Questions tagged [quasicoherent-sheaves]
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206
questions
4
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Global version of an ideal being "finitely generated up to radical"
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a finite type quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (...
2
votes
1
answer
50
views
Definition of an $\mathcal{O}_X$-module generated by its global sections in Liu
In Liu's Algebraic Geometry and Arithmetic Curves, the author says that an $\mathcal{O}_X$-module is generated by its global sections at $x \in X$ if
the canonical homomorphism $\mathcal{F}(X) \...
16
votes
2
answers
902
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Reference request: Bundles in Algebraic Geometry
I heard many times that quasi-coherent sheaves of $\mathcal O_X$-modules are morally the same thing as the sheaves of sections of a bundle $V\to X$ over $X$. We think of a ring $A$ as of the ring of ...
2
votes
0
answers
91
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On the right adjoint of the derived pushforward of a proper birational morphism of Noetherian quasi-separated schemes
Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(...
1
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0
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87
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Vakil's Generalization of QCQS Lemma
In the most recent notes of Vakil, this is problem 15.4.Y (which I think is just Hartshorne Lemma 5.14):
Let $X$ be qcqs, $\mathscr{L}$ invertible sheaf on $X$, $s\in \Gamma(X, \mathscr{L})$, and $\...
0
votes
2
answers
100
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If $U\cong\operatorname{Spec} B\subset \operatorname{Spec} A$ is an affine open, then $\widetilde{M}(U)\cong M\otimes_A B$
I want to prove the theorem.
Let $A$ be a ring and $X = \operatorname{Spec} A$. Furthermore, let $M$ be an $A$-module and $F$ an $O_X$-module that is associated to $M$. Then, for any affine open ...
7
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3
answers
1k
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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 $...
2
votes
1
answer
58
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Closed immersion of locally ringed spaces vs closed immersion determined by ideal sheaf
Let $f: X \to Y$ be a closed immersion of locally ringed spaces, that is,
$f$ is a homeomorphism onto a closed subset of $Y$,
$f^{\#}:\mathscr{O}_Y \to f_*\mathscr{O}_X$ is surjective,
$\mathscr{I} = ...
4
votes
0
answers
67
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On free direct summand locus of finitely generated modules over commutative Noetherian rings
Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. Assume that there exists an injective $R$-linear morphism $f: R\to M$. Consider the sets $$U:=\{\mathfrak p\in \text{Spec}...
0
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0
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59
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Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?
Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
0
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0
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60
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Quasi-coherent sheaves on a complex manifold form an abelian category?
Quasi-coherent modules on a ringed space is defined in Stacks Project. Is there a complex manifold $X$ such that the category of quasi-coherent $O_X$-modules is not an abelian subcategory of the ...
2
votes
1
answer
53
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$\det (f_{*} \mathcal{O}_D) \simeq \mathscr{L} (f_{*} D)$ (Hartshorne Exercise IV.2.6)
I am stuck at the end of exercise IV.2.6 and I would appreciate a hint on how to conclude.
We have $f:X\rightarrow Y$ a finite morphism of degree $n$ between curves and $D$ an effective divisor over X....
3
votes
2
answers
426
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Line bundles over $\Bbb P^1_k$
I am trying to understand the line bundle $O(1)$ over $\Bbb P^1_k$ and why
$$ O(1)^{\otimes n} = O(n)$$ from Vakil's notes, p398, line 6.
His explanation is rather long, so I took a screen shot. ...
3
votes
1
answer
51
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Do quasi-coherent sheaves form a reflective subcategory?
Let $X = $ Spec $A$ be an affine scheme.
I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.
It seems to me ...
4
votes
1
answer
352
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Simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point
Need to prove that simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point.
"Simple object" means that there is no non-trivial subobject.
I encountered ...
1
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1
answer
136
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The stalk of a point on a scheme is a localization of ring of affine open?
Let $(X,\mathcal O_X)$ be a Noetherian scheme. For every affine open subset $U$ of $X$, it holds that $U=\text{Spec}(\mathcal O_X(U))$. Let $x \in X$, and let $U$ be an affine open subset of $X$ ...
0
votes
1
answer
117
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Is relative spectrum a left adjoint?
Given a scheme $X$ and a quasicoherent sheaf of algebras $\mathscr{R}$ on it. Vakil's FOAG, section 17.1.2, page 470 says that the relative spec $\beta: Spec \mathscr{R} \to X$, representing the ...
8
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2
answers
280
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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 ...
3
votes
1
answer
190
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Classify all coherent sheaves on $\mathbb{A}^1_k$
Given a field $K$, I want to classify all coherent sheaves on $\mathbb{A}^1_k$, and moreover saying if there exist locally free sheaves that are not free on $\mathbb{A}^1_k$.
