# 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 $$\text{Spec }Z$$:

let’s look at the residue classes of $$Z$$:

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

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

…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.