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.

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Hartshorne problem III.10.5

Here is a problem I thought I solved but now I think it can't be right. The problem is as follows: Let $X$ be a scheme and $\mathcal{F}$ a coherent sheaf such that every $x\in X$ has an étale ...
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2 votes
1 answer
58 views

Is the "sheaf of derivations" locally free?

Let $k$ be a field. We require all algebras to be associative commutative, and when unital we require morphisms between them to respect the identity element. Let $X$ be a topological space, equipped ...
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Is the hom complex acyclic in negative degrees?

The question in the title originates from this issue. Consider the category of finitely generated $A$-modules over a Noetherian ring $A$, or equivalently the category of coherent sheaves over the ...
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Constructing a "sheaf of vector fields" for a flasque sheaf of $k$-algebras

Let $k$ be a field. We require all algebras to be associative and commutative. Unital algebra morphisms are required to preserve the multiplicative identity. Let $\mathcal{O}$ be a sheaf of unital $k$...
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2 votes
2 answers
137 views

For what base spaces is the sheafification adjunction unit always an isomorphism on sufficiently small opens?

Let $X$ be a topological space. We use this as the base space for all presheaves below. Since sheafification $F \mapsto \tilde{F}$ is left adjoint to the inclusion of sheaves into presheaves, this ...
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Does an exact sequence of commutative group schemes define an exact sequence of sheaves?

Suppose we are given an exact sequence $$ 0\to A\to B\to C\to 0$$ of commutative group scheme over a field $k$. Then does the sequence define an exact sequence of étale sheaves on a $k$-scheme $X$? I ...
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0 answers
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Is direct limit of injective étale sheaves injective?

Is the following statement true? if so where can I find a proof for reference purposes? Direct limit of injective étale sheaves of abelian groups on a Noetherian scheme is injective. This goes back ...
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1 answer
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Characterization of sheaves

Let $X$ be a topological space, and denote by $O(X)$ the poset category of the open sets; it is known that a presheaf on $X$ amounts exactly to a contravariant functor $\mathcal F:O(X)\to \mathbf {Set}...
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Structure induced by a sheaf

Let $n\geq 1$ and $F$ be a sheaf on $\mathbb{R}^n$ of continuous functions. Assume for every open subsets $U,V$ of $\mathbb{R}^n$ and every $f\in F(U)$, $g\in F(V)$, $g\circ f\in F(U\cap f^{-1}(V))$. $...
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1 answer
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Why is the generator of the Picard group of $\mathbb{P}^1$ isomorphic to $\mathcal{O}(1)$?

Assume $\mathbb{P}^1_k$ is the projective line over an algebraically closed field. In Hartshorne, Chapter II.6, Corollary 6.17, Hartshorne claims that the generator of $\text{Pic}(\mathbb{P}^1_k)$ is ...
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Question about a particular property of sheaves

Let $F$ be a Hausdorff topological field. Let us require all algebras to be associative commutative unital, and algebra morphisms to be unital. Let us say a sheaf $\mathcal{O}$ of $F$-algebras on a ...
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Does this notion of "continuity" for a sheaf of algebras make sense?

Let $X$ be a topological space, let $R$ a commutative ring. All algebras are required to be commutative unital associative, and algebra morphisms are required to be unital. Let us say a sheaf $\...
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1 answer
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Does invertibility of a section $f$ (of a sheaf of rings) in every open set containing $x$ imply invertibility of $f$ in the stalk at $x$?

Let $\mathcal{O}$ be a sheaf of commutative rings on a topological space $X$. Let a point $x \in X$ and a global section $f \in \mathcal{O}(X)$ be given. Suppose that for every open $U \subset X$ with ...
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1 answer
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Connecting two definitions of $Sh(X)$

I'm currently at the very start of Sheaves in Geometry and Logic and the authors have showed that every sheaf is actually a sheaf of cross-sections. I'll describe the definitions so I can state my ...
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2 votes
0 answers
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Structural sheaf of a submanifold

I am currently studying some topics in Algebraic Geometry, Algebraic Surfaces to be precise, and I have some doubts about the identity which I am going to write down. To fix notations, let $X$ be some ...
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homotopies between cofibrant resolutions

Let $X$ be a finite-dimensional noetherian separated scheme, let $\mathcal{U}=\{U_i\}$ be an affine cover and $\mathcal{N}$ the nerve of this cover. For any coherent sheaf $\mathcal{F}$ on $X$, we ...
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Which name receives this "pulled-back" sheaf?

