Let $H^\*$ and $K^\*$ be two cohomology theories, and $X$ a reasonable spectrum. Here, I'm thinking that $H^\*$ is singular cohomology (and for my purposes, rational cohomology will suffice), and $K$ is a Morava K-theory.


Suppose that I understand $H^\*(X)$ and $K^\*(X)$. Is there any way to use this (or other!) information to get a handle on $H^\*(L\_{K} X)$?, the $H$-cohomology of the Bousfield localisation of $X$ with respect to $K$?


I know that $L\_K X$ may sometimes be defined as the totalisation of a cosimplicial spectrum $K^{\bullet +1} \wedge X$. Does the associated Eilenberg-Moore spectral sequence have good convergence properties? This seems doubtful in the case where $H$ is rational cohomology and $K$ Morava K-theory; I think that the $E\_1$ term of the spectral sequence is $0$, but it's certainly the case that some $K$-localisations have torsion free homotopy.



Homology of localisations of spectra
I'm going to phrase in terms of $H$-homology instead of cohomology.


* If $H$ is a finite spectrum, smashing with it always commutes across the limit. More generally, if you express $H$ as a (homotopy) colimit of finite spectra -- eg, by taking a cellular filtration -- then you get a directed system of spectral sequences, and the groups you're looking for are the direct limits of the abutments. However, in most circumstances you can't compute this via the direct limit of the spectral sequences and need to be sensitive to interchanging the direct and inverse limits. This might work if you can get a very tight computational handle on things.
* If $H$ is a spectrum of finite type (connective and with finitely generated homology groups, or equivalently having a CW model with only finitely many cells in each dimension), then smashing with $H$ commutes across inverse limits of towers where there is some $r$ so that all the terms are $r$-connected. This doesn't apply in your circumstance, but would often apply when $K$ is connective.
* Let's take an opposing case where $H = H\mathbb{Q}$ and $K = H\mathbb{F}\_p$. Then the spectral sequence you're describing is the Adams spectral sequence and you certainly can't pass rationalization in through the spectral sequence. However, you *can* often get at the rationalization of the target because you're localizing with respect to an operation (multiplication by p) which lifts to an operation in the spectral sequence that shifts filtration (sometimes called multiplication by $h\_0$). A paper of Miller's ("On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space") introduces these kinds of techniques, and they also showed up in Mahowald-Ravenel-Shick's paper on the telescope conjecture. Unfortunately integral homology can't be expressed as one of these,




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Now let's get to some details about the more specific version that you asked.


Since $K(n)$-localizations are always $E(n)$-local, and $E(n)$-localization is smashing, $H \wedge L\_{K(n)} X \simeq L\_{E(n)} H \wedge L\_{K(n)} X$.


Further, the map $E(n)\_\* H\mathbb{Z} \to E(n)\_\* H\mathbb{Q}$ is an isomorphism: the former is the universal ring where formal group law of $E(n)\_\*$ obtains a logarithm, and that ring is rational. (Otherwise the ideal $(p)$ would give a nontrivial quotient, but that quotient is $E(n)\_\* H\mathbb{F}\_p = 0$.) Moreover, $H\mathbb{Q}$ is $E(n)$-local, and so $L\_{E(n)} H\mathbb{Z} \simeq H\mathbb{Q}$.


EDIT (some elaboration): The spectrum $E(n) \wedge H\mathbb{F}\_p$ has a complex orientation inherited from $E(n)$, and one from $H\mathbb{F}\_p$. This gives the coefficient ring $E(n)\_\* H\mathbb{F}\_p$ two formal group laws which must be isomorphic. The first has $p$-series satisfying $[p](x) = v\_k x^{p^k}$ mod $(v\_0,\cdots,v\_{k-1})$; $v\_0$ is $p$ by convention and $v\_n$ is a unit. The second has $p$-series $[p]'(x) = 0$. A strict isomorphism $f$ implies $[p](x) = f^{-1} [p]' f(x) = 0$. This inductively implies that $v\_k$ maps to zero for all $k$, and thus the unit $v\_n$ is zero; the entire ring $E(n)\_\* H\mathbb{F}\_p$ must be zero. 


EDIT (correcting accidental switch to cohomology): Therefore, the integral homology of the spectrum you're asking about is actually $H\mathbb{Q}\_\* L\_{K(n)} X$, and the universal coefficient theorem implies that the cohomology is given by applying $Ext(-,\mathbb{Z})$ to this (with a shift). This reduces the problem to the chromatic splitting conjecture that Eric described (i.e. you may want to stop working on it).


