Questions tagged [quasicoherent-sheaves]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
5 votes
2 answers
133 views

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}...
  • 2,656
2 votes
1 answer
71 views

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 ...
  • 523
0 votes
0 answers
45 views

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
64 views

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 ...
3 votes
1 answer
49 views

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....
  • 622
0 votes
0 answers
12 views

Which types of quasi-coherent sheaves have total spaces?

Let $X$ be a scheme. Let $\mathcal F$ be a quasi-coherent sheaf on $X$. Consider the functor $$\operatorname{S}_{\mathcal F}:\;(\mathrm{Sch}/X)^{\mathrm{op}}\to \mathrm{Set},\quad (f:T\to X)\mapsto \...
  • 1,251
0 votes
0 answers
26 views

Cokernel of canonical skyscraper map not generated by global sections.

I am trying to solve the following exercise (Görtz and Wedhorn Exercise 7.7): Let $A$ be an integral domain with field of fractions $K$, set $X=\operatorname{Spec} A$ and let $\eta \in X$ be the ...
4 votes
1 answer
94 views

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 ...
  • 952
1 vote
0 answers
30 views

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
65 views

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 ...
9 votes
1 answer
280 views

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 ...
  • 2,656
5 votes
1 answer
88 views

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 ...
  • 2,656
1 vote
1 answer
77 views

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
171 views

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 vote
0 answers
25 views

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}...
  • 11
0 votes
0 answers
62 views

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
64 views

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 ...
  • 87
0 votes
1 answer
91 views

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 ...
  • 87
0 votes
1 answer
106 views

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
103 views

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
77 views

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 ...
  • 10.1k
0 votes
0 answers
71 views

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
70 views

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)...
2 votes
0 answers
92 views

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 ...
  • 1,834
0 votes
1 answer
189 views

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 vote
1 answer
212 views

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 $M$ ...
  • 1,480
1 vote
1 answer
120 views

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 $...
  • 1,480
1 vote
1 answer
59 views

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 ...
0 votes
0 answers
86 views

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.
  • 653
1 vote
0 answers
40 views

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) \...
  • 4,987
0 votes
1 answer
76 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 ...
  • 2,126
3 votes
1 answer
216 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 ...
  • 1,301
0 votes
0 answers
216 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....
1 vote
1 answer
203 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 $\...
  • 151
1 vote
1 answer
118 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}...
2 votes
0 answers
71 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)...
  • 179
1 vote
1 answer
111 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}...
1 vote
1 answer
190 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\;:\;\...
  • 1,301
2 votes
1 answer
102 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 ...
  • 1,274
0 votes
0 answers
91 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 $\widetilde{M\otimes_{A}B}$? ...
  • 141
1 vote
0 answers
131 views

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}_{...
0 votes
1 answer
173 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 \...
3 votes
2 answers
187 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 ...
  • 1,274
0 votes
0 answers
186 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:\...
6 votes
1 answer
215 views

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\...
3 votes
1 answer
471 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. ...
  • 2,126
2 votes
1 answer
77 views

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 ...
1 vote
0 answers
50 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 ...
3 votes
0 answers
209 views

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 votes
1 answer
132 views

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 &...
  • 9,219