# Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum

We fix $$p$$ prime and $$n$$ a natural number. We let $$K(n)$$ be the $$2(p^{n}-1)$$-periodic Morava $$K$$-theory, i.e. $$K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$$ with $$|v_n|=2(p^n-1)$$. I distinctly recall that we should have $$\pi_*(L_{K(n)}BP)\cong (v_n^{-1}BP_*)^{\wedge}_{I_n}$$, yet I am unable to find an explicit reference in the literature to this fact. Do you have any idea where I can find the proof of such computation?

Also, I was wondering if we apply additional localizations with respect to these Morava $$K$$-theories this behavior continues. E.g. for $$m do we have $$\pi_*(L_{K(m)}L_{K(n)}BP)\cong (v_{m}^{-1}(\pi_*(L_{K(n)}BP)))^{\wedge}_{I_m}$$ and so on?

See Lemma 2.3 of the following paper, and the surrounding discussion:

@incollection {MR1320994,
AUTHOR = {Hovey, Mark},
TITLE = {Bousfield localization functors and {H}opkins' chromatic
splitting conjecture},
BOOKTITLE = {The \v{C}ech centennial ({B}oston, {MA}, 1993)},
SERIES = {Contemp. Math.},
VOLUME = {181},
PAGES = {225--250},
PUBLISHER = {Amer. Math. Soc., Providence, RI},
YEAR = {1995},
MRCLASS = {55P42 (55N20 55N22 55P60)},
MRNUMBER = {1320994},
DOI = {10.1090/conm/181/02036}}


Essentially the same argument shows that if $$E$$ is complex-orientable then $$L_{K(n)}E=\text{holim}_k v_n^{-1}E/I(k)$$, where the $$I(k)$$ are a sequence of ideals of the form $$(v_0^{i_0},\dotsc,v_{n-1}^{i_{n-1}})$$. Thus, if the sequence $$v_0,\dotsc,v_{n-1}$$ is regular on $$\pi_*(E)$$, we find that $$\pi_*(L_{K(n)}E)=(v_n^{-1}\pi_*E)^{\wedge}_{I_n}$$. This can be applied recursively to calculate the homotopy groups of $$L_{K(n_1)}\dotsb L_{K(n_r)}BP$$ whenever $$n_1<\dotsb.