Questions tagged [quasicoherent-sheaves]

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Is $\theta^\sharp:\mathscr O_{\operatorname{Spec}f_*\mathscr O_X}\to \theta_*\mathscr O_X$ isomorphic?

Suppose $f:X\to Y$ is a quasi-compact and separated morphism of schemes, $\pi: \operatorname{Spec}f_*\mathscr O_X\to Y$ is the canonical affine morphism such that $\pi_*\mathscr O_{\operatorname{Spec}...
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58 views

Why is the pullback (between affine varieties) of a quasi coherent sheaf quasi coherent?

Let $\phi:A\to B$ be a ring homomorphism inducing $f :\operatorname{Spec}(B) \to \operatorname{Spec}(A)$ on spectra. Let $M$ be an $A$-module and $\widetilde{M}$ be the corresponding quasi coherent ...
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1answer
33 views

Applications of fpqc descent of quasicoherent sheaves

I have been learning about fibered categories and stacks from Vistoli's notes. One of the main results in the notes is the statement that the fibered category of quasicoherent sheaves over a scheme $X$...
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27 views

Proof that direct image of quasi-coherent module is quasi-coherent

I was looking at the following proof (source) but I'm having trouble understanding why we have that $\mathcal{F}(f^{-1}V)$ is equal to the kernel of the map. Can someone please explain this to me?
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1answer
41 views

Question related to a quotient sheaf being quasi-coherent

Let $X$ be a scheme. Let $Q$ be an $O_X$-ideal (no additional assumptions). Let $F$ be the quotient sheaf $O_X/Q$. I would like to prove that there exists an affine open cover $\{ U_i \}$ of $X$, ...
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1answer
56 views

Motivation for Quasi Coherent Sheaf

I have some background in vector bundles in the context of differential geometry and I have seen how vector fields form a module over smooth functions on a smooth manifold. Recently I came across ...
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29 views

Cech - cocycle on the projective line over $\mathbb{F}_2$

I want to solve the following exercise. Consider the projective line $X = \mathbb{P}^1_{\mathbb{F}_2} = \rm{Proj}_{\mathbb{F}_2}\mathbb{F}_2[t_0,t_1]$ over the field $\mathbb{F}_2=\mathbb{Z}/2\...
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14 views

Pullback of Serre's twisted sheaf via projection $X\times_{\rm{Spec}(R)}\rm{Spec}(R')\rightarrow X$

The problem is due to exercise 2 of section 9.2 of Bosch's Algebraic Geometry book. The setting is as follows: Given a graded algebra $A$ over a ring $R$ (commutative with unit) and a morphism of ...
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38 views

Morphism between $\mathcal{O}_X$ - modules is locally an isomorphism

Given two quasi - coherent $\mathcal{O}_X$ - modules $\mathcal{F}$ and $\mathcal{G}$ on a scheme $X$ and a morphism $f:\mathcal{F} \rightarrow \mathcal{G}$ and a point $x\in X$ such that $f_x:\mathcal{...
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1answer
40 views

Ideal sheaf generated by local sections of an ideal sheaf

Let $X$ be a scheme and $\mathscr{I}$ a sheaf of ideals. Let $U$ be an open set of $X$ and $s_1,\dots,s_n \in \Gamma(U,\mathscr{I})$. I am seeking a clarification of what we mean by "the ideal sheaf $\...
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32 views

Corollary 5.10 in Hartshorne II

In Hartshorne II corollary 5.10 he claims the following: If $X=\operatorname{Spec}A$ is an affine scheme, then there is a 1-1 corresopndence between ideals $\mathfrak{a} \in A$ and closed subschemes $...
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30 views

On the definition of quasi-coherent sheaves

In Wikipedia it states that a quasi-coherent sheaf on a ringed space $(X, \mathcal O_X)$ is a sheaf $\mathcal F$ of $\mathcal O_X$-sheaf of modules which has a local presentation, that is, every point ...
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120 views

defining an ideal sheaf of a scheme

Let $X$ be a scheme. Let $A$ be an ideal of $\Gamma(X, O_X)$. I am wondering what does it mean by $Q$ is the $O_X$ ideal generated by $A$? Also how does one show that ideal sheaf defined in such a way ...
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41 views

Quasicoherent sheaves on relative spec, Vakil's Exercise 17.1.E

This is Exercise 17.1.E, p449 of Vakil's rising sea. Let $\mu: \mathcal{Spec B} \rightarrow X$ the relative spectrum construction for $\mathcal{B}$ a quasicoherent sheaf of algebras on $(X, \...
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37 views

