Localization of BP-module spectra with respect to BP-related homologies (Q1062313)

The author makes a technical study of localizations with respect to the homology theories \(v_ n^{-1}BP_*(\cdot)\) (n\(\geq 0)\) associated to Brown-Peterson homology at a prime p. The reasons for wanting such results can be found \textit{D. C. Ravenel}'s paper [Am. J. Math. 106, 351- 414 (1984)]. One has a tower \(X\to L_{\infty}X\to...\to L_ nX\to...\to L_ 0X\) for each CW-spectrum X, \(L_ nX\) being its localization with respect to \(v_ n^{-1}BP_*(\cdot)\) (and \(L_{\infty}\) with respect to \(\oplus_{n}v_ n^{-1}BP_*(\cdot))\). A CW-spectrum X is said to be harmonic if \(X=L_{\infty}X\), and s-harmonic (a stronger condition) if \(X=\lim_{\leftarrow}L_ nX\). The main results here concern properties of s-harmonic spectra. As a sample, it is proved that an associative BP- module spectrum E is s-harmonic if hom dim\({}_{BP_*}E_*\) is finite.