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 $|X|_0 \geq 2$ (i.e. $X$ is a finite type 0 spectrum other than a sphere). 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...

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...