Questions tagged [sheaf-theory]

Question 1

Let $\mathcal{C}$ be a Grothendieck site, and form a cartesian square with $f:X\to S$, $g: S'\to S$ to get $X'=X\times_S S'$, $f': X'\to S'$ and $g': X'\to X$. Let $\mathcal{F}$ be a sheaf of abelian ...

Question 2

Let us consider a cartesian diagram of schemes $$ \require{AMScd} \begin{CD} X'=X \times_S S' @>{g'} >> X \\ @VVf'V @VVfV \\ Y' @>{g}>> Y \end{CD} $$ and let $F$ a sheaf on $X$. ...

Question 3

Suppose $ j :U \to X $ is an open immersion of topological spaces. We know that $ j_{!}F $ is a subsheaf of $ j_{*}F$ for any sheaf $ F$ on $U$. If $ F = j^{-1}G $ for a sheaf $ G $ on $ X$, then I ...

Question 4

In a site $\mathcal{C}$ with coverage $J$, an object $P \in \mathcal{C}$ is $\textbf{local}$ if, for every covering $\{ U_i \rightarrow U \}$ and every morphism $P \rightarrow U$, there exists some $i$...

Question 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 $ev: E \hookrightarrow E^{**}$. ...

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Questions tagged [sheaf-theory]

Trivial base change theorem in a Grothendieck site

Construction of Base Change/Comparison Map $g^f_F \to f'_* g'^*F$

The natural map $j_{!} \hookrightarrow j_{*}$ on sheaves

What are the local objects in $k\mathrm{-alg}^{\text{op}}$ with the canonical coverage?

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

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