Questions tagged [quasicoherent-sheaves]
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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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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
58 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
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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
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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
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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 ...
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1answer
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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: ...
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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 ...
2
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1answer
69 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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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....
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1answer
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Where is the finiteness of product used in this proposition from Hartshorne?
See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related ...
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Tensor product of sheaves over weighted projective spaces
I have read that if we have a weighted projective space $\mathbb{P}$ in general it is not true that:
$$
\mathcal{O}_\mathbb{P}(n)\otimes \mathcal{O}_\mathbb{P}(m)\simeq \mathcal{O}_\mathbb{P}(n+m).
$$
...
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1answer
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Quasicoherent sheaves on the groupoid of vector bundles on a surface
Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$\mathcal L\in QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a ...
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1answer
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Is this exact: $0\to\mathcal{O}^{hol}\to\mathcal{O}(p)\to \Bbb{C}_p,$
Let $X$ be a Riemann surface and $p\in M$ some point.
Let $\mathcal{O}(p)=\mathcal{O}((-p))(U)=\{f\in \mathcal{O}^{hol}(U)\mid f\text{ has a zero of order atleast 1 at p}\}$
I.e. we have the divisor ...
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1answer
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Why are the noetherian objects in a category of quasicoherent sheaves just the coherent ones?
The question says it all really. Let $X$ be a noetherian scheme. Let $\mathcal{A}$ be the category of quasicoherent sheaves on $X$. I want to show that an object $\mathcal{F}$ in $\mathcal{A}$ is ...
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$f_* f^* \mathcal{G}= \mathcal{G}$ and $f^* f_* \mathcal{F}= \mathcal{F}$ for Quasicoherent Sheaves
Let $f: X \to Y$ a morphism between schemes, $\mathcal{F}$ a quasicoherent $\mathcal{O}_X$ module, $\mathcal{G}$ a quasicoherent $\mathcal{O}_Y$ module.
My question is what are the weakest possible ...
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Sheaves of abealian groups and base change
Let $k$ be a field of characteristic $p > 0$ and $R_1$ and $R_2$ be two $k$-algebras.
Let $X$ be a scheme over $k$. Let $f: R_1 \rightarrow R_2$ be a morphism of $k$-algebras induces the morphism ...
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1answer
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Counter-examples for quasi-coherent, coherent, locally free and invertible sheaves
I'm trying to find at least one counter-example for each of these concepts to feel more comfortable with understanding the ideas behind them but I cannot even get started :( Please help me find ...
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1answer
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Gaga and quasicoherent sheaf
Let $X$ be a complete algebraic variety over $\mathbb{C}$. Ad Serre GAGA stases, its analytification $X^{an}$ is compact and the analytification functor induces an equivalence of categories between $...
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QCQS lemma for modules?
The regular qcqs lemma states that if $X$ is a qcqs (quasi-compact quasi-separated) scheme and $f \in \Gamma(X, \mathcal{O}_X)$ then
$$\Gamma(X, \mathcal{O}_X)_f \cong \Gamma(D(f), \mathcal{O}_X)$$
...
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On the support of sheaf of modules or quasi-coherent sheaves over ringed spaces
This question Characterize commutative rings over which any module $M$ satisfies $\operatorname{Supp}(M)=V(\operatorname{Ann} M)$ asks to characterize commutative rings over which support of modules ...
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1answer
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Free module after tensoring with a flat local ring
Let $R$ and $R'$ be two local rings. Let $f:R \rightarrow R'$ be a local ring morphism and $R'$ is flat over $R$. Now, let $M$ be a finitely generated module over $R$. Suppose $ R' \otimes_R M$ is a ...
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1answer
85 views
Why is every sheaf of $\mathcal{O}_{X}$-modules not generated by global sections?
Let $(X, \mathcal{O}_{X})$ be a ringed-space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-modules. For any section $s \in \Gamma(X, \mathcal{F})$, we define a morphism,
$$
\mathcal{O}_{X} \...
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221 views
Why are the global sections of structure sheaf of Proj$S$ just the homogenous elements of $S$?
Let $A$ be a ring and define $S = A[x_{0}, x_{1}, \ldots , x_{r}]$. Let $X = \text{Proj }S$. I would like to show that $\Gamma(X, \mathcal{O}_{X}(n)) = S_{n}$. This is Proposition II 5.13 in ...
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162 views
Showing properties of inverse and direct image of quasicoherent sheaves via Yoneda's lemma
Disclaimer: A question about the same result has been asked here previously. See my comment at the bottom for why I think this question is unique.
Let $\phi: A \longrightarrow B$ be a morphism of ...
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How to get explicit descriptions of morphisms of sheaves from a known adjunction of functors?
This question is borne out of the following claim: Let $X = \text{ Spec }A$ be an affine scheme with $M$ and $N$ $A$-modules. Then $(M \otimes_{A} N)^{\sim} \simeq \widetilde{M} \otimes_{\mathcal{O}_{...
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1answer
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Extending the adjunction between the global section functor of a quasicoherent sheaf with module sheafification
So adjoint functors are one thing that has caused me endless pain since trying to learn algebraic geometry. They still haven't really "clicked" like they should have. I've been told that it makes it ...
