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."
Best Answer
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.