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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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Sheaf of Differential Forms Intuition [duplicate]

I have a question about the intuition behind Kähler differentials: Let $f:X \to S$ be a morphism between schemes and consider the diagonal embedding $\Delta: X \to X \times _S X$. It is a well ...
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Why is the Grassmannian functor representable by a scheme?

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...
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What is the correct definition of the inverse image functor on sheaves of modules?

There seem to be two different definitions of the inverse image functor on sheaves of modules in the literature and I just wanted to make sure I am understanding properly. Suppose $f: X \rightarrow Y$ ...
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Two questions about softness criterion for inverse image sheaf

Definition. Let $X\overset{f}{\to}Y$ be a continuous map. Given a sheaf $G$ on $Y$ define $f^\nleftarrow G=\varinjlim_{V\supset fU}GV$ where the limit is taken over open $V\subset Y$ containing $fU$. ...
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Interpretation of higher direct images

In my algebraic geometry course the higher direct images $R^i f_* \mathcal{F}$ of a sheaf of abelian groups $\mathcal{F}$ on a topological space $X$ were introduced as the right-derived functors of ...
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+100

Direct image of vector bundle under projection map

Let $\pi: Y \to X$ be a smooth projection map between manifolds, and assume the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~ \forall x \in X$. Given a smooth map ...
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Does adjunction between direct and inverse image preserve surjectivity or injectivity?

Let $X$ and $Y$ be a topological space and let $f \colon X \to Y$ be a continuous map. And let $\mathscr{F}$ be a sheaf on $X$ and let $\mathscr{G}$ be a sheaf on $Y$. Then it is well-known that there ...
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1answer
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Why is the direct image sheaf a left exact functor and not just exact?

This might be a rather stupid question, however, I cannot make sense of what is going wrong. Let $f:X\to Y$ be a morphism of ringed spaces, and $\mathcal{F,G,H}$ sheaves of abelian groups on $X$. ...
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Example of a pushforward sheaf

This is a doubt I have regarding the question: Example: Push-Forward Sheaf Let $f: X \to Y$ be a continuous map and $\mathscr{F}$ a sheaf on $X$. We have that $f_*\mathscr{F}$ is the sheaf on $Y$ ...
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Are invertible sheaves on projective line $\mathbb{P}_k^1$ all twisting sheaves $\mathcal{O}_{\mathbb{P}_k^1}(n)$?

Suppose $k$ is a field. Invertible sheaves on $\mathbb{P}_k^1$ are of the form $\mathcal{O}_{\mathbb{P}_k^1}(D)$ for some divisor $D$. For simplicity we may assume $D=x$ for some closed point $x\in |\...
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Monomorphism into separated sheaf

Let $X, Y : \mathbb{C}^\text{op} \to \mathbf{Set}$, where $\mathbb{C}$ is a small category, and assume the natural transformation $\alpha : X \to Y$ is a monomorphism. If $Y$ is separated with respect ...
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160 views

Upper Semi-Continuity of the Rank of the Fibre of a Sheaf

I've been trying to prove the following result: Let $X$ be an affine variety over an algebraically closed field $k$, $A=\Gamma(X,\mathcal{O}_X)$, $M$ an $A$-module of finite type, $\mathcal{F}=\...
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Exact sequence of module sheaves

In Bosch's textbook, Algebraic Geometry and Commutative Algebra, he claims, on pages 258-9 in Prop 4 that the functor $M\mapsto\tilde{M}$ from $A$-modules to $\mathcal{O}_X$-modules is exact ($\tilde{...
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1answer
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Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $E$ on an integral Noetherian scheme $X$ s.t. for every $x\in X$ and every non-...
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Definition of the Fibre of a Sheaf

I'm working through an exercise and have a few questions about the following construction. If anyone thinks they should be split into separate posts please let me know, but they seem related and I'm ...
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Is $\mathcal{O}_{\mathbb{P}^1 \sqcup \, \mathbb{P}^1}$ a non-ample line bundle, and if so, how to show this?

