Stable homotopy category and the moduli space of formal groups

Question

The usual disclaimer applies: I'm new to all this stuff, so be gentle.

It seems like the spectrum, as defined by Balmer, of the stable homotopy category of finite complexes is something like $M_{FG}$, the stack of formal groups (that is, $Spec L/ G$ where $L$ is the Lazard ring and $G$ acts by coordinate changes). I'm not actually sure if that's true, I don't think I've seen it written quite like that, but the picture of the spectrum in Balmer's paper looks an awful lot like how I'd imagine $M_{FG}$ looking.

If the above is right, then there's another tensor triangulated category with the same spectrum, namely the derived category of perfect complexes on $M_{FG}$ (whatever that means for stacks...).

So my question is:

Just how far away is the stable homotopy category from actually being equivalent to this derived category? Is there a theorem to the effect that it can't be equivalent to such a thing? Do we even know that it's not equivalent?

I've heard that chromatic homotopy theory is about setting up a rough dictionary between algebro-geometric terminology regarding $M_{FG}$ and the stable homotopy category, so I guess the question is about whether or not we can make the dictionary into a proper functor.

Answer 1

One useful thing to keep in mind is that the cohomological functor from the stable homotopy category to the category of quasi-coherent sheaves on the moduli stack $\mathcal{M}$ is not essentially surjective. For example, if you fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ consisting of formal groups over $\mathbb{F}_p$ having height $\geq n$. A standard problem in stable homotopy theory is to try to cook up finite spectra which map to the structure sheaf of $\mathcal{M}^{\geq n}$. You can generally only do this when $p$ is large compared with $n$. For small values of $p$ you generally have to make do with finite spectra whose image is the structure sheaf of some nilpotent thickening of $\mathcal{M}^{\geq n}$. These can always be found (a deep result of Devinatz-Hopkins-Smith) and this is what gives you such a strong connection between the topology of $\mathcal{M}$ and the "spectrum" of the stable homotopy category. But you have to work hard for it, and the connection is much weaker (closed subsets of $\mathcal{M}$ have an interpretation in the stable homotopy category, rather than closed substacks) than what you would expect if Adams-Novikov spectral sequences were to degenerate.

Answer 2

The relationship between the derived category of $M_{FG}$ and the stable homotopy category $\mathcal{S}$ is somewhat like the relationship between $D(gr_I(R))$ and $D(R)$, where $R$ is a commutative ring, and $I$ is an ideal in $R$, and $gr_I(R)=\bigoplus_nI^n/I^{n+1}$ is the associated graded ring. Complex cobordism gives a homological functor from $\mathcal{S}$ to the abelian category of quasicoherent sheaves on $M_{FG}$, but there is no useful functor from $\mathcal{S}$ to the associated derived category $D(M_{FG})$. Computationally, the morphism groups in $\mathcal{S}$ are the target of an Adams-Novikov spectral sequence whose $E^2$ page can be described as morphism groups in $D(M_{FG})$.

Answer 3

It's definitely known that the derived category of ${\cal M}_{FG}$ and the stable homotopy category are not equivalent. This is an instance of

The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far from the actual answer.

(The naming is due to Ravenel; this quote is from Paul Goerss' "The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres".) There is often a feeling that stable homotopy theory always deviates from algebra as soon as is possible.

As Neil said, the Adams-Novikov spectral sequence starts with morphisms in the derived category and computes stable homotopy groups of spheres. Every place where this spectral sequence does not degenerate indicates a point where the stable homotopy category deviates from the algebraic approximation. This includes the following phenomena.

• Hidden additive extensions, such as the hidden additive extension making $\pi_3^s$ into $\mathbb{Z}/24$ rather than $\mathbb{Z}/12 \times \mathbb{Z}/2$.

• Hidden multiplicative extensions. In the (2-local) stable homotopy groups there are elements $\eta \in \pi_1^s$, $\nu \in \pi_3^s$, and $\sigma \in \pi_7^s$. My recollection is that such that $\eta^2 \sigma = \nu^3$ on the $E_2$-term, but Toda showed that this relationship doesn't hold on-the-nose in stable homotopy groups of spheres.

• Differentials. For any prime $p$, there is always a nontrivial differential in the Adams-Novikov spectral sequence, and the first differential is called the Toda differential.