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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 $\mathcal F(V)$, $...

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Johnstone, Topos theory: families of arrows inducing the same sheaf condition

Johnstone, Topos theory, 0.3, page 13, asserts that, given a Grothendieck pretopology $P$, if the equalizer condition on a presheaf $F$ is satisfied for a family of arrows $R=\{U_i\to U\}$, then it is ...
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Graded global sections of Proj(S) for S a polynomial ring and more general

Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}...
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When is it the case that all closed immersions of all irreducible components are flat?

Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,\ldots,X_r$ be the irreducible components of $X$ and let $f_i: ...
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Exact sequences of $\mathcal{O}_X$-modules, sections over X minus a point, and splitting

Let $X$ be a (let's say irreducible) scheme, let $x$ be a closed point, put $U = X - \{x\}$. Let $0\to\mathcal{F}\to\mathcal{G}\to\mathcal{H}\to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If ...
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36 views

Adjoint property of $f^{-1}$ and $f_*$.

Let $f: X \rightarrow Y$ be a continuous map. Then a standard exercise is to show that the functors $f_*: \text{Sh}(X) \rightarrow \text{Sh}(Y)$ and $f^{-1}: \text{Sh}(Y) \rightarrow \text{Sh}(X)$ are ...
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1answer
24 views

Explicit sections after sheafification

Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $\mathscr{F}$ over a topological space $X$, the sheafification of $\mathscr{F}$, $\mathscr{F}^+$, as $$ \mathscr{F}^+(U) = \{ ...
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Is the total ring of fractions mod the regular functions flasque?

I want to show that $$0\to \mathcal O_{\mathbb{P}_k^1}\to \mathcal K\to \mathcal K/\mathcal O_{\mathbb{P}_k^1}\to 0$$ is a flasque resolution of $\mathcal O_{\mathbb{P}_k^1}$ with $k$ infinite, but ...
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1answer
29 views

Generators for category of abelian presheaves

Let $T$ be a site, and let $\mathcal{P}$ denote the category of abelian presheaves on $T$. Apparently $(Z_U)_{U\in T}$ is a family of generators for $\mathcal{P}$ defined by: $$Z_U(V)=\bigoplus_{\hom(...
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1answer
71 views

On $\operatorname{End}(\mathcal F)$ of a locally free sheaf of finite rank

$\underline{Background}$:Let,$X$ be a scheme(for simplicity we can assume it to be noetherian).Let $\mathcal F$ be a locally free sheaf of rank $r$ on $X$.Let, us also assume that {$\operatorname{Spec}...
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1answer
35 views

If restricted morphism of ringed spaces are equal, then they are actually equal

Given any two ringed spaces $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$, let $\{ U_\lambda\}_{\lambda \in \Lambda}$ be an open covering of the topological space $X$. If $$f,g: (X, \mathcal O_X) \to (Y,...
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33 views

Chow groups with coefficients in a local system

$\newcommand{\CH}{\mathrm{CH}} \newcommand{\F}{\mathscr{F}} $ Let $X$ be a smooth projective variety over a field $k$. Let $\F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski ...
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Does $\deg_k F(-D)/F = \deg_k D$ hold for effective divisors $D$ and coherent, torsion-free $\mathcal{O}_X$-modules $F$?

$\DeclareMathOperator{\F}{\mathcal{F}}\DeclareMathOperator{\o}{\mathcal{O}}$Let $X$ be a reduced, pure dimensional, projective curve over some field $k$. Let $\F$ be a coherent and torsion-free $\o_X$...
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1answer
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Sheaf cohomology and singular cohomology

Let $X$ be a manifold, $\mathcal{F}$ be an abelian sheaf on $X$. We can consider the etale space $F$ of $\mathcal{F}$, which is the disjoint union of stalks, whose topology is generated by local ...
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1answer
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Perversity and minimal extension functor

Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary. Question 1 : Is it true ...
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How does isomorphism of schemes induce maps on cohomology groups

Let $f:X \to X$ be an isomorphism of schemes. Let $\mathcal{F}$ be sheaf of abelian groups on $X$. When does $f$ induces a map $H^i(X, \mathcal{F}) \to H^i(X, \mathcal{F})$? I've seen similar ...
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Inclusion of closed subschemes.

Let $X$ be a scheme and $Y$ and $Z$ closed subschemes. What does it mean for $Y$ to be contained in $Z$? This question adresses the same question, but it covers only the affine case and it uses a ...
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Confusion about the definition of reduced scheme

I'm following Eisenbud-Harris Geometry of Schemes, and I am a little bit confused about how they define the notion of a reduced scheme. In the affine case, if $X = \text{Spec} \, R$, then setting $X_{...
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Closed embedding in flat topology

Let $j:Y \to X$ be a closed embedding of schemes. Then we know there is an exact sequence in the Zariski topology $0 \to I \to \mathcal{O}_X \to j_{*} \mathcal{O}_Y \to 0$. Now consider the sheaf $\...
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Is it possible to define schemes without using sheaves?

