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Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (Schemes) on $CRing^{op}$. That is, taking the full subcategory of $Psh(CRing^{op})$ which consists of objects $S$ such that $S$ is a sheaf in the global Zariski topology and $S$ has a cover by representables in the induced topology on $Psh(CRing^{op})$. This is the category of schemes. Then, one takes this category and equips it with the etale topology and repeats the construction of locally representable sheaves on this site (Sch with the etale topology) to get the category of algebraic spaces.

Can we "skip" the category of schemes entirely by putting a different topology on $CRing^{op}$?

My intuition is that since every scheme can be covered by affines, and every algebraic space can be covered by schemes, we can cut out the middle-man and just define algebraic spaces as locally representable sheaves for the global etale topology on $CRing^{op}$. If this ends up being the case, is there any sort of interesting further generalization before stacks, perhaps taking locally representable sheaves in a flat Zariski-friendly topology like fppf or fpqc?

Some motivation: In algebraic geometry, all of our data comes from commutative rings in a functorial way (intentionally vague). All of the grothendieck topologies with nice notions of descent used in Algebraic geometry can be expressed in terms of commutative rings, e.g., the algebraic and geometric forms of Zariski's Main theorem are equivalent, we can describe etale morphisms in terms of etale ring maps, et cetera. What I'm trying to see is whether or not we can really express all of algebraic geometry as "left-handed commutative algebra + sheaves (including higher sheaves like stacks)". The functor of points approach for schemes validates this intuition in the simplest case, but does it actually generalize further?

The main question is italicized, but feel free to tell me if I've incorrectly characterized something in the motivation or the background.

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There's an article by Kai Behrend on "Localization and Gromov-Witten invariants" where he introduces algebraic stacks and algebraic spaces in the way you want I think, though given Jim's answer doesn't seem to be what you wanted, maybe you could clarify your question a little? –  Kevin McGerty Jan 9 '10 at 19:52
    
A book I'm reading now, Donald Knutson's "Algebraic Spaces", could be another useful reference. –  Vinoth Jan 10 '10 at 1:14
    
The standard approach uses etale equivalence relations, while the functor of points approach applies a general step with the data of representables and a grothendieck topology. We could call this single general step taking the "locally representable sheaves on a category". –  Harry Gindi Jan 10 '10 at 3:37
    
@unknown(google): Sure, if you think that being pedantic about terminology might answer the question, then by all means, go ahead.. –  Harry Gindi Jan 8 '11 at 19:30

2 Answers 2

up vote 22 down vote accepted

Yes. The category of algebraic spaces is the smallest subcategory of the category of sheaves of sets on Aff, the opposite of the category of rings, under the etale topology which (1) contains Aff, (2) is closed under formation of quotients by etale equivalence relations, and (3) is closed under disjoint unions (indexed by arbitrary sets). An abstract context for such things is written down in "Algebraization of complex analytic varieties and derived categories" by Toen and Vaquie, which is available on the archive. Toen also has notes from a "master course" on stacks on his web page with more information. It might be worth pointing out that their construction of this category also goes by a two-step procedure, although in their case it's a single construction performed iteratively (and which stabilizes after two steps). This is unlike the approach using scheme theory in the literal sense, as locally ringed topological spaces, where the two steps are completely different. After the first step in T-V, you get algebraic spaces with affine diagonal. Also worth pointing out is that their approach is completely sheaf theoretic. The only input you need is a category of local models, a Grothendieck topology, and a class of equivalence relations. You then get algebraic spaces from the triple (Aff, etale, etale). But the general machine (which incidentally I believe is not in its final form) has nothing to do with commutative rings. I think it would be interesting to plug opposites of other algebraic categories into it.

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This answer seems to me to answer the question you ask. Perhaps you can clarify your question a bit? –  Chris Schommer-Pries Jan 9 '10 at 20:28
    
I wasn't familiar with this "equivalence relation" viewpoint, but after looking it up, I'll accept the answer. The functor of points view is much simpler though. I assume that he means that the category of algebraic spaces are the locally affine sheaves over the etale topology on CommRing^op? –  Harry Gindi Jan 10 '10 at 3:35
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I'm not exactly sure what you mean. I would say that what I wrote is the functor of points point of view. The answer to your other question, in its most literal interpretation, no. The reason is that you also need to say something about how you glue the affine pieces together, i.e. about the equivalence relation. You could take the quotient of any affine scheme by some random equivalence relation, which would have no reason to be an algebraic space. cont'd –  JBorger Jan 10 '10 at 4:03
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For instance, on just about any affine scheme (e.g. the affine line), the colimit of all nilpotent neighborhoods of the diagonal is ind-algebraic but not algebraic. Therefore the quotient (the so-called de Rham space discussed recently on MO) is also not algebraic. Yet it is locally affine in the sense that it is covered by affine schemes, namely the original one you started with. –  JBorger Jan 10 '10 at 4:06
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OK, let me try again. The T-V approach just means this: an algebraic space with affine diagonal is the same as a sheaf X on Aff which is covered by affines U_i such that each U_ij := U_i \times_X U_j is affine and etale over U_i and U_j. A general algebraic space is the same except that you only require that each U_ij is an algebraic space with affine diagonal (instead of being affine). –  JBorger Jan 10 '10 at 4:31

A Deligne--Mumford stack is an étale-locally ringed topos that is locally equivalent to the étale-locally ringed topos of an affine scheme. A Deligne--Mumford stack is an algebraic space if its diagonal is an embedding.

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Would you humor me and tell me what that means in terms of functors of points? –  Harry Gindi Feb 7 '10 at 18:35
    
A Deligne-Mumford stack is locally affine stack in terms of functor point –  Shizhuo Zhang Feb 7 '10 at 21:55

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