Questions tagged [quasicoherent-sheaves]

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Doubt on Lemma II.5.3 Hartshorne

I'm studying this lemma, and I do not understand the green boxed part, namely $M_i:=M\otimes_B A_{g_i}$ is an $A_{g_i}-$module.
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How to construct $S(q^* E) \rightarrow T(L) $ for morphism to projective bundle?

This is 4.2.2 in EGA II. Let $q : X \rightarrow Y$ be a morphism of schemes. Let $L$ be an invertible sheaf on $X$ and let $E$ be a finite type quasicoherent sheaf on $Y$. Let $\phi : q^*(E) \...
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29 views

Rank of a quasicoherent sheaf $\mathscr{F}$

In Ravi Vakil (Foundations of Algebraic Geometry), $\S 13.7.4$, the rank of a quasicoherent sheaf $\mathscr{F}$ at a point $p$ of $X$ is defined as dim$_{k(p)}\mathscr{F}_p/m_p\mathscr{F}_p$. A first ...
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44 views

Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project. First, Modules Definition 8.1 states that a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is locally generated by sections if for all ...
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25 views

Specific proof that the pullback of quasi-coherent sheaf is quasi-coherent

In Görtz-Wedhorn, the argument used to prove that the pullback of a quasi-coherent sheaf $\mathcal{G}$ by the map $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a quasicoherent sheaf is in remark 7....
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9 views

Showing that quasicoherence is affine-local

Let $\operatorname{Spec}R$ be an affine scheme. For an ideal $I$ of $R$, we obtain a ring homomorphism $R \to R/I$, and a corresponding scheme morphism $\alpha:\operatorname{Spec}(R/I) \to \...
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29 views

Cohomology of external tensor product of sheaves

Let $\mathcal F$ and $\mathcal G$ be sheaves on topological spaces $X$ and $Y$ respectively. The external tensor product of $\mathcal F$ and $\mathcal G$ is the sheaf on $X\times Y$ defined as $\...
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Ravi Vakil's book on Algebraic Geometry , Exercice 19.2.A

In what follows, $C$ will be a projective, geometrically regular, geometrically integral curve over a field $k$, and $\mathcal{L}$ is an invertible sheaf on $C$. $19.2.A.$ EXERCISE. Suppose $\mathcal{...
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70 views

Presheaf of $\mathcal{O}_X$-modules with restriction given by localization

Let $X$ be a scheme, and consider the distinguished affine open base of topology on $X$. That is, the data of all affine opens $\mathrm{Spec}(A)\subset X$ and inclusions only of the form $\mathrm{Spec}...
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35 views

Quasi Coherent and Coherent sheaves on formal schemes.

Let $A$ be a noetherian ring complete with respect to a principal ideal $(\pi)$: $A\simeq\lim_\leftarrow A/(\pi^n)$. Denote by $X$ the formal scheme $Spf(A)$ and by $X_{n-1}$ the scheme $Spec(A/(\pi^n)...
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30 views

Sheaf associated to global sections of a quasicoherent sheaf is the pushforward by the structure morphism

Let $A$ be a commutative unital ring and $X$ a separated, quasi-compact scheme over $A$ where the structure map is denoted $f:X\to \text{Spec }A$. Suppose $\mathcal{F}$ is a quasicoherent $\mathcal{O}...
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42 views

Injective map of sheaves preserved

I have the following question and I hope you can help me. Let us suppose that we have two schemes $X$ and $Y$, and an open immersion $f:X\hookrightarrow Y$. Let $\mathcal{F},\mathcal{G}$ two qc $\...
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42 views

Closed subschemes and quasi-coherent sheaves of ideals

Let me quote two results from Qing Liu's Algebraic Geometry and Arithmetic Curves: Lemma 2.2.23: Let $X$ be a ringed space, $\mathcal{J}$ be a sheaf of ideals on $X$, $V(\mathcal{J}) = \{x\in X\;:\;\...
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41 views

Quasi-coherent sheaf which is a vector bundle on curves.

This question is inspired by this question. Given a quasi-coherent sheaf on a smooth variety $X$ such that its restrictions to curves are finite dimensional vector bundles. Does it follow that the ...
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48 views

Inverse image of a sheaf associated to a module

Let $f:\operatorname{Spec}B \rightarrow \operatorname{Spec}A$ be a morphism of spectra, how do I show, for any $A$-module $M$, we have $f^*(\tilde{M})$ is isomorphic to $\widetilde{M\otimes_{A}B}$? ...
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Is the global sections functor exact for a SES of O_X-modules on an affine scheme when the middle is quasicoherent?

