# Does virtual Morava K-theory have an Eilenberg-Moore spectral sequence?

In a recent question, Tim Campion was interested in analyzing the Morava $$K$$–theory of a space $$X$$ by dissecting the space into connective and coconnective parts: $$X(m, \infty) \to X \to X[0, m].$$ It is a consequence of work of Ravenel and Wilson that this dissection is "stable": for a fixed height $$d$$ and for $$m \gg d$$, the natural maps $$X(m, \infty) \to X[m, \infty)$$ and $$X[0, m] \to X[0, m)$$ induce $$K$$–equivalences, hence the decomposition is independent of the precise value of $$m$$. We refer to these respectively as the "top" and "bottom" parts of (the $$K$$–homology of) $$X$$.

The bottom part is very amenable to study: for instance, Hopkins, Ravenel, and Wilson give a hefty set of tools for its analysis. The top part is quite a bit more curious, but a handful of things about it are known. As part of exploring the $$K$$–theory of infinite loopspaces, Kuhn dubs the top part as the "virtual $$K$$–homology of $$X$$", and he shows (among other things) that if $$X$$ is an infinite loopspace, there is a relationship between its virtual homology and the $$K$$–homology of the free $$E_\infty$$–ring on (the connective spectrum associated to) $$X$$.

In a subsequent question, Tim asked about the behavior of $$K$$–theory on loopspaces. Although it has a negative answer as stated, that answer draws from this same circle of ideas. Bauer addressed the behavior of applying the loopspace functor to the bottom part: provided that $$\pi_{d+1} X = 0$$ and that $$\pi_* X[0, m]$$ is finite, there is a natural convergent spectral sequence $$\operatorname{Cotor}_{*, *}^{K_* X[0, m]}(K_*, K_*) \Rightarrow K_* \Omega(X[0, m]).$$ (The negative answer then comes about from falsifying these hypotheses.)

In an effort to pair these two threads of inquiry, let's grant these two conditions on the homotopy of $$X$$, but let's then try to analyze $$K_* \Omega X$$ rather than $$K_* \Omega(X[0, m])$$. A decomposition of $$X$$ into connective and coconnective parts induces a decomposition of the loopspace: $$\begin{array}{ccccc}\Omega(X(m, \infty)) & \to & \Omega X & \to & \Omega(X[0, m]) \\ || & & || & & || \\ (\Omega X)(m-1, \infty) & \to & \Omega X & \to & (\Omega X)[0, m-1].\end{array}$$ Bauer's result allows us to visit information about the bottom part of $$X$$ upon the bottom part of $$\Omega X$$. What about the respective top parts?

Question: Take $$X$$ to satisfy $$X = X(m, \infty)$$ for $$m \gg d$$. Is there a convergent spectral sequence of signature $$\operatorname{Cotor}_{*, *}^{K_* X}(K_*, K_*) \Rightarrow K_* \Omega X?$$ (That is: given information about $$K_* X(m, \infty)$$, is there an analogous mechanism for visiting it upon $$K_* (\Omega X)(m-1, \infty)$$?)

I'm happy for the answer to be "no"; I'm not stellar at computing $$\operatorname{Cotor}$$, and it would be nice to see a carefully worked counterexample. I’d be happy to assume, as Tim does, that $$X$$ has a delooping (or several deloopings) if that were to change things. If "yes", then I'd also be happy to hear about the ensuing extension problem, but I wouldn't know what to even formulate about it without understanding the behavior of the highlighted question first. (I’m also happy to get an answer to a question I ought to be asking instead, should a reader be able deduce that from context.)

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- To head off any confusion, I believe that "virtual homology" is not actually a homology theory: the virtualization of the cofiber (resp. coproduct) need not agree with the cofiber (resp. coproduct) of the virtualizations. I haven't carefully checked the effect of either of these on $$K$$–theory, though—it could be that all these difference somehow end up being $$K$$–acyclic, but this would be seriously startling and is not at all my expectation.