Identifying the stacks in Devinatz-Hopkins-Smith

Question

I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what they did was a sensible mode of proof. I intend to go back and figure some of these conceptual pieces out; this question is a step in that direction.

The proof of smash nilpotence is, on the face of it, accomplished by a sequence of interpolations. The original statement is:

(Smash nilpotence:) If a map off a finite spectrum induces the zero map on $MU$-homology, then the map is in fact smash nilpotent.

Without any hassle, they reduce this to the following statement:

(Special case of Hurewicz nilpotence:) Suppose $\alpha$ is an element of the homotopy $\pi_* R$ of an associative ring spectrum $R$ of finite type. If $\alpha$ is in the kernel of the map $\pi_* R \to MU_* R$ induced by smashing $R$ with the unit map $S \to MU$, then $\alpha$ is nilpotent.

To address this question, they interpolate between the sphere spectrum and $MU$ in two ways. First, they produce a sequence of spectra $X(n)$, each given by the Thom spectrum associated to the composite $\Omega SU(n) \to \Omega SU \to BU$, sort of a restriction of the Bott map. The colimit of the $X(n)$ is $MU$, and hence $X(N)_* \alpha$ is zero for some sufficiently large $N$, where $\alpha$ is considered as a map $S^t \to R$. Since $X(1)$ is the sphere spectrum, the new goal is to show that nilpotence in $X(n+1)$-homology forces nilpotence in $X(n)$-homology for any $n$, then to walk down from $X(N)$ to the sphere spectrum. To move between $X(n+1)$ and $X(n)$, they interpolate between these spectra by pulling $X(n+1)$ apart using the filtered James construction; this results in an increasing sequence of $X(n)$-module spectra $F_{n, k}$ satisfying $F_{n, 0} \simeq X(n)$ and converging to $X(n+1)$ in the limit.

The rest of the argument falls into two pieces:

1) If $\alpha: S^t \to R$ is nilpotent in $X(n+1)$-homology then it induces the zero map in $F_{n,p^k-1}$-homology for some large $k$ --- that is, our argument continues somewhere in the approximating tower between $X(n+1)$ and $X(n)$.

2) If it induces the zero map in $F_{n, p^k-1}$-homology it also induces the zero map in $F_{n, p^{k-1}-1}$-homology.

To prove part 1, they investigate the $X(n+1)$-based Adams spectral sequence $$\mathrm{Ext}^{*, *}_{X(n+1)_* X(n+1)}(X(n+1)_*, X(n+1)_* F_{n, p^k-1} \wedge R) \Rightarrow (F_{n, p^k-1})_* R.$$ The key is the existence of vanishing lines in these spectral sequences, where the slope of the vanishing line can be made small by making $k$ large. In order to establish these vanishing lines, they perform a sequence of approximations, finishing with spectral sequences with the following $E_2$ terms: $$\mathrm{Ext}^{*, *}_{\mathbb{F}_p[b_n]}(\mathbb{F}_p, \mathbb{F}_p\{1, \ldots, b_n^{p^k-1}\}),$$ $$\mathrm{Ext}^{*, *}_{\mathbb{F}_p[b_n^{p^k}]}(\mathbb{F}_p, \mathbb{F}_p).$$

My question is: Is there a geometric interpretation for the stacks associated to the Hopf algebroids above? ( --or any of the other Hopf algebroids involved which I haven't listed.) Or: what's the geometric content of Part II of D-H-S?

For instance, it's well-known that there's a spectral sequence computing the homotopy groups of real K-theory whose input corresponds to the moduli stack of quadratics and translations. This stack is supposed to parametrize the available multiplicative groups over some non-algebraically closed field, which provides some geometric insight into the problem. I'd like to know if there's some kind of geometry that corresponds to the controlling stacks that sit at the bottom of the D-H-S argument.

Note: Of course, a positive answer to this question as phrased might not mean much. The process that D-H-S uses to reduce to this much smaller Ext calculation is an extremely lossy one with the very clear intention of just getting at the existence of a vanishing line. The geometry of these bottom stacks may have very little to say about the geometry of the stacks we started with.

Answer 1

The best I can do at this point is give you some interpretation of the Hopf algebroids associated to $X(n)$.

We can't really give a complete description of the stacks associated to these Hopf algebroids for the simple reason that their homotopy groups are pretty uncomputable. However, we can say a little about the role these play in homotopy theory.

First, no matter what $R$ is, the Hopf algebroid in spectra $(R, R \wedge R)$ attempts to recover the sphere from its cobar construction, and if $R_*R$ is a flat $R_*$-module we get a Hopf algebroid $(R_*, R_* R)$ attempting to do the same. In the cases of $X(n)$ these spectral sequences converge, so in some sense we should think of $(X(n)_*, X(n)_* X(n))$ as just another presentation of the moduli of formal groups.

In a little more detail, let's consider $MU_* X(n)$. This is a subring $MU_*[b_1, b_2, \ldots, b_n] \subset MU_* MU$. The latter ring parametrizes pairs of a formal group law together with a strict isomorphism $f(x) = \sum b_i x^{i+1}$ out to another formal group law; the former ring is the subring parametrizing a formal group law together with the truncation with a strict isomorphism out determined only up to the first $n$ coefficients. This is a natural description in terms of the $MU_* MU$-coaction, and so you might think of $X(n)$ as being associated to a moduli $\mathcal{M}_{fg}(n)$ of "formal groups equipped with a coordinate, determined up to the n'th stage".

Even further, you can consider the map of Hopf algebroids $$(MU_*, MU_* MU) \to (MU_*, \pi_* (MU \wedge_{X(n)} MU)).$$ The latter computes the homotopy groups of $X(n)$ and is definitely the Hopf algebroid associated to the cover $Spec(MU) \to \mathcal{M}_{fg}(n)$.

The $X(n)$-based Adams spectral sequence is then something you might think of as being associated to a Cartan-Eilenberg type spectral sequence associated to a composite sequence of covering stacks $$ Spec(MU) \to \mathcal{M}_{fg}(n) \to \mathcal{M}_{fg}. $$