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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2
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1answer
42 views

Sheafification of coker-presheaf of exp-map

This question seems so obvious to me, that I think there could be a answer to it already. If so, I would appreciate a link! Let $X$ be a complex manifold. Let $\exp:\mathcal{O}_X \to \mathcal{O}_X ^\...
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1answer
33 views

Sets as a category of sheaves on the category of finite sets

Can one give a Grothendieck topology $J$ on $\mathbb F$, the category of finite sets such that $Sh(\mathbb F,J)\simeq \mathcal Set$? I have the following observations: $[\mathbb F^\text{op},\mathcal ...
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1answer
31 views

Is the morphism $\mathcal F^\dagger \to \mathcal G^\dagger$ induced by $f$ injective?

Let $f: \mathcal F\to \mathcal G$ be an injective morphism of presheaves, is the morphism $\mathcal F^\dagger \to \mathcal G^\dagger$ induced by $f$ injective?
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35 views

Sheafification of presheaf with support is equal to sheafification with support?

Let $X$ be a topological space, $Z \subseteq X$ a closed subset and $\mathcal F$ a presheaf on $X$ with sheafification $\mathcal F^+$. We can define a subpresheaf $\mathcal H^0_Z(\mathcal F^+)$ of $\...
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0answers
69 views

Sheaf of nilpotent elements

Let $k$ an algebraically closed field and $X$ be a connected projective curve (= a $1$-dimensional, proper $k$-scheme). consider a closed subscheme $Z = V(\mathcal{J})$ of $X$ defined by nilpotent ...
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1answer
31 views

Support of a section behaves well with quotients

Let $A$ be reduced Noetherian, $f \in A$, $\mathcal{q}$ a prime ideal of $A$, and $\overline{f} = f + \mathcal{q}$ the image in the quotient. Is it true that $$ \operatorname{Supp}(\overline{f}) = \...
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1answer
58 views

Isomorphism of sheaf

Let $S$ be a quasi compact scheme and $X = \mathbb{P}(\mathcal{E})$ be its associated projective bundle corresponding to locally free sheaf $\mathcal{E}$ of finite rank $r$ on $S$ and $f : X \...
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Why is : $ H^p ( X ( \mathbb{C} ) , \mathbb{Q} ) = \mathrm{Hom}_{ D^b ( \mathrm{pt} ) } ( \mathbb{Q} , ( a_X )_* \mathbb{Q}_X [p] ) $?

There is a paragraph on page $ 133 $ appearing in the following link : https://www-fourier.ujf-grenoble.fr/~peters/Books/Motieven/PureMotives-final.pdf that i don't understand. The paragraph says : ...
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1answer
49 views

Stalk of structure sheaf is a local ring, where $\mathcal{O}_X$ is the sheaf of real valued functions

I come across this proof: Suppose $X$ is a manifold and let $\mathcal{O}_X$ be the sheaf of real valued functions. We check that $(X,\mathcal{O}_X)$ forms a locally ringed space. Given $x \in X$, let $...
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2answers
63 views

Geometric interpretation of sheaves defined with equalisers

In their book "Sheaves in geometry and logic", Mac Lane and Moerdijk gave the following definition of sheaves of sets: The definition makes "categorical" sense to me, but how do I interpret it ...
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1answer
40 views

Is this morphism $\mathcal O_Y\to f_*\mathcal O_X$ the same as $f^\sharp$?

