Question

Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the stable homotopy category) consisting of the $p$-acyclic spectra is generated (as a localizing subcategory) by the finite $p$-acyclic spectra? Is there a reference for this?

To be clear, I'm talking about Bousfield localization with respect to $\pi_*(-) \otimes {\mathbb Z}_{(p)}$. It is definitely a smashing localization.

Recall that a spectrum is $p$-acyclic if $\pi_*(X) \otimes {\mathbb Z}_{(p)} = 0$. Thanks for your attention.

Answer 1

The fiber of $S\to S_{(p)}$ is equivalent to $\mathrm{hocolim} \Sigma^{-1}M(k)$, where the limit is taken over the poset of natural numbers prime to $p$ (using the relation of divisiblilty, so $M(k)\to M(kk')$). Thus, if $X$ is a spectrum such that $X_{(p)}=0$, then $X\approx \mathrm{hocolim} \Sigma^{-1}M(k)\wedge X$.

Thus, if $X$ is $p$-localization acyclic, this proves that $X$ is in the localizing subcategory generated by the $M(k)$, as Tyler suggests, while the converse is obvious.