Full and faithful functor from quasi-coherent $\mathcal{O}_S$-modules to $S$-vector bundles

Let $S$ be a scheme. I want to show that the composition $\text{Spec}_S(-)\circ \text{Sym}(-)$ is a full and faithful contravariant functor from the category of quasi-coherent $\mathcal{O}_S$-...
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1answer
97 views

Line bundles over $\Bbb P^1_k$

I am trying to understand the line bundle $O(1)$ over $\Bbb P^1_k$ and why $$ O(1)^{\otimes n} = O(n)$$ from Vakil's notes, p398, line 6. His explanation is rather long, so I took a screen shot. ...
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50 views

Possible typo in “FGA explained”?

In chapter 3 ("Fibred categories") of FGA explained (Fantechi et al.), there is an analysis of the fibred category of quasi-coherent sheaves. However, there seems to be, in my opinion, a typo ...
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96 views

Zero section of quasi-coherent bundle

Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...
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103 views

Tensor product of quasi-coherent $\mathcal{O}_X$-modules on a ringed space

Let $(X,\mathcal{O}_X)$ be a ringed space. Show that the tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ of two quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}, \mathcal{G}$ is ...
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47 views

The Grassmannian of a quasi-coherent module

In Goetz' and Wedhorn's Algebraic Geometry Chap (8.6) page 214 is introduced the concept of generalized Grassmannian's. One remark I can't understand. (8.6) The Grassmannian of a quasi-coherent ...
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60 views

Well-definedness of conormal sheaf

In the definition of the conormal sheaf, we are given a locally closed immersion $X \to Y$, which factors through some closed subscheme $Z$ so that we have $X \to Z \to Y$, where the first map is a ...
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46 views

Surjectivity of homomorphism $\mathcal E \otimes_{O_X} \mathbb S(\mathcal E) \to \mathbb S(\mathcal E)(1)$

That should be an easy question, but probably I misunderstood something and just can't understand how that is true. So suppose we have scheme $X$, quasicoherent sheaf $\mathcal E$ and $\mathbb S(\...
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1answer
99 views

How to prove $\mathcal O_X(m)\otimes_{\mathcal O_X} \mathcal O_X(n)= \mathcal O_X(m+n)$?

Let $B$ be a graded ring and $X=\mathrm {Proj}\: B$. Let $B(m)$ denote the twist of $B$ (i.e. a graded $B$-module such that $B(m)_d=B_{m+d}$,) $\mathcal O_X(m)$ denote the quasi-coherent sheaf $\tilde{...
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222 views

Quasi-coherent sheaf on $Proj\ S$

Given a graded ring $S$ and a quasi-coherent sheaf $\mathcal{F}$ on $Proj\ S$, does there exist a graded $S$-module $M$ such that $\mathcal{F}\cong \widetilde{M} $? I know the result is true when $S$...
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1answer
96 views

Quasi-coherencity of the annihilator ideal sheaf of the sheaf associated to an A-module M

I am trying to find an example which shows that the annihilator ideal sheaf, denoted by $\mathrm{Ann}(\mathcal F)$, of a quasi-coherent sheaf $\mathcal F$ on a locally-noetherian scheme $X$, is not ...
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88 views

Direct sums of invertible sheaves commuting with global sections

I am looking at the Stacks Project's treatment of the functor of points for projective space. Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
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33 views

Regarding Sheafification functor

Assume we are in the category of sheaves of $\mathcal{O}_X$-modules. Suppose two presheaves maps to the same sheaf under the sheafification functor. Does it imply that two presheaves were same? I am ...
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1answer
89 views

Stalk of the product or power of quasi-coherent ideal sheaves

Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $\mathfrak I$ and $ \mathcal J$ be two quasi-coherent sheaf of ideals on $X$. Is it true that for every $x\in X$, we have $(\mathfrak I \mathcal J)...
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1answer
66 views

Inducing one point closed subset with a closed subscheme structure so that the stalk of the subscheme is a field

Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $x\in X$ be a closed point and $Y:=\{x \}$ . Is it always possible to make $Y$ into a scheme such that $(Y, \mathcal O_Y)$ is a closed subscheme of $...
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35 views

Some stalk of non-zero sheaf , on locally ringed space, is non-zero?

