Replacing Spectrum with Valuations of a Field - An Alternative to Schemes? - MathOverflow

Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

2010-07-21T20:20:28Z

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."

Answer by Bugs Bunny for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

2010-07-21T20:41:28Z

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.

Answer by Anweshi for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

2010-07-21T21:02:05Z

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.

Answer by Richard Borcherds for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

2010-07-21T21:09:02Z

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.

Answer by Eugene Eisenstein for Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

2010-07-21T21:14:20Z

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.