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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Global section defines a map from structure sheaf

Let $X$ be a smooth projective scheme over an algebraically closed field. Let $F$ be a coherent torsion-free sheaf on $X$. A global section $f$ of $F$ defines a morphism $O_X\rightarrow F$ given by: ...
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27 views

Pullback of a local homeomorphism is a local homeomorphism

Suppose we have pullback diagram of topological spaces: I want to prove: If $g:Y\to Z$ is a local homeomorphism (etale map), then $p_1:P\to X$ is a local homeomorphism. My idea: First of all $...
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13 views

Sheaves of Simplicial Rings?

Could someone provide me a reference for a treatment of sheaves of simplicial commutative rings? As in simplicial sheaves with a ring structure.
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65 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
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10 views

Sheaf on Base Extends Uniquely to a Sheaf — Prove via Etale Space

The theorem is well-known: Suppose $\{B_\alpha\}_{\alpha\in I}$ is a base of topology for $X$, and $\mathscr{F}^B$ is a sheaf on base on $X$. Then there exists a sheaf $\mathscr{F}$ extending $\...
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Pushforward of sheaves on the blowup of $\mathbb A^2$ to $\mathbb P^1$

In http://arxiv.org/abs/1210.2564 Example 4.12 it is written that for $Y$ the blowup of $\mathbb A^2$ at the origin (i.e. $Y \cong \mathrm{Tot} \, \mathcal O_{\mathbb P^1} (-1)$), and $π \colon Y \to \...
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43 views

Question on the second part of the definition of sheafification

I do not understand part $(2)$ of Proposition-Definition 1.2 on page $64$ of Hartshorne's Algebraic Geometry: The original texts are: 'Given a presheaf $F$ ... $F^{+}(U)$ is the set of functions $s$ ...
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66 views

How to show $Hom(V,V)\rightarrow Hom(V_x,V_x)$ is injective, V being semi-stable

Let $V$ be a semi-stable vector bundle over a smooth irreducible projective curve of genus $g\geq 2$. Let $x\in X$. How do we show that the canonical map $Hom(V,V)\rightarrow Hom(V_x,V_x)$ which ...
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66 views

Multiplication map between sheaves on $\operatorname{Proj}A$

Let $A$ be a graded ring, $X=\operatorname{Proj}A$ and let $f$ be an element of degree $d>0$. I have come across the phrase "Let $\mu\colon\mathcal{O}_X \to \mathcal{O}_X(d)$ be the map given ...
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38 views

Tensor product of the stricture sheaf with the function field

Consider an irreducible scheme $X$ with function field $K(X)$. Then define the presheaf $$U\mapsto \mathscr O_X(U)\otimes_{\mathscr O_X(U)} K(X)$$ for every open set $U\subset X$. Is this presheaf ...
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37 views

When the sheafification map is a presheaf epimorphism

Claim: Suppose $\mathcal{F}$ is a glueable presheaf on a paracompact hausdorff space. Then the sheafication map on global sections $\mathcal{F}(X) \to \tilde{\mathcal{F}}(X)$ is surjective. (Note ...
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51 views

restrictions of rational sections of an invertible sheaf.

Let $(X,\mathscr O_X)$ be an integral scheme with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf on $X$. Moreover suppose that $\{U_i\}$ is an open covering of $X$ such that $\...
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46 views

Sheafification by the small object argument

Is the sheafification functor constructed as a double application of the plus construction a special case of the small object argument? I thought it might be since I can't think of any other general ...
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182 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
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1answer
62 views

Given a locally ringed space there is a bijection between open and closed sets of $X$ and idempotent elements of $\mathcal{O}_X(X) $

This is a problem from Gortz and it does NOT assume that the underlying space is the spectrum of the ring or anything like that. Now I proved easily that given a clopen set of $X$ there is an ...
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46 views

Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
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1answer
40 views

Support of quotient sheaf of ideal sheaves with same support

I'm not very sure about this argument. Let $\mathscr{I},\mathscr{J}$ two ideal sheaves (you can think about ideal sheaves over a projective variety or even the projective space itself) and assume that ...
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60 views

Sheaf hom of coherent sheaves

Let $(X,\mathscr O)$ be a locally ringed space, and let $\mathscr F$, $\mathscr G$ be $\mathscr O$-modules. Consider the following facts: $\newcommand{\sF}{\mathscr F}\newcommand{\sG}{\mathscr G}\...
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16 views

Sheaf definition vs “Mayer Vietoris”

Let $F$ be a presheaf on a space $X$ and say that $F$ has property MY if for all $U, V$ open in $X$ we have an exact sequence $$0 \to F(U \cup V) \to F(U) \oplus F(V) \to F(U \cap V)  $$ Is this ...
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22 views

If $H^j(U,\mathcal{F})$ vanishes, stalk of the presheaf of cohomology groups of $\mathcal{F}$ also vanishes at $x \in U$

The following statement appears in these notes I'm trying to read: $\mathcal{F}$ be a sheaf of $R-$modules on a topological space $X$. Corresponding to $\mathcal{F}$ we can construct a presheaf $H^j(\...
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34 views

Vanishing set of a pullback section

Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$. If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and ...
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1answer
84 views

Twisted sheaf isomorphic to invertible sheaf associated to 1-cocycle

Let $X=\operatorname{Proj}A$ for some graded ring $A$ such that the irrelevant ideal $A_+$ is generated by the degree one part $A_1$ of $A$. Then Bosch's $\textit{Algebraic Geometry and Commutative ...
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2answers
42 views

References for the functor of points vision of schemes.

I recently discovered the idea of the functor of points. I would like to find a reference where the different visions of scheme are presented. It seems to me that the classical texts emphasize the ...
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1answer
48 views

Push-forward of quasi-coherent sheave on affine scheme is quasi-coherent

Let $X=$ Spec$R$, $Y=$ Spec$S$, $f:X \to Y$ be a morphism of schemes. Let $M$ be a $R$-module, and let $\mathcal{F}=\tilde{M}$ be the sheaf on $X$ induced by $M$. How can I show that the pushforward ...
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34 views

How to deduce the usual definition of Quasi-coherent Module over a scheme from the general definition over ringed Spaces

Quasi-Coherent Modules over a Ringed Space : Let $(X,\mathcal O_X)$ be a ringed space. A sheaf of modules $F$ over $(X,\mathcal O_X)$ is called quasi-coherent if for every point $x\in X$ $\exists U\...
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1answer
25 views

Showing a class of objects is $F$-acyclic

There's a lemma from homological algebra that I'm using in sheaf cohomology, and I can't remember where else I've seen it. Where are some other key places it is applied? Lemma: Let $F:\mathcal{C} \...
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52 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
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1answer
45 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 fixed-point-...
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22 views

Continuously variable *space* [closed]

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
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1answer
142 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 ...
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1answer
107 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 $...
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38 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 ...
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1answer
31 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
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1answer
64 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 $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
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102 views

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 ...
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67 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 $\pi_U :Z_U \to U$. ...
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1answer
79 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
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1answer
62 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 $...
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39 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 A,B$\...
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1answer
38 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 ...
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1answer
31 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 ...
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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 $X$...
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78 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 $X(\mathbb C)$...
2
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0answers
25 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 $\...
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1answer
59 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 ...
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34 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 ...
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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: ...
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1answer
60 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
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1answer
42 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 ...