Picard Groups of Some Local Categories

Abstract

For each $p$-local spectrum $E$, ${\mathcal{L}}_E$ denotes the full subcategory consisting of $E$-local spectra of the category of $p$-local spectra. The Picard group $\mathrm{Pic}({\mathcal{L}}_E)$ is the collection of isomorphism classes of invertible spectra in ${\mathcal{L}}_E$. 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 $⟨E⟩\ge ⟨H\mathbf{Z}/p⟩$ of the Bousfield classes, then $\mathrm{Pic}({\mathcal{L}}_E)=\mathbf{Z}$. In particular, $\mathrm{Pic}({\mathcal{L}}_E)=\mathbf{Z}$ if $E$ is connective.