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

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

In fact, this is often possible. The natural map

$\displaystyle X \rightarrow \mathrm{Tot}( R^{\otimes \bullet + 1} \otimes X)$

turns out to be an equivalence if ${X}$ is “nice” (e.g., connective with finitely generated homology) and ${R = MU}$. In other words, the spectrum ${X}$ can be recovered from the cosimplicial module ${MU^{\otimes \bullet + 1} \otimes X}$ over the cosimplicial ${E_\infty}$-ring ${MU^{\otimes \bullet + 1}}$. This, combined with the homotopy spectral sequence for a cosimplicial object, offers an approach to calculating ${\pi_* X}$ in terms of the ${MU^{\otimes \bullet + 1}}$-homology of ${X}$. What is this approach? It’s the Adams-Novikov spectral sequence! If we replace ${MU}$ with ${H \mathbb{Z}/2}$, we get the classical Adams spectral sequence and the above ${\mathrm{Tot}}$ is the 2-adic completion of ${X}$ (which is all that one can hope for, as a homotopy limit of ${H \mathbb{Z}/2}$-modules).

So let’s take the picture from the previous section. Given a spectrum ${X}$, under nice hypotheses we can recover ${X}$ from the ${MU}$-modules ${MU^{ \otimes (s +1)} \otimes X}$. In particular, we had

$\displaystyle X \simeq \mathrm{Tot}( MU^{\bullet + 1} \otimes X).$

Now, whenever you have a cosimplicial spectrum ${Y^\bullet}$, there is a homotopy spectral sequence

$\displaystyle \pi^s \pi_t Y^\bullet \implies \pi_{t-s} \mathrm{Tot}(Y^\bullet),$

where ${\pi_t Y^\bullet}$ defines a cosimplicial abelian group, and ${\pi^s}$ is the homotopy of that cosimplicial abelian group (i.e., the cohomology of the associated complex under Dold-Kan). In the present case, this means that we get a spectral sequence

$\displaystyle \pi^s MU^{ \otimes (t + 1)}_* (X) \implies \pi_* X,$

and where the ${\pi^s}$ is of the complex

$\displaystyle MU_*(X) \rightarrow (MU \otimes MU)_* X \rightarrow (MU \otimes MU \otimes MU)_* X \rightarrow \dots.$

But a nice thing happens since we are working with ${MU}$. Since ${MU}$ satisfies a flatness condition (${MU_*MU}$ is flat over ${MU_*}$), we have that

$\displaystyle \pi_*MU^{\otimes (s+1)} \simeq MU_*(MU) \otimes_{MU_*} \dots \otimes_{MU_*} MU_*MU \otimes_{MU_*} MU_*(X),$

and the above complex is precisely the cobar complex for computing the ${\mathrm{Ext}}$ group

$\displaystyle \mathrm{Ext}^{s,t}_{MU_* MU}(MU_*, MU_* X)$

in the category of ${MU_*MU}$-comodules. In particular, we find the spectral sequence runs

$\displaystyle E_2 = \mathrm{Ext}^{s,t}_{MU_* MU}(MU_*, MU_* X) \implies \pi_* X,$

which is precisely the ANSS.

3. The stacky language

The Adams-Novikov ss has been around a long time, but one of the newer developments that gives it an extra conceptual edge is the stacky language. Namely, let’s go back to the fact that we had a cosimplicial ${E_\infty}$-ring ${MU^{\otimes (s + 1)}}$, and that we recovered a given spectrum from the cosimplicial module ${MU^{\otimes (s+1)} \otimes X}$ on it. Taking homotopy groups everywhere, we have a cosimplicial ring

$\displaystyle \pi_* MU^{\otimes (s+1)}$

and a cosimplicial module

$\displaystyle \pi_* ( MU^{\otimes (s+1)} \otimes X).$

The cosimplicial module is “compatible” with cosimplicial ring structure in the strong sense that any coface or codegeneracy map in the cosimplicial module comes from tensoring up with the corresponding coface or codegeneracy map in the cosimplicial ring. Stated another way, we have a Hopf algebroid, and a comodule over the Hopf algebroid.

Now, a cosimplicial ring (for instance, that corresponding to a Hopf algebroid) is the same thing as a simplicial affine scheme. Moreover, a simplicial affine scheme gives a presentation of a stack, and a simplicial quasi-coherent sheaf on a simplicial affine scheme such that the face and degeneracy maps are cartesian is the same thing as a quasi-coherent sheaf on the associated stack.

So what does this tell us? Well, the Hopf algebroid ${\pi_* MU^{\otimes (s+1)}}$ presents a stack, and the comodule ${\pi_* ( MU^{\otimes (s+1)} \otimes X)}$ presents a quasi-coherent sheaf on it. One more thing: everything here takes values in the graded category of rings and modules. This grading is the same as an action of the multiplicative group scheme ${\mathbb{G}_m}$, and taking the stacky quotient gives a stack ${M_{FG}}$ and a quasi-coherent sheaf on ${M_{FG}}$.

The amazing result is that ${M_{FG}}$ is actually a natural object in algebraic geometry.

Theorem 2 (Quillen) ${M_{FG}}$ is the moduli stack of formal groups (i.e., one-dimensional smooth formal group schemes).

This is essentially Quillen’s theorem on the formal group law of ${MU}$. One of the things I’ve wondered about is whether there is a “derived” version of all this: as I heard from a topologist once, take the geometric realization of ${\mathrm{Spec} MU^{\otimes (s+1)}}$ in the category of derived stacks. Is this a concrete derived stack? Not knowing about derived stacks, I can’t offer much here.

Anyway, the point was: the descent cosimplicial diagram ${MU^{\otimes (s+1)}}$ has homotopy groups forming a cosimplicial ring, which presents a stack ${M_{FG}}$ classifying formal groups. Given a spectrum ${X}$, the homotopy groups of the cosimplicial diagram ${MU^{\otimes (s+1)} \otimes X}$ (equivalently, ${MU_* X}$ together with the comodule structure) describe a quasi-coherent sheaf on the stack of formal groups. We get a functor:

$\displaystyle \mathbf{Spectra} \rightarrow \text{Quasi-coherent sheaves on } M_{FG} ,$

which is a fancier way of saying that any spectrum ${X}$ gives an ${MU_*}$-module ${MU_*X}$ with a comodule structure over ${(MU_*, MU_* MU)}$.

Finally, we have:

Theorem 3 Let ${\mathcal{F}(X)}$ be the quasi-coherent sheaf defined by ${X}$ on ${M_{FG}}$. The ${E_2}$ page of the ANSS can be described as ${H^s(M_{FG}, \mathcal{F}(X))}$.

Why is that? Again, as before, the ${E_2}$ page of the ANSS is described as the cohomology of a cobar-type complex, or as ${\mathrm{Ext}}$ in the category of comodules over ${(MU_*, MU_* MU)}$; but if we identify comodules over this Hopf algebroid with quasi-coherent sheaves on the stack, the ${\mathrm{Ext}}$ groups become cohomology groups.

This language becomes really powerful once you start doing geometry with ${M_{FG}}$, rather than sticking to algebra with comodules. There’s a very nice structure theory for $M_{FG}$ in terms of the theory of heights (various theorems of Lazard) that let you “visualize” $M_{FG}$ via a stratification–which, remarkably, turns out to be reflected in stable homotopy! More on this later.