Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [sheaf-theory]

For questions about sheaves on a topological space. Usually you think of a sheaf on a space as the data of functions defined on that space, although there is a more general interpretation in terms of category theory. Use this tag with the broader (algebraic-geometry) tag.

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 $\in\mathcal F(X)$.

2962 questions
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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 sheaves on $X$) if and only if the induced map of…
Zhen Lin
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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 $\mathcal{F} (U) \otimes \mathcal{G}(U)$ for an open…
Tony
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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 sheaf $\mathcal{G}$, then they are equal if and only…
Federico
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Locally constant sheaf but not constant

I am searching for a locally constant sheaf which is not constant. Locally constant means that there exists an open covering of the total space such that the sheaf restricted to each open set in the cover is isomorphic to a constant sheaf. But the…
MathStudent
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Sheafification of a presheaf through the etale space

I have some problems to show that the following construction defines a sheafification: Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, with $\pi:Et(\mathcal F)\rightarrow X$ that is the…
Dubious
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example of a presheaf

I'm now trying to understand some of category theory, I think I can understand the concept of sheaf but I can not understand the difference between sheaves and presheaves. I asked someones about it and they told me that it could be helpful if I can…
Luis GC
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Prove extension by zero is a special case of lower shriek?

The lower shriek functor is defined by $$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$ On the other hand, if $j:V\subset X$ is the inclusion of an open set, the extension by zero…
user153312
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3 answers

Sheafification of the constant presheaf

Let $A$ be an abelian group, and $X$ a topological space. Define the constant sheaf $\mathcal{A}$ on $X$ determined by $A$ as follows: for any open set $U \subset X$, $\mathcal{A}(U)=$the group of continuous functions from $U$ to $A$, where $A$ is…
Radu Titiu
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Stalks of Skyscraper Sheaf

Here is Rotman's definition of the skyscraper sheaf: Let $A$ be an abelian group, $X$ a topological space, and $x \in X$. Define a presheaf by $x_*A(U) = \begin{cases} A & \text{if } x \in U,\\ \{0\} & \text{otherwise.} \end{cases}$ If $U \subseteq…
Robert Cardona
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etale space v. covering space

On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic to) $Y$ if and only if $Y \to X$ is a covering…
Justin Campbell
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Sheaf morphisms, images, cokernels and stalks

Let me first put down a couple definitions, two of which have terminology I make up for this post. If you already know about sheaf theory, you can safely skip Definitions 1-3 and 7-8, and the Construction. Definitions 4-6 introduce notation and…
MickG
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Why are sections of sheaves called sections?

Generally, a section is a right inverse. On the other hand, if $F$ is a ($\mathsf{Set}$-valued) sheaf, then the elements of $FU$ are usually also called sections. Why is this terminology justified and how is it related to right inverses? Please do…
user153312
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1 answer

Flasque Constant Sheaf

It is easy to show that if $X$ is an irreducible topological space, then the constant sheaf $\mathbb{Z}$ is flasque. Is the converse true?
user127542
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Prove that the support of a section of a sheaf is a closed set

Let $\mathcal{F}$ be a sheaf, and let $s\in\mathcal{F}$ be a section. Show that the support of $s$ is a closed subset of $X$. Let us consider the sheaf of all measurable functions on $\Bbb{R}$. Then the support of the measurable function:…
user67803
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1 answer

Presheaf quotient that is not a sheaf

Let $ \mathcal F $ be a sheaf (of, say, abelian groups), and $ \mathcal G $ be a subsheaf of $ \mathcal F$. It is not true that, in general, the quotient presheaf $ U \mapsto \mathcal F(U) / \mathcal G(U) $ is a sheaf. Could someone show me a…
Ervin
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