I am following Gathmann's ...
1
vote
1
answer
79
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Double dual of locally free $\mathcal{O}_X$-module of finite rank
This question has been asked by many people and the answers are mostly suggesting verifying the isomorphism on stalks. But I really have no idea how to proceed:
we define $\alpha:\mathcal{E} \to (\...
2
votes
1
answer
164
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Hartshorne proposition II.5.4
I am really confused on the proof of Proposition II.5.4 on Hartshorne's algebraic geometry book. Especially, I am not sure how does the previous Lemma II.5.3 work on this proposition. Let me state the ...
0
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0
answers
28
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Relative version of "On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules"
On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules (See Stacks Project). I want to consider the following generalization.
Let $f:X\to S$ be a (...
4
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4
answers
656
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Why quasi coherent sheaves?
So i was wondering why one considers quasi-coherent sheaves in algebraic geometry. I have read a lot that they are closely linked to the geometric properties of the underlying space. This means that ...
2
votes
1
answer
507
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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\;:\;\...
17
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2
answers
3k
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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.
...
4
votes
1
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376
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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 ...
3
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1
answer
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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 ...
5
votes
2
answers
155
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What is the universal property of the thickening $Y[\varepsilon]$?
Given an $S$-scheme $Y$, let $Y[\varepsilon]$ denote thickening $Y[\varepsilon]=Y \times_S D_S$ of $Y$. Here $D_S$ is the $S$-scheme $D\times_\mathbb Z S\to S$ where $D = \operatorname{Spec} \mathbb{Z}...
9
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1
answer
1k
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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 ...
2
votes
1
answer
136
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Example of a morphism of schemes whose kernel sheaf is not quasi coherent
I am trying to think of an explicit example of a morphism $\varphi: X\longrightarrow Y$ of schemes for which the kernel sheaf $ker(\mathcal{O}_Y\longrightarrow \varphi_*\mathcal{O}_X)$ is not quasi ...
0
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0
answers
72
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Quasi-coherent sheaves on projective space.
I would like to better understand the characterization of quasi-coherent sheaves on $X=\mathbb{P}_A^d = \operatorname{Proj}A[x_0,...x_d]$, where I call $B = A[x_0,...x_d]$
We define $\mathcal{F}[k] = \...
3
votes
1
answer
95
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Quasi-coherent $\mathcal{F} \cong \tilde{M}$ on $\operatorname{Proj}B$ does not determine $M$
I would like to understand in detail the fact that given $\tilde{M}\cong \tilde{N}$ on $\operatorname{Proj}B$, then it is not always true in general that $M \cong N$.
I know this has to do with ...
6
votes
1
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92
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Reconciliation of definition of presentable module/quasi-coherent sheaf with the fact that all modules have free presentation
Presentable module is defined in nLab (https://ncatlab.org/nlab/show/presentable+module) as “the cokernel of a homomorphism of free modules”.
What I’m confused about is that, according to https://en....
2
votes
1
answer
114
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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 $\mathcal{O}_S$-...
4
votes
1
answer
174
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Hartshorne Chapter 2 Exercise 5.6. d)
In this exercise 5.6. d) in chapter 2 from "Algebraic Geometry" by Hartshorne one has to show that for any ideal $\mathfrak a \subset A$ of a noetherian ring $A$ and $A$-module $M$ the sheaf ...
1
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0
answers
120
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How to show saturation map (w.r.t quasicoherent sheaves) isnt always injective / surjective
This question is motivated by problem 15.4.D(a) in Vakil, but to give some setup since the notation / terminology may differ: let $S_\bullet$ be a nice graded algebra (finitely generated, generated in ...
2
votes
1
answer
104
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Internal hom of equivariant sheaves
Let $G$ is affine algebraic group over $\mathbb{C}$ acting on a smooth scheme $X$ over $\mathbb{C}$, let $\mathcal{F},\mathcal{G}$ be two quasi coherent equivariant sheaves on $X$. Is there a natural ...
5
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1
answer
129
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sheaves on a scheme
I work within the framework of Demazure & Gabriel`s book. (My schemes are all functors). Let $X$ be a scheme. I can define an underlying topological space $|X$| of $X$. Its points are equivalence ...
0
votes
1
answer
265
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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 ...
1
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1
answer
301
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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}]...
2
votes
1
answer
429
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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, \...
1
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0
answers
29
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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}...
0
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0
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68
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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 ...
0
votes
0
answers
98
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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 ...
0
votes
1
answer
163
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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 ...
0
votes
1
answer
269
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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 ...
3
votes
0
answers
149
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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 ...
0
votes
0
answers
110
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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 ...
0
votes
0
answers
168
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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(\...
3
votes
0
answers
87
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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)...