Picard Groups of Some Local Categories

Abstract

For each p-local spectrum E, ℒ_E_ denotes the full subcategory consisting of E-local spectra of the category of p-local spectra. The Picard group Pic(ℒ_E_) is the collection of isomorphism classes of invertible spectra in ℒ_E_. If this is a set, it is a group with multiplication defined by the smash product. We show that if a spectrum E satisfies a relation (E) ≥ H_ℤ/p of the Bousfield classes, then Pic(ℒ_E) = ℤ. In particular, Pic(ℒ_E_) = ℤ if E is connective.