Let $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$ be a morphism of schemes, then we have $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$, it induces $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal ...
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1answer
20 views

Relatively prime elements of a local ring stay relatively prime in nearby local rings

Let $M$ be a complex manifold, $V\subset M$ be an irreducible analytic hyper surface and $p,q\in V$. I wonder why relatively prime elements of $\mathcal{O}_{V,p}$ stay relatively prime in $\mathcal{O}...
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54 views

Isomorphism involving push forward of sheaf

Let $S$ be any quasi-compact scheme and $\mathcal{E}$ be a locally free sheaf of rank $r$ on $S$ and $X = \mathbb{P}(\mathcal{E})$ be its associated projective bundle with structure map $f : X \...
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33 views

Koszul exact sequence of vector bundle

Let $S$ be any quasi-compact scheme and $\mathcal{E}$ be a locally free sheaf of rank $r$ on $S$ and $X = \mathbb{P}(\mathcal{E})$ be its associated projective bundle with structure map $f : X \...
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0answers
35 views

Is the sequence of $\mathcal O_Y$-modules $0\to f_*\mathcal I\to f_*\mathcal O_X\to f_*(\mathcal O_X/\mathcal I)\to 0$ exact?

Let $f: X\to Y$ be a morphism of schemes and $\mathcal I$ a quasi-coherent sheaf of ideals of $\mathcal O_X$, then we have an exact sequence of $\mathcal O_X$-modules $$0\to \mathcal I\to \mathcal O_X\...
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0answers
127 views

Annihilator of a coherent sheaf, on affine subsets.

I am trying to solve the exercice 1.9 page 173 in Liu's Algebraic Geometry and I cannot find my way in question c). Let $\mathcal{F}$ be a coherent sheaf on a locally Noetherian scheme $X$. We ...
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1answer
81 views

Monodromy of local system is well defined

I want to understand, why local systems (= locally constant sheafs) with values in a vector space $V$ define a monodromy representation $\rho: \pi_1(X) \to \operatorname {GL}(V)$. I already know, ...
3
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1answer
66 views

Vanishing Ext Group of Torsion-free and Skyscraper Sheaves?

Let $X$ be a projective 3-dimensional variety with mild singularities (rational double points). Is there some general result showing that $$Ext^{1}(\mathcal{F}, \mathcal{O}_{p}) =0$$ where $\...
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About the dual sheaf of an invertible sheaf

Recently I am quite confused about invertible sheaves while reading Hartshorne. In Chapter 2, exercise 5.1, it is discussed that for a locally free $O_X$-module of finite rank, denoted by $\mathcal{F}...
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1answer
69 views

Understanding the sheaf $\mathcal{O}_x(U)$ over an open $U\subseteq X$ as a subset of the Cartesian product $\prod_{P\in U} R_P$.

I am reading from the red book of varieties and schemes of David Mumford about sheaves: Let $R$ be a commutative ring, $1\in R$, $X=\text{Spec}R$ the set of all prime ideals $P \nsubseteq R$ and a $U\...
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1answer
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Sheaves In the Category $[\text{Ring}^{op}, \text{Set}]$.

I thought I heard something like this theorem somewhere, but I may be wrong: Theorem: (Informally stated) Sheaves in the category $[\text{Ring}^{op}, \text{Set}]$ the same as filtered colimits of ...
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1answer
96 views

Exact sequence on $\mathbb{P}^3$ obtained from Euler sequence

In the article of title: On the singular scheme of codimension one holomorphic foliations in $\mathbb{P}^3$ the author states that the sequence on $\mathbb{P}^3$ $$0\longrightarrow \mathcal{F}\oplus\...
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Is the presheaf defined by $U\mapsto (\mathcal I(U))^2$ a sheaf of ideals of $\mathcal O_X$?

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal I$ a sheaf of ideals of $\mathcal O_X$, is the presheaf defined by $U\mapsto (\mathcal I(U))^2$ a sheaf of ideals of $\mathcal O_X$?
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1answer
79 views

some examples of the Soft sheaves but not fine

As we know,a fine sheaf is also soft.So,I need some examples of the sheaves that are soft but not fine.Can the holomorphic sheaf $\mathcal O(X)$ be one?Any help and comments are accepted.Thanks a lot!
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Two confusions about torsion-free coherent sheaves

