# p-divisible Groups

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: ${p}$-divisible groups.

Here’s what I’ve been up to. The moduli stacks of ${p}$-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 ${p}$-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 ${K(n)}$-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 1 A ${p}$-divisible group over a scheme ${S}$ is an ind-scheme of the form

$\displaystyle 0 = \mathbb{G}_0 \rightarrow \mathbb{G}_1 \rightarrow \mathbb{G}_2 \rightarrow \dotsb$

with each ${\mathbb{G}_i}$ a finite, flat, commutative group scheme over ${S}$ of constant rank ${p^{ih}}$ for some fixed ${h}$, the maps closed immersions, and with ${\mathbb{G}_i}$ the kernel of the multiplication-by-${p^i}$ map ${[p^i]:\mathbb{G}_{i+1} \rightarrow \mathbb{G}_{i+1}}$. The number ${h}$ is called the height of the ${p}$-divisible group.

The example I prefer to keep in mind is ${A(p) = \varinjlim_i A[p^i]}$, the group of ${p}$-power torsion points of an abelian scheme ${A}$. Thinking about this immediately demonstrates one basic but useful point: each ${\mathbb{G}_i}$ is precisely the kernel of ${[p^i]}$ on any ${\mathbb{G}_j}$ with ${j\ge i}$, and so you can recover the individual finite group schemes from the ${p}$-divisible group. In other words, this specific colimit presentation is functorial for maps of group ind-schemes, making it kosher to think of the ${p}$-divisible group in terms of the individual finite groups.

Of course, any formal group law ${F}$ on ${R[[x_1,\dotsc,x_d]]}$ gives rise to a ${p}$-divisible group, with ${(\mathbb{G}_F)_i = R[[x_1,\dotsc,x_d]]/[p^i](x_1,\dotsc,x_d)}$. If you do homotopy theory, these are the ${p}$-divisible groups you’re primarily interested in. This raises the immediate question: why study ${p}$-divisible groups rather than just formal groups? The answer is that the height of a ${p}$-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 ${p}$-divisible groups separates into disjoint pieces, one for each height ${h}$. 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 ${h}$,’ but not every height ${h}$ formal group law on a ring, for instance, gives you a map to this substack. Instead, they tend to ‘spread out’ over ${\mathcal{M}_{fg}}$ itself, hitting not only the height exactly ${h}$ 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 ${p}$-divisible group ‘look like’? Well, over a point of the base scheme ${S}$, any connected component of ${\mathbb{G}}$ will actually be an affine formal scheme ${\mathrm{Spf}\, A}$, that is, ${\varinjlim \mathrm{Spec}\, A/I_n}$ where ${A}$ is a topological ring and ${I_n}$ a decreasing sequence of open ideals whose intersection is ${(0)}$, so that ${A = \varprojlim A/I_n}$. If the base field is algebraically closed, then ${A = k[[x_1,\dotsc,x_d]]}$ for some ${d}$, the group structure comes from a ${d}$-dimensional formal group law in the usual sense, and ${I_n = [p^n](x_1,\dotsc,x_d)}$. So the usual theory of formal group laws appears here. Of course, ${\mathbb{G}}$ 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 2 Let ${\mathbb{G}}$ be a ${p}$-divisible group over a complete noetherian local ring. Then there is a natural exact sequence of ${p}$-divisible groups

$\displaystyle 0 \rightarrow \mathbb{G}^0 \rightarrow \mathbb{G} \rightarrow \mathbb{G}^{et} \rightarrow 0$

where ${\mathbb{G}^0}$ is an affine formal scheme and ${\mathbb{G}^{et}[p^i]}$ is étale for each ${i}$. If ${S = \mathrm{Spec}\, k}$ with ${k}$ a perfect field, then this sequence naturally splits.

This follows immediately from the finite case, by constructing the connected-étale exact sequence for each ${\mathbb{G}_i}$. The splittings of these sequences over a perfect field are natural, so we get a splitting for ${\mathbb{G}}$ 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 ${\mathbb{G}^{et} \cong \underline{{\mathbb Q}_p/{\mathbb Z}_p}^n}$ for some ${n}$.

The dimension of ${\mathbb{G}}$ is the dimension of ${\mathbb{G}^0}$, or equivalently of any ${\mathbb{G}^0[p^i]}$.