There is a result which states that: A line bundle $L$ on a scheme $S$ is ample if and only if there exists an $n \in \mathbb{N}$ and global sections $\sigma_1, \dots, \sigma_n \in \Gamma(S, L^{\...
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Identifying sheaves of sections of fiber bundles

Given a topological space $X$, there's a fundamental category equivalence between local homeomorphisms to $X$ and sheaves of sets over $X$. One direction takes a local homeomorphism to its sheaf of ...
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Functor $\mathrm{Sh}_R(X)\to \mathrm{Ab}$ representable if and only if transforms colimits in limits

In the following let $R$ be a commutative ring, $X$ a topological space, $Ab(X)$ the category of abelian sheaves on $X$ and $\mathrm{Sh}_R(X)$ the category of sheaves of $R$-modules on $X$. For any ...
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Hartshorne Ex. II 1.16 b) Flasque sheaves and exact sequences

the exercise states that when we have an exact sequence $0\to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0$ of sheaves (say of Abelian groups) over a topological space $X$, and when $\mathcal{F}'$ ...
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Deducing that $R^kf_*(F \otimes L^{\otimes n})=0$ for large enough $n$ from a statement about sheaf cohomology

$\newcommand{\spec}{\mathrm{Spec}} \newcommand{\ra}{\rightarrow} \newcommand{\oh}{\mathcal{O}} \newcommand{\P}{\mathbb{P}}$ Let $f : X \ra \spec(A)$ be a projective morphism (Hartshorne's ...
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1answer
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Question about motivation for upgrading varieties to schemes (nilpotency)

In the Wikipedia article on schemes where we give a motivation of nilpotence for transitioning from varieties to schemes, it says that the ring of regular functions (i.e. coordinate ring) of the ...
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Why does exactness of Cech complex not make Cech cohomology trivial?

In my class notes we proved that the (Cech?; I'm not sure what it's usually called) complex $\underline{C}^\bullet((U_i), \mathcal{F})$ (Hartshorne Section III.4) is an exact cochain complex, where $(...
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Showing that $R^pf_*\mathcal{F} \cong \widetilde{H^p(X, \mathcal{F})}$

$\newcommand{\oh}{\mathcal{O}} \newcommand{\QCoh}{\mathsf{QCoh}} \newcommand{\ra}{\rightarrow} \newcommand{\F}{\mathcal{F}} \newcommand{\Mod}{\text{-}\mathsf{Mod}}$I'm working through the proof of ...
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Closed immersion property

I am having fundamental struggles in understanding the proof on stacks projects. Ironically, what I don't understand is the "clearly" part of the statement. More precisely: Lemma 25.4.6 Let $X,Y$ ...
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Canonical Divisors versus Principal Divisors

Perhaps this is so obvious a question that no one has asked it before, but can someone provide a simple example of a canonical divisor which is not a principal divisor or conversely on a Riemann ...
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Flabby sheaf is “obviously” soft?

Given a topological space $X$. A sheaf $\mathcal{F}$ on $X$ is flasque if for any open set $U\subseteq X$, $\Gamma(X,\mathcal{F})\to\Gamma(U,\mathcal{F})$ is surjective. And a sheaf $\mathcal{G}$ is ...
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1answer
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Sheafification and restriction to open subset

Let $X$ be a topological space and $\mathcal{F}$ be a presheaf on $X$. We denote by $\mathcal{F}^+$ the sheafification of $\mathcal{F}$. Let $U\subset X$ be an open subset. We denote by $\mathcal{F}|...
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Representability criterion for schemes

I am trying to understand. Lemma 25.15.4 Let $F$ be a contravariant functor the category of schemes with values in the category of sets. Suppose that $F$ satisfies the sheaf property. ...
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Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and $...
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Writing global sections of sheaves on $\mathbb{P}^n$ as morphisms

$\newcommand{\Pe}{\mathbb{P}} \newcommand{\oh}{\mathcal{O}} \newcommand{\F}{\mathcal{F}} \newcommand{\ra}{\rightarrow} $I know that for a global section $\sigma \in \Gamma(\Pe^n, \oh(-1)) $ this is ...
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1answer
48 views

Restriction of a structure sheaf to the closed subset

Working with the definitions and proving properties of restriction/pushforward/pullback of sheaves was okay until I realized that I can't do calculation of a simple example: Let $X=\text{Spec} (k[...
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Invertible ideal sheaf.

