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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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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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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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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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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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44 views

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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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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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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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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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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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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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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87 views

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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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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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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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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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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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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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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1answer
26 views

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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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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$X_f$ 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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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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73 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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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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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
31 views

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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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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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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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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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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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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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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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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1answer
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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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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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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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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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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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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 $\mathcal{O}_{X}(n)|_{D+(x_{i})}$ ...
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Sections of a quasi-coherent sheaf along the non-vanishing set of a section of a line bundle.

Let $(X,\mathcal O)$ be a quasicompact, quasiseperated scheme, $\mathcal L$ a line bundle on $X$ and $\mathcal F$ any quasi-coherent sheaf on $X$. Let $s \in \Gamma(X,\mathcal L)$ be any global ...
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Closure of category of sheaves under inverse limits.

How do I show that the category of sheaves on a space $X$ taking values in , a category $K$ admitting inverse limits, admits inverse limits? The limit is a presheaf, I tried to show that it is also ...
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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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Transition functions of the dual sheaf

Let $(X, \mathcal{O})$ be a ringed space and $\mathcal{F}$ an $\mathcal O$-module on $X$, which furthermore is assumed to be locally free of some finite rank $n \in \mathbb N$. Then the dual sheaf $\...