Construction - Spherical height n dimension

Let $X\in {\mathcal{S}}^{p-sm}$ be a p-small space, as in p-adic K(n) Euler Characteristic.

Lemma: the $K(n)$-local spectrum ${\int }_X{\mathbb{S}}_{K(n)}\simeq L_{K(n)}({\Sigma }_{+}^{\infty }X{\otimes }_{\mathbb{S}}{\mathbb{S}}_{K(n)})$ is dualizable in $S{p}_{K(n)}$. (Where the integral sign left denote colimit of a constant diagram indexed by $X$.)

Proof The monoidal unit ${\mathbb{S}}_{K(n)}$ is dualizable. Ambidexiterity and stability imply dualizable objects are closed under $\pi$-finite colimits, and in fact, all colimits.

action: verify this carefully. if this is indeed the case, this means in this case, at finite height dualizability is a localization.

Construction (K(n)-local spherical dimension)

$\widetilde{{\chi }_n}(X):=Tr(I{d}_{{\int }_X{\mathbb{S}}_{K(n)}})\in {\pi }_{0}En{d}_{S{p}_{K(n)}}({\mathbb{S}}_{K(n)})\simeq {\pi }_0({\mathbb{S}}_{K(n)})$

Actions: compute this in special cases and compare to p-adic K(n) Euler Characteristic