In the last two posts, we made some basic constructions with finite group schemes (which are always assumed flat and commutative), and discussed their decompositions into connected, étale, multiplicative, and unipotent parts. Today we’ll bootstrap those constructions up to our real objects of interest: -divisible groups.

Here’s what I’ve been up to. The moduli stacks of -divisible groups chatter at my mind like winged monkeys, and so this series of posts is going to continue, slowly, into the indefinite future. I’m particularly interested in various ways of describing -divisible groups, so you should expect to hear from me about the Dieudonné correspondence, Dieudonné crystals, and possibly even Thomas Zink’s theory of ‘nilpotent displays.’ There’ll be a few asides on Witt vectors as well.

We’ve been doing a summer seminar on Goodwillie calculus at Northwestern, and I recently gave a talk about Nick Kuhn’s cool theorem that Goodwillie towers in spectra split after finite -localization. I’m in the process of making some fairly detailed notes about this paper, which should appear here shortly. I’m also going to a summer school in Oregon on infinity-categories, and will probably end up posting some notes on Thom spectra or whatnot as a result.

**1. Definitions **

Definition 1A-divisible groupover a scheme is an ind-scheme of the form

with each a finite, flat, commutative group scheme over of constant rank for some fixed , the maps closed immersions, and with the kernel of the multiplication-by- map . The number is called the

heightof the -divisible group.

The example I prefer to keep in mind is , the group of -power torsion points of an abelian scheme . Thinking about this immediately demonstrates one basic but useful point: each is precisely the kernel of on any with , and so you can recover the individual finite group schemes from the -divisible group. In other words, this specific colimit presentation is functorial for maps of group ind-schemes, making it kosher to think of the -divisible group in terms of the individual finite groups.

Of course, any formal group law on gives rise to a -divisible group, with . If you do homotopy theory, these are the -divisible groups you’re primarily interested in. This raises the immediate question: why study -divisible groups rather than just formal groups? The answer is that the height of a -divisible group is better behaved than that of a formal group — most importantly, it’s invariant under base change. In highbrow geometric terms (which I hope to explain in later posts, if they’re unfamiliar), the moduli stack of -divisible groups separates into disjoint pieces, one for each height . On the other hand, there’s just a single moduli stack of formal groups, which is filtered by open substacks representing ‘formal groups of height .’ So you can obviously pick out a locally closed substack of ‘formal groups of height exactly ,’ but not every height formal group law on a ring, for instance, gives you a map to this substack. Instead, they tend to ‘spread out’ over itself, hitting not only the height exactly point, but lower heights as well. At the end of this post, I’ll show an example of this in action.

Okay, so what does a -divisible group ‘look like’? Well, over a point of the base scheme , any connected component of will actually be an affine formal scheme , that is, where is a topological ring and a decreasing sequence of open ideals whose intersection is , so that . If the base field is algebraically closed, then for some , the group structure comes from a -dimensional formal group law in the usual sense, and . So the usual theory of formal group laws appears here. Of course, can have more than one connected component. The most precise thing we could say is that the connected-étale exact sequence from last time generalizes:

Proposition 2Let be a -divisible group over a complete noetherian local ring. Then there is a natural exact sequence of -divisible groups

where is an affine formal scheme and is étale for each . If with a perfect field, then this sequence naturally splits.

This follows immediately from the finite case, by constructing the connected-étale exact sequence for each . The splittings of these sequences over a perfect field are natural, so we get a splitting for in this case too.

If we’re over an algebraically closed field, étale group schemes are constant, and an examination of ranks shows that we must have for some .

The **dimension** of is the dimension of , or equivalently of any .

**2. The multiplication-by- map and Cartier duality **

One of the most useful facts about finite flat commutative group schemes is that their orders are multiplicative in exact sequences. As an application, given a -divisible group , we have an exact sequence

and the image of on the -torsion is clearly -torsion, so we in fact have

But the orders of these group schemes are respectively , , and , so the sequence must be exact on the right as well. In particular, maps each surjectively to , with kernel the finite -scheme — thus, it’s a surjection from to itself with finite kernel, also called an **isogeny**. A -divisible group can equivalently be defined as a group object in the category of ind-schemes on which is an isogeny.

