This is sort of a two part question:


1) In one construction of $BP$, the Brown-Peterson spectrum, one uses this idempotent map of Quillen's, $g:MU\_{(p)}\to MU\_{(p)}$ and from what I can tell, the image of this map is $BP$. I have not worked through the details of this carefully. I guess the main idea is just using Brown representability on the cohomology theory $g^\ast(MU^\ast\_{(p)}(-))$. Anyway, what happens if, instead of looking at Quillen's idempotent map, we just look at the localization map $MU\to MU\_{(p)}$? Would the cohomology theory and formal group law thereby produced give us the same information as $BP$ but just in a much more unwieldy form?


2) In general, what effect does localization (at a general homology theory) have on complex orientability, and are there interesting cases in which such orientability is preserved and produces interesting formal groups?



Formal group laws arising from localizations of MU
Regarding (1), since the localisation map from $\mathbb{Z}$ to ${\mathbb{Z}}\_{(p)}$ is injective, and $MU\_\ast$ is free over $\mathbb{Z}$, localisation $MU \to MU\_{(p)}$ will be injective, and so the image of the localisation map will again be $MU$.


Regarding $(2)$, localisation does preserve complex orientations -- a class $u \in E^2(\mathbb{C} P^\infty)$ is an orientation if its restriction to $S^2$ is a unit in $E^2(S^2) = E\_0$. Since the map from a ring into a localisation of the ring carries units to units, the same will hold for any localisation of $E$.


This procedure doesn't in some sense make the cohomology theory more interesting, but rather less interesting, as localisation usually kills some of the difficulties in the algebra of the original cohomology theory. But that in itself is perhaps interesting.


