Places and schemes

Question

In my algebraic number theory, we're currently studying local fields, and we've noted that given a number field $K$, its normalized discrete valuations correspond exactly to primes of $O_K$, and we are supposed to think of its archimedean absolute values as "primes at infinity". This terminology suggests that the set of places (equivalence classes of absolute values modulo homeomorphism) is a sort of projectivization or compactification of $\operatorname{Spec}(O_K)$. Is there a natural geometric structure, such as the structure of a scheme or locally ringed space, that can be put on this set? If so, what is its relation to the affine scheme of $O_K$? Finally, do we get something similar when we extend to arbitrary global fields, or even to function fields of complex algebraic curves? Any information or references would be appreciated.

Answer 1

The global fields beside algebraic number fields are function fields $F$ of transcendence degree $1$ over a finite field $K$. In this case every valuation of $F$ is trivial on $K$ and therefore discrete. The set $S$ of all these valuations can be equipped with the cofinite topology and becomes a scheme by defining the sheaf of rings on $S$ through

$ \mathcal{O}_S(U):=\bigcap\limits_{v\in U}O_v $

for any open set $U\subseteq S$. Here $O_v$ denotes the valuation ring of the valuation $v$. The scheme $(S,\mathcal{O}_S)$ is then a projective integral algebraic curve without singularities and $F/K$ is its function field.

Considering function fields $F/\mathbb{C}$ a serious problem arises: by a rseult of E. Weiss the absolute value function on $\mathbb{C}$ cannot be extended to $F$.