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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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
18 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 ...
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53 views
+100

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 ...
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20 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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50 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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31 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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46 views

The sheaf (of stalks) of meromorphic functions, why don't we use a more natural definition?

If $A$ is a commutative ring with $1$, let's denote with $R(A)$ the set of regular elements of $A$. Let $(X,\mathcal O_X)$ be a locally Noetherian scheme, then the sheaf (of stalks) of meromorphic ...
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1answer
50 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
38 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 ...
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25 views

Quotient Sheaves

Let $X$ be a ringed space, and $J$ be a sheaf of ideals of the structure sheaf. Define, $Y = \{x\in X ~ | ~ J_x \not = \mathcal{O}_x\}$, this is a closed set. We have an inclusion $i:Y\to X$. Is there ...
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31 views

When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
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1answer
24 views

Representation of an effective Cartier divisor: is there an imprecision in Liu's book?

I will use the notation of Liu's book which is also the standard one. If you don't feel comfortable with it please ask more information in the comments Let $X$ be a scheme, and consider a Cartier ...
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2answers
47 views

Sheaf associated to a Cartier divisor

This question is motivated by a construction, unclear for me, related to Cartier divisors. But, in the end it can be reduced to a question involving only sheaves on topological spaces. Let $X$ be a ...
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1answer
42 views

Twisting an exact sequence of sheaf gives an exact sequence

Reading through Hartshorne's Algebraic Geometry, and reading different notes about cohomology of sheaves, I have often seen the argument that if you have an exact sequence of coherent sheaves over a ...
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2answers
61 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this ...
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35 views

Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object ...
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24 views

Commutation of pullbacks with disjoint unions in sites?

From what I understand, in superextensive sites disjoint unions commute with pullbacks. I know this holds in every topos sense pullback has a right adjoint, and I have heard something along the lines ...
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0answers
32 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
3
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1answer
154 views

Alternative description of the sheafification

For me the sheafification of a given presheaf is this: Proposition-Definition: Given a presheaf $\mathscr{F}$, there is a sheaf $\mathscr{F}^+$ and a morphism $\theta \colon \mathscr{F} \to ...
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1answer
52 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self ...
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0answers
25 views

Prove that for $\mathcal{F}$ a sheaf on manifold $M$ we have $H^0(\{U\},\mathcal{F})=\mathcal{F}(M)$

I want to prove that that for $\mathcal{F}$ a sheaf on manifold $M$ we have $H^0(\{U\},\mathcal{F})=\mathcal{F}(M)$, where $\{U\}$ is any locally finite open cover of $M$. (Note that I think we really ...
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41 views

Categories of defintion for sites on spaces and sites on schemes

Starting with example 2.26 in Vistoli's Notes he defines topological sites on $\mathsf{Top}$, while sites on schemes are always defined on a slice category $\mathsf{Sch}/X$. Why is this?
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33 views

Are representables on the étale site on topological space sheaves?

The gluing lemma says representables on the classical site on $\mathsf{Top}$ are sheaves. Basic scheme theory says the same is true for the small Zariski site of a scheme. Are representables on the ...
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1answer
52 views

Is Coherent sheaf acyclic?

I am no expert in sheaf theory so the following question may be trivial. Let $X$ a complex manifold, and let $\mathcal{F}$ a coherent sheaf on $X$. Is $\mathcal{F}$ acyclic? If not: can you give a ...
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64 views

For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $\mathcal{O}_X$-module $F$ has the structure of a $G$-representation

Reading Huybrechts & Lehn's "The geometry of moduli spaces of sheaves" I am stuck with a particular statement that they make in the chapter on GIT without explanation. Namely, they say that for an ...
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1answer
27 views

A confusion with the Definition of 'Skyscraper Sheaf' from 'Stack Project'

According to Hartshorne, (Chapter, Ex 1.17): Let $X$ be a topological space, let $P$ be a point, and let $A$ be an abelian group. Define a sheaf $i_p(A)$ as follows: $i_P(A)(U)=A$ if $P\in U$ and ...
3
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1answer
41 views

sheaf defined on the support of a sheaf

Let $X$ be a topological space and $\mathcal{F}$ a sheaf on $X$. Let $Y$ be the support of $\mathcal{F}$ (Hartshorne, exercise 1.14), i.e., $Y = \left\{ P \in X: \, F_P \neq 0\right\}$. Is it true ...
3
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1answer
41 views

Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
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1answer
26 views

$a\in X$. Show that the stalk $\mathscr{F}_a$ is isomorphic to the stalk of $\mathscr{F}|_U$ at $a$ on the topological space $U$.

