Questions tagged [quasicoherent-sheaves]
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142
questions
-2
votes
0answers
22 views
Is $\theta^\sharp:\mathscr O_{\operatorname{Spec}f_*\mathscr O_X}\to \theta_*\mathscr O_X$ isomorphic?
Suppose $f:X\to Y$ is a quasi-compact and separated morphism of schemes, $\pi: \operatorname{Spec}f_*\mathscr O_X\to Y$ is the canonical affine morphism such that $\pi_*\mathscr O_{\operatorname{Spec}...
1
vote
1answer
58 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 ...
3
votes
1answer
33 views
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 $X$...
0
votes
1answer
27 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?
1
vote
1answer
41 views
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 $X$, ...
1
vote
1answer
56 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 ...
1
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0answers
29 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\...
0
votes
0answers
14 views
Pullback of Serre's twisted sheaf via projection $X\times_{\rm{Spec}(R)}\rm{Spec}(R')\rightarrow X$
The problem is due to exercise 2 of section 9.2 of Bosch's Algebraic Geometry book. The setting is as follows: Given a graded algebra $A$ over a ring $R$ (commutative with unit) and a morphism of ...
0
votes
0answers
38 views
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{...
1
vote
1answer
40 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 $\...
0
votes
0answers
32 views
Corollary 5.10 in Hartshorne II
In Hartshorne II corollary 5.10 he claims the following: If $X=\operatorname{Spec}A$ is an affine scheme, then there is a 1-1 corresopndence between ideals $\mathfrak{a} \in A$ and closed subschemes $...
0
votes
1answer
30 views
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 ...
1
vote
1answer
120 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 ...
0
votes
0answers
41 views
Quasicoherent sheaves on relative spec, Vakil's Exercise 17.1.E
This is Exercise 17.1.E, p449 of Vakil's rising sea.
Let $\mu: \mathcal{Spec B} \rightarrow X$ the relative spectrum construction for $\mathcal{B}$ a quasicoherent sheaf of algebras on $(X, \...
2
votes
0answers
37 views
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 $\mathcal{O}_S$-...
3
votes
1answer
97 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. ...
1
vote
0answers
50 views
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 ...
1
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0answers
96 views
Zero section of quasi-coherent bundle
Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...
2
votes
0answers
103 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 ...
1
vote
0answers
47 views
The Grassmannian of a quasi-coherent module
In Goetz' and Wedhorn's Algebraic Geometry Chap (8.6) page 214 is introduced the concept of generalized Grassmannian's. One remark I can't understand.
(8.6) The Grassmannian of a quasi-coherent ...
3
votes
1answer
60 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 ...
0
votes
0answers
46 views
Surjectivity of homomorphism $\mathcal E \otimes_{O_X} \mathbb S(\mathcal E) \to \mathbb S(\mathcal E)(1)$
That should be an easy question, but probably I misunderstood something and just can't understand how that is true.
So suppose we have scheme $X$, quasicoherent sheaf $\mathcal E$ and $\mathbb S(\...
2
votes
1answer
99 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{...
9
votes
1answer
222 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 $S$...
2
votes
1answer
96 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 ...
0
votes
0answers
88 views
Direct sums of invertible sheaves commuting with global sections
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
1
vote
0answers
33 views
Regarding Sheafification functor
Assume we are in the category of sheaves of $\mathcal{O}_X$-modules. Suppose two presheaves maps to the same sheaf under the sheafification functor. Does it imply that two presheaves were same?
I am ...
4
votes
1answer
89 views
Stalk of the product or power of quasi-coherent ideal sheaves
Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $\mathfrak I$ and $ \mathcal J$ be two quasi-coherent sheaf of ideals on $X$.
Is it true that for every $x\in X$, we have $(\mathfrak I \mathcal J)...
4
votes
1answer
66 views
Inducing one point closed subset with a closed subscheme structure so that the stalk of the subscheme is a field
Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $x\in X$ be a closed point and $Y:=\{x \}$ .
Is it always possible to make $Y$ into a scheme such that $(Y, \mathcal O_Y)$ is a closed subscheme of $...
3
votes
0answers
35 views
Some stalk of non-zero sheaf , on locally ringed space, is non-zero?
If $\mathcal F$ is a non-zero sheaf of Abelian groups on a locally ringed space $(X,\mathcal O_X)$, then is it true that some stalk of $\mathcal F$ is non-zero ?
If this is not true in general, then ...
2
votes
0answers
27 views
Structure of quasi-coherent sheaf of ideals
Let $\mathcal J$ be a sheaf of ideals on a scheme $(X,\mathcal O_X)$ . Is it true that $\mathcal J$ is quasi-coherent if and only if for every affine open subset $U=Spec A$ of $X$, there is an ideal $...
4
votes
1answer
54 views
Relating the sheaf associated to a cyclic module over an affine scheme to the structure sheaf
Let $(X,\mathcal O_X)$ be the affine scheme of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R$. From the sheaf $\mathcal O_X$ and the closed subset $Z:=V(J)$ of $X$ , how do we recover ...
2
votes
0answers
107 views
What is the correct definition of the inverse image functor on sheaves of modules?
There seem to be two different definitions of the inverse image functor on sheaves of modules in the literature and I just wanted to make sure I am understanding properly. Suppose $f: X \rightarrow Y$ ...
0
votes
1answer
88 views
About Hom and global section functor for $O_X$ modules
Is it true for any $O_X$-module $F$ that
$$
Hom_{O_X}(F,O_X)\cong F(X)= \Gamma(X,F)
$$
And is it in general true that the sheafification of $F(X)$ is $F$ ?
I think that it is true for quasi-...
2
votes
1answer
55 views
Finite Morphism between Regular Curves induces Category Equivalence
Let $C,D$ curves (so $1$-dimensional proper $k$-schemes). Assume futhermore that they are also regular and $f:C \to D$ is a finite morphism.
It is known that in this case the pushforward functor $f_*:...
3
votes
0answers
48 views
Local behavior of sheaf of ideals given by a closed immersion
I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals
$I_Y(U) = $ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$}
is quasi coherent. I sort of understand ...
1
vote
0answers
33 views
Broad question on morphisms of stalks of quasi-coherent sheaves on schemes
This question was inspired by reading about a criterion for a morphism into projective space (over an algebraically closed field) to be a closed immersion based on local rings. It got me thinking ...
2
votes
0answers
72 views
Questions about Hartshorne Proposition II.5.9
My question is about the first part of the proof.
Let $X$ be a scheme. For any closed subscheme $Y$ of $X$, the corresponding ideal sheaf $I_Y$ is a quasi-coherent sheaf of ideals.
Proof: ...
2
votes
0answers
46 views
Left Exactness of global sections functor over quasi compact sheaves [duplicate]
Let $0 \rightarrow E' \rightarrow E \rightarrow E''$ be a short exact sequence of quasi coherent sheaves on a scheme X. Show that the sequence $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ ...
2
votes
0answers
92 views
Projective space $Proj$
We let $S$ be a graded algebra.
I have a 2 questions regarding the Proj construction.
It seems to me that we do not know what $O_{Proj S}Proj(S))$ is ?
How is the map $S_0 \rightarrow \Gamma(Proj S,...
1
vote
0answers
28 views
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 ...
1
vote
0answers
70 views
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,...
0
votes
1answer
210 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}...
1
vote
1answer
17 views
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 ...
1
vote
1answer
21 views
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 ...
2
votes
1answer
86 views
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 ...
3
votes
1answer
120 views
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: ...
4
votes
0answers
246 views
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 ...
3
votes
1answer
225 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 $\...
0
votes
1answer
58 views
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....
