When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its rationalization together with its $p$-completion for all $p$ (and a bit of information about how they fit together). This method is used to further refine $p$-local homotopy theory using the chromatic filtration. In this approach, we filter the category of finite $p$-local spectra by the Morava K-theories $K(n)$, so that we have a sequence of thick subcategories $L_n\operatorname{Sp}_{(p)}$ which give rise to an Adam-Novikov spectral sequence and can be studied using geometric information about formal groups.
From a categorical point of view, the reason this works is because these subcategories are exactly the elements of the Balmer spectrum $\operatorname{Spc}(\operatorname{Sp}_{(p)})$. What I'm wondering, though, is why we use the $p$-local category instead of the $p$-complete category. It certainly works, since we can quite easily compute $p$-completions of $p$-local finite abelian groups, but it seems like it would be more natural to consider the Balmer spectrum of $\operatorname{Sp}_p^{\wedge}$. If we look at Barthel and Beaudry's description in their chapter of the Handbook, for instance, they construct a direct comparison map exhibiting $\operatorname{Spec}(\mathbb{Z})$ as a retract of $\operatorname{Spc}(\operatorname{Sp})$, which they use to describe the Morava K-theories at a prime $p$ as the elements of the fiber over $p$. This isn't literally true unless we work in the $p$-complete category.
So, what actually is this Balmer spectrum? Is it the same as the spectrum of the p-local category? Is it unknown? And is there any reason not to work with this category directly?
