Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be [Morava $K$-theory](https://en.wikipedia.org/wiki/Morava_K-theory) of height $h$. Recall that the coefficients $K(h)\_\ast$ are a graded field, and $K(h)\_\ast(X)$ is a finite-dimensional vector space over this graded field. Denote $|X|\_h := dim\_{K(h)\_\ast} K(h)\_\ast (X)$.


**Question:** If $h \leq h'$, then do we have $|X|\_h \leq |X|\_{h'}$?


In some special cases, the answer is *yes*:


* When $X = \mathbb S\_{(p)}$ the answer is *yes*: we have $|\mathbb S\_{(p)}|\_h = 1$ for all heights $h$.
* When $h' = \infty$, the answer is *yes*. Here, $K(\infty) = H\mathbb F\_p$ and $|X|\_\infty$ counts the number of cells of $X$. The "yes" answer is straightforward to show by induction on the number of cells.
* If $|X|\_{h'} = 0$, then the answer is *yes*. That is, $|X|\_{h'} = 0 \Rightarrow |X|\_{h} = 0$ for $h \leq h'$. This follows from the [thick subcategory theorem](https://ncatlab.org/nlab/show/thick+subcategory+theorem) (though I'm pretty sure this is in fact one of the easier observations which goes into the proof of that theorem).
* As $h \to \infty$ it's well-known (I think it's a simple connectivity argument?) that $K(h)\_\ast(X)$ is eventually just $(H\mathbb F\_p)\_\ast(X) \otimes\_{\mathbb F\_p} K(h)\_\ast(X)$. So the sequence $|X|\_h$ is eventually constant at the value $|X|\_\infty$. So for fixed $X$, there are at most finitely many exceptions to a "yes" answer.


So the simplest case I'm not sure about is when $h = 0$, $h' < \infty$, and $X$ is not a sum of shifts of spheres. For instance, probably the simplest 2-cell type 0 spectrum is $\Sigma^\infty \mathbb C \mathbb P^2$. In this case, we have $|\Sigma^\infty \mathbb{CP}^2|\_0 = |\Sigma^\infty \mathbb{CP}^2|\_\infty = 2$, so the question is whether $|\Sigma^\infty \mathbb{CP}^2|\_h$ ever drops to 1, i.e. whether $\Sigma^\infty \mathbb{CP}^2$ is ever $K(h)$-locally invertible...



Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$?
Yes, this is true, and appears in the early literature, although I do not immediately remember exactly where; I'd guess work of Wilson and/or Ravenel. I'll assume that $h>0$ for notational simplicity. There is a homology theory $E$ with $E\_\*=\mathbb{F}\_p[v\_h,v\_{h+1}^{\pm 1}]$, and this is a discrete valuation ring in the graded sense. It follows that when $X$ is finite, $E\_\*(X)$ is a finite sum of $n$ terms like $E\_\*$ and $m$ terms like $E\_\*/v\_h^k$, for some $n,m\geq 0$. It follows that $v\_h^{-1}E\_\*(X)$ is isomorphic to a direct sum of $n$ copies of $v\_h^{-1}E\_\*$. Using the fact that all formal groups of strict height $h$ become isomorphic after faithfully flat extension, we find that $|X|\_h=n$. On the other hand, we have a cofibre sequence $\Sigma^{|v\_h|}E\to E\to K(h+1)$, giving a short exact sequence relating $K(h+1)\_\*(X)$ to the cokernel and kernel of multiplication by $v\_h$ on $E\_\*(X)$. This gives $|X|\_{h+1}=n+2m$.