My understanding is that smooth manifolds can be defined either in the usual way (using smooth charts and atlases), or using sheaves (as a locally ringed space which is locally isomorphic to the sheaf ...
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1answer
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Sheaves of abelian groups and tensor product

Let $X$ be a topological space and $\mathcal{F}$ and $\mathcal{G}$ two sheaves of abelian groups. Now let me define a presheaf $\mathcal{F} \otimes_{\mathbb{Z}} \mathcal{G}$ such that $\mathcal{F} \...
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1answer
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Why is this map continuous? Group operation on étalé space of a sheaf

Let $X$ be some topological space and suppose we have a presheaf of abelian groups, $F\colon \mathrm{\textbf{Op}}(X) \to \textbf{Ab}$. Construct the corresponding étalé space as follows: a) For each $...
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How to show $\operatorname{supp}(\mathcal O_Y/\mathcal I)\supset\operatorname{supp}(f_{*}\mathcal O_X)$?

I think the sentence underlined has a type error, it should be $Z=\operatorname{supp}(\mathcal O_Y/\mathcal I)=\operatorname{supp}(f_{*}\mathcal O_X)$. However, I can only show $\operatorname{supp}(\...
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1answer
34 views

Exact sequences of $\mathcal{O}_X$-modules and splitting

Let $X$ be a scheme (a curve over a field for example). Exactness of a sequence $\mathcal{F}\to \mathcal{G}\to \mathcal{H}$ of $\mathcal{O}_X$-modules can be checked on stalks : it is exact if and ...
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Pulling-back functions that vanish of order one respectively two in $x$ yields a commutative diagram

Let $X$ be a complex manifold, $\sigma:\hat{X}\to X$ is the blow up of $X$ at $x$. Define $E:=\sigma^{-1}(x)$. $\mathcal{I}_{\{x\}}$ is the ideal sheaf at $x$. we compare the two exact sequences $0\...
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1answer
43 views

Locally free $\mathcal{O}_{X}$ modules are not projective

Let $(X , \mathcal{O}_{X})$ be a locally ringed space. I know that in general it is not true that locally free $\mathcal{O}_{X}$ modules are projective in the category of $\mathcal{O}_{X}$ modules (...
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$\sigma$ algebra as sheaf and what is its direct image?

Let $X$ be a measurable space.(i.e. $X$ is equipped with sigma algebra $R$.) Furthermore, note that $R$ really defines a sheaf of sigma algebra via $R\cap U$ for any open $U\subset X$ where ...
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A question about pullback bundle and sheaf

Let $X$ be a compact complex manifold, $\sigma:\hat{X}\to X$ is the blow up of a point $x\in X$. Let $E:=\sigma^{-1}(x)$ and $L\to X$ be a line bundle, then how to give a rigorous proof to show that ...
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35 views

espace etale of the sheaf

This is exercise I-8 from Geometry of schemes by Eisenbud and Harris. Let $\mathscr{F}$ be a sheaf, then topologize the union $\overline{\mathscr{F}}=\bigcup_{x \in X}\mathscr{F}_x$ by taking basic ...
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Set theoretic issues in the definition of a site in Stacks Project

I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a ...
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1answer
35 views

Confusion about sections generating a sheaf of modules

Let $(X,\mathcal O_X)$ be a ringed space and $M$ an $\mathcal O _X$-module. The stacks project defines what it means for a family of global sections of $M$ to generate it by asking the associated ...
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1answer
51 views

Sheave of sets, what does $\{f_i \} \mapsto \{f_i \mid_{U_i \cap U_j}\}$ mean?

... for an open covering $U = \bigcup U_i$, an $I$-indexed family of functions $f_i : U_i \to \Bbb{R}, \ i \in I$, is an element of the product set $\prod_i CU_i$, while the assignments $\{f_i\} \...
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1answer
53 views

Is the tensor product of a locally free sheaf with its dual always globally free?

Suppose $(X,\mathcal O_X)$ is a locally ringed space. Let $\mathcal F$ be a locally free $\mathcal O_X$-module of some fixed rank $n \in \mathbb N$. Then the dual $\mathcal F^{\lor} := \mathcal{Hom}_{...
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Does the geometric version of Nakayama's lemma hold for smooth manifolds?

Consider the following geometric formulation of Nakayama's lemma. Proposition. Let $F$ be a quasi-coherent sheaf locally of finite type on a scheme $X$. Consider the quotient map $\pi:F_x\to F_x\...
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1answer
58 views

The cokernel of $H^0(\hat{X},\sigma^*L^k)\to H^0(E,\mathcal{O}_E)\otimes L^k(x)$

Let $X$ be a compact manifold. $\sigma:\hat{X}\to X$ is the blow up of $X$ of $x\in X$. Denote $\sigma^{-1}(x)$ by $E$. $L^k\to X$ is a very ample line bundle. $L^k(x)$ is a fiber. And by $L^k\to X$ ...
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1answer
48 views

Does sheafification of bundles have a right adjoint?