If $\mathcal F$ is a non-zero sheaf of Abelian groups on a locally ringed space $(X,\mathcal O_X)$, then is it true that some stalk of $\mathcal F$ is non-zero ? If this is not true in general, then ...
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27 views

Structure of quasi-coherent sheaf of ideals

Let $\mathcal J$ be a sheaf of ideals on a scheme $(X,\mathcal O_X)$ . Is it true that $\mathcal J$ is quasi-coherent if and only if for every affine open subset $U=Spec A$ of $X$, there is an ideal $...
4
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1answer
54 views

Relating the sheaf associated to a cyclic module over an affine scheme to the structure sheaf

Let $(X,\mathcal O_X)$ be the affine scheme of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R$. From the sheaf $\mathcal O_X$ and the closed subset $Z:=V(J)$ of $X$ , how do we recover ...
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107 views

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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88 views

About Hom and global section functor for $O_X$ modules

Is it true for any $O_X$-module $F$ that $$ Hom_{O_X}(F,O_X)\cong F(X)= \Gamma(X,F) $$ And is it in general true that the sheafification of $F(X)$ is $F$ ? I think that it is true for quasi-...
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1answer
55 views

Finite Morphism between Regular Curves induces Category Equivalence

Let $C,D$ curves (so $1$-dimensional proper $k$-schemes). Assume futhermore that they are also regular and $f:C \to D$ is a finite morphism. It is known that in this case the pushforward functor $f_*:...
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48 views

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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33 views

Broad question on morphisms of stalks of quasi-coherent sheaves on schemes

This question was inspired by reading about a criterion for a morphism into projective space (over an algebraically closed field) to be a closed immersion based on local rings. It got me thinking ...
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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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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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92 views

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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28 views

Is a locally free sheaf of finite rank finitely presented?

I need to prove that a locally free sheaf of finite rank is finitely presented. Being homework, I'm not asking for a proof but rather for an explanation of why my counterexample is wrong. Here is the ...
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70 views

Cech cohomology of a quasi-coherent sheaf on an affine scheme and Leray acyclicity Theorem.

Let $X$ be an affine scheme, $\mathcal{F}$ a quasi-coherent sheaf on $X$. Let $\mathcal{U}=\{U_i\}_{i \in I}$ be an affine covering of $U$ (not necessarely made up of principal open subsets). Moreover,...
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1answer
210 views

Confusion about Exercise II .5.15 in Hartshorne

I'm a bit confused about Exercise II 5.15 in Hartshorne's Algebraic Geometry, especially part (b) and (c) which are (b) Let $X$ be an affine noetherian scheme, $U$ an open subset, and $\mathscr{F}...
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1answer
17 views

If $Y$ is an affine scheme, and $G$ is an $O_Y$-module, then why is $\operatorname{Hom}_{\operatorname{Mod}(Y)}(O_Y, G)\cong G(Y)$?

Question is entirely captured by the title. This has just been stated in a proof, and despite writing things out on paper I have not been able to see exactly why it should always be true. Potentially ...
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1answer
21 views

Why is it that two O_Y-modules are isomorphic if their Hom-sets to any fixed O_Y-module are always isomorphic?

For more context, in case this is necessary, let $\phi:R \to T$ be a morphism of rings. Let $(X, O_X) = (Spec(R), O_{Spec(R)})$ and let $(Y, O_Y) = (Spec(T), O_{Spec(T)})$ be the corresponding affine ...
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1answer
86 views

A doubt on Proposition 5.1.12 of Liu's Algebraic geometry and arithmetic curves.

Let $X$ be a scheme. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If two of them are quasi-coherent, then so is the third. This is ...
3
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1answer
120 views

Relative Spec (the structure map)

Given a scheme $S$ and a quasi coherent sheaf $\mathcal{F}$ of $\mathcal{O}_S$ algebras, we want to define a scheme $X = \mathrm{Spec}(\mathcal{F})$ over $S$. To do so, we define it in three stages: ...
4
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246 views

Morphisms with connected fibers

Let $f\colon X\to Y$ be a morphism of schemes. I am interested in the following property (too long for the title): $$ f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y} \quad \text{(P)}$$ Under very reasonable ...
3
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1answer
225 views

The kernel of a morphism of quasi-coherent sheaves on a scheme $(X,\mathcal{O}_X)$ is quasi-coherent.

Let $(X,\mathcal{O}_X)$ be a scheme. I know that an $\mathcal{O}_X$-module $\mathcal{F}$ is quasi-coherent if for each $x \in X$ there exists an open neighborhood $U$ of $x$ and an exact sequence of $\...
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1answer
58 views

Lemma about quasi-coherent modules

I am trying to understand quasi-coherent modules from https://stacks.math.columbia.edu/tag/01BD . Let me state partially the lemma 17.10.5 about the construction of an example of quasi-coherent module....