One of the really nice pictures in homotopy theory is the “chromatic” one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups). A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen’s theorem on the formal group of $MU$. I didn’t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I’ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes of Haynes Miller.)

1. Descent

Let’s start with some high-powered generalities that I don’t really understand, and then come back to earth. Consider an ${E_\infty}$-ring ${R}$; the most important examples will be ${R = H \mathbb{Z}/2}$ or ${R = MU}$. There is a map of ${E_\infty}$-rings ${S \rightarrow R}$, where ${S}$ is the sphere spectrum.

Let ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a module over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \otimes X}$, where the smash product of spectra is written ${\otimes}$. In fact, we have an adjunction

$\displaystyle \mathrm{Mod}(S) \rightleftarrows \mathrm{Mod}(R)$

between ${R \otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \otimes X}$ together with the “descent data” on ${X}$. The benefit is that while the homotopy groups ${\pi_* X}$ may be intractable, those of ${R \otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.

Let’s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of

$\displaystyle M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B.$

How does one imitate this construction in homotopy? One then has a cosimplicial ${E_\infty}$-ring given by the cobar construction

$\displaystyle R \rightrightarrows R \otimes R \dots .$

The ${\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram

$\displaystyle R \otimes X \rightrightarrows R \otimes R \otimes X \dots .$ (more…)