Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra

Abstract: We investigate several versions of the telescope conjecture on localized categories of spectra, and implications between them. Generalizing the "finite localization" construction, we show that on such categories, localizing away from a set of strongly dualizable objects is smashing. We classify all smashing localizations on the harmonic category, HFp-local category and I-local category, where I is the Brown-Comenetz dual of the sphere spectrum; all are localizations away from strongly dualizable objects, although these categories have no nonzero compact objects. The Bousfield lattices of the harmonic, E(n)-local, K(n)-local, HFp-local and I-local categories are described, along with some lattice maps between them. One consequence is that in none of these categories is there a nonzero object that squares to zero. Another is that the HFp-local category has localizing subcategories that are not Bousfield classes.