Let $L\_E$ denote Bousfield localisation with repsect to the cohomology theory $E$. I am trying to follow through some calculations in Hovey-Strickland's paper [Morava $K$-theories and localisation](http://www.math.rochester.edu/u/faculty/doug/otherpapers/kn.pdf)


Claim 7.10(e) is that


$$L\_{K(n)}X = \underset{\leftarrow}{\text{holim}}\_I L\_{E(n)}X \wedge S/I$$


where the homotopy limit is over a tower of generalised Moore spectra (the Moore spectra $S/I$ is defined in 4.12 and the tower in 4.22)


Define
$$E\_\*^{\vee}:=\pi\_{\*} (L\_{K(n)} (E \wedge X))$$


the $K(n)$-local version of Morava $E$-theory (where, I believe, Morava $E$-theory here is what I might call a completed Johnson-Wilson theory, but I don't believe it really matters).


The claim (8.04) is then that we can extract a Milnor exact sequence from this:


$$0 \to \varprojlim\_I {}^1 (E/I)\_{\ast+1}(X) \to E\_\*^\vee X \to \varprojlim\_I (E/I)\_\*X \to 0 $$


I'm not sure how to show this. I would like to think that we can get a sequence


$$0 \to \varprojlim\_I {}^1 \pi\_{\ast+1}(L\_{E(n)}(E \wedge X) \wedge S/I)\to E\_{\ast}^\vee X \to \varprojlim\_I \hspace{1mm} \pi\_\* (L\_{E(n)}(E \wedge X) \wedge S/I) \to 0 $$


and then if you drop the $E(n)$-localisation it seems to work, but I'm not really sure about this. 



Milnor exact sequence in $K(n)$ local Morava $E$-theory
In this case, this follows because $E(n)$-localization is a smashing localization. Specifically,
$$
L\_{E(n)}(X) = L\_{E(n)}(\mathbb{S}) \wedge X
$$
for any $X$. In particular, this means that the smash product of any spectrum with an $E(n)$-local spectrum is already $E(n)$-local.


The fact that $E(n)$-localization is smashing was one of the Ravenel conjectures; specifically, conjecture 10.6 in his paper "[Localization with respect to certain periodic homology theories](http://www.math.rochester.edu/u/faculty/doug/mypapers/loc.pdf)". I don't have a copy of his orange book handy and so can't give you a reference for the theorem statement.


As a result, as the Morava $E$-theories are $E(n)$-local already, so are the smash products $E \wedge X$ in your equation.