(EDIT: Beware, my definition of torsion-free is incorrect below. The correct one is to say $\mathcal{F}$ is torsion-free if for all $x \in X$, and all $s \in \mathcal{O}_{X,x}-\{0\}$, the ...
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29 views

Very Explicit Unit/Counit of the Inverse/Direct Image Adjunction

I've based the title on this question here. There, for sheaves $\mathcal{F}$ and $\mathcal{G}$ on toplogical spaces $X$ and $Y$ respectively, given a continuous map $\varphi:X\to Y$ they construct the ...
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1answer
109 views

Pull back of twisted sheaf under a regular map associated to a base point free linear system

Let $D$ be a divisor on a normal projective variety $X$ and $V$ be a subspace of the global section of $\mathscr O_X(D),$ L is a base point free linear system and $\phi_L:X\overset{(g_0:\cdots:g_n)}\...
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1answer
42 views

Presheaves of Functions

Let $X$ be a topological space, $\mathcal{F}$ a sheaf on $X$ and $K=\coprod_{x\in X}\mathcal{F}_x$. Then let $\mathcal{G}$ be the sheaf of functions from $X$ to $K$. We can define a map of sheaves $i:...
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0answers
19 views

Sheaves of rings on finite spectral spaces

Let $X$ be an integral Noetherian non-separated scheme whose underlying space is finite. By 1.41 of the thesis of Tedd it is at most one-dimensional and since integral zero-dimensional schemes are ...
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1answer
50 views

Inverse/Direct Images and Sheafification

Let $X$ and $Y$ be varieties, $\varphi:X\to Y$ a morphism, and $\mathcal{G}$ a sheaf on $Y$. Let $s$ denote the sheafification functor. If $\mathcal{F}$ is a sheaf on $X$, let $\varphi_*\mathcal{F}$ ...
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1answer
31 views

Understand a result about sheaf of modules of finite type

In this Stacks Project entry: Section 17.9: Modules of finite type, the authors define a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ (in which $(X,\mathcal{O}_X)$ is a ringed space) to be of finite ...
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1answer
95 views

Different definitions of Cartier divisor and when they agree

On a scheme $X$, the most general definition of Cartier divisor is a global section in $\Gamma(X, \mathcal{K}^{*}/\mathcal{O}^{*})$, where $\mathcal{K}^{*}$ is the sheaf of invertible elements of the ...
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0answers
53 views

Structure sheaf of the spectrum of the integers

I am learning about schemes and my level is still very basic. Currently, I would, as an exercise, like to solve the following problem: Let $X = Spec(\mathbb{Z})$: Describe the structure sheaf $\...
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1answer
32 views

Subgroup of global sections with support contained in a closed subset

For a closed subset $Z$ of a topological space $X$ and for a sheaf of abelian groups $\mathcal F$ on $X$, let $\Gamma_Z(X,\mathcal F)$ be the subgroup of $\Gamma (X,\mathcal F)$ consisting of all ...
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1answer
30 views

Taking stalks and right derived functors of a special type of additive left exact functors

Let $X$ be a Noetherian scheme and $Sh(X)$ denote the category of sheaf of abelian groups on $X$.Let $x\in X$ be a closed point. Let $\mathfrak F : Sh(X) \to Sh(X)$ be an additive left exact functor ...
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1answer
71 views

When do global sections of a $\mathcal{O}_X$-Module $\mathcal{F}$ that generate $\mathcal{F}_x$ also generate $\mathcal{F}$ in a neighborhood of $x$?

