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Questions tagged [sheaf-theory]

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 $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

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Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
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3answers
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Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
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Tensor product of invertible sheaves

Given two invertible sheaves $\mathcal{F}$ and $\mathcal{G}$, one can define their tensor product, but in this definition $\mathcal{F} \otimes \mathcal{G} (U)$ is (apparently) not simply equal to $\...
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1answer
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Modules over a functor of points

I have a question on the ''functor of points''-approach to schemes and $\mathcal{O}_X$-modules. Please let me first write up a defintion. Let $Psh$ denote the category of presheaves on the opposite ...
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Examples of surjective sheaf morphisms which are not surjective on sections

Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of ...
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tensor product of sheaves commutes with inverse image

Let $f : X \to Y$ be a morphism of ringed spaces and $\mathcal{M}$, $\mathcal{N}$ sheaves of $\mathcal{O}_Y$-modules. Then one has a canonical isomorphism $f^*(\mathcal{M} \otimes_{\mathcal{O}_Y} \...
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Sheaves and complex analysis

A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or $\mathbf{...
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Why doesn't Hom commute with taking stalks?

I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, G)(U)=Mor(...
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Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space

Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- ...
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When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
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Why tensor product of two sheaves of modules is a sheaf of modules?

Let $(X, \mathcal O_X)$ be a ringed topological space. Consider two $\mathcal O_X$ modules, $\mathcal F$ and $\mathcal G$. First we define the tensor product presheaf $\mathcal F \otimes_{p,\mathcal ...
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Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
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Sheafs and closed immersion

Let $f:X \rightarrow Y$ be a continuous map of topological spaces, such that it is closed immersion. Let $\mathfrak{F}$ and $\mathfrak{G}$ be sheafs on $X$ and $Y$ respectively. How to show, that ...
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Proving that morphism of sheaves is iso iff induced morphism on stalks is iso

Is the following proof sound/does anyone have another more elegant (categorical) proof? The direction $\Rightarrow$ is obvious the "family of stalks"-functor is a functor and functors take isos to ...
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Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
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Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
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Sheafification of a presheaf through the etale space

I have some problems to show that the following contruction defines a sheafification: Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, ...
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On the definition of the structure sheaf attached to $Spec A$

Let $A$ be a ring (commutative with $1$),if $X=Spec A$ we want to attach a sheaf of rings to $X$. If $f\in A$, $D(f)=X\setminus V(f)$ is an element of the base and we define $$\mathcal O_X(D(f)):=A_f$...
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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, ...
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1answer
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Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have $H^i(X,\...
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When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
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Exact sequence of sheaves with non exact sequence of global sections

Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms $$\mathcal{F}_1\longrightarrow \mathcal{F}_2\...
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1answer
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Double dual of torsion free sheaves are locally free?

I was reading Nakajima's "Lectures on Hilbert schemes of points on surfaces". In Chapter 2 he says that if $E$ is a torsion-free sheaf then $E^{**}$ is locally free. I guess that by `torsion-free' ...
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1answer
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Separated scheme

How to show, that the affine line with a split point is not a separated scheme? Hartshorne writes something about this point in product, but it is not product in topological spaces category! Give the ...
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1answer
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Example of epimorphisms such that the product is not an epimorphism in the category of sheaves

I've heard that in the category of sheaves over a topological space $X$, products of epimorphisms are not epimorphisms. I think that it's equivalent to saying that $\mathbf{Sh}(X)$ does not satisfy ...
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1answer
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Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
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Sheafs of abelian groups are the same as $\underline{\mathbb{Z}}$-modules

In Vakil's notes page 76, he claims that a sheaf of abelian groups is the same as a $\underline{\mathbb{Z}}$-module, where $\underline{\mathbb{Z}}$ is the constant sheaf associated to $\mathbb{Z}$. ...
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2answers
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Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ D\...
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2answers
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An exercise in Liu regarding a sheaf of ideals (Chapter II 3.4)

I'm fairly certain there is both a typo and an omission in this exercise. It reads "Let $X$ be a scheme and $f \in \mathcal{O}_X(X)$. Show that $U \mapsto f|_U \mathcal{O}_X(U)$ for every affine open ...
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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 ...
5
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1answer
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Existence of acyclic coverings for a given sheaf

Let $\mathcal{F}$ be a sheaf over $X$ and $\mathcal{U}=\{U_i\}_{i\in I}$ a covering of $X$. I say that $\mathcal{U}$ is acyclic for $\mathcal{F}$ if $H^k(U_{i_0 \ldots U_n}, \mathcal{F}|_{U_{i_0 \...
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1answer
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A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...
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3answers
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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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3answers
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A sheaf whose stalks are zero outside a closed subset

Let $X$ be a topological space. Let $Y$ be a closed subset of $X$. Let $f\colon Y \rightarrow X$ be the canonical injection. Let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Suppose $\mathcal{F}...
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1answer
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$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 $X$. ...
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1answer
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Induced sequence of global sections

I'm reading Differential Analysis on Complex Manifolds by Raymond O. Wells. It states the following in the beginning of section 3 of chapter 2 on page 51: Consider a short exact sequence of sheaves: ...
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1answer
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Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
5
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1answer
179 views

Proof of a certain proposition on sheaves without using cohomology

Definition Let $\mathcal F$ be a sheaf of abelian groups on a topological space $X$. We say $\mathcal F$ is quasi-flasque if it satisfies the following condition. For every exact sequence of sheaves $...
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1answer
471 views

Functor of section over U is left-exact

I am trying to prove $\Gamma(U,\cdot)$ is a left-exact functor $\mathfrak{Ab}(X)\to\mathfrak{Ab}$. This is Exercise 1.8 in Hartshorne, Chapter II or exercise 2.5.F of Vakil's notes (Nov 28,2015 ...
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1answer
253 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
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1answer
297 views

Presheaf image of a monomorphism of sheaves is a sheaf

EDIT: The original version regarding coker is false. Consider a morphism of sheaves of abelian groups $\Phi:\mathscr{F}\to\mathscr{G}$. It is not in general true that Im$_{pre}\Phi$ is a sheaf. ...
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1answer
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Will $i^*$ pull back injectives to injectives?

Let $i: Z \rightarrow X$ be a closed embedding. Need $i^*$ of an injective sheaf of abelian groups be injective? Need $i^*$ of a flabby sheaf be flabby? Thanks :D! Also (maybe should be a separate ...
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1answer
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The idea behind the notion of dualizing sheaf

Well, studying sheaf cohomology, I've faced the notion of dualizing sheaf on a projective scheme over a field $k$. Recall that a dualizing sheaf on $X$ (according to Hartshorne) is a coherent sheaf $\...
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2answers
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Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq X$...
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1answer
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Differentiable manifolds as locally ringed spaces

Let $X$ be a differentiable manifold. Let $\mathcal{O}_X$ be the sheaf of $\mathcal{C}^\infty$ functions on $X$. Since every stalk of $\mathcal{O}_X$ is a local ring, $(X, \mathcal{O}_X)$ is a locally ...
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1answer
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Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have the ...
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2answers
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Canonical sheaf projective bundle

Given a smooth variety $Y$, and a vector bundle $E$ of rank $r+1$ defined on it, call $X$ the variety associated to the projective bundle given by $E$. Call $\pi: X \rightarrow Y$ the natural map. ...
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1answer
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Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
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3answers
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Sheaf Theory for Complex Analysis

I've recently been reading Complex Analysis in One Variable by Narasimhan and Nievergelt. I've done enough work in complex analysis prior that I find the majority of the text quite understandable and ...
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1answer
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Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...