This is exercise 3.24 from Gathmann's notes. Let $\mathscr{F}$ be a sheaf on a topological space $X$, and let $a\in X$. Show that the stalk $\mathscr{F}_a$ is a local object in the following ...
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1answer
38 views

Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...
1
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1answer
36 views

Stalks of ringed space

Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that ...
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11 views

Differences between locally free sheaves and local systems

Is every local system with fiber a vector space a locally free sheaf? What are the main differences between these two concepts? I was playing with the sheaf of sections of $Mo \to S^1$ ($Mo=$Möbius ...
3
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0answers
27 views

Computing the monodromy of a local system $\mathcal{L}$

I was trying to learn a little bit about local systems and their monodromy. In the notes I'm following they define the monodromy of a local system in the following way: Let $X$ be a topological ...
2
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1answer
63 views

Pullback of an invertible sheaf through an isomorphism

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback. Is it true ...
2
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0answers
27 views

Flatness of a counit for the inverse/direct image adjunction for a finite map of (locally Noetherian?) schemes

Let $f:X\to Y$ be a finite map of locally Noetherian schemes. In fact it's not clear to me whether the hypotheses of finiteness or local Noetherianity are ultimately relevant for my question (I would ...
1
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1answer
60 views

Preimage of diagonal subscheme is a closed subscheme

Let $\alpha: X\to S$ and $\beta:Y\to S$ be $S$-schemes and let $\Delta\subseteq Y\times_S Y$ be the diagonal subscheme defined as follows (following Eisenbud-Harris): for each affine open subscheme ...
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29 views

Local Systems and covering spaces.

Given a covering space $\varpi:Y \twoheadrightarrow X$ (with $X$ connected) if we consider the sheaf $\Gamma$ of sections of this bundle we can show that $\Gamma$ is a local system. I would like to ...
3
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135 views

Pullback of sheaf of a divisor by a desingularization mod torsion.

Let $f:\tilde{Z}\to Z$ be a proper birational map between (irreducible) varieties over $\mathbb{C}$ with $\tilde{Z}$ smooth and $Z$ normal. Let $D$ be a Weil divisor on $Z$. I am reading a paper ...
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0answers
59 views

Is the pullback of an ample divisor always nef?

I have read that if $f:X\to Y$ is a morphism of projective varieties and $D$ is an ample Cartier divisor on $Y$ then $f^*D$ is nef. The proof is that ample implies semi-ample, which means that some ...
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23 views

what can we say if we just know the global section has a given universal algebra structure?

Given an universal algebra A, how to reconstruct a topological space X with some universal (and natural) property (e.g initial,terminal...)in the category of topological spaces whose global sections ...
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3answers
83 views

Extension by zero not Quasi-coherent.

Hartshorne's Example 5.2.3 in Chapter 2 states that if $X$ is an integral scheme, and $U$ is an open subscheme with $i:U \rightarrow X$ the inclusion, then if $V$ is any open affine not contained in ...
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0answers
66 views

A question in the proof of equivalence of Cech cohomology and Sheaf cohomology

While proving the equivalence of Sheaf cohomology (defined by using injective resolutions), Tennison (in his book 'Sheaf Theory' on page 146) says : If we let $S$ be a presheaf such that the ...
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1answer
47 views

why is the inverse image defined by the right kan extension instead of the left?

Why don't we define the inverse image of a sheaf to be the left kan extension and then take the sheafification?
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0answers
29 views

Which kind of covers preserve the smooth structure?

Suppose I have a topological space $X$ and a cover $S_i$ of $X$. We say $X$ is coherent with respect to the cover $S_i$ if $X$ has the weak topology with respect to $S_i$, that is, if $S \subset X$ is ...
3
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1answer
41 views

Presheaf with same global sections as associated sheaf

My question is very easy to state : if a presheaf has the same global sections as its associated sheaf, is it a sheaf ? I imagine its false but havent found a counter example.
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26 views

Image of presheaf morphism

I know the definition of the image of a morphism in a category, but I'm wondering if in my specific case there is a nice simplification. Let ...
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23 views

Linked (sheaf, cosheaf)-pair on Alexandrov spaces

Sometimes a topologial space (X, $T_{open}$) has the property that its closed sets form a topology $T_{closed}$ too (Alexandrov space). In addition one has a sheaf $\mathcal{S}_{open }$ with respect ...
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1answer
51 views

Intuitive way to understand identity/gluing axioms of sheaf

Is there an intuitive way to understand the identity and gluing axiom of a sheaf, specifically in the setting where the source category is affine schemes? What is the motivation for such a definition? ...
4
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2answers
60 views

Why isn't every $\mathcal O_X$-module quasi-coherent?

This might be a stupid question, but I don't understand an easy fact. Let $(X,\mathcal O_X)$ a ringed space. We know that every module $M$ over a ring $R$ has a free presentation, so why isn't every ...