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 Spec L iso to 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 $Spec HZ \simeq Spec Z$, 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 $Spec L$ (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.

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