This post is speculative, and is meant to attempt to understand what the sphere spectrum is from this derived point of view.

$\DeclareMathOperator{\spec}{Spec}$ The sphere spectrum $S$ is absolutely essential to everything that’s done in homotopy theory – it’s the whole reason the field was created! $\newcommand{\Sp}{\mathrm{Sp}}$ It acts on everything, and in fact, modules over $S$ are exactly spectra. $\newcommand{\Mod}{\mathrm{Mod}}$ Categorically, we can write this as

$\newcommand{\QCoh}{\mathrm{QCoh}}$ We know another interpretation of the left hand side, coming from algebraic geometry; morally, it should be $\QCoh(\spec S)$. It is quite reasonable to ask what $\spec S$ is. For instance, what are its points? What is the structure sheaf?

# The thick subcategory theorem

$\newcommand{\cc}{\mathcal{C}}$ The thick subcategory theorem is a theorem that’s part of the Ravenel conjectures, that was proved by Hopkins-Smith. We’ll recall a proof of this result here. $\newcommand{\FF}{\mathbf{F}}$ Let’s recall a definition from homological algebra.

Definition: A subcategory of a triangulated category is thick if it’s closed under retracts, weak equivalences, and satisfies the 2-out-of-3 property for cofiber sequence.

Theorem (Hopkins-Smith): The only thick subcategories of the category of $p$-local finite spectra are the categories $\cc_n$ of $p$-local spectra that are $K(n-1)$-acyclic.

Proof: Let $\cc$ be any thick subcategory of the triangulated category of $p$-local finite spectra. We’ll show that any object $X$ in $\cc$ that’s in $\cc_n$ (with $n$ minimal) gives an inclusion $\cc_n\subseteq \cc$. Let $X$ be any other $K(n-1)$-acyclic spectrum that is in $\cc$, and let $DZ$ denote its Spanier-Whitehead dual. The identity on $X$ adjuncts to a map $S\to X\wedge DX$ (everything is $p$-local), and for $m\geq n$, we therefore have an injection $K(m)_\ast(S) \to K(m)_\ast(X)\otimes_{\FF_{p^m}[v_m^{\pm 1}]}K(m)_\ast(X)^\vee$. If $F$ is the fiber of $S\to X\wedge DX$, then this tells us that $K(m)_\ast(F) \to K(m)_\ast(S)$ is zero if $m\geq n$.

Now, the composite $F\to S\to Z\wedge DZ$ is zero on $K(m)$-homology for $m\geq n$, but this argument. Since $Z$ is $K(n-1)$-acyclic – this implies that it is $K(m)$-acyclic for $% $ – it follows that $K(m)_\ast F \to K(m)_\ast Z\otimes_{\FF_{p^m}[v_m^{\pm 1}]} K(m)_\ast DZ$ is zero for $% $. But if $K(n)_\ast(F)\to K(n)_\ast(Z\wedge DZ)$ is null for all $n$, it follows from nilpotence that $F^{\wedge k}\to (Z\wedge DZ)^{\wedge k}$ is null for some $k\gg 0$. The evaluation map from $(Z\wedge DZ)^{\wedge k}\to Z\wedge DZ$ therefore shows that $F^{\wedge k}\to Z\wedge DZ$ is null, i.e., the map $F^{\wedge k} \wedge Z\to Z$ is null. The cofiber of this map is thus $Z\wedge \text{cofib }(F^{\wedge k} \to S)\simeq Z\vee(Z\wedge \Sigma F^{\wedge k})$,

We know that $\text{cofib }(F\to S) = X\wedge DX$ is in $\cc_n$, so we use the cofiber sequence

to show by induction that $\text{cofib }(F^{\wedge i} \to S)\wedge Z$ is in $\cc$ for all $i$. Since $Z$ is a retract of this, we have $Z\in \cc$.

It’s easy to see that $\cc\subseteq \cc_n$ (since otherwise, $\cc$ would contain some spectrum that’s $K(m)$-acyclic for $% $, which contradicts minimality). This concludes the proof.

# What does this mean for us?

$\newcommand{\Sp}{\mathrm{Sp}}$ Let $\spec \Sp$ denote the the collection of thick subcategories $\cc$ of $\mathcal{D}$ which are prime ideals, in the sense that $C\otimes D\in\cc$ if $C\in \cc$ and $D\in\mathcal{D}$, and if $C\otimes D\in\cc$, then either $C\in\cc$ or $D\in\cc$. This theorem shows that the only elements in $\spec \Sp$ are the thick subcategories $\cc_{n,p}$, for every integer $n$ and every prime $p$. $\newcommand{\Z}{\mathbf{Z}}$ This means that we would expect $\spec \Sp$ to be $\mathbf{N}\times \spec \Z$ as sets.

