Bousfield localization of ghost maps (Q1688707)

A map is said to be a \textit{ghost map} if it induces the zero map on all stable homotopy groups. For a Bousfield localization functor \(L\) as the homotopy-theoretic analogue of a localization functor for rings and modules, a spectrum \(X\) is said to be \textit{\(L\)-ghost-preserving} if the \(L\)-localization of every ghost with source \(X\) is also a ghost map. The localization functor \(L\) is said to be \textit{ghost-preserving} if all spectra are \(L\)-ghost-preserving, and \(L\) is said to be \textit{weakly ghost-preserving} if all finite spectra are \(L\)-ghost-preserving. In this paper, the authors consider the Bousfield localization of ghost maps, and observe that \(J\)-localization is ghost-preserving and \(J\)-completion is weakly ghost-preserving, but not ghost-preserving, where \(J\) is a set of primes. They also show that if \(E\) is a connective spectrum, then the associated Bousfield localization functor \(L_E\) is weakly ghost-preserving. The authors prove that the functor \(L_n\) (localization with respect to \(K(0) \vee \cdots \vee K(n)\)) is not weakly ghost-preserving for \(n \geq 1\) and that \(L_{K(1)}\) is not weakly ghost-preserving, where \(K(n)\) is the \(n\)th Morava \(K\)-theory. Finally, the authors consider spectra \(X\) such that the localization of any ghost into \(X\) is a ghost as the interesting dual notion of the \(L\)-ghost-preserving spectra.