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A scheme is defined to be a sheaf which is locally isomorphic to the spectrum of a ring. The idea behind this is that given an affine coordinate ring of a variety over an algebraically closed field, we can recover the variety, i.e. the geometric object, by looking at the maximal ideals of this affine coordinate ring. Including the prime ideals (which add in the irreducible subvarieties), we get the notion of a scheme, which is something which is gotten essentially from the spectrum of a ring.

Another way to recover a variety from the algebra associated with it is to consider the valuations of its function field. Specifically, the points of a non-singular complete variety correspond to the valuations on the function field of the variety. We can actually define the variety as the set of valuations. If $K$ is the field and $v(K)$ denotes the set of valuations on $K$, then we declare $\{v \in v(K) \mid v(x) > 0\}$ for each $x \in K$ to be closed, giving a topology on the set of valuations. Finally, we can define the local ring at each point to be the valuation ring for that valuation. My question is, what if, instead of looking at spectra of rings, we defined a new object, which is locally the set of valuations of a field? For Dedekind rings, these seems to give something similar to the spectrum of the given Dedekind domain. Is this interesting in other contexts? Can one gain something by looking at it from this perspective?

Edit: Although valuations do not give varieties up to isomorphism, our new object could still be something along the lines of "variety up to birational equivalence."

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Is this something like what you mean? sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i –  Qiaochu Yuan Jul 21 '10 at 20:31
    
This seems similar. Also, is this the way in which the "spectrum" from algebraic geometry connects to the "spectrum" from functional analysis? –  David Corwin Jul 21 '10 at 20:49
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David, although Grothendieck hated valuation theory (and then had to concede its usefulness when Serre pointed him in the direction of valuative criteria) and thought that Tate's dream of $p$-adic analytic spaces was ridiculous (until Tate showed what he could do with it), the role of valuation-theoretic methods in algebraic geometry has undergone somewhat of a reivival in recent years. No way does it replace schemes, but these crazy spaces (and their quasi-compactness) are a powerful tool for some constructions in the spirit of Zariski. See my comment on the rabbit's answer below. –  BCnrd Jul 21 '10 at 22:42
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There are some inaccuracies in the question. 1. A scheme is not a sheaf. It is a topological space together with a sheaf of rings (such that...). 2. We cannot actually define a variety of dimension greater than one as the set of valuations of $K/k$: this ignores the existence of birationally equivalent, but nonisomorphic, smooth projective varieties. –  Pete L. Clark Jul 22 '10 at 5:54
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C'mon folks, a big plus of Grothendieck topologies and topoi is that we can "identify" geometric objects with functors they represent, and then ask if those functors are fppf sheaves. Anyone who's ever said that a functor on schemes is "represented by an algebraic space" instead of "is an algebraic space", or "identifies" a scheme with the fppf or etale sheaf functor it represents (e.g., when using finite flat commutative group schemes, doing descent theory, using constr. etale sheaves, etc.) has engaged in such abuse of terminology. This is completely standard, and convenient. No problem. –  BCnrd Jul 22 '10 at 15:07
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This is an old approach to finding models for varieties, introduced by Zariski in 1944 in his work on resolution of singularities. (See "The compactness of the Riemann manifold of an abstract field of algebraic functions", Bulletin of the American Mathematical Society 50: 683–691, doi:10.1090/S0002-9904-1944-08206-2, MR0011573) He defined a Zariski topology on a space of valuations, which seems to have inspired Grothendieck's definition of Zariski topology on a scheme. Much of Zariski's work on these spaces is rather similar to Grothedieck's work on the foundations of schemes. Zariski called the space of valuations the "Riemann manifold" of a variety, though it is now called the Zariski-Riemann space.

Volume 2, chapter VI section 17 of Zariski and Samuel's book on commutative algebra gives more details.

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I should add that in Elliptic Curves by Alain Robert, he says that the set of valuations "is going to play the role of 'spectrum' of the field K(V)." This book was written in 1972. –  David Corwin Jul 29 '10 at 17:33
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This "space of valuations" to me sounds like the Riemann-Zariski space. Generally this should be pretty nasty, for surfaces eg it is some kind of limit of the system of all possible blow-ups. It is an old idea.

