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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1answer
38 views

How to show the open sets $X_{s_i}$ cover $X$?

Let $X$ be a scheme and $\mathcal F$ an invertible sheaf on $X$ generated by $s_0,s_1,\cdots,s_n\in \mathcal F(X)$. How to show the open sets $X_{s_i}$ cover $X$?
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How “nice” is the functor giving relative sheaf cohomology?

Let $X$ be a topological space, $\mathcal{F}$ be a sheaf of abelian groups on $X$. If $i: A \hookrightarrow X$ is a closed subspace of $X$ and $j: X \setminus A \hookrightarrow X$ denotes the open ...
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190 views

Direct image of vector bundle under projection map

Let $\pi: Y \to X$ be a smooth projection map between manifolds, and assume the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~ \forall x \in X$. Given a smooth map ...
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1answer
60 views

Inducing a sheaf on an open subset, or any subset?

I am working out of Serre's FAC - if my definition of a sheaf is bizarre to you, that is because it is the older definition. In modern language this object is typically referred to as etale space (I ...
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1answer
28 views

Exercise on monomorphism of sheaves

The following is exercise 6, chapter 0 from Johnstone's Topos Theory: Suppose $F\rightarrowtail G$ a monomorphism of sheaves in $Shv(X)$ [$X$ being a topological space], and $\sigma\in G(U)$ for ...
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1answer
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Do we have $(\mathcal L\otimes_{\mathcal O_X}\mathcal L)(X)=\mathcal L(X)\otimes_{\mathcal O_X(X)}\mathcal L(X)$?

Let $X$ be a scheme and $\mathcal L$ an invertible sheaf on $X$, do we have $(\mathcal L\otimes_{\mathcal O_X}\mathcal L)(X)=\mathcal L(X)\otimes_{\mathcal O_X(X)}\mathcal L(X)$?
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1answer
129 views

Explicit computations for derived functors

Let $F$ be a left exact functor from the category of sheaves of abelian groups to the category of abelian groups, $\mathscr{F}$ a sheaf of abelian groups on a topological space $X$. Since injective ...
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1answer
78 views

The etale locale of a sheaf?

It's well-known that sheaves over a topological space are equivalent to etale spaces over the same space. Now if we replace "topological space" by "locale", we can still define sheaves over a locale, ...
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112 views

Hartshorne Proposition 2.7.7

There are 2 claims in this Proposition that I can't see. In the proof of part b) he writes $(f) \geq -D_0$ thus $f$ gives a global section of $\mathcal L(D_0)$ How does this follow? I think having ...
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154 views

Alternative definition of “sheaf”

Let $(X,\tau)$ denote a topological space and $\mathcal{O}$ denote a presheaf on this space with codomain $\mathbf{Set}$. We can take the category of elements of $\mathcal{O}$, which consists of a ...
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45 views

Is there a name for a “continuous space of sequences”?

Is there a name for a "continuous space of sequences" in the following sense? i.e. let $f:X\mapsto X$ be a continuous surjective but not necessarily injective function over some compact toplogical ...
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1answer
194 views

Hartshorne Exercise II. 1.18

Exercise 1.18 of chapter II asks to show that given a map of topological spaces $f:X\to Y$, the functors $f_* : \mathsf{Sh}_Y \longrightarrow\mathsf{Sh}_X$ and $ f^{-1}: \mathsf{Sh}_X\longrightarrow \...
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2answers
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Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent?

Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.
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2answers
122 views

About the kernel of the structure map of a morphism of schemes

Let $f:X\to Y$ be a morphism of schemes, let $\mathcal{K}$ be the kernel of the structure map $\mathcal{O}_Y\to f_*\mathcal{O}_X$. Do we have $$\mathrm{Supp}(\mathcal{O}_Y/\mathcal{K})=\overline{f(...
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38 views

Do we have $\operatorname{Ker}f^\sharp =\operatorname{Ker}g^\sharp$?

Let $f: Y\to X$ and $g:Z\to X$ be two closed immersions of locally ringed spaces. If $Y\simeq Z$, do we have $\operatorname{Ker}f^\sharp =\operatorname{Ker}g^\sharp$?
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2answers
175 views

Alternative Construction of Sheaf from Sheaf on a Base

In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too. Suppose ...
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1answer
94 views

Confusion about elements of the stalk of the direct image sheaf

$\DeclareMathOperator{\F}{\mathcal{F}}$Let $\F$ be a sheaf on $X$ and $\pi : X \rightarrow Y$ a continuous map. Then the the direct image sheaf $\pi_* \F$ is a sheaf on $Y$. An explicit definition of ...
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1answer
209 views

Showing that the skyscraper sheaf is a sheaf.

