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How do we define quasi-coherent sheaves on schemes?

Say we start by defining the category of affine schemes Aff as CRing$^{op}$ (the opposite category of unitary commutative rings). In this context we have an obvious way to define quasi-coherent sheaves:

A quasi-coherent sheaf on an affine scheme X=Spec A is just an A-module.

If we now define schemes as presheaves on Aff (satisfying some condition), how do we define what a quasi-coherent sheaf is? The same question applies also to the operations of pushforward and pullback, which in Aff have obvious definitions.

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Also, it would be a good thing to define directly quasi-coherent sheaves, without having to define them as sheaves satisfying some quasi-coherence property (thus eliminating sheaves of a non-algebraic nature). Furthermore, it would be great if the definition were "natural" enough so to extend obviously to algebraic stacks. –  babubba Feb 25 '10 at 18:13
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nice questions, actually, quasi-coherent sheaves are actually not sheaves but presheaves because we do not need to impose specialized grothendieck topology when define it. check out the paper by Orlov "quasi coherent sheaves in commutative and noncommutative geometry" and Kontsevich-Rosenberg "noncommutative stacks". You will get what you want –  Shizhuo Zhang Feb 25 '10 at 18:23

6 Answers 6

up vote 6 down vote accepted

The corresponding nlab page has several approaches to the definition of quasicoherent sheaves of O-modules including some in functor of points approach, in various degrees of abstractness. All these definitions while simultaneously applicable define equivalent categories. This works not only for (qcoh modules) over schemes but over more general functors *(e.g. stacks) on Aff. Look also for some quoted references there, including Orlov's paper mentioned by Zhang in the comment above.

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A stack is not a functor. –  David Zureick-Brown Feb 25 '10 at 19:55
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@David:Skoda used the "general"functor to mean stack. It is pseudo functor –  Shizhuo Zhang Feb 25 '10 at 20:09
    
Ok. I read it as more general functor than the functor of points of a scheme', but still a functor. But I guess saying more general' to mean `generalization of' is fine. –  David Zureick-Brown Feb 26 '10 at 0:35
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A quasicoherent sheaf of modules on a scheme is precisely a morphism of stacks (=transformation of 2-functors) on $CRing^{op}$ from the scheme to the stack of modules. More generally, a quasicoherent oo-stack of modules on an oo-scheme is a morphism of oo-stacks (transformation of oo-functors) from that oo-scheme to the oo-stack of oo-modules. Details are, as Zoran says, at that nLab entry. –  Urs Schreiber Mar 4 '10 at 20:23

You can define a quasicoherent sheaf on a functor $X : \mathrm{AffSch^{op}} \rightarrow \mathrm{Set}$ as a choice of a module $R$ module $F_x$ for every $x \in X(\mathrm{Spec}R)$ along with some compatibility isomorphisms. If $X$ is the functor of points of scheme, and $F$ is a an honest quasicoherent sheaf on this scheme, then $F_x$ is just the pullback to $\mathrm{Spec} R$ via the map $x$. The compatibility isomorphisms that we require are the ones that naturally arise from pseudo-functoriality of the pullback. The details are given in the second page of the following notes of a lecture by Jacob Lurie http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf.

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You can define the category of quasicoherent sheaves on a scheme or stack (or an arbitrary functor) as the limit over all affines Spec R mapping to the stack of the categories R-Mod. You will need some descent theorem however to compare this with a more usual definition. The same definition works for the ($\infty$-categorical refinement of) the derived category of quasicoherent sheaves. (I learned this from Toen's survey on Higher and Derived Stacks).

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I think what you mean is pseudo functor $F:CAff\rightarrow Cat$,where $Cat$ is 2-category of $R-mod$.Or, equivalently a fibered category $\mathfrak{G}=$($\mathfrak{F},\mathfrak{F}\rightarrow \mathfrak{E})$,where $\mathfrak{E}$ is $CAff$. Then you take the $Lim\mathfrak{G}$,right? Yes,this definition can be generalized to arbitrary functor and even category over category. It is equivalent to the definition of Kontsevich-Rosenberg(there, they use $(Lim\mathfrak{G})^{op}$,it is the cartesion section of the fibered category. This definition can also applied to triangulated category. –  Shizhuo Zhang Feb 26 '10 at 9:06
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This POV is also used in Beilinson-Drinfeld. They consider cartesion section of fibered category of derived category of D-modules over affine schemes as definition of derived category of quasicoherent D-modules(If I understand correctly) –  Shizhuo Zhang Feb 26 '10 at 9:14
    
I would be very surprised if this works for triangulated categories - e.g. descent certainly doesn't. Usually you enhance them in some way (dg, $A_\infty$, stable $\infty$) to get such operations to work (which is eg what Beilinson-Drinfeld do). –  David Ben-Zvi Feb 26 '10 at 12:28
    
Regarding "functor", I was thinking in the -context where I literally do mean functor (ie the limit is taken in the ($\infty$,1)−category of (presentable, stable) ($\infty$,1)−categories)− sorry if I abuse terminology in the non−$\infty$ setting. –  David Ben-Zvi Feb 26 '10 at 12:33
    
@Ben-Zvi: I might not understand what you mean by "descent"doesn't works in triangulated category(without enhancement). Actually, for Karoubian triangulated categories, we have Barr-Beck's theorem. The proof is lending this story via Verdier abelianization to abelian categories, using Beck's theorem there and go back. –  Shizhuo Zhang Feb 26 '10 at 15:43

For the non-experts out there, I've found this great expository article by Gomez http://arxiv.org/PS_cache/math/pdf/9911/9911199v1.pdf

Quasi-coherent sheaves are defined in Definition 2.45 (the same way Dinakar and Yuhao defined them)

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The definition in EGA I of a quasi-coherent sheaf is a little different, they define a quasi-coherent sheaf of modules as a sheaf of modules which locally has a presentation. You can then prove that on an affine scheme there exists a global presentation and use this to recover the `M-tilde' definition. This generalizes to any site (and in particular to algebraic stacks, where it is equivalent to something in the spirit of Dinakar's answer); check the stacks project notes for more.

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Thanks for reminding me of the stacks project, I tend to forget it. I'm aware of the definition in EGA0. Still, it would be nice to have a definition which didn't stop to define arbitrary sheaves. –  babubba Feb 25 '10 at 19:34
    
I'm still confused though: if I want to define (quasi-coherent) sheaves on an algebraic stack X, what is the site to which I should apply the machinery you talk about? Is it the comma category AlgSt Aff/X ? Or the category of open embeddings into X? Something else? –  babubba Feb 25 '10 at 19:34
    
For a Deligne-Mumford stack, you want the etale site. For an Artin stack, you want the lisse-etale site. You have to think a bit about what `locally has a presentation' means here. –  David Zureick-Brown Feb 25 '10 at 19:54

Seems like people answered this question with many high-brow point of views. But I think it is still worth mentioning that given a Functor F from Schemes^{op} to Sets, a down to earth way to talk about it is to give a q.coh. sheaf on F is equivalent to give a q.coh. sheaf on X for any element of F(X), in a compatible way such that pull-backs give compatible data and some cocycle conditions are satisfied.

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Oh, seems like this is the same as what Dinakar Muthiah said above. –  Yuhao Huang Feb 26 '10 at 6:39
    
Hi Hao! I ever asked related question here: mathoverflow.net/questions/15223/… I am now working a little bit on checking all the definitions of qcoh.(high-brow POV and down-to-earth POV) to see whether they are exactly the same –  Shizhuo Zhang Feb 26 '10 at 9:21

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