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Trying to understand vector bundles on manifolds via locally free sheaves
My background is primarily in algebra and topology/geometry with my primary interest lying in algebraic geometry. I am learning about locally free sheaves in the context of schemes, and they always ...
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Quasi coherent sheaves restricted to affine open subsets
This is a proposition in Hartshorne's Algebraic Geometry.
Let $X$ be a scheme. Then an $\mathcal{O}_X$ module $\mathcal{F}$ is quasi coherent then for every open affine subset $U=\text{Spec}(A)$ of ...
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1answer
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Global sections on quasi coherent sheaves on affine scheme
This is a lemma from Hartshorne's Algebraic Geometry.
Let $X=\text{Spec}(A)$ be an affine scheme $f\in A, D(f)\subseteq X$.
Let $\mathcal{F}$ be a quasi coherent sheaf on $X$.
If $s\in \...
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1answer
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Invertible sheaf such that for any point there exists a global section not vanishing at that point
Let $\mathcal{L}$ be an invertible sheaf on a quasicompact scheme $X$. I want to prove that there exists an epimorphism $\mathcal{O}_{X}^{\oplus I}\rightarrow\mathcal{L}$ if and only if for any point $...
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1answer
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If a finite type quasicoherent sheaf is globally generated at a point, it is globally generated in a neighborhood of that point.
Let $\mathcal{F}$ be a quasi-coherent sheaf of finite type on a scheme $X$. Suppose that we have a morphism of sheaves $\varphi:\mathcal{O}_{X}^{\oplus I}\rightarrow \mathcal{F}$ such that $\varphi_{p}...
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Two definitions of $\mathcal{O}_{\mathbb{P}^{n}}(l)$
On the one hand, we may define $\mathcal{O}_{\mathbb{P}^{n}}(l)$ as the invertible sheaf with trivializing cover $\{D(X_i): i\in \{0,...,n\}\}$ and transition functions $\left(\frac{X_i}{X_j}\right)^l$...
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2answers
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Structure of locally free $\mathcal{O}_{X}$-module on affine open set
Suppose $X$ is a scheme. I have been studying (finite rank) locally free $\mathcal{O}_{X}$-modules, and more generally, quasi-coherent sheaves on $X$ mainly from Ravi Vakil's excellent notes as well ...
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1answer
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Taking ideal sheaf which is not locally generated by sections, where does it fail to construct a closed subscheme
Let $\mathcal{I}$ be a sheaf of ideals on the scheme $X$ which is not necessarily locally generated by sections.
Following the construction of a closed subscheme where we set $Z = \text{supp}(\...
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1answer
233 views
(Quasi)coherent sheaves on smooth manifolds, and their applications
Since a smooth (real) manifold is canonically a locally ringed space, we can define (quasi)coherent sheaves over smooth manifolds in the usual manner. But is the category of (quasi)coherent sheaves on ...
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Converse to “free module on locally constant sheaf is quasi-coherent”
Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then one can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the "free module over $\mathcal{E}$", as the sheafification of the ...
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1answer
647 views
Category of quasicoherent sheaves not abelian
Wikipedia mentions that the category of quasicoherent sheaves need not form an abelian category on general ringed spaces. Is there a `naturally occurring' example of this failing, even for locally ...
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2answers
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Tensoring by locally free sheaves preserves quasi-isomorphism
Let X be a scheme. Given a quasi-isomorphism of complexes of quasi-coherent sheaves $f:\mathcal{F}^.\to \mathcal{G}^.$ and a locally free sheaf $\mathcal{E}$ we could consider the induced map $f\...
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1answer
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Morphisms between quasi-coherent sheaves
Let $X$ be a noetherian scheme and let $F,G$ be coherent sheaves. Let $Z$ be a closed subscheme of $X$ not containing any associated point of $X$. Let $F|_U\to G|_U$ be a morphism of sheaves of ...
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2answers
291 views
Riemann-Roch equation for proper integral curves (not necessarily non-singular!)
In Qing Liu - Algebraic Geometry and Arithmetic Curves, chapter 7.3, Theorem 3.26 the author states the Theorem of Riemann-Roch as follows:
Let $f: X \rightarrow \text{Spec }k$ be a projective ...
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1answer
333 views
Clarification on Ravi Vakil's AG notes, Exercise 13.3.H
Let me first write what this exercise is:
[...] Suppose $X$ is a quasicompact quasiseparated scheme, $\mathscr{L}$ is an invertible sheaf on $X$ with section $s$, and $\mathscr{F}$ a quasicoherent ...
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2answers
198 views
Hartshorne Lemma II.5.3 Proof
The question I have is from the proof of the lemma above, but it is actually a more general statement about quasi-coherent sheaves on an affine scheme. Suppose $X= \text{Spec }A$ for some ring $A$, ...
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How to deduce the usual definition of Quasi-coherent Module over a scheme from the general definition over ringed Spaces [duplicate]
Quasi-Coherent Modules over a Ringed Space : Let $(X,\mathcal O_X)$ be a ringed space. A sheaf of modules $F$ over $(X,\mathcal O_X)$ is called quasi-coherent if for every point $x\in X$ $\exists U\...
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1answer
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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 $...
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1answer
216 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 ...
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2answers
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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
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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
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1answer
134 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 $f^{*}\mathcal{G}$...
3
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1answer
221 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 ...
