Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$?

Question

Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be 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 (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...

Answer 1

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 $|X|_{h+1}=n+2m$.

Answer 2

For a finite $p$-local complex you can get an upper bound on the dimension of $K(n)_*X$ as a $K(n)_*$-module by calculating the number of $\mathbb{F}_p$-generators in $H_*(X; \mathbb{F}_p)$ as a module over $\Lambda(Q_n)$, where $Q_n$ is the Adams-Margolis in the Steenrod algebra dual to $\tau_n$ (or $\xi_n$ at the prime $2$). This is because of an Adams spectral sequence computing $k(n)_*X$ using the fact that $$H^*(k(n);\mathbb{F}_p) = A//E(Q_n),$$ where $k(n)$ is the connective cover of $K(n)$ and $A$ is the Steenrod algebra. It is clear that when $n$ is chosen large enough so that the $|Q_n|$ is slightly bigger than the difference in dimension of the top and the bottom cell of $X$, then $Q_n$ will act trvially on $H_*(X; \mathbb{F}_p)$ and the Adams SS will collapse at the $E_2$-page leading to $$ dim_{K(n)_*} K(n)_*X = dim_{\mathbb{F}_p} H^*(X; \mathbb{F}_p)$$ as desired.