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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18 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
4
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0answers
33 views

Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space ...
1
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0answers
107 views

Homology and Hyperhomology

Let $X$ be a non-singular variety over $k$(algebraically closed). Suppose we have the following complexes (not exact sequence) of $\mathcal{O}_X$-modules. $0 \longrightarrow A^2 \longrightarrow A^1 ...
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3answers
184 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
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1answer
16 views

Sheaf of sections of vector bundle over a manifold is an $\mathcal O_M$-module

Section 13.1.2 of Ravi Vakil: "Fix a rank $n$ vector bundle $\pi:V\rightarrow M$. The sheaf of sections $F$ of $V$ is an $\mathcal O_M$-module - given any open set $U$, we can multiply a section over ...
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2answers
151 views

Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. ...
3
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0answers
39 views

Vector bundles in Ravi Vakil's notes on quasicoherent sheaves

In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free ...
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56 views

Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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1answer
43 views

Is this assignment of the topos of sheaves functorial?

Let $\mathcal{C}$ be a site and for any object $X$ of $\mathcal{C}$ denote by $\text{Sh}(X)$ the category of sheaves on the site $\mathcal{C}/X$. My question is, what can we say about this assignment? ...
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41 views

Quasicoherent-sheaves and pushfoward

How to prove the proposition in the picture below? It seems to be easy, but I am a beginner. Thanks in advanced for your help!
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29 views

Ramanan's definition of differentiable function

In his book Global Calculus, Ramanan defines a differential manifold as follows: What is meant by condition (b)? Is $\mathcal A$ simply a subsheaf of the sheaf of real valued continuous functions ...
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0answers
48 views

The induced map on stalks is well-defined

$\require{AMScd}$ Let $\phi:\mathscr{F\to G}$ be a morphism of sheaves on $X$, let $\mathscr F_P$ be a stalk of $\mathscr F$ at $P\in X$, and let the stalk map $\phi_P:\mathscr F_P\to\mathscr G_P$ ...
3
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1answer
55 views

The “Hartshornian” sheafification of a sheaf

Given a sheaf $\mathscr F$ on $X$, how does one show that its sheafification (in the sense of Hartshorne) is isomorphic to it? The most obvious thing to do is a universal property argument: since ...
2
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1answer
104 views

Text book for sheaf theory

Is there any nice text book for sheaf theory for an under gradute student? Tennison's sheaf thory was too hard for me, Please help me, Thanke you very much.
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0answers
33 views

What is the sheaf of endomorphisms?

I saw this term or symbol “End$(\mathcal{F})$", where $\mathcal{F}$ is a quasi-coherent sheaf, in some places, e.g. First Ext group of a sheaf. But I found nothing on Google when I typing "sheaf of ...
4
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1answer
83 views

Uniqueness of the structure sheaf

Given a ring $A$ and his Spectrum $X=Spec(A)$ seen as a topological space with the Zariski topology, it's possible to build a sheaf on $X$ satisfying the conditions $O_X(X)=A$ $O_X(X_f)=A_f$ where ...
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0answers
34 views

Can I define a site as a category endowed with a pretopology instead of a topology?

If $K$ is a pretopology on a category $\mathcal{C}$ and $J$ the topology it induces, are the Grothendieck toposes $\text{Sh}(\mathcal{C},K)$ and $\text{Sh}(\mathcal{C},J)$ the same in general? As I ...
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45 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
4
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53 views

Prove extension by zero is a special case of lower shriek?

The lower shriek functor is defined by $$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$ On the other hand, if $j:V\subset X$ is the ...
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1answer
23 views

Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves?

It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero. Is there any ...
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0answers
40 views

What is meant by $\mathbb{C}^*$?

In Brylinski (Loop Spaces, Characteristic Classes and Geometric Quantization), the symbol $\mathbb{C}^*$ is used quite a lot, and I'm really not sure what it means. I've no shortage of ideas: the ...
1
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1answer
16 views

Restriction of a sheaf equals zero

I'm reading online lectures notes on sheaves, and I'm confused about the meaning of the equation $F|_U=0$ below. $F|_U$ is defined as the inverse image sheaf $f^\ast F$ along the inclusion of some ...
2
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0answers
52 views

What are some properties of the sheaf of distributions?

In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, ...
4
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1answer
70 views

Additivity of the first Chern class

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr ...
2
votes
1answer
57 views

Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?

I'm wondering whether $\mathsf{PSh}(\{x\})$ or $\mathsf{Sh}(\{x\})$ are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the ...
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0answers
93 views

Simple question in “Sheaves in geometry and logic”

There's an argument I don't understand in "Sheaves in geometry and logic" by Mac Lane and Moerdijk, that seems a priori easy but I can't see it. Page 174, diagram (10) (involving the power ...
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0answers
32 views

Normal cone and specialization

This question is from the Kashiwara and Schapira's book: Sheaves on Manifolds Let M be a closed submanifold of X and let S be a locally closed subset of X, prove that C$_M$(S) = ...
3
votes
1answer
70 views

Finite flat pushforward of a constant sheaf

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also ...
2
votes
1answer
56 views

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal ...
15
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1answer
209 views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
3
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1answer
35 views

Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$

Let $P$ be a $\mathsf{Set}$-valued presheaf and let $f^\ast:\mathsf{PSh}(Y)\rightarrow \mathsf{PSh}(X)$ be the (topological) inverse image sheaf functor, defined on objects as the filtered colimit ...
5
votes
0answers
76 views

How do the direct and inverse image sheaf functors interact with homotopy?

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 $f^{-1}$) form a well-known adjunction for ...
6
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0answers
104 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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32 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 ...
5
votes
0answers
55 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
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1answer
37 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
65 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
votes
3answers
74 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
11 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 $X\times X$
1
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1answer
37 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
102 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 ...
0
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2answers
45 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
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0answers
67 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
64 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
40 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
60 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 ...
7
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92 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
52 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 ...