Let $f:X \to Y$ be a continuous map of topological spaces, and let $\mathcal{F}_Y$ be a subsheaf of the sheaf of germs of continuous functions over $Y$, i.e. $\mathcal{F}_Y \subset \mathcal{C}^0_Y$. I'...
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Cotangent sheaf of smooth manifold

I was trying to carry out the algebraic geometry construction of the tangent sheaf in the case of smooth manifolds following the ideas outlined in the answer of this question, but I have a couple of ...
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1 vote
1 answer
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What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?

In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". ...
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4 votes
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Motivating Grothendieck Toposes generated by Souslin Objects

I am currently reading about the independence of the continuum hypothesis from ZFC following the topos theoretic proof given in Chapter VI.3 of MacLane Moerdijk's Sheaves in Geometry and Logic. The ...
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1 answer
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Surjective Morphism of sheaves

I am currently working on surjective morphisms of sheaves and trying to understand the subtleties connected to the need for sheafification of the presheaf image. By definition (e.g. in Hartshorne) $\...
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Factorization of a morphism of LRS

The questions, in particular, are the two written in italics, but any other correction is welcome. Thank you in advance. Let $(f,f^\#):(Y,\mathcal O_{Y})\to (X,\mathcal O_{X})$ be a morphism of LRS, ...
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How to generate Grothendieck Topologies from families of sieves

If $\mathcal{C}$ is a small category with pullbacks, a basis for a Grothendieck topology on $\mathcal{C}$ is a function which assigns to each object $C$ a collection $K(C)$ of families of morphisms ...
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1 vote
1 answer
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Closed immersions of schemes vs closed immersions of LRS

In my course (that vaguely follows Liu's Algebraic Geometry) the definition of closed (resp. open) immersion $f:Y\to X$, is that $f$ induces a homeomorphism onto $f(Y)$, $f(Y)$ is closed (resp. open) ...
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3 votes
0 answers
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Is the ideal product presheaf a sheaf?

Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\mathcal{I},\mathcal{J}\subset\mathcal{O}_X$, we define the ideal product presheaf $\mathcal{I}\cdot_p\mathcal{J}$ as the ideal presheaf $$ ...
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2 votes
1 answer
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Why is the image of $H_1(X,\mathbb{Z})\rightarrow (\Omega^1)^*$ a lattice in $(\Omega^1)^*$? (for Albanese varieties)

Albanese varieties are described here, and provide motivation for this question, although are not central to it. Let $X$ be an algebraic variety and $\Omega^1$ the space of everywhere regular ...
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1 vote
1 answer
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Sheaf with an inverse is locally free of rank 1.

Let $(X,\mathcal O)$ be a ringed space. I am trying to prove that any $\mathcal O$-module that has an inverse with respect to the tensor product is locally free of rank $1$. I've found in Hartshorne a ...
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1 vote
0 answers
29 views

Understanding the representations of stabilizer groups of points in a stack

I'm trying to learn about stacks, but I'm sure I'm misunderstanding something about sheaves on them. Let $\mathcal X$ be a DM stack over $\mathbb C$, and let $F$ be a sheaf of $\mathcal{O}_{\mathcal X}...
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3 votes
1 answer
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Why should the target category of a sheaf be complete?

The categorical definition of sheaf given on Wikipedia (https://en.wikipedia.org/wiki/Sheaf_(mathematics)#Complements) requires that the target category $\mathcal{C}$ of the functor $\mathcal{F}:\...
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2 votes
0 answers
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A question about Flasque Sheaves on a site..

We define a Flasque Sheaf on a site as one whose first Čech Cohomology vanishes for every covering of every object of the site. I know this definition is non-standard, but Lei Fu's 'Étale Cohomology ...
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1 vote
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Grothendieck trace formula over $\mathbb{F}_{p^{n}}[[t]]$?

Let $n\in\mathbb{N}$, $k=\mathbb{F}_{p^{n}}$, $S=\operatorname{Spec}(k[t])$, $F$ the Frobenius automorphism on $S$ and $\mathcal{E}$ a sheaf of modules over $X$ such that $\mathcal{E}_{0}$ is of ...
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0 votes
0 answers
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Definition of Constant presheaf in Hartshorne versus other sources

Let $X$ be topological space. The Hartshorne defines constant presheaf to be a presheaf $F$ whose section at every non empty open set is abelian group $A$ and section at empty set is trivial(zero) ...
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1 vote
1 answer
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$\mathbb{P}^n$-bundles over a regular noetherian scheme Hartshorne Exercise II.7.10(b).

The exercise is quoted below and here is what I have done so far. First off, (a) is just a definition there is nothing to show. For (b), the exercise is a straightforward check of the definitions ...
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3 votes
1 answer
113 views

Grothendieck Group of a Nonsingular Curve (Hartshorne Exercise II.6.11).

