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0answers
34 views

Base-point free linear sheaf and extending the base field

This is a question from Ravi Vakil's notes I've been stuck on, namely 18.2.I. Let $X$ be a scheme over a field $k$, and let $K/k$ be any field extension. Let $\mathcal{L}$ be an invertible sheaf on ...
1
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1answer
51 views

Conormal bundle of Cartier divisors

Given any closed immersion of schemes $i:Z\to X$ defined by a sheaf of ideals $\mathcal{I}$ on $X$, apparently the conormal bundle is $\mathcal{C}_{Z/X}:= {\mathcal{I}}/{\mathcal{I}^2}$ "seen as a ...
4
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2answers
60 views

Why isn't every $\mathcal O_X$-module quasi-coherent?

This might be a stupid question, but I don't understand an easy fact. Let $(X,\mathcal O_X)$ a ringed space. We know that every module $M$ over a ring $R$ has a free presentation, so why isn't every ...
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0answers
18 views

Ideal Sheaf: Making sure I got it right

I want to know if I unwrapped the definition of ideal sheaves correctly. Let $\mathscr{O}$ be a sheaf of rings on topological space $X$. The definition of an ideal sheaf is as follows: The ideal ...
3
votes
1answer
49 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
3
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1answer
50 views

Why are Grothendieck's and Hartshorne's definitions of quasi-coherence equivalent?

Hartshorne's Algebraic Geometry defines an $\mathcal O_X$-module $\mathscr F$ to be quasi-coherent if there is an open affine cover $(U_i=\operatorname{Spec} A_i)_{i\in\mathcal I}$ of $X$ such that ...
1
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1answer
74 views

$\Gamma(X,-)$ for Quasi-Coherent/Coherent Sheaves Maps to R-modules/Finitely Generated R modules

I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the ...
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0answers
46 views

Pushforward of a quasicoherent sheaf on a noetherian scheme is quasicoherent

I'm reading through and trying to understand the proof of Proposition 5.8 in Chapter II of Algebraic Geometry by Hartshorne that the pushforward of quasicoherent sheaf $\mathcal{F}$ by a morphism ...
1
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0answers
52 views

Some propreties about $ \mathfrak{Coh}_X $ and $ \mathfrak{QCoh}_X $.

I would like to know : why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $ \mathfrak{QCoh}_X $ of quasi ...
2
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1answer
56 views

Turning a geometric vector bundle into a free $\mathscr{O}_X$-module

Given a geometric vector bundle $V\to X$ on a scheme $X$, consider the sheaf of sections $\mathscr{I}(V/X)$ defined by $U\mapsto \operatorname{Hom}_U(U,V_{|U})$. The goal is to equip the sheaf of ...
0
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1answer
64 views

Checking quasicoherence on a qcqs scheme

Let $(X,\mathscr{O}_X)$ be a scheme and $\mathscr{F}$ be an $\mathscr{O}_X$-module. It can be shown that $\mathscr{F}$ is quasicoherent iff for every affine open $U = \operatorname{Spec} A$ and $s\in ...
1
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1answer
73 views

Adjunction of pushforward and pullback

Let $f:(X,\mathscr{O}_X)\to (Y,\mathscr{O}_Y)$ be a morphism of ringed spaces. Then for any $\mathscr{O}_X$-module $\mathscr{F}$ and $\mathscr{O}_Y$-module $\mathscr{G}$, there is a natural bijection ...
3
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0answers
99 views

Pullback along the Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then $F^*(\mathcal{L}) \cong ...
1
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0answers
53 views

Reference Request: Quasi-Coherent Sheaves, Picard Group, etc.

I'm a rookie in algebraic geometry, trying to learn. Recently I noticed that I really don't understand the following topics very well. I am asking you if there is comprehensive references (one maybe ...
6
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2answers
277 views

Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. ...
3
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0answers
74 views

Vector bundles in Ravi Vakil's notes on quasicoherent sheaves

In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free ...
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0answers
101 views

Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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0answers
44 views

Quasicoherent-sheaves and pushfoward

How to prove the proposition in the picture below? It seems to be easy, but I am a beginner. Thanks in advanced for your help!
1
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0answers
62 views

When are direct products exact in the category of quasi-coherent sheaves?

I would like to know if there is a description (or at least some sufficient condition known) of a (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. I ...
7
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0answers
117 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
4
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0answers
67 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
1
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1answer
36 views

Map between free sheaf modules not arising from a “matrix”

My question is about this example in the Stacks project. For convenience and completeness, here is the relevant part. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$ of the real line ...
2
votes
1answer
56 views

The higher cohomologies of a quasi-coherent sheaf on the intersection of two affine open subsets.

It is well-known in algebraic geometry that if $X$ is affine and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $$ H^i(X,\mathcal{F})=0,~ \forall ~i\geq 1. $$ Now let $X$ be an arbitrary ...
4
votes
1answer
75 views

Is the push-forward of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
0
votes
1answer
58 views

Definition of a $\mathcal{O}(a,b)$?

