Quasi-coherent sheaves in the Functor-of-points approach - MathOverflow

Quasi-coherent sheaves in the Functor-of-points approach

2010-02-25T18:12:17Z

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.

Answer by Dinakar Muthiah for Quasi-coherent sheaves in the Functor-of-points approach

2010-02-25T19:06:12Z

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.

Answer by David Zureick-Brown for Quasi-coherent sheaves in the Functor-of-points approach

2010-02-25T19:08:30Z

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.

Answer by Zoran Škoda for Quasi-coherent sheaves in the Functor-of-points approach

2010-02-25T19:14:16Z

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.

Answer by David Ben-Zvi for Quasi-coherent sheaves in the Functor-of-points approach

2010-02-26T03:27:34Z

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).

Answer by Yuhao Huang for Quasi-coherent sheaves in the Functor-of-points approach

2010-02-26T06:37:52Z

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.

Answer by babubba for Quasi-coherent sheaves in the Functor-of-points approach

2010-03-08T22:44:31Z

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)