Let’s look at the curves and . They’re pretty simple, so naturally, we’re going to use some big tools to analyze him. Today the plan is to look at these curves by studying how functions behave on them locally.
1. Vanishing at a point
Every point on is of the form . Let’s pick a point . Here’s some of the usual objects:
- The ring of polynomial functions
- The ideal corresponding to is
As is the common practice, we’ll let denote the ring
This ring (and its properties) are going to be our main tool for studying the local properties of functions.
This ring contains a special ideal:
This ideal is the unique maximal ideal in , simply because it contains every non-unit in (the only way can fail to be a unit is if , which is equivalent to ).
We now consider the following question: what can we say about the generators of this ideal? Perhaps we can convince ourselves that
but, in fact, is actually a principal ideal. Indeed, we have the relation
which implies (since is a unit in ) that
Since every element of is either a unit or is in , and is now known to be a principal ideal, we know that every element of can be written in the form
(for instance, we may take or ).
We define a map which sends . This map is a homomorphism of the (multiplicative) monoid to the (additive) monoid . This map extends to the field of fractions of , whose elements are of the form without any restrictions on . Here we have
Here’s a simple example, still on our curve . Consider the function
At the point we have
indicating no vanishing or poles at this point. At the point we have
indicating vanishing to order at this point. At the point we have
indicating a pole of order at this point.
2. Discrete Valuation Rings
A ring is said to be a discrete valuation ring (or DVR) if there is a surjective group homomorphism . In the example above, we saw that is a discrete valuation ring with the map .
Does this always happen for curves? As it happens, no; a necessary and sufficient condition for to be a DVR is that must be a smooth point of the curve (this is a consequence of Nakayama’s lemma, which allows us to take a basis for the -dimensional vector space and lift it to a generator for the ideal ).
We can see an example of why smoothness is necessary. Consider the curve . Here we know that the curve is smooth everywhere besides . We’ll now see that is a DVR everywhere except at .
Our basic trick will be the following: in order for to be a DVR, we need the ideal to be principal. This will only happen when .
Here the situation is pretty simple. Let be a point on , so that . Then
which gives us a relationship between the two generators of . We now note that
so that is principal precisely when .