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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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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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1answer
46 views

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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39 views

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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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
39 views

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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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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$S$-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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72 views

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: ...
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1answer
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Covering scheme by affines, $X = \bigcup X_f$

I am reading lemma 27.27.3 in stacks project. In the proof it seems it seems to claim: If $X$ is a scheme, where $f_1,\ldots, f_n \in \Gamma(X, O_X)$ generates the ring. Then $X= \bigcup X_{f_i}$...
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Stalks, Germs and Localisation

I've been trying to prove the following proposition: Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety over an algebraically closed field $k$. ...
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Given a scheme $X$ and invertible sheaf $L$ on $X$, show that $\mathcal{Hom}(L, \mathcal{O}_X)$ is invertible

The question is that given a scheme X and invertible sheaf L on X given by cocycles {$ \varphi_{ij}$}, show that $\mathcal Hom_{\mathcal O_X}(L, \mathcal O_X)$ - the sheaf associated with $U \mapsto \...
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1answer
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Irreducible Subsets of Ringed Spaces

Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety. Is it true then that $(X,\mathcal{O}_X)$ is always locally isomorphic to an irreducible ...
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Inclusion and pullback sheaf

Let $X$ be a topological space and let $S \subset X$ be a subspace with induced topology (not necessarily open or closed). Let $i : S \to X$ be the inclusion map. Assume moreover that for any ...
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Restriction commutes with pullback of sheaves of modules

Let $f:(X, O_X) \rightarrow (Y, O_Y)$ be a morphism of ringed spaces. Let $G$ be a sheaf of $O_Y$-module. $U \subseteq Y$ an open subset. Is it true that $$f^*G|_{f^{-1}(U)} \simeq (f|_{f^{-1}(U)}...
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1answer
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$X_f$ of locally ringed space $(X, O_X)$.

Let $(X, O_X)$ be a locally ringed space. $f \in \Gamma(X,O_X)$ be a global section. $$X_f:= \{ x \in X \, ; \, f_x \text{ is invertible in } O_{X,x} \} $$ It is claimed that $X_f$ is an open ...
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1answer
300 views

Chern Character of Dual Coherent Sheaf?

Let $X$ be a $n$ (complex) dimensional variety and let $\mathcal{F}$ be a coherent sheaf on $X$. Using the sheaf hom which I denote $\mathcal{H}om$, we can define the dual coherent sheaf as $$\...
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1answer
36 views

sheaf isomorphisms does not necessarily glue to be isomorphism

For an isomorphism of sheaf on X $f : \mathscr{F} \to \mathscr{G}$, suppose it is "locally isomorphic," that is, there is an open cover $U_i$ such that $f|U_i: \mathscr{F}_{U_i} \to \mathscr{G}_{U_i}$ ...
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1answer
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Pullback of sheaves topology and sites

Let $T = Top$ be the site of topological spaces with the usual open covering and let $\mathcal{F}$ be a sheaf on the site $T$. Naturally, for a space $X \in Ob(T)$, the sheaf $\mathcal{F}$ induces a ...
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Is sheafification monoidal?

My question has two parts, one specifically about sheafification, and the second one is whether there is an "abstract-nonsensification" of it. Let $X$ be a fixed topological space, $(-)^a : \mathbf{...
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1answer
633 views

Is a coherent locally free sheaf isomorphic it's dual?

Hartshorne chapter II problem 5.1 a) is to prove that the double dual of a coherent locally free sheaf $\mathscr{E}$ over a ringed space $(X,O_X)$ is isomorphic to $\mathscr{E}$. This can be done by ...
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Understanding surjective morphism of sheaves

Given a surjective morphism of sheaves $\varphi:\mathcal{A}\to \mathcal{B}$ on topological space $X$, it may not be surjective on sections,i.e., on an open set $U$, $\varphi(U):\mathcal{A}(U)\to \...
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Alternative formulation of Grothendieck topology

In Mac Lane and Moerdijk's Sheaves in Geometry and Logic there is a reformulation of the Grothendieck topology conditions in terms of arrows, namely a Grothendieck topology on a small category $\...
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1answer
61 views

Why aren't the “higher twist” Möbius bands distinct line-bundles over $S^1$?

It is well known (using for instance sheaf cohomology) that there exist only two possible one-$\mathbb R$-dimensional vector bundles over $S^1$: the trivial bundle and the Möbius bundle. But what ...
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Is every restriction map in the sheaf theory surjective?

