Let’s try to define a few words I’ve thrown around for a few weeks. What is a -divisible group, and how do we know what its height is? I’m going to do two things that will either make this easier to understand or way more confusing. We will be working with group schemes. To keep from repeating everything twice with the word “formal” in front of everything I will rarely specify whether I mean formal group scheme or group scheme. Obviously some things are different in the formal case, but not so much. The other thing I’ll do is assume our group schemes are affine to simplify notation, but if you want to do this more generally you can.
The point of these posts should be to give an overview of how these things fit together. When trying to learn about this stuff, there are so many hundreds of terms and papers and details in the papers that it is really easy to forget what is going on. For instance, basically any reference on -divisible groups will get very caught up in all the technical details of Dieudonne theory, and I want to massively downplay this aspect for the purpose of defining this invariant called the height of a variety.
On to the definition. A -divisible group is just an inductive system of group schemes satisfying two properties. First there must be an so that the order of is . Second, they must fit into an exact sequence . All this says is that if we look at the map that is multiplication by on , the kernel of this is the copy of that sits inside via .
This might seem like a strange set of conditions at first, but really the two most natural examples of forming inductive systems of group schemes already satisfy both of these. The first one is to take an abelian variety of dimension . Then we have the isogenies multiplication by , multiplication by , etc. The kernels of these are all group schemes and it is well-known that they are isomorphic to , , etc. We just take the maps to be the inclusions and the .
The other main example is to do the same trick with by taking successive kernels of multiplication by . We just get the inductive system . In this case the orders are just , so . We’ve suggestively labelled this number which is the height of the -divisible group. This one is usually denoted .
There are lots of easy properties of -divisible groups that can be verified mentally. For instance, you can put any two of and into an exact sequence . A slightly harder property to check is that under mild base assumptions we have an equivalence of categories between -divisible groups and divisible formal Lie groups. Under this equivalence we get that corresponds to the one-dimensional Lie group with group law , so our earlier notation of calling this makes sense because it comes from .
Since ultimately we are concerned with computing heights, we should see if there is a way to figure out the height without computing orders. Let’s denote the Cartier dual of a group scheme by . We can check that taking Cartier duals everywhere, we get another -divisible group. I.e. is an inductive system satisfying the properties of a -divisible group. This is often called the Serre dual. If we denote the whole inductive system by , then it is customary to write for the Serre dual.
Note that duals can be quite different from the original group. In particular, the dimension can be different. In the case of we get a dimension etale group scheme, but it’s dual is and hence a dimension connected (with nilpotent structure) group scheme. If we add these two dimensions we get , the height. This is true in general. We have the formula . So we only need to know the dimensions of the group and its dual.
Another (much harder for calculating, but sometimes handy in theory) way to determine the height is to use the Dieudonne module we defined last time. If we take it is a -module, but also a free of finite rank -module. The rank of this module turns out to be the height of . In a similar fashion, you can form another module out of called the Tate module. By definition, multiplication by is a map , and hence we get an inverse system. We define to be the Tate module. It is a free -module. The rank of this is the height of .
That is about the sketchiest crash course on -divisible groups you can get, but I think it mentions enough to get to the next definition: the height of a variety in positive characteristic.