$\DeclareMathOperator{\res}{res}$Let $X$ be a topological space, $p \in X$ a point, $U \subset X$ an open subset covered by $\bigcup_{i \in I}U_i$, and $S$ a set (or an abelian group). I'm trying to ...
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85 views

How to prove by universal properties that image sheaf is the sheafification of the image presheaf

I'm working on an exercise from Vakil's notes. Exercise 2.6.C Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ is the ...
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1answer
36 views

Leary cover and good cover

On most textbooks, the definition of Leray cover is: Let ${\displaystyle {\mathfrak {U}}=\{U_{i}\}}$ be an open cover of the topological space $X$, and ${\mathcal {F}}$ a sheaf on $X$. We say that ${\...
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1answer
88 views

Is $\mu_n$ a locally constant sheaf for the Zariski topology?

For a scheme $X$, the sheaf $\mu_n$ is defined by $\mu_n(U) = $ $n$-th roots of unity in $\mathscr O_X(U)$. Assuming that $n$ is coptime to the characteristic of $X$, $\mu_n$ is certainly locally ...
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1answer
119 views

Vanishing of Chern characters in degree below codimension of support

Let $\mathcal{E}$ be a coherent sheaf on a smooth variety $X$ of dimension $n$. We know that in such a case there is a locally-free resolution of length $n$: $$E_{n} \to \ldots \to E_{1} \to \...
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1answer
39 views

Confused about definition: “Pointwise Equalizer”

I am reading these notes on topos theory, and I have a small confusion about Proposition 1.16 on page 12. What is the difference between a "pointwise equalizer" ($K$ in proposition 1.16) and the ...
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1answer
244 views

Does the category of sheaves have enough injectives?

Let $\mathcal F$ a sheaf of a category $C$ over some space $X$. There is an injection $$\mathcal F \hookrightarrow \Pi_{x\in X} \ \mathcal F_x.$$ It seems like if the category $C$ has enough ...
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69 views

About gluing of sheaves on a cover

Suppose we are given a cover $\{U_i\}_{i \in I}$ of a space $X$ and a gluing data $ ( \mathcal{F}_i, \psi_{ij} )_{i,j \in I}$ for the sheaves of sets with respect to this covering. I want to show ...
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2answers
82 views

Relative spectrum of a quasicoherent sheaf of algebras (affine case)

I'm trying to understand the concept of the relative spectrum of a sheaf of quasicoherent algebras. Here is the situation: We are given a scheme $X$ and a quasicoherent sheaf of $\mathscr{O}_X$-...
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1answer
68 views

What is the meaning of “support propre”?

At the top of page 278 in Deligne's Weil I paper, he refers to the cohomology group $H_c^i(X,\mathcal{F})$ as having "support propre". This literally translates to "proper support", but in context it ...
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0answers
32 views

Maps between two Leray spectral sequences

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
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1answer
39 views

Counter-example: If $U \subseteq X$ is open and we have a S.E.S of Abelian sheaves, the corresponding S.E.S of rings/modules may not be exact [duplicate]

I'm looking for a counter-example for the problem in title when $F_1$ is not flabby. We know that if $0\to F_1 \to F_2 \to F_3\to 0$ is a short exact sequence of Abelian sheaves and $F_1$ is flabby ...
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1answer
115 views

Counter-examples for quasi-coherent, coherent, locally free and invertible sheaves

I'm trying to find at least one counter-example for each of these concepts to feel more comfortable with understanding the ideas behind them but I cannot even get started :( Please help me find ...
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82 views

Stalk at a point of the sheaf of open sets

Fix a base topological space $X$. For every open subset $U$ of $X$, define $\mathcal F(U)$ to be the set of open subsets of $U$. The restriction map $\mathcal F(V) \to \mathcal F(U)$ sends $W$ to $W ...
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1answer
160 views

The twisting sheaf

The definition given of the twisting sheaf on $\textrm{Proj}S$ for some graded ring $S$ is $\mathcal O_S(n) = \widetilde{S(n)}$. I'm having trouble seeing what this sheaf is though, I'm not even ...
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59 views

Direct sums of presheaves and basis of topology

Let $X$ be a topological space, and $\mathcal{B} = (U_i)_i$ be a basis for its topology. Let $F := \underset{i}{\bigoplus} F_i$ be a direct sum of presheaves on $X$. We denote by $F_{\mathcal{B}}$ its ...
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1answer
148 views

degree of line bundle as an integral

We consider the compact Riemann surface $\mathbb{P}^1$ (my notation for the Riemann sphere). I make an open cover $\mathcal{U} = \{U_1, U_2\}$ where $U_1 = \{z \in \mathbb{C} | |z| < 1+ \epsilon\}$ ...
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1answer
53 views

Ideal sheaf which is locally generated by sections.

