Questions tagged [quasicoherent-sheaves]

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Transition functions on invertible sheaves

I'm studying "Foundations of algebraic geometry" by Ravi Vakil. Chapter 14.1 is about invertible sheaves on $\mathbb{P}^1_k$. It says between two affine subsets $\operatorname{Spec}k[x_{1/0}]...
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Vanishing of Ext groups of Coherent sheaves over Noetherian regular scheme

Let $(X,\mathcal O_X)$ be a Noetherian regular scheme of dimension $1$. Then, for any coherent sheaf $\mathcal F$ and any quasi-coherent sheaf $\mathcal G$, it holds that $\mathcal Ext^i(\mathcal F, \...
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Globalisation of varieties

I am trying to understand the way to globalize the notion of variety. Let $\mathcal A$ be a quasi-coherent sheaf locally of finite type over an affine variety $X$. We define a variety $Y=\mathrm{Specm}...
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How to think about the two quasi-coherent algebras $\bigoplus\limits_{n}\mathscr{E}^{\otimes n}, \,\, \bigoplus\limits_{n}\mathscr{E}(n)$?

Let $X$ be a scheme and $\mathscr{E}$ be a quasi-coherent $\mathcal{O}_X$-module. Take $X$ to be projective with a very ample bundle $\mathcal{O}_X(1)$ for a structure morphism $\pi: X \to S$, so that ...
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Linear system of conics passing through $3$ points

I would like to understand why the set of classes of divisors corresponding to the conics of $\mathbb P^2$ passing through $3$ non-colinear points is a linear system of dimension $2$. Thank you very ...
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Euler characteristic of a curve

Let $C$ be an irreducible curve on a surface $S$. We have the classical exact sequence $0\rightarrow\mathcal O_S(-C)\rightarrow\mathcal O_S\rightarrow\mathcal O_C\rightarrow 0$. Why can we deduce that ...
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Is a sheaf, which is flasque when restricted to any open affine necessarily flasque?

This is something I got stuck thinking about while trying to solve a problem in Hartshorne: if $\mathcal{F}$ is a quasi coherent sheaf on a noetherian scheme $X$ such that $\mathcal{F}|_U$ is flasque ...
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Hartshorne problem II.8.7

Again I'm stuck on a problem in Hartshorne. The situation is as follows: Let $X=\text{Spec}(A)$ be an affine, non-singular scheme which is finite of some field $k$. Let also $\mathcal{F}$ be a ...
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injective sheaves on a projective scheme cannot be coherent [duplicate]

Let $X$ be a projective scheme. If it helps (e.g. gives way to a short/elegant answer) fix a base field $k$ and assume smoothness. It is often said that the category of coherent sheaves over $X$ does ...
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It seems to me that quasi-coherent sheaf is a sheaf of sheaves.

In Wikipedia, I was looking for the definition of quasi-coherent sheaf. However I got confused. It seems to me that quasi-coherent sheaf is a sheaf of sheaves. Though, indeed, it says quasi-coherent ...
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Proving that the stalk of the sheaf of homomorphisms of a module of finite presentation is isomorphic to the module of homomorphism of stalk

Consider $\mathscr{F}$ to be a module of finite presentation over a ringed space $X$. I want to prove that, for any sheaf of modules $\mathscr{G}$ and $x \in X$, the canonical morphism $\mathscr{H}om(\...
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Quasicoherent sheaves over manfiolds

For any locally ringed space $\left(X,O_X\right)$, a quasicoherent sheaf is a sheaf of $O_X$-modules which are locally the quotient of free modules. Considering a manifold (Haussdorf, second-countable)...
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Hypersurfaces of Proj and quasi-coherent sheaf

Let $F$ be a homogeneous polynomial of degree $d$ and consider the closed subscheme $ X = Proj(k[x_0, \dots, x_n] / (F) ) \subset \mathbb{P}^n_k$. Now we have an exact sequence: $ 0 \longrightarrow \...
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Equivalent definitions of flat morphism

Suppose $\pi : X \to Y$ satisfies that pullback on quasi-coherent sheaves is exact, how do I prove that $\pi$ is flat via the local definition; i.e stalkwise $O_{X,p}$ is a flat $O_{Y,q}$ module ...
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Completion of a Stalk of Pushforward/ Direct Image of a $\mathcal{O}_X$-module by Finite morphism

Let $f : X \to Y$ be a finite morphism of integral projective schemes over fixed field $k$. Let $x \in X, y=f(x)$ and $\mathcal{F}$ a quasicoherent $\mathcal{O}_X$-module. Since $f$ is quasi-finite, ...
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Theorem 7.4.16 (Riemann–Hurwitz formula) in Qing Liu's Algebraic Geometry and Arithmetic Curves

I have a question about a part from the proof of Theorem 7.4.16 on page 290 from Liu's "Algebraic Geometry and Arithmetic Curves". The claim is Theorem 7.4.16 Let $f : X \to Y$ be a finite ...
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$\mathcal{D}$-module direct image under affine open immersion

I am learning about the direct image functor for $\mathcal{D}$-modules. In particular, unlike the inverse image, the underlying $\mathcal{O}$-module is not the direct image of $\mathcal{O}$-modules. I ...
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Tensor product of quasi-coherent sheaves on an affine scheme

Given an affine scheme $X = \text{Spec}(A)$, then from an $A$-module $M$ we can form the associated sheaf $\tilde{M}$, where $$ \tilde{M}(D_f) = M_f = M \otimes_A A_f. $$ Now, given $A$-modules $M$ ...
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Alternative definition of sheaf associated to a module?

Given an affine scheme $X = \text{Spec}(A)$, then from an $A$-module $M$ we can form the associated sheaf $\tilde{M}$, where $$ \tilde{M}(D_f) = M_f = M \otimes_A A_f $$ This agrees with the presheaf $...
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Locus where $v: \mathscr{E} \to \mathscr{F}$ is surjective

Let $\mathscr{E}, \mathscr{F}$ be quasicoherent modules over a scheme $S$, $\mathscr{F}$ of finite type, and $v: \mathscr{E} \to \mathscr{F}$ a homomorphism. Consider the functor $F: (Sch/S)^{op} \to ...
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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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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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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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  • 1,208
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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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1 answer
100 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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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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2 votes
0 answers
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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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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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1 answer
121 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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2 votes
1 answer
69 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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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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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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3 votes
2 answers
160 views

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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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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6 votes
1 answer
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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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2 votes
1 answer
288 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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2 votes
1 answer
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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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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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3 votes
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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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3 votes
1 answer
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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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1 vote
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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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2 votes
0 answers
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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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3 votes
0 answers
180 views

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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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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  • 2,493
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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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2 votes
1 answer
425 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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5 votes
1 answer
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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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