I have copied the exercise below for reference. I was able to figure out how to do (a) and (d), so let me focus on (b) and (c). Please do not provide me with full solutions, but hints (and if ...
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1 vote
1 answer
51 views

Transition functions on invertible sheaves

I'm studying "Foundations of algebraic geometry" by Ravi Vakil. Chapter 14.1 is about invertible sheaves on $\mathbb{P}^1_k$. It says between two affine subsets $\operatorname{Spec}k[x_{1/0}]...
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-1 votes
1 answer
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Porperties of Presheafs, that extend globally.

So i have the following statements: I can prove 2 and 4 are true, but i struggle to decide whether 1 and 3 are true and false. I would say neither of them are true because thats the whole point to ...
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0 votes
1 answer
39 views

F is a sheaf, iff $Hom(Y,F(-))$ is a sheaf

So i have a problem with the following task: Here is what I have tried so far: Let $U=\bigcup_{i\in I}U_i$ for opens $U_i$. We want to show that the following holds for each $a\in C$: $Hom(a,F(U))\...
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0 votes
0 answers
34 views

Global section of a locally constant sheaf

Assume that $k$ is a field , $X$ a connected topological space, $L$ locally isomorphic to the constant sheaf $k_X$ and $\Gamma(X;L)\neq 0$. I want to show that $L$ is actually isomorphic to the ...
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3 votes
1 answer
82 views

Stalks (of sheaves) preserve exactness

Let $\mathcal F\xrightarrow{\alpha}\mathcal G\xrightarrow{\beta}\mathcal H$ be an exact sequence of abelian sheaves on the topological space $X$, so that $\operatorname{ker}\beta\cong \operatorname{...
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2 votes
1 answer
137 views

How to prove 2 morphism of sheaves induces the same maps on stalks?

How to prove that for two morphisms $f,\psi: F \rightarrow G$ of sheaves of sets on a topological space $X$ if $f = \psi$ if and only if $f, \psi$ induce the same maps on stalks. Deduce that a ...
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2 votes
2 answers
58 views

Omission of locality axiom in some definitions of sheafs

I'm currently reading about sheaf theory and am confused about possible conflicting definitions of a sheaf. The common one I've come across in courses on Algebraic Geometry and also on Wikipedia is ...
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0 votes
0 answers
13 views

Is the Lefschetz Hyperplane Theorem true for analytic spaces?

The LHPT is usually stated as something like the following. Given an $n$-dimensional quasiprojective variety $X\subset \mathbb{C}\mathbb{P}^n$, and a generic hyperplane section Y, the natural map on ...
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The sets $\mathcal{F}_d(m,n,k) = \Big\{ x \in (m, n) : x^2 = k^2 \pmod d\Big\}$ seem to have relationships with each other. What is their structure?

For $m,n,k,d\in \Bbb{Z}, m\leq n$, define for interval of integers $(m,n)$: $$ \mathcal{F}_d(m,n,k) := \Big\{ x \in (m,n): x^2 = k^2 \pmod d\Big\} $$ Then $\mathcal{F}_d(m,n, i)\cdot\mathcal{F}_d(m', ...
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2 votes
0 answers
34 views

Is local freeness open for curves?

Let $X$ be a nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, flat over $S$ (via the projection). So my question is the following: is ...
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  • 353
2 votes
0 answers
43 views

Singularity set of coherent analytic sheaf

I am trying to understand section 5.5 of the standard reference Differential geometry of complex vector bundles (S. Kobayashi). Let me set up some notation: let $X$ be a complex manifold, $x_0\in X$ ...
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2 votes
0 answers
35 views

Monodromy representation of etale $\pi_1$ of projective line minus 3 points

Consider the following setting. I have a map from the Legendre family $Y$ to $\mathbb{P}_1\setminus \{0,1,\infty\}(\mathbb{Z})$. Call this map $f$, and I am trying to understand the monodromy ...
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1 vote
0 answers
59 views

Higher direct image of a locally constant sheaf

I am trying (struggling) with understanding the local system structure of the higher direct image of a locally constant sheaf. Say I have a locally trivial fibration $f: X\longrightarrow B$, with ...
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2 votes
0 answers
46 views

Comparison of cohomologies on two sites

Let $C$ be a category with two different topologies $\tau_1, \tau_2$ such that $\tau_1$ is stronger, i.e. any covering in $\tau_2$ is also a covering in $\tau_1$. Denote the corresponding sites by $...
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  • 1,022
2 votes
2 answers
87 views

Is the limit of a family of sheaves a sheaf?

So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if ...
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0 votes
0 answers
29 views

Sheafification of presheaves with values in an abelian category

Let $\mathcal{A}$ be an abelian category, and let $X$ be a topological space. Can we always define a sheafification for any $\mathcal{A}$-valued presheaf over $X$? More precisely: let $\mathsf{PSh}_\...
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