groups on a topological space. I would like to show that given two sheaves ...
2
votes
1answer
44 views

when a presheaf is a sheaf

I've seen a very natural definition when a presheaf $F:C^{op}\rightarrow Set$ is actually a sheaf. This definition used the functors $hom(-,-)$ and $F$ and notions of injective and surjective maps ...
1
vote
1answer
23 views

morphism betweem sheaves that is an isomorphism in local sections of a basis

Let $\{U_{\alpha}:\alpha \in A\}$ be a basis of open sets for the topological space $X$. Let $\mathscr{F},\mathscr{G}$ be sheaves over $X$. Suppose that there exist a morphism $\phi: \mathscr{F} \to ...
1
vote
1answer
29 views

Map between free sheaf modules not arising from a “matrix”

My question is about this example in the Stacks project. For convenience and completeness, here is the relevant part. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$ of the real line ...
2
votes
1answer
65 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
2
votes
1answer
30 views

Inclusion of quotient sheaves restricted to open subset

When introducing sheaf cohomology following for example Chapter 8 of Kempf's book on Algebraic Varieties, we make the following standard definitions. If $\mathcal{F}$ is a sheaf of abelian groups on ...
0
votes
1answer
54 views

A torsion-free sheaf of rank 1 on a surface

Let $X$ be a surface and $E$ be coherent sheaf on $X$. Now there is always a natural map $\mu:E\longrightarrow E^{\vee\vee}$. The kernel of this map is precisely the torsion subsheaf of $E$. Now if ...
0
votes
1answer
48 views

Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
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votes
0answers
20 views

Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
0
votes
1answer
57 views

(Non-)Isomorphism of (pre-)sheaves

I want to prove the following statement: Let $f:F \to G$ be a homomorphism of sheaves. Let $(U_i)_{i \in I}$ be an open coverage of $X$, such that $f|{U_i}: F|U_i \to G|U_i$ is an isomorphism for all ...
3
votes
0answers
43 views

Axiom of glueing: direct limit of sheaves in a noetherian topological space. [duplicate]

I'm trying to prove that in a noetherian topological space the following property is satisfied: Consider a direct system of sheaves and morphisms $\{ \cal{F}_t, f_{ij} \}_t$. Consider the presheaf ...
1
vote
2answers
70 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
2
votes
0answers
67 views

First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.

Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a ...
3
votes
1answer
62 views

Proof of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is ample $\iff$ $a,b >0$.

I would like some help understanding the proof in $(\impliedby)$ direction. Hartshorne on page 156, Example 7.6.2 says: If $\mathcal{L}$ is an invertible sheaf on $\mathbb{P}^1 \times \mathbb{P}^1$ ...
2
votes
1answer
46 views

Inductive limit of sheaves over noetherian topological space

Let $X$ be a topological space. Let $I$ be a poset and let $\mathcal F_i$ for $i\in I$ be sheaves on $X$, and $\{\pi_{ij}\colon \mathcal F_i \rightarrow \mathcal F_j\}_{i,j\in I}$ be an inductive ...