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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 $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

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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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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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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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31 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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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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233 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 ...
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On Hartshorne Chapter II exercise 1.13

I am trying to solve the question about espace etale of a presheaf on Hartshrone (Chap II ex 1.13): Given a presheaf $\mathcal{F}$, we define a topological space $$\mathrm{Spe(\mathcal{F})}=\...
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How to obtain the identity of $\mathcal B(U)$ from the above definition?

Let $(S,\mathcal O_S)$ be a scheme. An $\mathcal O_S$-algebra $\mathcal B$ is an $\mathcal O_S$-module together with a morphism $\varphi:\mathcal B\otimes_{\mathcal O_S}\mathcal B\to \mathcal B$ which ...
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Which page can I find the definition of $\mathcal O_S$-algebra in “Stacks Project”?

Let $(S,\mathcal O_S)$ be a scheme. Which page can I find the definition of $\mathcal O_S$-algebra in "Stacks Project"?
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How to show the two definitions of $\mathcal O_S$-algebra are equivalent?

Let $(S,\mathcal O_S)$ be a scheme. What's the definition of $\mathcal O_S$-algebra? I got the following answer: It's a sheaf $\mathcal{A}$ of $\mathcal{O}_S$-modules on $S$ which is also a sheaf ...
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What's the meaning of $\mathcal O_S$-algebra?

Let $(S,\mathcal O_S)$ be a scheme. What's the definition of $\mathcal O_S$-algebra?
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Weak Nullstellensatz in locally ringed spaced?

Is there a version of the weak Nullstellensatz valid in general locally ringed spaces? The statement I imagine is something like this: Let $(X, \mathcal{O}_X)$ be a locally ringed space (possibly ...
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looking for a proof or a reference of classical result

Let $X$ be a complex manifold of dimension $n$, $\Omega_{X}$ the sheaf of top degree forms, $\mathcal{D}_{X}$ the sheaf of holomorphic differential operators of infinite order and $q$ the natural ...
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109 views

Global section of tensor product of sheaves

I want to know if there is an easy proof for the following statement: Let $X$ be a non-singular, projective variety ( maybe it is easier Surfaces=X). Let $E$ be a locally free sheaf; then there ...
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Categories for which every contiuous sheaf is representable

I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor $$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$ is representable. Is there a name for such ...
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proving that a functor is an equivalence of categories

I would like to prove that a functor is an equivalence of categories. I tried to use that equivalence of categories is the same as fully faithful plus "surjective". I found a difficulty verfying this ...
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isomorphism between category of sheaves and morphisms of abelian groups

I am working on theory of category and I found this exercise. I tried a lot but I didn't know how I could do. Let $A$ a discrete valuation ring. Show that the category of sheaves of abelian groups on $...
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Difference between Derived Functors $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$

Let $F,G$ be quasicoherent sheaves of $\mathcal{O}_X$modules on a scheme $X$. What is exactly the difference between the derived functors $\mathcal{Ext}^i(F,G)$ and $Ext^i_X(F,G)$? By definition $...
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Extension Classes $Ext^1 _X (\mathcal{F},\mathcal{G})$ of Sheaves

Let $k$ be a field and $X$ be a proper $k$ scheme. Futhermore let $\mathcal{H}$ be a coherent $\mathcal{O}_X$-module. Grothendieck's Finiteness Theorem says that for all $i \ge 0$ the cohomology ...
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Why is the bounded functions not a sheaf?

I know that the presheaf of bounded functions is not a sheaf but I don't see why. I checked in wikipedia they say that this presheaf does not verify the axiom of "Glue". For me it verifies this axiom. ...
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Unique Way to Extend a Sheaf by 0? Serre's FAC

I am working on understanding the proof of the following proposition given in Serre's FAC. The following is verbatim from page 11. Proposition 6. Let $Y$ be a closed subspace of $X$, and let $\...
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Morphism between Coherent Sheaves $\mathcal{F}_x \to \mathcal{F}^{\vee \vee}_x$ [duplicate]

Let $X$ be a surface (therefore $2$-dimensional, proper k-scheme). Assume that $X$ is regular and let $\mathcal{F}$ be a coherent sheaf(therefore locally finite generated as $\mathcal{O}_X$-module) of ...
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Is the sheafication of a “presheaf of $\mathcal{O}_X$-modules” an $\mathcal{O}_X$-module?

Let $(X,\mathcal{O}_X$) be a ringed space. A presheaf of $\mathcal{O}_X$-modules is a presheaf $\mathcal{F}$ of abelian groups on $X$ such that $\mathcal{F}(U)$ is an $\mathcal{O}_X(U)$-module for ...
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Not Following Serre's Argument: Extension/Restriction of a Sheaf, Continuity

I am working through the proof of proposition 5 in Section 5: Extension and restriction of a sheaf in FAC by Serre. FAC can be found in english here, and in particular this question arises on page 11,...
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Stein Factorization

I have a question about an argument in Hartshorne's "Algebraic Geometry" (see p 280); here the excerpt: Here we introduce the Stein factorization of projective morphism (btw.: I think that by Chow's ...
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1answer
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Locally constant sheaves and homotopy equivalences

I know that, if $X$ is a locally arcwise connected and locally simply connected topological space, then the restrictions of any locally constant sheaf $\mathcal{F}$ on $X$ corresponding to inclusions $...
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1answer
108 views

Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
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119 views

Flat sections of flat vector bundles

Let $E\to X$ be a vector bundle with flat connection $\nabla$. Is there a canonical way to construct a vector subbundle of $E$ from the flat sections ($\nabla \sigma=0$)? More specifically, I would ...
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38 views

Compatibility of $f^*$ and $\mathbb{V}$.

