Topologized objects in algebraic topology

I attended MIT’s Talbot workshop last week, which was super fun and – at times – even productive. I had a couple facts about Morava $E$-theory come up repeatedly in side-conversations, and I think it’s worth sharing those with the rest of you now.

Here’s the point: continuous Morava $E$-theory is the only sort of Morava $E$-theory. Let me explain by taking a moment to recall where it comes from. The central thesis of chromatic homotopy theory is that the stable category is tightly bound to the moduli stack of $1$-dimensional, commutative, smooth formal groups via the homology theory $MU_*$ of complex bordism. Generally, to any ring spectrum $E$ we can associate a simplicial scheme given by the following fancy construction:

The assignment $X \mapsto E_* X$ can be thought of as sending $X$ to a quasicoherent sheaf over this simplicial scheme. In the case of $MU$, the associated simplicial scheme is equivalent to the moduli of formal groups, and various niceness results in algebraic topology say that this assignment is not too lossy — in many ways, the stable category behaves like $\mathrm{QCoh}(\mathcal{M}_{\mathbf{fg}})$.

Number theorists know quite a bit about formal groups, and we can shamelessly piggyback on their hard work to learn things about the stable category. For instance, after localizing at a prime, there are countably many geometric points on the moduli of formal groups, belonging to the Honda formal groups and enumerated by “height”. In stable homotopy theory, these geometric points are reflected by the following theorem:

(One of the Hopkins-Smith periodicity theorems:) There is a sequence of homology theories called Morava $K$-theories, which are complex oriented with formal group given by these Honda formal groups. Moreover, these give “exhaustive” list of field spectra, in the sense that every homology theory with natural Künneth isomorphisms is some kind of field extension of a Morava $K$-theory.

The next place to look for instruction from the number theorists after the points of $\mathcal{M}_{\mathbf{fg}}$ is in its local geometry. This analysis was carried out by Lubin and Tate, who found the following result:

(Lubin-Tate:) Every geometric point on $\mathcal{M}_{\mathbf{fg}}$ is smooth. The infinitesimal deformation space of the Honda formal group with height $n$ has dimension $(n-1)$.

This, too, is reflected in stable homotopy theory: associated to any formal group $G$ over a perfect field there is a spectrum $E_n$, called Morava $E$-theory, whose coefficient ring is given by the infinitesimal deformation space of $G$.

Morava $K$-theories have lots of nice properties, largely owing to their Künneth isomorphisms, but Morava $E$-theories also have lots of nice properties from the perspective of structured ring spectra, largely owing to the fact that they are mixed- rather than positive-characteristic. Of course, the $K$- and $E$-theory of a particular formal group are closely related to one another, and so a lot of algebraic topology gets done by using whichever is appropriate and transferring results back and forth as necessary.

This all sounds great, but there’s a wrinkle: everything in the above story works perfectly when studying cohomology theories, but there are two sorts of Morava $E$-homology people consider: ordinary and continuous. These are defined respectively by the following two formulas:

The left-hand formula is exactly what you’d expect, so why are we studying the right-hand formula? After all, it’s not even properly a homology theory — localization is not expected to preserve infinite wedges!

The literal meaning “homology theory” aside, the right-hand formula is in fact the one we want. To start to see why, let’s recall the following formula for $K$-localization: if $X$ is an $E$-local spectrum (and $X \wedge E$ is always so local), then we have the formula

where $M^0(I)$ is a finite spectrum with the property $BP_* M^0(I) = BP_* / (p^{I_0}, v_1^{I_1}, \ldots, v_{n-1}^{I_{n-1}})$. It is, again, a consequence of the periodicity theorems that such spectra exist for $I \gg 0$, along with the expected quotient maps used to build the inverse system.

This formula indicates what’s really going on: the infinitesimal deformation space of a point is always defined as a formal scheme, i.e., as the formal colimit of a directed system of finite schemes, each of which captures a “nilpotent of degree $m$” piece of the deformation space, with $m$ growing large. In the case of a smooth point, the deformation space is given by a power series ring, and this colimit amounts to remembering the adic topology on this ring. This is actually a huge deal — the formal spectrum of a power series ring (i.e., the spectrum with this topology taken into account) is what you’d expect: a single point, with structure sheaf at the point given by the power series ring. On the other hand, just the spectrum of a power series ring is terrifyingly large and repugnant — it has all sorts of prime ideals that you wouldn’t expect, on account of the existence of transcendental power series.