Can any one tell me what is the definition of this notation $\mathcal{O}(a,b)$. I know $\mathcal{O}(a)= \widetilde{S}(a)$ for some ring $S$. Can $\mathcal{O}(a,b)$ be defined in the same way. thanks ...
2
votes
1answer
157 views

Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules

The question is basically like this: Prove that if $S_{\cdot}$ is a finitely generated (in degree 1) graded ring over a field $k$ and $M_{\cdot}$ is finitely generated, then the saturation ...
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3answers
591 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
3
votes
0answers
74 views

Property of pullback of quasi-coherent sheaves

In Hasrtshorne the pullback $f^{*}\mathcal{F}$ of a sheaf $\mathcal{F}$ on $Y$ via a map $f:X \rightarrow Y$ is defined as $f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$. It is quite ...
4
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1answer
97 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
2
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1answer
66 views

Non-injective affine quasi-coherent module induced by injective module over the global sections??

Recently, I came across this answer on MO, which (allegedly - I have trouble understanding EGA/SGA, so I have not checked the reference) provides a reference for an injective $A$-module, $A$ a ...
1
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1answer
163 views

Extension of coherent sheaf from open subspace.

I'm trying to solve following problem: Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there ...
1
vote
1answer
82 views

Question on the definition of sheaves.

When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties. ...
4
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1answer
144 views

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
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0answers
150 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
1
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1answer
182 views

About globally generated Sheaves

On Vakil's Lecture Notes, he puts an important exercise that says: ''Suppose $\cal{F}$ is a finite type quasicoherent sheaf on a scheme $X$. Show that$\cal{F}$ is globally generated at $p$ if and ...
3
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0answers
70 views

Tensor product of structure sheaves of subvarieties

Suppose I have a complex variety $X$ and two closed subvarieties $A$ and $B$, with closed immersions $i:A\to X$ and $j:B\to X$. Then we have two $\mathcal O_X$-modules $i_*\mathcal O_A$ and ...
4
votes
1answer
133 views

Hom functor of quasi-coherent sheaf maybe not quasi-coherent

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give ...
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1answer
177 views

Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
3
votes
1answer
149 views

Quasi-coherent sheaves on varieties

I am reading Kempf's book "Algebraic varieties". On page 55 the author considers a sheaf of rings ${\mathcal{A}}$ (commutative, unital) on a topological space $X$. An ${\mathcal{A}}$-module ...
3
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0answers
116 views

Question from Liu, Chapter 5 Ex 1.16

I have a question from Liu's book on Algebraic Geometry, doing Chapter 5 Question 1.16: Let $f:X\rightarrow S$ be a morphism of schemes, with the following base change. $$\require{AMScd} \begin{CD} ...
2
votes
0answers
244 views

Relative spectrum of a quasi-coherent algebra.

I'm working out the notion of spectrum of a quasi-coherent algebra over a scheme. $\require{AMScd}$ Let $(X,\mathcal O_X)$ be a scheme and $\mathcal A$ a quasi-coherent $\mathcal O_X$-algebra, i.e. a ...
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1answer
421 views

Deligne's formula

Let $M$ be some $A$-module and $f \in A$. Why do we have an isomorphism $$\varinjlim_n \hom_A(f^n A,M) \cong M_f \text{ ?}$$ Background. Let $X$ be a scheme, $U$ an open subscheme, and $F,G$ ...
4
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1answer
302 views

Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
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1answer
284 views

Quasicoherent ideal sheaves on open subschemes

Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since ...
7
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1answer
250 views

Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
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0answers
156 views

Example of a coherent sheaf on an open subset where extension isn't trivial?

Motivation: I am working on problem II.5.15 in Hartshorne's Algebraic Geometry, which is to prove that, given a noetherian scheme $X$, an open subset $U\subset X$, and a coherent sheaf $\mathscr{F}$ ...
4
votes
1answer
1k views

The projection formula for quasicoherent sheaves.

I am looking for a certain way of proving the following : Let $f. X \rightarrow S$ be a morphism of schemes. Suppose that f is quasiseparated and quasicompact, or that X is noetherian. Let ...
9
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2answers
854 views

Inverse image of the sheaf associated to a module

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows ...
3
votes
1answer
110 views

Tensor product of the structure sheaf with itsself

Let $\pi : X \to S$ be a morphism of schemes. Is the $\mathcal{O}_X$-module $\mathcal{O}_X \otimes_{\pi^{-1} \mathcal{O}_S} \mathcal{O}_X$ quasi-coherent? Here, $\mathcal{O}_X$ acts on the tensor ...
3
votes
1answer
141 views

How to compute $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1})$ and $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1}(n))$

Let $\mathcal{O}_{\Bbb P^1}$ be the structure sheaf of the projective line $X=\Bbb P^1_k$ over some field $k$ (algebraically closed of characteristic $0$). What is a good (and, preferably, easy) way ...