@Akhil no problem! I'm pretty sure I was sitting behind you at the Dan Quillen memorial conference! :-)

Well, that's pretty much it, so I guess I should delete this question, haha.

Hey @Akhil I don't really know why, but the proof/explanation is in appendix B (specifically Cor. B.13) of Hovey and Strickland's "Morava K-theoriesand Localisation" which I believe can be found here: hopf.math.purdue.edu/Hovey-Strickland/kn.pdf This is also discussed briefly in Example 3.5.4(a) of Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" (math.rochester.edu/people/faculty/doug/otherpapers/…) although I'm sure there are people on this site who can explain it perhaps intuitively.

@Fernando okay great thanks! I sort of thought that might be the one I needed :-)

Thanks @Fernando! Is this somewhere in Neeman's book? Also, do you happen to know which set-theoretic principles must be assumed?

Haha, okay yeah, obvious. (Wondering what Ind category of coherent modules looks like....)

But one more question: you seem to be talking about only the finite spectra. Can this be extended to say, suspension spectra, or harmonic spectra?

Also thanks for that reference for the Smith paper! I couldn't find a copy of it anywhere.

Oh thanks!! yeah, I got the projective dimension situation from another theorem I was getting this mixed up with.

Presumably you mean that the objects $F(b)$ for $b$ an object of $B$ are all objects of $D$. Do you mean the same for the morphisms?

Hrm. Good point. Too bad. I do believe that the $K(n)$ localization of $X$ is equivalent to the $K(n)$ localization of $M_nX$, but I'm not certain that map can be lifted...

So in particular, it seems that any harmonic spectrum $X$ is weakly equivalent to a wedge product of its monochromatic slices?

I'm not entirely sure what "moduli of curves theory" is, but if I'm not mistaken, there are some pretty amazing applications of the moduli stack of elliptic curves in physics.