Let $X$ be an affine scheme. Suppose we have an exact sequence of $\mathcal{O}_{X}$-modules \begin{equation*} 0 \to \mathcal{F}_{1} \xrightarrow{\phi} \mathcal{F}_{2} \xrightarrow{\psi} \mathcal{F}_{...
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92 views

Canonical bundle and skyscraper sheaf

Let $Y$ be a smooth projective variety, $\omega_Y$ its canonical bundle and $k(y)$ the skyscraper sheaf. In a proof in Huybrechts' book, he uses the "restriction" map $r_{y_1,y_2} \colon \...
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Čech cohomology of quasi-coherent sheaves and Leray's acyclicity theorem

My professor told us that if $X$ is a separated scheme and $\mathcal{F}$ a quasi-coherent scheme over $X$, then for any affine cover $\mathcal{U}=\left\{U_{i}\mid i\in I\right\}$, $\bigcup_{i\in I}U_{...
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Restricting split short exact sequence of quasi-coherent sheaves to their coherent sub-sheaves.

Let $X$ be a projective variety, $Z$ a hypersurface section and $U \overset{def}= X \setminus Z$ its complement, an open affine subscheme of $X$. Let $i:U \hookrightarrow X$ be the corresponding open ...
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62 views

If a global morphism of sheaves induces isomorphisms on fibers, then is it an isomorphism?

Let $X$ be a Noetherian scheme (regular if needed), and let $\mathcal{E}$ be a locally free sheaf of rank $2$ on $X$. Let $\pi:\mathbb{P}(\mathcal{E})\to X$ be the natural morphism, and let $f:\...
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How to prove $ ((S\times_A T)(1))_{(f\otimes g)}\cong S(1)_{(f)}\otimes_{S_{(f)}}(S\times_A T)_{(f\otimes g)}\otimes_{T_{(g)}} T(1)_{(g)}$?

I'm currently trying to solve Exercise 5.11 in chapter 2 of Hartshorne: Let $S,T$ be $\mathbb{Z}_{\geq 0}$-graded rings with $S_0=T_0=A$, and define their Cartesian product $S\times_A T$ to be $$ S\...
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106 views

Quasi-coherent sheaves $\supset$ locally free sheaves?

This question has been completely reformulated following the guidelines in the comments below, to make it clearer where I was able to get and where I can't get out of. Such comments helped me a lot. ...
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Homomorphism of quasi coherent sheaves [closed]

I was trying to solve the problems of Liu's book and wanted to show that if both $F$ ,$G$ are coherent then $Hom(F,G)$ is also coherent... but I realised that I really need to understand the meaning ...
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35 views

product of quasi-coherent sheaves is not a quasi-coherent sheaf [duplicate]

Reading the basics of $\mathcal{O}_{X}$ modules, where $\left(X,\mathcal{O}_{X}\right)$ is a fixed ringed space, i understood that arbitrary direct products of quasi-coherent $\mathcal{O}_{X}$ modules ...
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Scheme theoretically, when the union of the interserction is the intersection of the union

We have the definition: Definition. Let $X$ be a scheme. Let $Z,Y⊂X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\mathcal{I},\mathcal{J}⊂\mathcal{O}_X$. The scheme theoretic ...
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Explicit description of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a line bundle

I understand the construction of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a sheaf on $\Bbb{P}_\Bbb{C}^1$, but I'm trying to understand how exactly does this define a line bundle and why people call this the &...
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42 views

On global section of Hom-functor on quasi-coherent sheaves on a quasi-affine scheme

Let $R$ be a commutative Noetherian ring and $U$ be an open subscheme of the affine- scheme $X=\text{Spec}(R)$ such that $\Gamma_U(\mathcal O_U)\cong R$. If $\mathcal E, \mathcal F$ are quasi-coherent ...
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Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
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Hartshorne 5.11 twisted sheaf

Question 5.11 of Hartshorne Let $S$ and $T$ be two graded rings with $S_0=T_0=A$. We define the Cartesian product $S\times_A T$ to be the graded ring $\bigoplus_{d\geq 0}S_d\otimes T_d.$ If $X= \...
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$f_*\mathcal{F}$ is quasi-coherent if $\mathcal{F}$ is quasi-coherent and $f$ is affine

Let $f:X\to Y$ be an affine morphism of schemes. Prove that if a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent, then $f_*\mathcal{F}$ is also quasi-coherent. I think the first ...
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63 views

Proving that a quasi-coherent $\mathcal{F}$ is coherent by looking at one affine covering

Let $X$ be a noetherian scheme. I'm going through the proof that to check that a quasi-coherent $\mathcal{F}$ is coherent, it suffices to check that $\Gamma(U_i, \mathcal{F})$ is a finite $\Gamma(U_i,...
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58 views

Show that $\oplus\widetilde{M}_{\alpha}\cong \widetilde{\oplus M_{\alpha}}$ with $M_{\alpha}$ $\mathcal{O}_{X}(X)$-modules.

Let $X=\operatorname{Spec}(A)$ be an affine scheme, and let $M_{\alpha}$-be $A$-modules. I want to show that $\oplus\widetilde{M}_{\alpha}\cong\widetilde{\oplus M_{\alpha}}$. Let $D(f)$ be a ...
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159 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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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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36 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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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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160 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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30 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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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
75 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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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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159 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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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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113 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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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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260 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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1answer
113 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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1answer
139 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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1answer
348 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
145 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 ...