one of the periodicity theorems is that a spectrum lies in the category of K(n)-acyclics iff it has a v(n+1)-self-map. since the sphere is rationally nontrivial, it itself doesn't have vn elements in its homotopy groups
but it does have elements which are themselves interrelated by things that look like vn maps. a type n spectrum is built out of some cells and some attaching maps, and the fact that it carries a vn-self-map is a statement about some kind of interrelationship between those attaching maps and vn
well, no no, don't talk about the ASS. like, what is it that you don't want to talk about?
i've been thinking about whether or not one can define a topology on "spec(S)" (whatever that is) by looking at ring spectra S--->R for which the R-ANSS converges.
i.e. thinking of that as a definition of "flat" cover
but then, i'm not sure... like, there are some weird things going on, for instance, what precisely the ANSS converges to.
because if it's the R-localization, then that sort of makes sense... but the R-completion... well, wtf?
on cohomology there's a surjection $H^* HF_2 \to \Sigma^{-n} H^* K(F_2, n)$ - I think you're just quotienting out by the monomials with too much excess