The Cartier duality functor is exact, so the dual of the surjective map is a closed immersion . The -torsion of is the subscheme of maps to that factor through , or equivalently, through along — but this is precisely . Thus, the diagram

where the arrows are , defines a -divisible group, called the **Cartier dual** or **Serre dual** of .

**3. Frobenius and Verschiebung **

The assignment is functorial for schemes over a field of characteristic , and the Frobenius map is natural. Thus, the Frobenius maps of the schemes define a Frobenius map , where is the -divisible group with -torsion . Likewise, the Verscheibung map is natural for group schemes over , so there’s a Verschiebung map . Moreover, Cartier duality interchanges and for -divisible groups, just as it did for finite groups. We have , as well as in characteristic .

These definitions and conclusions easily generalize to -divisible groups. Of course, we have to replace ‘étale’ by ‘ind-étale’ and so on, but I usually won’t say the ‘ind-‘ part.

Theorem 3Let be a -divisible group over a perfect field of characteristic , and its Frobenius and Verschiebung maps. There’s a natural decomposition

as above. We can identify as , as , and as .

All this is somewhat trivial — just an extension of the theory of finite group schemes. I’ll conclude with a nontrivial theorem, followed by an example.

Theorem 4The height of is the sum of the dimension of and the dimension of .

*Proof:* Since , there’s an exact sequence of finite group schemes

Of course, is just , which is a finite group scheme of order . Last post’s structure theorem tells us that is the connected part of , and thus of order . Thus is order . But is the dual of the cokernel of ; it’s also the cokernel of . This is a map of finite group schemes of the same order, so its cokernel has the same order as its kernel, which is where . Thus .

**4. An example **

The height of a -divisible group is invariant under base change, but the height of a formal group is not. This is the main argument for working with -divisible groups rather than formal groups. The following example was shown to me by Paul Goerss, and caused me enough grief that I think it’s worth going into in detail.

Let be the Lubin-Tate ring representing deformations of a height 2 formal group over . The universal deformation of this formal group is a -typical formal group law with -series

To make things a bit easier, let’s replace with and with

Notice that this is a height 2 formal group law. One can check this via the language of -divisible groups by noticing that is free of rank over — indeed, this is true after modding out by the maximal ideal , and so it’s true over by Nakayama’s lemma.

Now let be the field of fractions of . Base changing to , we now have that where is a unit in the power series ring. Thus, is free of rank , so the resulting formal group law is height 1.

Where did the missing height go? The answer, of course, is ‘into the étale piece,’ and to figure out what this means, we have to base change as a -divisible group as opposed to as a formal group — that is, we have to base change each -torsion term one at a time. It looks like we did this in the previous paragraph, but we didn’t quite do it right! The trick is that inverting the elements of (which is a kind of colimit) doesn’t commute with the limit defining the power series ring. Thus, is not actually , but rather, this latter ring is one of the connected components of the true tensor product. To find the true tensor product (taking for simplicity’s sake), we use the Weierstrass preparation theorem to identify

This now has a rank- subalgebra generated by , which is clearly étale because the derivative of the defining polynomial is the unit . The quotient Hopf algebra is the aforementioned , which is precisely the connected piece of the -torsion of the -divisible group, base changed to .

The same phenomenon appears on each -torsion. The -series of over is a unit times a polynomial of the form

with each non-leading coefficient a multiple of ; with inverted, the subalgebra generated by is étale of rank , and the quotient Hopf algebra is isomorphic to , which is connected of rank . This gives us a -divisible group with a height 1 étale part and a height 1 formal part.

In the next post, I’ll prove a nice big theorem: the Serre-Tate equivalence between connected -divisible groups and formal groups satisfying a certain property on .