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The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" question may be (rather then stating that the answer is 42;))

Is there a purely categorical procedure that takes the category of commutative rings as input and produces the category of schemes (over $\mathbf{Z}$) as output?

A possible place to start would be to consider a scheme $X$ as a functor from the category $CommRing$ of commutative rings to the category of sets: $A\mapsto Hom_{Sch}(Spec(A),X)$ where $A$ a commutative ring. If we instead of $Spec(A)$'s we consider all schemes, then we simply get the Yoneda embedding. But some questions arise.

  1. Does this give a fully faithful functor from schemes to functors from commutative rings to sets? Or loosely speaking, do $Spec(A)$-valued points ($A$ a commutative ring) suffice to determine a scheme? (My guess is that the answer is yes and this is classical.)

  2. Is there a way to characterize those functors that actually come from schemes? For example one can introduce a Grothendieck topology on $CommRing$ (or its opposite) and require that the functor should be a sheaf in that topology. But in that case, can one describe the topology without referring to the fact that the objects of $CommRing$ are commutative rings? (Here my guess is the first question is probably too complicated but there are some necessary conditions.)

  3. Regardless whether the answer to 2. is positive or negative, is there a way to describe algebraic spaces or stacks as presheaves on $CommRing^{op}$ that satisfy some conditions?

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A very satisfactory answer to #3, under mild finiteness hypotheses, is Artin's criteria (see his paper "Versal deformations & algebraic stacks", & his earlier stuff on alg. spaces), coupled with "converse" results of Olsson, and Artin's results on remarkable stability properties of these kinds of objects (preservation under fppf quotients, contraction results, etc.): these imply the concepts of alg. space/stack can be checked in interesting abstract settings, unlike for schemes, and that there's natural sense in which these concepts need no further generalization for "geometric" purposes. –  BCnrd May 31 '10 at 0:22
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Thanks, BCnrd! This and your other comments were very helpful. Would you consider posting them as an answer? –  algori May 31 '10 at 15:57
    
@algori: Glad to be of help. I'll take a pass on reposting. –  BCnrd May 31 '10 at 18:43
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6 Answers

  1. The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of affine schemes $Aff = CommRing^{op}$ into $Sch$; the topology induced on $Aff$ by that on $Sch$ is also the Zariski topology. The comparison lemma ([SGA4] III, 4.1) then says that, because any object in $Sch$ can be covered by objects in $Aff$, the categories of sheaves on both sites are equivalent. In particular, representable sheaves in $Sch$ (i.e., schemes) are determined by their values on affine schemes.
  2. For a sheaf $F$ on $Aff$ to be represented by a scheme it is enough that it be covered by affine schemes, i.e., that there exist affine schemes $U_i$ together with open immersions $U_i \to F$ (you have to define what this means, of course) such that $\coprod_i h_{U_i} \to F$ is an epimorphism of sheaves. Actually, you can take this as a definition of schemes. The compatibility of the gluings in the classical definition is taken care of here by the sheaf condition.
  3. Algebraic spaces can be similarly defined. While I was writing this, Harry beat me to giving the reference to the excellent notes of Bertrand Toën from a course of his on algebraic stacks.

In 2, you also ask if you can construct schemes from $Aff$ without actually using the fact that you are dealing with commutative rings. I think not. The categorical nonsense can get you only so far: at some point you have to introduce the geometry itself, and that is given by the $Aff$ with its topology. If you replace $Aff$ by the category of open sets in some $\mathbb{R}^n$ with open immersions you would end up defining manifolds. This is what Toën calls geometric contexts.

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+1 for good taste! –  Harry Gindi May 30 '10 at 23:08
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Thanks, Alberto! Yes, I was hoping there is a procedure which would extract the Zariski topology (or other) from $Aff$ just by looking at it as a category. But maybe this is too much to hope for. –  algori May 31 '10 at 16:00
    
@algori: What do you mean by "just looking at the category"? –  Harry Gindi May 31 '10 at 16:59
    
Harry -- I was thinking that maybe there is a condition on the category that would imply the existence of a preferred topology on it and which will pick the, say Zariski topology when applied to $Aff$. Then it would be interesting to see to which other categories this is applicable and what the result would be. –  algori May 31 '10 at 18:27
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Toen's notes on stacks construct the category of schemes as the category of etale sheaves (presheaves satisfying descent in the etale topology) on CRing^op with a jointly surjective cover by smooth monomorphisms (exercise: show that smooth monomorphisms of affines are etale) of representable functors (i.e. affines).

http://www.math.univ-toulouse.fr/~toen/m2.html

He constructs algebraic spaces in a similar way, then constructs algebraic stacks using the same approach after a digression into homotopical descent theory (which generalizes readily to the approach taken in Toen-Vezzosi (Homotopical Algebraic Geometry).