My question is about this Example in the stacks project. Let $(X,\mathcal{O}_X)$ be a locally ringed space with a sheaf of $\mathcal{O}_X$-ideal $\mathcal{I}$. Then the support of the $\mathcal{O}_X$...
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1answer
109 views

Equivalence between two definitions of sheaves

I am reading "Notes on Grothendieck topologies, fibered categories and descent theory", version of october 2, 2008, from Angelo Vistoli, he introduces the definition of sheaf on page $31$, definition $...
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1answer
55 views

Fine sheaf and exact form

I can't see why since $\xi$ is an element of $Z^2(U,\epsilon)$ and $\epsilon$ is fine, there exists a $\tau\in C^1(U,\epsilon)$ so that $\delta\tau=\xi$.
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1answer
143 views

When is a sheaf determined by its stalks

Let $\mathcal{F}$ be a sheaf on a topological space. Of course, it is not true in general that $\mathcal{F}$ is entirely determined by its stalks, that is, there can be another sheaf $\mathcal{G}$ ...
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17 views

Differential operators of finite order

$(X,\mathcal{O}_X)$ is a complex manifold or an algebraic variety over $\mathbb{C}$ with his structure sheaf. Let $\delta : X \to X \times X$ the diagonal morphism and $\mathcal{J} \subseteq \mathcal{...
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1answer
125 views

$\tilde{H}^n(X,\mathbb{Z})\cong {H}^n(X,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}$?

Let $X$ be a differentiable manifold. Consider the homomorphism: $\phi: H^n(X,\mathbb{Z})\to H^n(X,\mathbb{R})$ induced by the inclusion of the constant sheaves $\mathbb{Z}\subset\mathbb{R}$. Let $\...
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1answer
49 views

The criterion for conic sheaves

In Kashiwara and Schapira's book , chapter 3.7 It says if $X$ is a locally compact space, $F\in Ob(\mathbf{D}^+(\mathfrak{Mod}(A_X)))$, $\mu$ an action of $\mathbb{R}^+$ on $X$, consider the map $\mu,...
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1answer
177 views

Meaning of Grothendieck quote: recover a topological space by its category of sheaves

In "Récoltes et Semailles"(- Grothendieck), there is a moment when the author talks about the idea of sheaves of sets over a topological space, then taking the category of sheaves (of sets over a ...
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1answer
219 views

Sheaves with the same stalks are not necessarily isomorphic

I know that for two sheaves having the same stalks in a necessary but not sufficient condition to be isomorphic. However, I also know that if I have two subsheaves $\mathcal{F},\mathcal{F}'$ of a ...
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0answers
49 views

What does it mean that a prime number “$l$” is invertible in a scheme X?

What does it mean that a prime number "$l$" is invertible in a scheme X? Thanks you all.
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2answers
115 views

Exact sequence of sheaves in Serre's “Faisceaux algebriques coherents”

I hope this is not a duplicate but I could not find any reference to what I'm about to ask in the other questions. I am studying Serre's article, and in particular the 1st chapter, in view of an exam, ...
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1answer
51 views

Comparing cohomologies of $\mathcal{O}_X$ and $\mathcal{O}_{X_\text{red}}$

Let $X$ be a noetherian scheme. If $X_\text{red}$ is affine, then $H^i(X, \mathcal{O}_{X}) \rightarrow H^i(X,\mathcal{O}_{X_\text{red}})$ is an isomorphism for all $i>0$. Is this true even if $X_\...
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2answers
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Theorem of Formal Functions Hartshorne

I have following question in the proof of thm in Hartshoren's AG (II, page 277): Since $X$ is projective then of cource we have an embedding $i:X \to P^N _Y$. But what about the coherent sheaf $\...
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1answer
75 views

Pullback of Globally Generated Invertible Sheaf

Let $f: S \to X$ be a surjective morphism between surfaces (so so $2$-dimensional, proper $k$-schemes) and $\mathcal{L} \in Pic(X)$ a globally generated invertible sheaf. Therefore for every $x \in X$ ...
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2answers
76 views

Theorem on Formal Function Hartshorne

Let consider the Thm on Formal Functions [see Hartshorne, III.11.1]. We have with $f:X \to Y$ a projective morphism of noetherian schemes, $\mathcal{F}$ be a coherent sheaf on $X$ and let $y\in Y$. ...
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1answer
195 views

Do any authors take the sheaf-theoretic viewpoint on multivalued functions and/or indefinite integrals?

It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves. For example: The real square-root function can be viewed as the sheaf $\mathcal{F}$ defined on the ...