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

TL;DR

This paper explores various forms of the telescope conjecture in localized spectra categories, classifies smashing localizations, and describes Bousfield lattices, revealing structural properties and differences among these categories.

Contribution

It generalizes finite localization, classifies all smashing localizations in specific categories, and describes Bousfield lattices, highlighting their structural differences and properties.

Findings

All smashing localizations are away from strongly dualizable objects.

Bousfield lattices are explicitly described for several localized categories.

No nonzero object squares to zero in these categories.

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.