# Field theory and valuations of codimension 2?

1. May 6, 2009

### Hurkyl

Staff Emeritus
When studying a field, might be interested in the discrete valuation rings (DVR). In the case of a global field, one can naturally (in some sense) think of the field as a curve, and the DVR's as corresponding to the points on the curve. To each DVR there is a corresponding discrete valuation on the field, and a corresponding non-Archmedian absolute value. One can generalize discrete valuation (respectively non-Archmedian absolute value) to valuation (resp. absolute value), and when such things exist - e.g. for number fields -- they are very important to have.

It strikes me that I've never really seen the codimension 2 case mentioned from the field perspective, even in passing, e.g. for the fields k(x,y) or Q(x). Certainly these still have discrete valuations which correspond to the irreducible curves lying in the projective plane over k, or in $spec \, \mathbb{Z}[x]$, but what about the points? Obviously, we want to consider the regular local rings of dimension 2 contained in the field, but what is the right notion corresponding to "valuation" or "absolute value", and what would the Archmedian ones look like?