# Questions tagged [quasi-coherent-modules]

The quasi-coherent-modules tag has no usage guidance.

23
questions

**4**

votes

**0**answers

280 views

### Compact quasi-coherent sheaves

Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.

**3**

votes

**2**answers

234 views

### Obstructions to abelian sheaf being quasi-coherent

Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher ...

**13**

votes

**1**answer

812 views

### Does every sheaf embed into a quasicoherent sheaf?

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...

**7**

votes

**0**answers

341 views

### Free modules generate all quasi-coherent modules

The following statement is true* and not hard to prove.
Let $X$ be a quasi-compact and separated scheme. Then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a free $\mathcal{O}_X$-...

**4**

votes

**0**answers

202 views

### Quasi-coherent module of (global) finite presentation

If $\mathscr{X}$ is a stack over some base ring $k$ (if you are not familiar with stacks, read "schemes" here), we may consider it as a pseudofunctor $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (...

**8**

votes

**1**answer

194 views

### When are free modules on sheaves of sets quasicoherent?

This question was previously asked over at math.SE.
Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over ...

**1**

vote

**0**answers

287 views

### A criterion for purity

I have started reading the book "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn. This is a statement in this book at page no.3 the last line.
"$E$
is pure if and only if all ...

**2**

votes

**0**answers

137 views

### Lifting a local section to a global section along a homomorphism of quasi-coherent sheaves

If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property?
For every quasi-coherent sheaf $M$...

**5**

votes

**1**answer

651 views

### When are direct products exact in the category of quasi-coherent sheaves?

(This question is crossposted from MSE, since there the question did not recieve any attention whatsoever.)
I would like to know if there is a description (or at least some sufficient condition known)...

**1**

vote

**0**answers

98 views

### Do flat resolutions guarantee the existence of Tor (without enough projectives)?

Let $\mathcal A$ be an abelian category with a symmetric monoidal structure $\otimes$. Suppose that $\mathcal A$ does not have enough projectives, but every object has a flat resolution Then, is the ...

**5**

votes

**1**answer

375 views

### Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. And a repost from this MSE question.
The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is ...

**2**

votes

**1**answer

301 views

### Torsion theory for quasi-coherent sheaves?

In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies:
(1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$.
(2) If $Hom(T,F)=0$ for ...

**3**

votes

**0**answers

216 views

### What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?

It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...

**4**

votes

**0**answers

119 views

### Are injective modules flabby on basic open sets?

In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and $...

**7**

votes

**1**answer

1k views

### Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)

For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an ...

**7**

votes

**0**answers

372 views

### Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...

**2**

votes

**1**answer

419 views

### Projection formula for immersions

Let $i : Y \to X$ be a quasi-compact immersion of schemes and let $M$ be a quasi-coherent sheaf on $X$. There is a canonical homomorphism
$M \otimes i_* \mathcal{O}_Y \to i_* i^* M.$
Question: Is ...

**5**

votes

**2**answers

521 views

### Basics(?) about quasi-coherent modules on projective schemes

EDIT. (05-04-12) I have revised and improved the questions.
Let $A$ be a commutative $\mathbb{N}$-graded $R$-algebra, which is finitely generated by $A_1$ as an $A_0$-algebra. You may also assume ...

**2**

votes

**1**answer

236 views

### Is the pushforward via a proper map of a finite presentation module of finite presentation?

It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are ...

**3**

votes

**0**answers

221 views

### Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?

(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...

**1**

vote

**1**answer

257 views

### Flatness on the formal plane from flatness on lines through the origin?

Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...

**3**

votes

**0**answers

772 views

### quasi-coherent modules outside algebraic geometry?

Let $X$ be a ringed space. A quasi-coherent module on $X$ is a module which has locally a presentation, i.e. locally on $X$, it is the cokernel of a map between free modules. If $X$ is a scheme, then ...

**19**

votes

**2**answers

2k views

### What is the center of Qcoh(X)?

The center of a category $C$ is the monoid $Z(C)=\mathrm{End}_{[C,C]}(\mathrm{id}_C)$. Thus it consists of all families of endomorphisms $M \to M$ of objects $M \in C$, such that for every morphism $M ...