2. The multiplication-by-${p}$ 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 ${p}$-divisible group ${\mathbb{G} = \varinjlim \mathbb{G}[p^i]}$, we have an exact sequence

$\displaystyle 0 \rightarrow \mathbb{G}[p^i] \rightarrow \mathbb{G}[p^{i+j}] \stackrel{[p^i]}{\rightarrow} \mathbb{G}[p^{i+j}],$

and the image of ${[p^i]}$ on the ${p^{i+j}}$-torsion is clearly ${p^j}$-torsion, so we in fact have

$\displaystyle 0 \rightarrow \mathbb{G}[p^i] \rightarrow \mathbb{G}[p^{i+j}] \stackrel{[p^i]}{\rightarrow} \mathbb{G}[p^j].$

But the orders of these group schemes are respectively ${p^{ih}}$, ${p^{(i+j)h}}$, and ${p^{jh}}$, so the sequence must be exact on the right as well. In particular, ${[p]}$ maps each ${\mathbb{G}[p^i]}$ surjectively to ${\mathbb{G}[p^{i-1}]}$, with kernel the finite ${S}$-scheme ${\mathbb{G}[p]}$ — thus, it’s a surjection from ${\mathbb{G}}$ to itself with finite kernel, also called an isogeny. A ${p}$-divisible group can equivalently be defined as a group object in the category of ind-schemes on which ${[p]}$ is an isogeny.

The Cartier duality functor is exact, so the dual of the surjective map ${[p]:\mathbb{G}[p^{i+1}] \rightarrow \mathbb{G}[p^i]}$ is a closed immersion ${\mathbb{G}[p^i]^\vee \rightarrow \mathbb{G}[p^{i+1}]^\vee}$. The ${p^i}$-torsion of ${\mathbb{G}[p^{i+1}]^\vee}$ is the subscheme of maps to ${\mathbb{G}_m}$ that factor through ${\mathbb{G}_m[p^i]}$, or equivalently, through ${\mathbb{G}[p^i]}$ along ${[p]}$ — but this is precisely ${\mathbb{G}[p^i]^\vee}$. Thus, the diagram

$\displaystyle 0 \rightarrow \mathbb{G}[p]^\vee \rightarrow \mathbb{G}[p^2]^\vee \rightarrow \dotsb,$

where the arrows are ${[p]^\vee}$, defines a ${p}$-divisible group, called the Cartier dual or Serre dual of ${\mathbb{G}}$.

3. Frobenius and Verschiebung

The assignment ${X \mapsto X^{(p)} = X \otimes_k^\sigma k}$ is functorial for schemes over a field ${k}$ of characteristic ${p}$, and the Frobenius map ${X \rightarrow X^{(p)}}$ is natural. Thus, the Frobenius maps of the schemes ${\mathbb{G}[p^i]}$ define a Frobenius map ${\mathbb{G} \rightarrow \mathbb{G}^{(p)}}$, where ${\mathbb{G}^{(p)}}$ is the ${p}$-divisible group with ${p^i}$-torsion ${\mathbb{G}[p^i]^{(p)}}$. Likewise, the Verscheibung map ${G^{(p)} \rightarrow G}$ is natural for group schemes over ${k}$, so there’s a Verschiebung map ${\mathbb{G}^{(p)} \rightarrow \mathbb{G}}$. Moreover, Cartier duality interchanges ${F}$ and ${V}$ for ${p}$-divisible groups, just as it did for finite groups. We have ${[p] = VF}$, as well as ${p = FV}$ in characteristic ${p}$.

These definitions and conclusions easily generalize to ${p}$-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 3 Let ${\mathbb{G}}$ be a ${p}$-divisible group over a perfect field ${k}$ of characteristic ${p}$, ${F}$ and ${V}$ its Frobenius and Verschiebung maps. There’s a natural decomposition

$\displaystyle \mathbb{G} \cong \mathbb{G}_{cu} \times \mathbb{G}_{eu} \times \mathbb{G}_{cm}$

as above. We can identify ${\mathbb{G}_{cu}}$ as ${\varinjlim \mathrm{ker}\, F^n \cap \mathrm{ker}\, V^n}$, ${\mathbb{G}_{eu}}$ as ${(\varinjlim \mathrm{ker}\, V^n)/\mathbb{G}_{cu}}$, and ${\mathbb{G}_{cm}}$ as ${(\varinjlim \mathrm{ker}\, F^n)/\mathbb{G}_{cu}}$.

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 4 The height of ${\mathbb{G}}$ is the sum of the dimension of ${\mathbb{G}}$ and the dimension of ${\mathbb{G}^\vee}$.