In turn, that’s what the extra $K$-localization is doing: it’s recalling that Lubin-Tate space (and hence Morava $E$-theory) is given as an ind-scheme (resp., as a pro-spectrum), and certainly we shouldn’t forget this crucial fact about our set-up. This extra step has all sorts of niceness consequences in algebraic topology; for instance, we have the following:

(Hovey-Strickland:) When $X$ is a spectrum such that $E^* X$ is even-concentrated, then $K^* X$ is given by reduction at the maximal ideal: $K^* X = E^* X / I_n$. Conversely, when $K^* X$ is even-concentrated, then $E^* X$ is pro-free with this same reduction property.

There is an analogue of this property for continuous $E$-homology $E^\vee_*$ but not for $E_*$. In fact, the failure of this for the noncontinuous version is dramatic: its Bousfield class is given by a wedge of Morava $K$-theories, with heights ranging from $0$ up to $n$. You can sort of see this coming — without continuity to control the image of the power series generators, you can imagine sending one of them to an invertible element in some target ring, thereby decreasing the height of the pulled-back formal group law. On the other hand, the Hovey-Strickland theorem says that the cohomological Bousfield classes of $K$- and $E$-theory coincide, as you would reasonably expect. That there is a Hovey-Strickland theorem for $K$-homology and continuous Morava $E$-homology says, among other things, that we’re specifically repairing this defect in the homological Bousfield class of $E$-theory.

OK, deep breath.

I think these are neat facts, coming from being careful about the insertion of formal geometry into algebraic topology. However, there’s a different fact that this perspective doesn’t quite jive with and which I would like to understand better: fracture squares. Chromatic fracture is a big deal in algebraic topology; its noble goal is to reconstruct higher chromatic spectra from chromatic layers and gluing data. It takes the form of the following pullback square:

implicitly noting that the Bousfield classes of $E$- and $K$-theory don’t depend upon the choice of formal group, but just of its height $n$. If you like, you can inductively expand this out entirely in terms of $K$-theoretic localizations, and you’ll find you’re taking a limit over a big punctured cube whose vertices are given by all the $K$-theoretic localizations applied in sequence, with variations on which localizations you skip. Moreover, this construction and theorem crucial — it’s how we chromatic homotopy theorists do everything, from understanding finite spectra to constructing $\operatorname{TMF}$.

That’s why it’s a puzzle that this doesn’t seem to play nicely with formal geometry. Suppose that $X$ is an $E$-theory; then one of the corners in this fracture cube takes the form $L_{K(t)} E_n$. The homotopy of this spectrum is computable, given noncanonically in choice of generators by:

What the hell is this thing? The first thing to say is that this doesn’t have an interpretation in terms of classical formal geometry: we’re inverting something in the maximal ideal (which is not that frightening; consider building $\mathbb{Q}_p$ from $\mathbb{Z}_p$) and then recompleting against a nonclosed ideal (which is frightening; it means the completion map is not continuous in the respective adic topologies). You might also worry that I claimed that this presentation is noncanonical in choice of generators, then inverted a particular element — but a surprising fact is that the recompletion erases this choice of inversion. Selecting different generators will give isomorphic rings after passing to the recompletion. This indicates that we are, in some sense, performing a geometric operation.

But what operation and what geometry?

I’ve spoken a little bit to Nat Stapleton about this, who studies (among other things) “transchromatic character maps,” which use in an essential way this height-jumping feature of this strange localization of $E$-theory. We have some ideas (more accurately: he has some ideas) about what moduli problem this ring presents, but our phrasing of this is seriously crippled by the lack of an algebro-geometric framework in which to work. I’m at a loss as to what to do — we’ve spoken to a few algebraic geometers about this, and the most promising thing they’ve suggested has been Huber’s theory of adic spaces, but even this doesn’t quite accomplish the task at hand.

Who knows.

To wrap it up, topologized objects appear elsewhere in stable homotopy theory as well. For instance, work of Ando-Morava-Sadofsky and of Ando-Morava use pro-systems of Thom spectra to do some pretty cool transchromatic things. The Tate construction can also be thought of in such a way, as can parts of Lin’s theorem / the Segal conjecture. So, these sorts things are all around, and given their collective mysterious nature, I feel that we really don’t have a good way of thinking about them yet.