Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete

Question

Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory.

Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (in the sense of Hovey-Strickland).

I looked at the results in Appendix $A$ of Hovey-Strickland, but I was not able to prove or disprove this statement. In particular using the fact that $K(n)$-localization is equivalent to taking limit over generalized Moore spectra, we get $$ \lim_I{}^1\pi_{*+1}(M\wedge S/I) \hookrightarrow \pi_*(M) \twoheadrightarrow \lim_I\pi_*(M\wedge S/I). $$ But, I am not sure why $\lim_I\pi_*(M\wedge S/I)$ and $\lim_I{}^1\pi_{*+1}(M\wedge S/I)$ are $L$-complete.

Any references or counter-examples are appreciated. Thank you.

Answer 1

The identity map of $S/I$ satisfies $I_n^r.1_{S/I}=0$ for $r\gg 0$, so $I_n^r.\pi_k(M\wedge S/I)=0$ for $r\gg 0$, so $\pi_k(M\wedge S/I)$ is $L$-complete. The spectra $S/I$ can be assembled into a tower, and for towers both $\lim$ and $\lim^1$ of $L$-complete modules are $L$-complete by Theorem A.6(g) of the cited memoir. Moreover, extensions of $L$-complete modules are $L$-complete, by Theorem A.6(e). It follows that $\pi_*(M)$ is $L$-complete.