Questions tagged [sheaf-theory]

Question 1

I'm reading MacLane and Moerdijk's "Sheaves in Geometry and Logic," and I'm having trouble understanding a description given in section 8 of chapter III of the supremum of a family of ...

Question 2

In Rotman’s An Introduction to Homological Algebra he defines an étale sheaf (of abelian groups) on p. 276 as follows: Definition. If $p: E \rightarrow X$ is continuous, where $X$ and $E$ are ...

Question 3

I have a question about the proof of the following result in Kashiwara, Schapira, Sheaves on Manifolds: Proposition 2.5.12 [Let $X$ be a Hausdorff and locally compact space.] Let $A$ be a ring, and ...

Question 4

Let $f : X \rightarrow Y$ be a continuous function between topological spaces. Let $F$ be a sheaf on $X$, and $y \in range(f)$ and $x$ be a preimage. Then, there is a natural map $(f_* F)_{y}\...

Question 5

Let $\mathcal{C}$ be a small category. $\operatorname{PSh}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\operatorname{op}}, \operatorname{Set})$ is the category of presheaves on $\mathcal{C}$. For ...

Question 6

This is a well-known fact, but why is the Brauer group of a Calabi-Yau threefold $X$ equal to $H^2(X,\mathbb{Z})_{\text{tors}}$? I understand that $\text{Br}(X) = H^2(X,\mathcal{O}_X^*)_{\text{tors}}$ ...

Question 7

Here's the statement on the book: Let $(\mathscr{O}_X, \mathscr{O}_X)$ be a ringed space. Let $0 \rightarrow \mathscr{F}' \rightarrow \mathscr{F} \rightarrow \mathscr{F}'' \rightarrow 0$ be an exact ...

Question 8

I am working through some first exercises in sheaves. Right now I'm interested only in sheaves of abelian groups (and/or sets) but I think the following can be asked with any value category that has ...

Question 9

A coverage on a category $C$ assignes to every object $c$ of $C$ a family of covers of $c$, and a cover $S$ of $c$ is just a collection of morphisms $S = \{f_i \colon c_i \to c\}$ with codomain $c$. ...

Question 10

I have a little problem in the proof of (4) in 0A6H. For an affine scheme $X=\text{Spec }A$, and $K, L\in D(A)$, we want to show that the cohomology sheaf $H^n(R\mathcal{Hom}(\tilde{K},\tilde{L}))$ is ...

Question 11

I want to know the following: Let $E$ be a coherent torsion-free sheaf on a surface $S$ and $f$ an endomorphism on $E$. As $E$ is without torsion, there is an inclusion $ev: E \hookrightarrow E^{**}$. ...

Question 12

Hi dear mathstack community. It seems to me there is a misprint in "Lecture notes on sheaves and perverse sheaves" by M.Goresky but I'm not quite sure if I understood everything correctly: I ...

Question 13

A closed immersion $T \to \overline{T}$ of schemes is called a square-zero extension with ideal sheaf $J$ if $J$ is the ideal sheaf of $T$ in $\overline{T}$ and $J^2 = 0$. Let $X$ be a scheme of ...

Question 14

Consider the following commutative diagram of ring homomorphisms: $$\begin{array}{ccccccc} A & \rightarrow & B & \longrightarrow & 0 \\ \downarrow & & \downarrow& \\ A' &...

Question 15

I am working on Exercise 5.1(a) in Hartshorne, and I am trying to understand why the isomorphisms between a locally free module of rank $k$,$\xi$, and its double dual glue together well, whereas the ...

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Questions tagged [sheaf-theory]

Supremum of Subsheaves

Why can’t the universal covering map $\mathbb{R} \to S^1$ be made into an étale sheaf of abelian groups?

Sections with compact support commutes with tensor over a locally compact Hausdorff space. Why can we reduce to the compact case?

Are stalks of a pushforward sheaf determined by stalks at the preimages?

How to show that if $D \hookrightarrow X$ is bold and $F$ is a sheaf, then $\operatorname{Hom}(X, F) \to \operatorname{Hom}(D, F)$ is a bijection?

Why is the Brauer group of a Calabi-Yau threefold equal to the torsion of the third integral cohomology group?

Some Questions on Görtz's Algebraic Geometry (Vol. 1), Exercise 7.24

Sheaf sections over $U$ as fibered product over an open cover of $U$

Problems with omitting stability-under-pullback condition for a site

Sheafification of hypercohomology

extension to double dual

Extension by zero of a constant sheaf

Obstructions $\omega(g)$ in $\mathrm{Ext}^1$ associated to a square-zero extension

A generalization of $\mathcal{I'}/\mathcal{I'}^2\rightarrow \Omega_{A'/A}\otimes_{A'} B'$

When is a bundle isomorphic to its dual?

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