The case of schemes is given a more general treatment in a fixed "geometric context", which is a category with a grothendieck topology and a fixed class of morphisms compatible with it. A scheme is then simply a "geometric variety" in the "algebro-geometric context", which is CRing^op equipped with the etale topology, where the fixed class of morphisms is the class of smooth morphisms of affines (morphisms corresponding to smooth morphisms in CRing).

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But this is just dressing up the "old-fashioned" ringed space definition in fancy language, so is there any purpose to it beyond obsfucation of the geometric origins of the subject? –  BCnrd May 30 '10 at 22:53
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Harry, I won't pass judgement on HAG since I know nothing about it, but I am suspicious of value of def'ns which don't allow to do something hard to express without them. For alg. space/stacks, the "proof of concept" is seen geometrically by accepting schemes as known and considering functors or fibered categories which admit "covering" by scheme for suitable topology: similar to atlas def'n of manifold. I don't know any advantage for alg. spaces/stacks which is attained by less geometric def'ns; for many serious foundational proofs one reduces to thms for (non-affine) schemes anyway. –  BCnrd May 31 '10 at 0:00
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Well, at least in the context of HAG (the tiny part of it that I've read), one doesn't have the option of using actual geometric things like locally ringed spaces, since one is not working with CRings. One instead works with an arbitrary symmetric monoidal model category, which forces one to define things like flatness, faithfulness, smoothness, unramifiedness, finite presentation, etc. directly by their functorial properties. I'm no expert, and I don't know about the value of it to algebraic geometers, so I'll defer to your expertise on that question. –  Harry Gindi May 31 '10 at 0:15
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@BCnrd, I am certainly not an expert either, but, I don't think Toen's definition is meant to REPLACE that of a scheme, but rather, translate it into a different language- that of category theory? Why? Because you can then "categorify it"- so you can, in some sense, talk about "schemes up to homotopy". Also, as Harry points out with Geometric contexts, since category theory is a universal language, such a definition allows you to extend ideas of algebraic geometry to other fields (e.g. topology and differential geometry). –  David Carchedi May 31 '10 at 0:17
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@BCnrd: In HAG II, T-V try out the theory with a notion of complicial algebraic geometry, where SSets are replaced with Complicial sets, and Brave New Algebraic geometry, where SSets are replaced with spectra (in the algebraic topology sense). These are (according to T-V) very interesting and very different from the ordinary and simplicial (derived) cases. –  Harry Gindi May 31 '10 at 0:41
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If $B$ is a site, then the presheaf category $\hat{B} = Set^{B^{op}}$ is also a site and the topology restricts to the one on $B$. See this related question.

If we take $B$ to be the dual of the category of rings with the Zariski topology ($R_i \to R$ is a covering iff the ring morphisms $R \to R_i$ are localizations at elements $f_i \in R$ such that the $f_i$ generate the unit ideal), consider the full subcategory of $\hat{B}$ consisting of all sheaves which are covered by representable functors, then we get the category of schemes with the Zariski topology.

Also if $B$ is the category of open subsets in some euclidean space, you can produce the category of manifolds.

Basically, this constructions allows you to construct global objects using local models.

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This isn't really a "characterization" of functors represented by schemes: as you know, it's just the usual definition of a scheme recast in more categorical language. The answer to question #2 is, sadly, "no" (as far as anyone knows). The raison d'etre for the theory of algebraic spaces is that there a characterization really is possible (assuming mild finiteness hypotheses, say); see my comments on David's answer above. –  BCnrd May 31 '10 at 12:42
    
? "This isn't really a "characterization" of functors represented by schemes: as you know, it's just the usual definition of a scheme recast in more categorical language." Yeah, and that was the question. And it does provide a construction without locally ringed spaces. –  Martin Brandenburg May 31 '10 at 13:05
    
This is a nice answer to part 1 of the question, but I think BCnrd is pointing out that this doesn't answer part 2, since this answer takes "the Zariski topology" as part of the data, presumably defined in the classical way, hence in the language of commutative rings. –  Peter LeFanu Lumsdaine May 31 '10 at 14:01
    
