I want to talk a bit about valuation rings. As a general topic, valuation rings are rather new to me, so hopefully my explanations go smoothly. The only prior knowledge I will assume is that of local rings and some elementary facts about integral elements.
To begin, let be a field. Then a valuation on is a map such that the following hold for all :
Notice that the first condition means that is a group homomorphism from the multiplicative group to the additive group , hence the image is a subgroup of the real numbers, which we call the valuation group of . We will set which is assumed to be larger than any real number. Now any such pair of a field and valuation on is called a valuation field.
Now let be any valuation field. We then easily see the following properties:
- for all
- for all
If we define , we see that is a subring of called the valuation ring of . Since is a subring of a field, it is an integral domain. If we set we see that and is the field of fractions of .
Notice that is an ideal of called the valuation ideal of . Now suppose that is a unit of Then , implies . But implying that . Therefore . Since the set of non-units is an ideal of we see that is a local ring. By property of local rings, is maximal and is a field, called the residue field of
We want to show one more interesting property of valuation fields before moving forward. Suppose for some . We then see that implying . Then must be a unit of since otherwise would imply that an obvious contradiction. From this we have
So we have proven that implies
Next we want to discuss the idea of equivalent valuations on . For two valuations on we say and are equivalent and write if there exists a positive real number such that for all . This then defines an equivalence relation on valuations on . It is easy to see that equivalent valuations given the same valuation ring, valuation ideal, and group of units. A valuation is called discrete if its value group is an infinite cyclic group, i.e., If the value group is equal to – not simply isomorphic to – the integers, then the valuations is called a normalized discrete valuation. Notice that every discrete valuation is equivalent to a normalized discrete valuation.
We want to focus for a bit on discrete valuations rings, or simply DVRs. So let be a normalized discrete valuation on on a field with valuation ring and valuation ideal . Then the following properties hold:
- Let be an element such that . Then any can be expressed as where and is a unit of .
- Any nonzero ideal of is of the form for some In particular, is a PID.
- is integrally closed.
Such an element in the above is called a uniformizing parameter of . From here we can see that two discrete valuations on a fixed field are equivalent if and only if they have the same valuation ideal. We finish this section with a result that gives a characterization of DVRs. It can be shown that the following two conditions on an arbitrary ring are equivalent:
- is a DVR.
- is a local PID.
Next time we discuss -adic valuations and complete valuation rings.