Proof: Since ${[p] = VF}$, there’s an exact sequence of finite group schemes

$\displaystyle 0 \rightarrow \mathrm{ker}\, F \rightarrow \mathrm{ker}\, [p] \rightarrow \mathrm{ker}\, V \rightarrow 0.$

Of course, ${\mathrm{ker}\, [p]}$ is just ${\mathbb{G}_1}$, which is a finite group scheme of order ${p^h}$. Last post’s structure theorem tells us that ${\mathrm{ker}\, F}$ is the connected part of ${\mathbb{G}_1}$, and thus of order ${p^d}$. Thus ${\mathrm{ker}\, V}$ is order ${p^{n-d}}$. But ${\mathrm{ker}\, V}$ is the dual of the cokernel of ${F^\vee:\mathbb{G}^\vee \rightarrow (\mathbb{G}^\vee)^{(p)}}$; it’s also the cokernel of ${F^\vee:\mathbb{G}_1^\vee \rightarrow (\mathbb{G}_1^\vee)^{(p)}}$. This is a map of finite group schemes of the same order, so its cokernel has the same order as its kernel, which is ${p^{d'}}$ where ${d' = \dim\mathbb{G}^\vee}$. Thus ${d' = n-d}$. $\Box$

4. An example

The height of a ${p}$-divisible group is invariant under base change, but the height of a formal group is not. This is the main argument for working with ${p}$-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 ${R = {\mathbb Z}_p[[u_1]]}$ be the Lubin-Tate ring representing deformations of a height 2 formal group over ${{\mathbb F}_p}$. The universal deformation of this formal group is a ${p}$-typical formal group law ${F}$ with ${p}$-series

$\displaystyle [p]_F(x) = px +_F u_1x^p +_F x^{p^2}.$

To make things a bit easier, let’s replace ${R}$ with ${{\mathbb F}_p[[u_1]]}$ and ${F}$ with

$\displaystyle [p]_F(x) = u_1x^p +_F x^{p^2}.$

Notice that this is a height 2 formal group law. One can check this via the language of ${p}$-divisible groups by noticing that ${R[[x]]/[p^i](x)}$ is free of rank ${p^{2i}}$ over ${R}$ — indeed, this is true after modding out by the maximal ideal ${(u_1)}$, and so it’s true over ${R}$ by Nakayama’s lemma.

Now let ${K = {\mathbb F}_p(u_1)}$ be the field of fractions of ${R}$. Base changing to ${K}$, we now have that ${[p]_F(x) = x^p\cdot u}$ where ${u}$ is a unit in the power series ring. Thus, ${S[[x]]/[p^i](x)}$ is free of rank ${p^i}$, 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 ${\mathbb{G}_F}$ as a ${p}$-divisible group as opposed to as a formal group — that is, we have to base change each ${p^i}$-torsion term ${R[[x]]/[p^i](x)}$ 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 ${R}$ (which is a kind of colimit) doesn’t commute with the limit defining the power series ring. Thus, ${R[[x]]/[p^i](x) \otimes_R K}$ is not actually ${K[[x]]/[p^i](x)}$, but rather, this latter ring is one of the connected components of the true tensor product. To find the true tensor product (taking ${i=1}$ for simplicity’s sake), we use the Weierstrass preparation theorem to identify

$\displaystyle R[[x]]/[p](x) \otimes_R S \cong R[[x]]/(u_1x^p + x^{p^2}) \otimes_R K \cong R[x]/(u_1x^p + x^{p^2}) \otimes_R K \cong K[x]/(u_1x^p + x^{p^2}).$

This now has a rank-${p}$ subalgebra generated by ${y = x^p}$, which is clearly étale because the derivative of the defining polynomial ${u_1y + y^p}$ is the unit ${u_1}$. The quotient Hopf algebra is the aforementioned ${K[x]/(x^p)}$, which is precisely the connected piece of the ${p}$-torsion of the ${p}$-divisible group, base changed to ${K}$.

The same phenomenon appears on each ${p^i}$-torsion. The ${p^i}$-series of ${F}$ over ${R}$ is a unit times a polynomial of the form

$\displaystyle u_1^{1 + \dotsb + p^{i-1}}x^{p^i} + \dotsb + x^{p^{2i}}$

with each non-leading coefficient a multiple of ${u_1}$; with ${u_1}$ inverted, the subalgebra generated by ${y = x^{p^i}}$ is étale of rank ${p^i}$, and the quotient Hopf algebra is isomorphic to ${K[x]/(x^{p^i})}$, which is connected of rank ${p^i}$. This gives us a ${p}$-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 ${p}$-divisible groups and formal groups satisfying a certain property on ${[p]}$.