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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66 views

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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32 views

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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119 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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95 views

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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46 views

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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61 views

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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54 views

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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141 views

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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57 views

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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96 views

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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51 views

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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87 views

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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109 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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129 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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113 views

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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101 views

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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101 views

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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47 views

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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47 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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142 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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150 views

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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104 views

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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32 views

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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48 views

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
76 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
104 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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107 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 $c \in \operatorname{ob}(C)$, ...
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36 views

Inductive limit of constant sheaves

Let $X$ be a topological space and let $U_n \subset U_{n+1}$ be an increasing sequence of open subsets of $X$. Let us note $U = \cup_n U_n.$ Do we have $$\varinjlim_{n \to + \infty} \mathbb{C}_{U_n} \...
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63 views

Is the definition of a sheaf in Serre's paper “Coherent Algebraic Sheaves” equivalent to the standard definition?

In Serre's paper "Coherent Algebraic Sheaves", the definition of a sheaf is so different from other book's I have ever seen. I want to know whether it is equivalent to the standard definition.
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44 views

Show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is an algebraic variety

I'd like to show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is a variety. Is it even true that $\mathbb{H} = \{ x + iy : y > 0 \}$ is an algebraic variety? There's no metric, so it's ...
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124 views

Restriction of sheaf on a closed subscheme is in general not quasi-coherent?

In Hartshorne 5.2.4, we have If $Y$ is a closed subscheme of a scheme $X$, then the sheaf $\mathcal{O}_{X|Y}$ is not in general quasi-coherent in $Y$. I'm having a hard time believing this, ...
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81 views

Do we have $f^*(\mathscr G\otimes_{\mathscr O_Y}\mathscr H)=(f^*\mathscr G)\otimes_{\mathscr O_X}(f^*\mathscr H)$? [duplicate]

Let $f:X\to Y$ be a morphism of schemes and $\mathscr G,\mathscr H$ two $\mathscr O_Y$-modules, do we have $f^*(\mathscr G\otimes_{\mathscr O_Y}\mathscr H)=(f^*\mathscr G)\otimes_{\mathscr O_X}(f^*\...
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1answer
86 views

Question about sheaf of modules induced by inclusion map of a closed subscheme of an affine scheme

For a close subscheme $\text{Spec}(A/\mathfrak{a})$ of an affine scheme $\text{Spec}(A)$, why does the inclusion map induces a sheaf of module that's isomorphic to sheaf associated with $A/\mathfrak{a}...
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52 views

Construction of “etale associated-sheaf” in Rotman's

On pg 282 Prop 5.68 Rotman makes the following construction given a presheaf $P$ of abelian groups over a space $X$. The construction is as follows: For $P^{et}:= (E^{et}, p^{et},X)$. I am ...
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1answer
134 views

Why is the etale map an open map?

In Proposition 5.59 (page 276) of his book An Introduction to Homological Algebra, Rotman states that an etale map is always an open map on sheaf space. (5.59iii) Proposition 5.59 Let $\mathcal{S}...
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60 views

How to show the morphisms $f_i$ glue to a morphism $f:X\to Y$.

$\textbf{Liu Qing, Proposition 5.1.31.}$ Let $Y=\operatorname{Proj}A[T_0,\cdots,T_d]$ be a projective space over a ring $A$, and let $X$ be a scheme over $A$. (b) Conversely, for any invertible ...
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65 views

Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
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1answer
93 views

Deformation theory formalism

I just started reading Hartshorne's "Deformation theory" book and occasionally I get confused by the way he treats various objects (and it seems like it is common in the literature overall, e.g. in ...
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1answer
83 views

Showing $d\pi$ is non-zero in module of Kähler differentials

While reading about Kähler differentials I came across the seemingly innocent statement that $d\pi \in \Omega_{\mathbb{R/Q}}$ in non-zero. This is kind of "obvious" as if $\alpha$ is algebraic then ...