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?



Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum
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 $n\_1<\dotsb<n\_r$.