Let $\mathcal{F}$ be a sheaf on an $n$-dimensional manifold $X$. More precisely, $\mathcal{F}$ is a contravariant functor, defined on the category of open subsets of $X$ to the category of (finite ...
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1answer
147 views

Line bundle on projective $A$-scheme is the difference between two very ample line bundles

Let $\mathcal{L}$ be an invertible sheaf on a projective $A$-scheme $X$. Then we can always find two very ample invertible sheaves such that $$\mathcal{L}= \mathcal{M} \otimes \mathcal{N}^*$$ (here ...
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1answer
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Sheaf of a Closed Subset

I’ve been given the following definition: Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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Sheaf Cohomology vs Singular Cohomology in Locally contractable Space

Let $X$ be a space and we denote by $\mathbb{Z}_X$ the sheaf of local constant sections on $X$. We are going to compare the sheaf cohomology $H^i(X,\mathbb{Z}_X)$ with singular cohomology $H^i(X,\...
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Torsion Free Quasi Coherent Module

Let $C$ be regular curve. Consider a finite and locally free morphism $f: C \to \mathbb{P}^1$. (the latter mean that $f_* \mathcal{O}_{C}$ is a free $\mathcal{O}_{P^1}$ module) Let $\mathcal{F}$ be ...
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31 views

Support of Torsion Sheaf

Let $\mathbb{P}^1$ the projective line (considered as scheme) and $\mathcal{F}$ a quasi corent sheaf of finite type on on it. Denote by $\mathcal{F}_T \subset \mathcal{F}$ it's torsion subsheaf. My ...
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Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand: In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ ...
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Soft sheaves on $T_1$-space

Suppose we have a topological space $X$ and an open subset $U \subseteq X$ with inclusion $j \colon U \hookrightarrow X$. If we have a sheaf $\mathcal F \in \text{Sh}(X)$, then we know that in ...
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methods for computing cohomology from data of an exact sequence

Let X be a topological space and $A$ be a ring. Suppose we have an exact sequence of sheaves of $A$-modules on X, $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, suppose we ...
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1answer
41 views

Direct limit of $\mathscr{F}(U)$ is the same as direct limit of $\mathscr{F}(X_f)$, where $P\in U$ and $f\notin P$

This question is from Mumford's The Red Book of Varieties and Schemes (Section I.4). Let $X\subseteq k^n$ be an irreducible algebraic set, $R$ its affine coordinate ring. Since $X$ is irreducible, $I(...
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47 views

Universal mapping property for projective schemes and globalising it to projective bundles

Let $Y = \text{Spec}A$ be a noetherian affine scheme. Let $S$ be a graded $A$-algebra which is finitely generated by $S_{1}$ as an $S_{0}$ algebra. In other words, $S$ looks like $$ S = A[x_{0}, x_{1},...
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Is the sheaf of rings $\mathscr{O}$ the sheafification of a presheaf?

This a paragraph from Hartshorne's Algebraic Geometry: Next we will define a sheaf of rings $\mathscr{O}$ on $\text{Spec }A$. For each prime ideal $\mathfrak{p}\subseteq A$, let $A_{\mathfrak{...
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1answer
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Morphism between Invertible Sheaves injective?

Let $X$ be a $k$-scheme and $\mathcal{L},\mathcal{N}$ two invertible sheaves on $X$. Assume that there exist a morphism $\mathcal{L} \to \mathcal{N}$ of $\mathcal{O}_X$-modules. Assume that $X$ has ...
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1answer
71 views

How to show that $\mathcal{F} \otimes \mathcal{F}^\vee \cong \mathcal{O}_X$

Let $\mathcal{F}$ be a rank $1$ locally free sheaf. If we define $\mathcal{F}^\vee = Hom_{\mathcal{O}_X}(F, \mathcal{O}_X)$, then how would one go about showing that $\mathcal{F} \otimes \mathcal{F}^\...
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Why are (Pre)sheaves more important than Co(pre)sheaves?

I'm learning Sheaf Theory, and this is an issue that's been bothering me. Fix a small category $\mathcal{C}$. A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor $F:\...
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Twisting sheaf of projective space

Let $A$ be a ring, $S=A[x_{0},...,x_{n}]$, and $X=$Proj $(S)=\mathbb{P}_{A}^{n}.$ Hartshorne defines the twisting sheaf $\mathcal{O}_{X}(n)=S(n)^{\thicksim}$. Since $\mathcal{O}_{X}(n)|_{D+(x_{i})}$ ...