Valuation rings can be defined in a variety of equivalent ways. They are integral domains:
- with a valuation to a totally ordered abelian group.
- where the set of ideals is totally ordered with respect to inclusion.
- where for every in the fraction field, either or (or both) lies in the ring.
- They are local rings that are maximal under the notion of domination with respect to local rings contained in a fixed field. We say that an inclusion is a domination if the maximal ideal of pulls back to that of .
It is a simple and useful exercise to show that these are equivalent and the stacks project does a fine job of it. Personally, I find it easiest and most useful to use definitions 1+2+3. The main lemma about valuation rings is:
Main Lemma: Suppose is a local ring contained in a field. Then there exists a valuation ring that dominates .
Before we move on to the proof, let me briefly explain why it’s important. This lemma is more or less equivalent to saying that on any scheme , a specialization can be realized by a map with a valuation ring whose maximal ideal maps to and generic point maps to .
This is, for instance, how valuation rings show up in the valuative criterion of properness or the arc topology of Bhargav-Mathew.