What does the sphere spectrum have to do with formal group laws?

This post assumes that you’re familiar with the definition of a prime ideal, a local ring, (R{(p)}), the sphere spectrum, (\mathbb{S}), and the Lazard ring, (L)._

During a talk Jacob Lurie gave at Harvard this April, he labeled the moduli space of (1-d commutative) formal group laws as (\text{Spec }\mathbb{S}).

Eric Peterson kindly explained why (\text{Spec }\mathbb{S} \simeq \text{Spec }L) and I found his answer so lovely that I wish to share (all mistakes are due to me).

Why is (\text{Spec }L) iso to (\text{Spec }\mathbb{S})?

This is part of the story of geometers working with higher algebra asking “what is an ideal of a ring spectrum?”

A ring (R) ——————-> category (Mod_R \supseteq Perf_R) (finitely presented)

Note that (Perf_R) is the category of perfect complexes of (R)-modules. A perfect complex of (R)-modules is a chain complex of finitely generated projective (R)-modules (P_i), and is thus of the form \(0 \to P_s \to … \to P_i \to 0\)

The ring spectrum (\mathbb{S}) ——————-> category (Mod_{\mathbb{S}} \supseteq Perf_{\mathbb{S}})

Note that (Mod_\mathbb{S} \simeq) Spectra, and (Perf_\mathbb{S} \simeq) Finite Spectra

A finite spectrum is a spectrum which is the de-suspension of (\Sigma^\infty F), where (F) is a finite CW-complex.

There’s a theorem by Balmer answering “what is an ideal in this context”, which points out this analogue:

(\text{Spec }R) as a space; (p) as a point (an element of (\text{Spec }R))

(Perf_R) as a space; (\mathcal{P}) as a point (a subcategory of (Perf_R))

satisfying that (\mathcal{P}) is:

  1. (\otimes)-closed against R-modules \(a \in Perf_R; b \in \mathcal{P} \Rightarrow a \otimes b \in \mathcal{P}\)
  2. a thick subcategory of (Perf_R) (i.e., it’s closed under cofiber sequences and retracts i.e., closed under extension)

A “prime ideal” of (Perf_R) is a “proper thick tensor-ideal” (P) ((\subsetneq Perf_R)) s.t.

\(a \otimes b \in \mathcal{P} \Rightarrow a \in \mathcal{P} \text{ or } b \in \mathcal{P}\)

So, if (K_(-)) is a homology theory with Künneth isomorphisms \(K_(X \wedge Y) \simeq K_(X) \otimes_{K_} K_*(Y))\)

(\Rightarrow \mathcal{P} = {X K_*(X) = 0}) must be a “prime ideal”.

Sanity check:

\begin{align} K_(X \wedge Y) & \simeq K_* X \otimes K_Y \ & \simeq 0 \otimes K_Y \simeq 0 \ \end{align*}

Here’s the surprising theorem that ties this prime ideal excursion into our original question (Periodicity Theorem: Hopkins and Smith):

  1. Any (C \subset Perf_{\mathbb{S}}) arises in this way
  2. All homology theories with Künneth isomorphisms are Morava K-theories including (Hk) where (k) is a field, which is just the infinite Morava K theory (K(\infty){(p)})._

The proof of this is currently beyond my grasp, so I’m afraid I can’t talk you through it.

Taking this theorem’s proof as a black box, we’ve scraped together enough context to parse the answer of why (\text{Spec }L \simeq \text{Spec }\mathbb{S}).

Let’s look at (\text{Spec }Z):

Screen Shot 2015-05-01 at 1.33.04 AM

let’s look at the residue classes of (Z):

Screen Shot 2015-05-01 at 1.33.00 AM

and at (\text{Spec }HZ \simeq \text{Spec }Z), where the ring spectra (HR) represent (H^*(-;R));

Screen Shot 2015-05-01 at 1.32.56 AM

By the nilpotence theorem, the ideals of (\mathbb{S}) are the Morava K’s (one for each height and each prime)…

2015-04-30 18.57.27

…so, (\text{Spec }\mathbb{S}) looks like ( \text{Spec }L) (by a theorem of Lazard, 1-d formal group laws over separably closed fields of char p are classified up to iso by their height).

To be absolutely clear: for (K(n)_{(p)}); ((p)) corresponds to the characteristic of the field (over which the formal group law is defined), while (n) corresponds to the height of the formal group law.

Afternote:

A comment of Lennert Meier’s on MO caught my interest. He mentioned that as the spectrum (\ell) (associated to a supersingular elliptic curve) is Bousfield equivalent to (K(0) \vee K(1) \vee K(2)) (with an implicit localization at a prime), we have (\ell_*(K(A,n)) = 0) for (A) finite abelian and (n \geq 3).

Note that (F) and (E) are Bousfield equivalent if for every spectrum (X: F_(X)) vanish iff (E_(X)). This is an equivalence relation on spectra.

Any elliptic cohomology is Bousfield equivalent to a wedge of Morava K-theories. Before we discard looking at individual elliptic cohomology theories, in favor of their “universal” counterpart with nice automorphisms, let’s look at the difference between (K(0) \vee K(1)) and complex K-theory, and try to lift these differences to those of (K(0) \vee K(1) \vee K(2)) and supersingular (\ell). It was pointed out to me that this is like comparing a local ring to its residue field.

To compute the Atiyah Hirzebruch spectral sequence of (E^*(X)), we need to know both the attaching maps in the space (X), and the attaching maps in the spectrum (E) (which I believe are called its Postnikov tower), both are hard (in most cases).

We currently only know how to compute the AHSS of (\ell^*(X)) when we have some map from (CP^\infty \to X) (since we define (\ell) using (CP^\infty)), for this map induces a map between spectral sequences.

Written on April 30, 2015