Morava modules and completed $E$-homology

Question

Let $E = E_n$ be the $n$-th Morava $E$-theory and let $\{ M_{I} \}$ be a tower of generalised Moore spectra. Then (see this previous question) there is a Milnor exact sequence

$$0 \to \varprojlim_I {}^1 (E \wedge M_I)_{\ast+1}(X) \to E_*^\vee X \to \varprojlim_I (E \wedge M_I)_*X \to 0. $$

Here we write $E_*^\vee X$ for $\pi_*L_{K(n)}(E \wedge X)$.

The functor $\mathcal{K}_*X := \varprojlim_I (E \wedge M_I)_*X$ is sometimes called the Morava module of $X$, and seems to have first been introduced by Hopkins, Mahowald and Sadofsky whilst studying the $K(n)$-local Picard group.

In Hopkins-Mahowald-Sadofksy, under the condition that $\mathcal{K}_*X$ is finitely-generated, the authors show that the $E_2$-term of the $K(n)$-local $E_n$-based Adams spectral sequence can be given by $H_c^\ast(\mathbb{G}_n,\mathcal{K}_*X)$, where $\mathbb{G}_n$ is the $n$-th (extended) Morava stabilizer group.

It is known that the $\varprojlim{}^1$ term of the Milnor exact sequence vanishes under suitable conditions; for example if $E_*^\vee X$ is finitely-generated, or pro-free, as an $E_*$-module. In these cases then $E^\vee_*X \simeq \mathcal{K}_*X$. I suspect that the above spectral sequence should have $E_2$-term $H_c^\ast(\mathbb{G}_n,E^\vee_*X)$. This leads to the following question:

If $\mathcal{K}_*X$ is finitely-generated is there an isomorphism $\mathcal{K}_*X \simeq E^\vee_*X$?

By the comments above it would suffice to show that $E^\vee_* X$ is finitely-generated. Hovey-Strickland Proposition 8.6 gives a number of equivalent conditions, including that $X$ is $K(n)$-locally dualisable or that $K_*X$ is finite.

Answer 1

Here are some initial steps, which may be useful.

First, consider $A=L_{K(n)}(E\wedge X)$, which is a $K(n)$-local $E$-module with $\pi_*A=E^\vee_*X$. It is not hard to see that $A\wedge M_I=E\wedge X\wedge M_I$. Given this, we see that we have a short exact sequence $$ \lim_I{}^1\pi_{*+1}(A\wedge M_I) \to \pi_*(A) \to \lim_I\pi_*(A\wedge M_I), $$ and we want to know whether the middle term is finitely generated, assuming that the last term is. All this makes sense for an arbitrary $K(n)$-local $E$-module $A$, and I suspect that it is easiest to work in that generality. In particular, we can replace the operation $A\mapsto A\wedge M_I$ with $A\mapsto A\wedge_EE/I$. This is advantageous because we can construct $E/I$ for any ideal of the form $(u_0^{i_0},\dots,u_{n-1}^{i_{n-1}})$, and it is easy to analyse the maps between these objects.

Next, recall the category of $L$-complete $E_0$-modules described in Appendix A of Hovey-Strickland. Results given there suffice to show that all $E_0$-modules arising in any plausible approach to the above problem will be $L$-complete. Note in particular that if $P$ is $L$-complete then the map $P=L_0P\to\widehat{P}=\lim_IP/IP$ is always surjective.

I am not sure exactly where to go from here, but I would suggest doing the case $n=1$ first. There we need to show that if $\lim_k\pi_*(A/p^k)$ is finitely generated, so is $\pi_*(A)$. The obvious approach is to compare the tower $\{\pi_*(A/p^k)\}$ with the towers $\{\pi_*(A)/p^k\}$ and $\{\text{ann}(p^k,\pi_{*-1}(A))\}$.