# 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. Continue reading

# 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, ${p}$-divisible groups, I want to say a few more words about the structure of finite group schemes in characteristic ${p}$. 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 ${p}$-divisible groups, which proves the interesting result that ${p}$-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 ${p}$-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.

# Finite flat commutative group schemes, with an eye towards p-divisible groups

I’ve been reading a lot recently about ${p}$-divisible groups, which are algebro-geometric approximations to homotopy theory much like formal groups, though perhaps a bit better behaved. Recall that we can get a formal group by taking the completion of an elliptic curve at the identity; over a field of characteristic ${p}$, this will be height 2 if the curve is supersingular, but only height 1 if it’s not. More generally, the sorts of base changes that invert things in the maximal ideal of your base ring tend to decrease the height of formal groups. The ${p}$-divisible viewpoint says that formal groups in positive characteristic naturally sit inside objects whose height is constant under base change — in particular, every elliptic curve will give you a height 2 ${p}$-divisible group, with some of this height being non-formal’ if the curve is ordinary. (And more generally, you get a height ${2n}$ ${p}$-divisible group from an abelian variety of dimension ${n}$ in characteristic ${p}$.)

In algebraic geometry and number theory, various results by Tate, Grothendieck, Messing and others argue the general point that the deformation theory of a grouplike object in characteristic ${p}$ is controlled by the deformation theory of its ${p}$-divisible group. This idea has been key to a number of recent developments in homotopy theory, — in general, we hope that various moduli objects of ${p}$-divisible groups are easier to study than the analogous moduli objects of formal groups, while offering the same level of access to the category of spectra. The key result in this field, due to Lurie, actually allows you to build spectra from 1-dimensional ${p}$-divisible groups, in much the same way that the Goerss-Hopkins-Miller theorem lets you build spectra from deformations of formal groups. Reaching the very boundary of things I can claim to understand, Lurie’s theorem is a key component of Behrens and Lawson’s construction of the topological automorphic forms spectra.

A ${p}$-divisible group is a certain sort of ind-scheme which can be canonically written as a colimit of finite flat commutative group schemes. So before we can say anything about ${p}$-divisible groups, we have to know some things about these gadgets (which I’ll generally just call ‘group schemes’), and this post will fill in that background. (Presumably this is all well-known to the average algebraic geometer, but it was new to me, and hopefully it’ll help out other budding homotopy theorists curious about this subject.)

# Countably generated abelian groups

The following theorem is one that many of us use practically daily.

Theorem 1 If ${A}$ is a finitely-generated abelian group, then ${A}$ can be written as a direct sum

$\displaystyle A \cong {\mathbb Z}^r \oplus \bigoplus_{p\text{ prime}} \bigoplus_{n=1}^\infty ({\mathbb Z}/p^n{\mathbb Z})^{e_{p,n}}$

in a unique way — that is, the exponents ${r}$ and ${e_{p,n}}$ are uniquely determined by ${A}$ (and all but finitely many of them are zero).

This theorem is really two important theorems in one. The first is a decomposition theorem, which lets us decompose our objects (finitely generated abelian groups) in a simple way (a direct sum of cyclic groups). The second is a uniqueness theorem, which tells us when two of our objects are the same (in this case, precisely when the exponents ${r}$ and ${e_{p,n}}$ are the same).

In particular, this theorem tells us all we’d ever need to know about finite abelian groups. When we go further, though, the situation becomes far more difficult. Here are a few radically different examples of infinite abelian groups (all of which are countable, even!):

• ${{\mathbb Z}}$, which we’ve already taken care of.
• ${{\mathbb Q}}$, the rational numbers.
• ${{\mathbb Q}/{\mathbb Z}}$.
• ${{\mathbb Z}_{(p)}}$, the set of rational numbers with denominators prime to ${p}$.
• ${{\mathbb Z}/p^\infty = {\mathbb Z}_{(p)}/{\mathbb Z} = \varinjlim {\mathbb Z}/p^n}$ along the maps that are multiplication by the obvious powers of ${p}$.

We’d like to generalize the above classification theorem to deal with infinite abelian groups. This is surprisingly difficult for high cardinalities, and as far as I can tell in my limited research, there’s no real hope for such a theorem for uncountable abelian groups. For countable abelian groups which are assumed to be torsion, however, there’s a rather nice theorem, due to Ulm in the 30’s [4], and in this talk we’ll attempt to prove it. More generally, we’ll exhibit a couple convenient ways of decomposing abelian groups.

Most of this is from Kaplansky’s little red book [2], which is a great and short read for people of all mathematical backgrounds. I’m going to omit the word abelian’ a lot. Additionally, all of this applies mutatis mutandis for countably generated modules over a PID.

# Monsky’s Theorem, or the genius of bizarre thinking

I was shown this by the illustrious Nir Avni, and it was so beautiful and bizarre that I presented it at our undergrad math seminar that week.  Now a friend of mine wants to turn it into art, and I’m writing it up in service of that lofty goal.  Kate, may your project join the slim annals of awesome math-themed visual art.

The question is: for what $n$ can you cut a square into $n$ triangles of equal area?  By slicing it into rectangles and cutting each rectangle into two triangles, it’s easy to see that you can get any even number, as the pictures below show.

So can you do it with an odd number of triangles?  Think about it a bit, and after the jump, we’ll think about it together.  (This post will be at a lower level than most of my other writing here — it should be understandable by math undergrads and ideally even the Laity.  Let me know how I’m doing!)

# Arithmetic Topology

One of my favorite viewpoints on math is that it is the study of metaphor.  By abstracting an existing metaphor, we invent new objects (groups were invented because Galois realized that ‘polynomials are like plane figures’); by inventing a new metaphor, we can import techniques from one field to another (the Italian school of algebraic geometry got a lot of mileage from ‘varieties are like manifolds’; the functor-of-points school, from ‘schemes are like functors on the category of rings’).  A poet might compare the evening to an etherized patient and leave it there, and in poetry this is fine, for in poetry we revel in mystery, allusion, in half-knowledge.  But in math, we can’t stand these things, and so we must grab our things and run to the nearest hospital, examining all the gurneys we can in the hope of better understanding the twilight.

One particular metaphor that’s captivated me recently goes by the unassuming name of ‘arithmetic topology,’ essentially due to Barry Mazur in an unpublished paper from 1963 (though as always, neverendingbooks has a more precise history of the idea).  The one-sentence idea, which should be all the convincing anyone needs to do math, is that the integers are like three-space and prime numbers are like knotted circles.  I know, I know.  And yes, it does go a lot further than that — more generally, there’s a fascinating story about Galois-group-looking things relating to the algebraic topology of honest-to-God topological things.  In the interest of always doing something that’s slightly too hard for you, I’m trying to understand this right now, and in doing so will hopefully learn a lot of algebraic geometry, algebraic number theory, and knot theory ‘by accident.’  I’ll write about it here as I do so, and I invite you to learn with me if you don’t know what I’m talking about / correct my errors and give me references if you do.  Below the fold, an introduction to what seems to be going on, and a rough outline of what I’ll probably be writing about.

# The Goodstein Sequence, or, using infinity to prove finitary results

I’d like to inaugurate this blog by demonstrating a wacky and somewhat radical proof technique.  If you feel threatened as you wade through a world of skeptics of infinity, I present to you your weapon.  It’s a cute little sequence of natural numbers called a Goodstein sequence.