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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?

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  • $\begingroup$ 2) is obvious if the localization map is a map of ring spectra. A complex orientation of $E$ is the same as a map of ring spectra $MU \to E$. $\endgroup$
    – Sean Tilson
    Commented Jul 6, 2012 at 16:30
  • $\begingroup$ @Sean, localization always preserves ring structure, so does that necessarily mean the localization map is a map of ring spectra? And I guess I'm asking if there are any interesting formal group laws that come from precisely such maps, though it appears not. $\endgroup$
    – Jonathan Beardsley
    Commented Jul 6, 2012 at 20:04

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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.

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  • $\begingroup$ @Craig Does a localization of the spectrum always induce an actual localization of the cohomology rings? $\endgroup$
    – Jonathan Beardsley
    Commented Jul 6, 2012 at 20:05
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    $\begingroup$ @Jon, do you mean the localisation of the homotopy rings, rather than cohomology rings (that's what we're doing above with $MU$)? If so, in the generality of your original question (i.e., localisation at a homology theory), I think that the answer is no. For instance, localisation at Morava K-theories tends to introduce unexpected negative homotopy groups, even to connective spectra. Localising at a prime, however, is much more reasonable, and has the desired effect on $MU$. Indeed, Sullivan's original localisation of spaces was constructed by localising their homotopy. $\endgroup$
    – Craig Westerland
    Commented Jul 7, 2012 at 21:36

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