I am a little bit confused with the idea of an invertible ideal sheaf. I cannot convince myself that there are invertible ideal sheaves on a scheme $X$ non isomorphic to the structure sheaf of $X$. I ...
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Question which is similar to sheaf property

Hartshorne, Chapter 3, Exercise 3.7(a): $A$ is Noetherian, $\mathfrak{a}$ is an ideal, $U=\operatorname{Spec} A\setminus V(\mathfrak{a})$. For any $A$-module $M$, $\Gamma(U,\tilde{M})\cong \varinjlim\...
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Question about Etale Sheaves

Let consider the one point field scheme $Spec(K)$ and denote by $G:=\text{Gal}(\overline{K}/K)$ the corresponding Galois group. We consider by $\mathbf{Sh}\big(\text{Spec }k)_{\text{et}}$ the ...
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Serres Vanishing Theorem II

I am having problem with the proof in Serre's Vanishing Theorem. If we were to translation line 9 of the proof of Lemma 29.3.1 in generality it seems to say that: Let $X$ be scheme, $I$ a sheaf ...
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Pullback of locally free sheaves is locally free

Lemma 17.4.3 states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module. Suppose that $G$ is a locally free $O_Y$...
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Simple question about this chain of sheaf morphisms

From 'Introduction to Algebraic Geometry' by Justin R. Smith: What is the role of that intermediate sheaf $\mathcal{A}_{\mathbb{C}}$ in the chain in this example ? Could we not have got from the ...
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Questions about Constant Sheaf

Consider this definition of the Constant Sheaf in 'Introduction to Algebraic Geometry', Justin R. Smith: So we have 2 groups here: the group $A$, which is the co-domain of the continuous functions, ...
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Local behavior of sheaf of ideals given by a closed immersion

I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals $I_Y(U) = $ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$} is quasi coherent. I sort of understand ...
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1answer
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Hartshorne, Exercise III 4.2 (a): A morphism $\mathcal{O}^r \to f_* \mathscr{M}$ that is an iso over the generic point.

I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let $f: X \to Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a ...
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1answer
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Bijective $\mathcal{O}_X$-Module Homomorphisms

Let $(X,\mathcal{O}_X)$ be a ringed space, $\mathcal{F}$ and $\mathcal{G}$$\,\,\,$$\mathcal{O}_X$-modules, and $\varphi:\mathcal{F}\to\mathcal{G}$$\,$ an $\mathcal{O}_X$-module homomorphism. If $\...
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1answer
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Idempotence of Lawvere-Tierney topology induced by Grothendieck topology

I'm hoping someone can elucidate a step in the proof of V.1.2 in Mac Lane & Moerdijk's Sheaves in Geometry and Logic. Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology. $J$ ...
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1answer
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Inverse image ideal sheaf and pullback of ideal sheaf

Assume that we are given a morphism $m: X\to Y$ of varieties and that $I\subset O_Y$ is an ideal sheaf defining some subscheme $T\subset Y$. Then we have two objects on $X$ associated to $I$. The one ...
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Projective space $Proj$

We let $S$ be a graded algebra. I have a 2 questions regarding the Proj construction. It seems to me that we do not know what $O_{Proj S}Proj(S))$ is ? How is the map $S_0 \rightarrow \Gamma(Proj S,...
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1answer
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Lemma for showing that presheaves are colimits of representables

$\newcommand{\PShv}{\text{PShv}}$ $\newcommand{\Fun}{\text{Fun}}$ $\newcommand{\C}{\mathcal{C}}$ $\newcommand{\Hom}{\text{Hom}}$ $\newcommand{\ra}{\rightarrow}$ $\newcommand{\op}{\text{op}}$ $\...
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Questions about Hartshorne Proposition II.5.9

My question is about the first part of the proof. Let $X$ be a scheme. For any closed subscheme $Y$ of $X$, the corresponding ideal sheaf $I_Y$ is a quasi-coherent sheaf of ideals. Proof: ...
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Looking for a criterion for split exact sequence of sheaves of modules similar to Miyata's theorem

Miyata's theorem states that: For a commutative Noetherian ring R, and a short exact sequence: $0 \rightarrow M \rightarrow P \rightarrow N \rightarrow 0$ if $P$ is isomorphic to a direct ...
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1answer
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$S$-local objects of presheaves are reflective and characterize local presentability

Let $PSh(\mathcal{C})$ be the category of presheaves on a small category $\mathcal{C}$. Let also $S$ be any fixed set of morphisms in $PSh(\mathcal{C})$. I say that an object $F\in PSh(\mathcal{C})$ ...
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Left Exactness of global sections functor over quasi compact sheaves [duplicate]

Let $0 \rightarrow E' \rightarrow E \rightarrow E''$ be a short exact sequence of quasi coherent sheaves on a scheme X. Show that the sequence $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ ...
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1answer
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Proof of Sheafification in Bosch

In Bosch's textbook, "Algebraic Geometry and Commutative Algebra," he introduces the sheafification through "a rigorous method derived from Cech cohomology." In particular, he proceeds as follows: ...