Let $Y$ be a scheme. For a locally free $\mathcal{O}_Y$-module $\mathcal{E}$ of rank $n$, the scheme $\mathbb{V}(\mathcal{E})$ is defined as $\mathbb{V}(\mathcal{E}) := \mathbf{Spec}(S^\bullet\mathcal{...
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Extending a property of coherent sheaf

Let $f: X \to Y$ be a morphism between $k$-schemes where $Y$ is irreducible with unique generic point $\eta$. Set $F := f^{-1}(\eta)$ as the generic fiber and consider a coherent sheaf $\mathcal{G}$ ...
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Presheaves are sheaves for the trivial topology

I've just started learning about toposes and I have a stupid question to ask. Suppose we are given a small category $\mathcal{C}$ with trivial topology $T$ on it, (where the trivial topology $T$ on ...
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Iterated integrals as pre-sheaves

Let $n \in \{ 1, 2, 3, \ldots \}$ be fixed and set $N = \{ 1, \ldots, n \}$. Let $X_1, \ldots, X_n$ be measure spaces and for $I = \{ i_1, \ldots, i_m \} \subseteq N$ set $X^I = X_{i_1} \times \cdots ...
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Weakening the gluing axiom for sheaves

The gluing axiom states that given an open cover $\{U_i\}$ of some open set $U$, if I have a section $s_i$ on each $U_i$ such that the restrictions agree on intersections, then there exists a unique ...
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Adjunction Formula for Sheaves

Let $f: X \to Y$ be a morphism of ringed spaces, $\mathcal{G}$ a $\mathcal{O}_Y$-module. I have a question about an argument used to construct a adjunct morphism $f^{\#}: \mathcal{G} \to f_*(f^*(\...
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Is $M$ an $A$-module with finite presentation?

Given an $A$-module $M$, then we have an $\mathcal O_{\operatorname{Spec}A}$-modules $M^\sim$. If $M^\sim$ is locally of finite presentation, is $M$ an $A$-module with finite presentation?
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Sheaf on a point space is always the skyscaper sheaf

Let $(X,\mathcal{T}_X)$ be a topological space, where $X=\{x\}$ is a point space, and hence $\mathcal{T}_X=\{\emptyset,\{x\}\}$. Is it true that any sheaf $\mathcal{F}$ on $(X,\mathcal{T}_X)$ is a ...
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46 views

How does one tensore two exact sequences

This is related to Ravi Vakil's notes on Foundations of Algebraic Geometry. In 15.3.B, we are asked to check that if quasicoherent sheaves $F$ and $G$ are globally generated at a point $p$, then so is ...
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139 views

Dualizing sheaf for local complete intersection

I am studying section III.7 (The Serre Duality Theorem) of Hartshorne's Algebraic Geometry and have some issues with the proof of Theorem 7.11. Let $X$ be a closed subscheme of $P = \mathbb{P}^N_k$ ...
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Can “relations” between topological spaces be composed?

(In what follows, $\Omega = \{0,1\}.$) In set theory, we can define that a relation $X \rightarrow Y$ is a function $X \rightarrow \mathcal{P}(Y)$. This is the same as a subset of $X \times Y$, by ...
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Prove image sheaf is the sheafification of image presheaf with universal properties

This is an exercise given by Vakil’s online notes on algebraic geometry. I wanted to solve the problem by universal properties, but I could not make it complete. I found a solution online as follows: ...
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Projective Nullstellensatz and regular rings

If I am not mistaken, and according to the projective Nullstellensatz, we have: $\mathbb{P}_{\mathbb{C}}^n = \mathrm{Proj} (\mathbb{C} [X_0 , \dots, X_n])$, by the correspondence: $ A \to \mathrm{Proj}...
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Vanishing locus of global section of invertible sheaf has codimension one

Let $X$ be a projective scheme over some field $k$. Let $\mathcal{L}$ be an ample invertible sheaf on $X$ and let $s \in \mathcal{L}(X)$ denote a global section of $\mathcal{L}$. Let $X_s = \{P \in X \...
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looking for flabby sheaf resolutions

I am looking for manipulable flabby resolution of the sheaf of top-degree forms (let say on a complex manifold $X$) which is not canonical, i.e. not the Godement resolution. Does it exist any ...
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Hereditary topological axioms: from X to its etale space

So, this a question that arose from reading the first few pages of Ramanan's (excellent) book "Global Analysis", which uses modern algebraic notions used mainly in Category theory, while avoiding any ...
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Presheaves saturated by monomorphisms

This is Corollary 1.3.10, p12. Note we are fixing a small Eilenberg Zilber category. The terminology are all given in the text. Tell me if the link is not accessible. 1.3.10: If a class of ...
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1answer
73 views

Left adjoint of evaluation functor

This is Theorem 1.10, pg5 (Kan) Let $A$ be a small category, together with a locally small category $C$ with has small colimits. For any functor $u:A \rightarrow C$, the evaluation at $u$, $...
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1answer
100 views

Further examples of stalks

I am currently learning about stalks for the first time. In my exploration online about the topic, I routinely run into the same three examples: In a constant sheaf associated with an abelian ...
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What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
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100 views

Left and right adjoint of the presheaf evaluation $\widehat{C} \to \mathbf{Set}$, $F \mapsto F(c)$

I would like to find the left and right adjoint of the presheaf evaluation functor $F \colon \widehat{C} \rightarrow \mathbf{Set}$, $X \mapsto X(c)$ for a fixed object $c \in \operatorname{ob}(C)$, ...