# Questions tagged [quasicoherent-sheaves]

The tag has no usage guidance.

142 questions
Filter by
Sorted by
Tagged with
22 views

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 ...
38 views

32 views

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 $\mathcal{O}_S$-...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
46 views

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 $S$...
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 ...
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 ...
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 ...
89 views

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 ...
27 views

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 ...
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 ...
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: ...
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 $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ ...
92 views

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 ...
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 ...
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 ...
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: ...
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 ...
### 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 \$\...