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

F. Luke Wolcott

Vol. 276 (2015), No. 2, 483–509

DOI: 10.2140/pjm.2015.276.483

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, $H{\mathbb{F}}_p$-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, $E\left(n\right)$-local, $K\left(n\right)$-local, $H{\mathbb{F}}_p$-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 $H{\mathbb{F}}_p$-local category has localizing subcategories that are not Bousfield classes.

Keywords

telescope conjecture, Bousfield localization, Bousfield lattice, harmonic category, smashing localization

Mathematical Subject Classification 2010

Primary: 55P42, 18E30, 55U35

Milestones

Received: 14 July 2014

Revised: 13 January 2015

Accepted: 19 January 2015

Published: 15 July 2015