Given a topological space $X$, let $\mathbf{Bundle}(X)=\mathbf{Top}/X$ be the category of bundles over $X$, and let $\mathbf{Sh}(X)$ be the category of sheaves over $X$. Then there's a sheafification ...
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In what sense is a constant sheaf of abelian groups with stalks G isomorphic to G?

In the English translation of Serre's FAC (link to the PDF can be found in this mathoverflow discussion in the top answer) Serre gives his first example of a sheaf of abelian groups, the constant ...
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1answer
35 views

Sheaf on basic open sets.

Starting from a $B$-sheaf $\hat{\mathcal{F}}$, a sheaf defined on the basic open sets of a topological space $X$, I am asked to prove that the construction $$\mathcal{F}(U) = \varprojlim_{V \subset U,...
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42 views

Canonical bundle on $\Bbb P^1$ questions

Let us consider the canonical bundle $K$ on $\Bbb P^1$. Writing an element in $\Bbb P^1$ as $[z_0:z_1]$ we have $U_0$ where $z_0\ne 0$ and $U_1$ where $z_1\ne 0$ as normal. I can write the ...
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2answers
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Is the canonical morphism of sheaves $f^{-1}f_*\mathcal F\to \mathcal F$ an isomorphism? [duplicate]

Let $f:X\to Y$ be a closed immersion of topological spaces. Let $\mathcal F$ be a sheaf of rings on $X$. Is the canonical morphism of sheaves $\varphi: f^{-1}f_*\mathcal F\to \mathcal F$ an ...
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What's the definition of the restriction of a sheaf of rings to a closed set?

For the ringed space $(\operatorname{supp}(\mathcal O_X/\mathcal I),(\mathcal O_X/\mathcal I)|_{\operatorname{supp}(\mathcal O_X/\mathcal I)})$, what's the definition of the restriction of $\mathcal ...
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1answer
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Is $\mathcal I$ a quasi-coherent sheaf on $X$?

Let $X$ be a noetherian scheme and $\mathcal I$ a sheaf of ideals of $\mathcal O_X$. Is $\mathcal I$ a quasi-coherent sheaf on $X$?
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First Isomorphism Theorem for Sheaves, Missing Step

The following excerpt is from an english translation of Serre's FAC (here). This is in Chapter I: Sheaves, $\S1:$ Operations on Sheaves, n$^{\circ}$ 7: Subsheaf and Quotient Sheaf and n$^{\circ}$ 8: ...
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1answer
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Property of presheaves morphism

I am trying to prove that whenever there is an injective morphism of presheaves $\mathcal{F} \rightarrow \mathcal{G}$, the morphism induced on the sheafifications is still injective; it's basically ...
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1answer
51 views

Image sheaf isomorphism : another proof? (Exercise 2.1.4 in Hartshorne)

Assume you have a sheaf morphism $f:\mathcal{F} \rightarrow \mathcal{G}$ and consider the sheafification of the presheaf $\textrm{im}(f)$. I want to prove that it's isomorphic to a sub sheaf of $\...
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1answer
28 views

Morphism of sheaves property

Two quick questions: Let $f$ be a morphism between sheaves $\mathcal{F}$ and $\mathcal{G}$. Then we know that $\textrm{im}(f)$ is in general a presheaf. I believe the inclusion $i:\textrm{im}(f)(U) \...
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1answer
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What changes in the sheaf theory of topological spaces with the “étale topology”?

The customary site structure on the category of topological spaces has covering families given by open covers. What "happens" if we refine this topology and let any jointly surjective family of local ...
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1answer
29 views

Is the morphism labeled by red rectangle injective?

Is the morphism labeled by red rectangle injective?
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Is the canonical morphism $\operatorname{Pre}(\operatorname{im}\theta) \to \operatorname{im}\theta$ injective?

Let $X$ be a topological space. Let $\mathcal F, \mathcal G$ be two sheaves of Abelian groups on $X$ and $\theta: \mathcal F\to \mathcal G$ a morphism. Denote the presheaf of the image by $\...
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36 views

Characterizing Morphism of Sheaves

Let $\mathscr{F}$ and $\mathscr{G}$ by sheaves over a topological space $X$. For convenience, say they are sheaves of abelian groups. To be precise, I am using the etale space definition of a sheaf. ...
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69 views

Fibers vs Stalks

In the passage below, in the last paragraph, what is the meaning for the parenthesized bit - "notice we do not say fiber!". Is what they mean "notice we did not say that the total space $E$ is the ...