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. 



Stable homotopy category and the moduli space of formal groups
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.


