@skd I'd like to give an argument something like the following: if it doesn't have at least 2^n cells, then its BP-homology couldn't be complicated enough to be v_0, ..., v_{n-1} torsion because of some kind of homological dimension argument. what I'd probably try to do is argue that BP<n-1>_* X is finite-dimensional and inductively try to argue that this couldn't be the case if the number of generators of H_* X wasn't a multiple of 2^n, using some kind of iterated Bockstein argument
however, i couldn't make that last argument fly easily and I didn't have any interesting enough applications to make it work
Actually, I think you can just apply Proposition 6.3 of Hovey-Strickland directly: If E were compact, then take X = L_{n}S^0 and Y = E, to deduce that E_* is countable, which it isn't
There is probably a direct way to see that it can't be a retract of such an L_nY though
@JonathanBeardsley I believe so, If you take your abelian category $A$, then chain complexes on $A$ form a symmetric monoidal model category $K(A)$, and the derived category is the homotopy category of $K(A)$. This is covered in Chapter 4 of Hovey.... provided this is what you meant?
I think there end up being questions about this localization preserving monoidal structure... but I imagine it works. I just feel like there are classical references for this I don't know.
@JonathanBeardsley It would be interesting to know for example when left or right Bousfield localisations preserve the monoidal structure. My guess is left localisations follows for basically the same reasons, but right localisations are probably trickier!