For a different direction towards compactifying $Spec(\mathbb{Z})$, you can read about Arakelov theory and look at the recent papers of Connes and Consani on the arXiv about geometry over the "field" with one element.

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The underlying spaces of schemes are also nasty if one stares at them too closely, but with experience they're not so bad. Ditto for Riemann-Zariski spaces, so they shouldn't be regarded as "pretty nasty". In fact, Berkovich spaces (and adic spaces) are like this too: also spaces of valuations on a ring (in the affinoid case), but with practice one gets accustomed to them by putting on foggy glasses. As for the recommendation in a "different direction", I recommend Borger's viewpoint. –  BCnrd Jul 22 '10 at 1:59
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+1 for recommending Borger's viewpoint. –  Chandan Singh Dalawat Jul 22 '10 at 3:36
    
Thank you Brian, you are absolutely right. I should have said "the topology can be different from what we are used to on schemes" instead of "nasty." Also thank you for mentioning Borger. –  Eugene Eisenstein Jul 25 '10 at 23:22
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This perspective is given in C. Chevalley, "Introduction to the theory of algebraic functions in one variable". There instead of considering smooth algebraic curves over the complex numbers(or Riemann surfaces), the author considers simply considers the function field. The author also constructs the curve with the topology from this description, as you mention in your post.

In general for other commutative rings, for example for the ring $\mathbb Z$, considering the valuations instead of the ideals is the perspective of Arakelov theory. I do not know anything on that subject; but I have seen it mentioned in Neukirch, "Algebraic Number Theory". The objective seems to be treating $\mathbb Z$ as a compactified projective curve, and the archimedean valuations play the role of the points at infinity. I have a recollection that that the theory of the infinite place is achieved by attaching a hermitian bundle to the geometric object we are considering. Perhaps you can read more in Serge Lang, "Introduction to Arakelov Theory", or Faltings, "Lectures on the arithmetic Riemann Roch theorem".

This theory seems to have been used in Faltings' proof of the Mordell conjecture. I am not aware of other applications. Perhaps experts can say more on this.

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You mean treating Spec Z as a compactified projective curve. –  Qiaochu Yuan Jul 21 '10 at 21:26
    
Yes, if you want it phrased thus. –  Anweshi Jul 22 '10 at 10:48
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You cannot define the variety as the set of valuations. Birational varieties will have the same set of valuations.

As far as I remember, logicians were looking at it for a while. If you have spare 250 bucks, you can learn something there.

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Fine, then our new object would be like a variety up to birational equivalence. –  David Corwin Jul 21 '10 at 20:45
    
Yeah, this is probably right, Doc. –  Bugs Bunny Jul 21 '10 at 21:07
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Bugs, there are more clever/useful ways to define a "Riemann-Zariski space" attached to a scheme, involving structure beyond function fields. Consider Michael Temkin's very creative use of a scheme-like version of Riemann-Zariski spaces in his recent work on semistable curve fibrations and higher-dimensional analogues. So the answer is a definite "yes" to the question of whether spaces of valuations can be useful in non-birational modern algebraic geometry; moreover, it has nothing to do with logic stuff. Grothendieck would spin in his grave over this, except he's still alive (it seems). –  BCnrd Jul 21 '10 at 22:38
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In one of the more extreme topologies defined by Voevodsky -- the $h$-topology -- the valuation rings whose fraction fields are algebraically closed play the role of local rings. This observation was a little useful to me once, but I don't know if it has ever been useful to anybody else. –  Tom Goodwillie Jul 22 '10 at 1:05
    
If $R$ is a normal subring of a field $K$, it is common to study $R$ by considering the set of valuations that are nonnegative on $R$. This will distinguish different subrings of $K$, so you can get more information than just the birational class. As far as I know, there is no difficulty extending this construction to non-affine normal schemes, but I don't know a reference for that. –  David Speyer Jul 22 '10 at 17:52
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