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

Question

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 $m<n$ 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?

Answer 1

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}}

There is a copy at https://people.math.rochester.edu/faculty/doug/otherpapers/chromatic-splitting.pdf

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 $n_1<\dotsb<n_r$.