Let $\newcommand{\m}{\mathcal}(X, \m{O}_X)$ be a ringed space, $\m{F}$ a $\m{O}_X$-Module and $x \in X$ a point. Now let $s_1, \dots, s_n \in \Gamma(X, \m{F})$ be such that $s_{1, x}, \dots, s_{n, x}$ ...
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1answer
62 views

The direct image sheaf

For a ringed topological space$(X,O_X)$,let $f:X\to Y \space $be a continuous map between topological spaces. We know that $f_*O_X$ defines a sheaf on Y. So is the stalk of $f_*O_X$ just the same as ...
5
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1answer
50 views

Isomorphism of sheaves of rings

Suppose we have locally ringed spaces $(X, F_1), (X, F_2)$. If $F_1(X)$ and $F_2(X)$ are isomorphic as rings, can we conclude that sheaves $F_1$ and $F_2$ are isomorphic? I think this is true ...
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0answers
40 views

What module is $\Bbb{C}[x,y](n)_{(x)}$? Serres twisting sheaf $\mathcal{O}(n)|_{D_+(x)}$

I am reading Hartshorne's textbook on algebraic geometry. I have just read the definition of the sheaf $\mathcal{O}(n)$ on $\text{Proj}(S)$ for a graded ring $S$. I want to first understand $\mathcal{...
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0answers
50 views

Why is the Dimension of a Stalk Well Defined?

Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety over an algebraically closed field $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. For ...
3
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2answers
71 views

What is the difference between the stalk of a sheaf at a point and the section of said sheaf over that point?

I'm asking this question with a particular regard to the geometric aspect(s) of the difference(s). Let there be a sheaf $\mathcal{F}: Top(X)^{op} \rightarrow \mathfrak{Ab}$, with $Top(X)$ being the ...
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0answers
46 views

Sheaf Associated with a presheaf as an inverse image

My question essentially comes from EGA I (the Springer version), of the item (3.5.6) of the Préliminaires. The ideia is that given a presheaf of a topological space $X$ going to a category $C$, we ...
2
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1answer
31 views

Sheaf cohomology vs Cohomology of sheaves

Do the two expressions "Sheaf cohomology" and "cohomology of sheaves" refer to the same thing or they have different meanings? I'm asking because I read in many texts the two expressions on the same ...
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0answers
31 views

Regarding Sheafification functor

Assume we are in the category of sheaves of $\mathcal{O}_X$-modules. Suppose two presheaves maps to the same sheaf under the sheafification functor. Does it imply that two presheaves were same? I am ...
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1answer
19 views

Scheme of finite Krull dimension, with a closed point, whose closed subsets are all comparable

Let $X$ be a scheme which contains a closed point and also assume that for every two closed subsets $Y_1$ and $Y_2$ of $X$, we have either $Y_1 \subseteq Y_2$ or $Y_2 \subseteq Y_1$. Also assume that $...
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1answer
42 views

Construction of Inverse Image Functor

Let $X$ and $Y$ be topological spaces, and $\varphi:X\to Y$ a continuous map. Let $\mathbf{X}$ and $\mathbf{Y}$ be the categories corresponding to the open sets of $X$ and $Y$ with arrows given by ...
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0answers
38 views

Is there a “natural” categorical description of (pre-)sheaves of modules?

I've been wondering about the following: Is there a $\textit{neat}$ description of the category $[\mathcal P]\mathcal{Mod}(\mathcal O)$ of [pre]sheaves of modules on a sheaf of rings $\mathcal O$? ...
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3answers
124 views

Why presheaves are generalized objects?

While self studying category theory (Yoneda lemma), I came across the statement that for any category $\mathsf{C}$ the functor category $\mathsf{Fun}(\mathsf{C}^{op}, \mathsf{Set})$ represents ...
4
votes
1answer
74 views

Stalk of the product or power of quasi-coherent ideal sheaves

Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $\mathfrak I$ and $ \mathcal J$ be two quasi-coherent sheaf of ideals on $X$. Is it true that for every $x\in X$, we have $(\mathfrak I \mathcal J)...
4
votes
1answer
56 views

Inducing one point closed subset with a closed subscheme structure so that the stalk of the subscheme is a field

Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $x\in X$ be a closed point and $Y:=\{x \}$ . Is it always possible to make $Y$ into a scheme such that $(Y, \mathcal O_Y)$ is a closed subscheme of $...