Take K theory and try killing p as a K-algebra. There’s a sseq that starts with the free associative algebra over K_* on a generator in degree 1, the differential takes that generator to p. The Leibniz rule says x^2 is a cycle, and it survives. So now you have this junk in degree 2, so you gotta kill that with a new E_1-cell! That puts new junk in degree 4... now keep doing that forever. Then you’ve built K(1) as a K-algebra with infinitely many E_1-cells; one in each even degree

By no coincidence at all, that’s also what the cell structure of CP^infty looks like (as we see by looking at the Thom spectrum construction)

regardless of the choices of the v_i's you make, is there a unique spectrum with homotopy K(n)_*? (and similarly for E(n)_*)

unique as a spectrum, that is, with no additional structure

Maybe I’m misinterpreting but couldn’t you just take a wedge of EM spectra?

ah, yes, of course. i probably want to ask for uniqueness as an MU-module

@DylanWilson Where can I read about this. In particular this spectral sequence, is it in HA somewhere?

Here for example: arxiv.org/abs/1805.07184

Also I should’ve said there’s cells in odd degrees- but it’s still modeled on CP^infty cuz these E_k stories have a shift by k...

@DylanWilson Thanks!

@skd As far as I understand you don't need to make a choice for the $v_i$'s in $BP$ and therefore in $E(n)$ (Johnson-Wilson), am I wrong?

we

(sorry about that, problematic keyboard...)