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: -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.

More on finite group schemes

In the last post, we discussed finite flat commutative group schemes (henceforth, ‘finite group schemes’ or even ‘finite groups’) and made some basic constructions. Before we get to the main objects of interest, -divisible groups, I want to say a few more words about the structure of finite group schemes in characteristic . The key point, which will be increasingly hammered into your brain as this series of posts continues, is that everything is controlled by the Frobenius and Verschiebung maps. Today, as a first step in this argument, I’ll discuss a few basic examples of finite group schemes, as well as what happens when you dualize the connected-étale exact sequence from last time.

By the way, I didn’t mention sources in my last post. I’ve found the following extremely helpful:

• These notes for material on finite group schemes and Witt vectors.
• Demazure’s Lectures on p-divisible groups, which has an interesting perspective on formal groups and proves the classification theorem.
• Tate’s paper on -divisible groups, which proves the interesting result that -divisible groups over integrally closed domains are controlled by their behavior over the general fiber (that is, the field of fractions). This is short but gives a good summary of things one should understand about -divisible groups. I’d go through the result here if I understood the requisite class field theory.
• Messing’s The Crystals Associated to Barsotti-Tate groups; with Applications to Abelian Schemes. This is higher-level than the other sources. It proves deformation-theoretic results that might tie in to Lurie’s theorem — I haven’t looked at it in enough detail yet.