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Enough injectives in the category of quasi-coherent sheaves on a stack

For a scheme $X$, I have a reference - https://stacks.math.columbia.edu/tag/077P - that says there are enough injectives in the category $\text{QCoh}(X)$. I am looking for a reference that says the ...
Arkadij's user avatar
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1 vote
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214 views

Vakil's Generalization of qcqs Lemma

(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma) In the most recent notes of Vakil, this is problem 15....
Teddy's user avatar
  • 29
3 votes
2 answers
349 views

Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated

Closely related to this question in MSE, but the difference is that we will set $X$ to be scheme and $\mathcal{F}$ to be quasi-coherent. Let $X$ be a locally ringed space. We say an $\mathcal{O}_X$-...
Z Wu's user avatar
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4 votes
0 answers
175 views

Confusion in identification of quasicoherent sheaves on BG and G -representations

I asked this question on MSE a few days ago, but didn't get a response and also managed to confuse a senior colleague with it since then. This is probably a stupid question, so please bear with me. ...
Sergey Guminov's user avatar
2 votes
1 answer
224 views

Is any "relative support" for (complexes of) quasi-coherent sheaves known?

Let $f:X\to S$ be a morphism of Noetherian schemes; in the case I am interested in $S=\operatorname{Spec}R$ is affine and $f$ is proper. For a complex $C$ a complex of quasi-coherent sheaves on $X$ I ...
Mikhail Bondarko's user avatar
1 vote
0 answers
217 views

Is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules quasi-coherent on a complex manifold?

On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there ...
Zhaoting Wei's user avatar
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1 vote
0 answers
46 views

Pullback of coherent sheaves on Stein manifolds

Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
Doug Liu's user avatar
  • 525
2 votes
0 answers
64 views

Relative version of "On a Noetherian scheme, every quasi-coherent module is the filtered colimit of its coherent submodules"

Also on MSE. 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 ...
Display Name's user avatar
1 vote
1 answer
184 views

flatness of restriction of structure sheaf over ring of global sections

Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$. But I want to prove it only by knowing the definition of structure sheaf ...
Hamed Khalilian's user avatar
2 votes
1 answer
433 views

Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms?

Let $f:X\to Y$ be an affine morphism of locally Noetherian schemes. By this, we know that $Rf_*\mathcal{F}=f_*\mathcal{F}$ for any quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules (the ...
Stabilo's user avatar
  • 1,479
6 votes
1 answer
398 views

Is Qcoh(X) locally presentable?

Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (...
PresentableQCoh's user avatar
4 votes
2 answers
821 views

Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
clarkkent's user avatar
  • 121
3 votes
0 answers
231 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 ...
user267839's user avatar
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5 votes
1 answer
706 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.
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5 votes
2 answers
376 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 ...
user avatar
16 votes
1 answer
1k 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 $\mathcal{E} \to \mathcal{E}'$?...
Ingo Blechschmidt's user avatar
7 votes
0 answers
356 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 $\mathcal{O}_X$-...
Martin Brandenburg's user avatar
4 votes
0 answers
227 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 $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (...
Martin Brandenburg's user avatar
9 votes
1 answer
322 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 ...
Ingo Blechschmidt's user avatar
1 vote
0 answers
384 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 ...
Anoop singh's user avatar
2 votes
0 answers
152 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 $M$...
HeinrichD's user avatar
  • 5,402
6 votes
1 answer
1k 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)...
Pavel Čoupek's user avatar
1 vote
0 answers
119 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 ...
derived101's user avatar
5 votes
1 answer
445 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 ...
bbnkttp's user avatar
  • 161
2 votes
1 answer
513 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 ...
stupidq75's user avatar
3 votes
0 answers
401 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)}(\...
Zhaoting Wei's user avatar
  • 8,727
4 votes
0 answers
169 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 $...
José Navarro's user avatar
8 votes
1 answer
2k 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 ...
John Salvatierrez's user avatar
9 votes
0 answers
633 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 ...
Martin Brandenburg's user avatar
3 votes
1 answer
652 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 ...
Martin Brandenburg's user avatar
5 votes
2 answers
588 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 ...
Martin Brandenburg's user avatar
2 votes
1 answer
332 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 ...
Yosemite Sam's user avatar
  • 1,869
3 votes
0 answers
236 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}...
Ben Webster's user avatar
  • 44.1k
1 vote
1 answer
268 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 ...
Ben Webster's user avatar
  • 44.1k
6 votes
0 answers
1k 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 ...
Martin Brandenburg's user avatar
21 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 ...
Martin Brandenburg's user avatar