@Martin: when one asks for a "characterization" of a property it is desired to give something not trivially equivalent to the initial definition (cf. Peter's comment). Could choose broader intent, but of limited use (e.g., is your "characterization" useful to represent Hilbert functors?). Was a huge problem in 1960's to give a characterization of functors represented by schemes. Grothendieck discovered necessary conditions (fpqc descent, algebraization, etc.), essential in Artin's solution via alg. spaces (e.g., new representability proofs for Hilb, Pic, etc., beyond Grothendieck's results). –  BCnrd May 31 '10 at 14:45
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The answer to the main question is undoubtedly "yes". I think there are going to be two lots of literature about this, one coming direct from the Grothendieck school (probably somewhere in the Demazure-Gabriel book), and another from the category-theorists, where I remember some work of Cole that puts the construction in a more general context. (I'm lying when I say "I remember"; I was certainly told about this at one point and filed away the information mentally.) I think the answer to Q1 is "yes and foundational", to Q2 is that representable functors are so well studied that the information is available, but being a scheme is rather subtle in practical terms. The connection with the flat topology here is well known.

As for Q3, even more out of my depth here.

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The answer that comes as "classifying topos of local rings" will doubtless be resisted by geometers; but see Mac Lane-Moerdijk Ch. 8 for all that. What I was alluding to can be traced in Johnstone, Topos Theory, from the index entry on "local ring". Basically adding in "with local ring homomorphisms", in passing from commutative rings to commutative local rings, is a step with general categorical meaning. But the work of Julian Cole referenced there seems ultimately unpublished. (Be careful what you wish for in the "purely categorical"!) Please can we have more expositions of SGA? –  Charles Matthews May 31 '10 at 9:48
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The answer to 1. is yes. The answer to 2.) is also yes. Look here: http://rigtriv.wordpress.com/2008/07/16/representable-functors/

For 3.) A "strong" algebraic (Artin) stack is a "geometric stack" on the category of affine schemes=opposite category of commutative rings equipped with the etale topology. http://ncatlab.org/nlab/show/geometric+stack. In other words, it a pseudofunctor (or fibred category) obtained by stackifying a strict 2-functor of the form $Hom(blank,G)$, where $G$ is a groupoid object in schemes (so technically to view these as geometric stacks, you should take your site to be all schemes, not just affine ones). There are some subtleties however:

*You need that the source and target map of the groupoid $G$ are smooth maps of schemes.

** Often, Artin stack means instead the stack associated to a groupoid object in algebraic spaces, rather than schemes.

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The link for #2 is disguised version of def'n of scheme. Artin gave non-tautological criteria on abstract functor which imply it's an alg. space (so can do "geometry" with it), whereas "criterion" in link amounts to constructing the scheme, so no deeper than Yoneda (i.e., useful but linguistics). There's no deep useful abstract criterion for representability by a scheme (except for trick with alg. spaces). It's a miracle that mild generalization to alg. spaces admits real sol'n to the problem. I find n-Lab impenetrable; is it's version of #3 better than "old-fashioned" definition of Artin? –  BCnrd May 30 '10 at 23:25
    
Well, I never claimed that #2 was that deep. It was merely a positive answer to the question "Is there a way to characterize those functors that actually come from schemes". My "#3" answer is equivalent to that of Artin's, just more compact. If you are given an atlas $X \to \mathbf{X}$ of an algebraic stack, then, the groupoid $G$ is $X \times_{\mathbf{X}} X \rightrightarrows X$. Conversely, given $G$, $G_0 \to G$ gives an atlas $G_0 \to St(Hom(blank,G))$. –  David Carchedi May 30 '10 at 23:56
    
@BCnrd: As for #2, now I see what you mean. I guess, it technically is an answer, but, maybe not one that "really" has something to do with the functor. A more intrinsic one would be preferable, I agree. As for #3: See userpage.fu-berlin.de/~nhoffman/allahabad.pdf Definition 3.6. They define Artin stacks the same way as me. BY the way, weak pullback of "two copies of" $G_0 \to G$ for any groupoid scheme is $G_1$- the scheme of arrows. Stackification commutes with finite limits, so this implies that $G_0 \times_{BG} G_0$ is $G_1$, hence a scheme. –  David Carchedi May 31 '10 at 0:27
    
@BCnrd: Aha! So, I guess there's two variants of Artin stacks in the literature and the one that I've described is slightly more strong. What you are saying amounts to just asking for groupoid objects in algebraic spaces, rather than schemes. I'll EDIT. –  David Carchedi May 31 '10 at 1:13
    
Thanks, David! Re the n-cat cafe: the material is organized there so that every page cites every other page, which makes it difficult (but not impossible) to get anything out of it. –  algori May 31 '10 at 17:01
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