$\newcommand{\Mfg}{\mathcal{M}_\textbf{fg}}$ This is an awful lot like $\Mfg$ – but it’s not, since

as triangulated categories. But perhaps this is true in the derived context?

In particular, I think the following proto-theorem should be true.

$\newcommand{\dMfg}{\mathfrak{M}_\textbf{fg}}$ Proto-Theorem: If a ‘‘derived moduli stack’’ $\dMfg$ of formal groups exist, then $\QCoh(\dMfg)\simeq \Sp$ as $\infty$-categories.

There is evidence against this theorem, though, as the legends tell us in this MO question. The Adams-Novikov spectral sequence (described below) not degenerating is supposed to be like $\dMfg$ differenting from the category of spectra.

Still, morally, one wants $\spec S$ to be as close as humanly (chromotopically?) possible to $\spec \Sp$, but if you’re to believe Lurie’s definition of a derived stack, presented in a previous blog post, the derived scheme $\spec S$ is given by the data of $\spec \mathbf{Z}$ – with the Zariski topology – and a structure sheaf of $\mathbf{E}_\infty$-rings. I think this is wrong: the underlying set of $\spec S$ doesn’t contain all the Morava $K$-theories, that should morally be there (by the thick subcategory theorem)!

$\newcommand{\GG}{\mathbf{G}}$ Let’s try to construct the derived moduli stack of formal groups. We can lift the infinitesimal neighborhood of a point; this is exactly the derived scheme $\text{Spf }E$ that we were considering in previous posts. Suppose that the proto-theorem was literally true, so that $\dMfg = (\Mfg,\mathcal{O}^\textbf{top})$. This would admit a map $\spec MUP \to \dMfg$, where $MUP = \bigvee_i \Sigma^{2i} MU$ is the spectrum of periodic complex cobordism. For any spectrum $X$, we’d then know that the (quasicoherent) sheaf $\pi_\ast (f_\ast(MUP\wedge X)\otimes \mathcal{O}^\textbf{top})$ on $\Mfg$ corresponds to the quasicoherent sheaf on $\Mfg$ associated to $X$, and that $\Gamma(\dMfg,f_\ast(MUP\wedge X)\otimes \mathcal{O}^\textbf{top}) \simeq X$. Then the descent spectral sequence for $X = S$ would run

$\newcommand{\Ext}{\mathrm{Ext}}$ A quasicoherent sheaf on $\Mfg$ is just a $(MU_\ast, MU_\ast MU) = (MUP_0, MUP_0 MUP)$-comodule, and the cohomology groups are then just $\Ext$ groups. In particular, the $E_2$-page would just be $\Ext^s_{MU_\ast MU}(MU_\ast, MU_\ast)$ – and now the spectral sequence is just the Adams-Novikov spectral sequence!

$\newcommand{\Eoo}{\mathbf{E}_\infty}$ Recall that maps $(R,\GG_R)\to (R^\prime,\GG_{R^\prime})$ over $\Mfg$ correspond to homotopy commutative maps $E_R \to E_{R^\prime}$, where $E_R$ and $E_{R^\prime}$ are the cohomology theories associated to $R$ and $R^\prime$ (I guess here I’m assuming that $\GG_R$ and $\GG_{R^\prime}$) are homotopy commutative. This is Proposition 1.2 in this paper of Paul Goerss. One feature of $\dMfg$ (that follows from the above discussion) would be that it’d classify maps between $\Eoo$-rings: maps $\spec A\to \spec B$ over $\dMfg$ would correspond to $\Eoo$-maps $B\to A$ – at least if $B$ and $A$ are $MU$-complete. But this is literally telling us that $\dMfg$ should be $\spec S$! This is some form of a functorial Landweber exact functor theorem.

If we were to go about trying to actually construct $\dMfg$, we would need to know that its underlying stack, i.e., $\Mfg$, is locally trivial in some Grothendieck topology. As we said in a previous post, the best topology to use when considering derived stuff is the etale topology. I don’t think it’s true, though, that $\Mfg$ is locally trivial in the etale topology. So:

Question: In what topology is $\Mfg$ locally trivial? What is the appropriate notion of a ‘‘derived stack’’ in that topology?

Here is a table that I made when trying to understand this story.

Note that with Lurie’s definition, the last line is not true, even for derived affine schemes! A fabulous reference for the right hand side, not in the derived context, is Goerss’ Quasicoherent sheaves over the moduli stack of formal groups, available on the arXiv.

In some sense, the whole goal of chromatic homotopy theory is to construct $\dMfg$.