A Note on Hopkins' Picard Groups of the Stable Homotopy Categories of $L_n$-Local Spectra

Abstract

For a stable homotopy category, M. Hopkins introduced a Picard group as a category consisting of isomorphism classes of invertible objects. For the stable homotopy category of $L_n$-local spectra, M. Hovey and H. Sadofsky showed that the Picard group is actually a group containing the group of integers as a direct summand. Kamiya and the author constructed an injection from the other summand of the Picard group to the direct sum of the $E_r$-terms $E_r^{r,r-1}$ over $r\ge 2$ of the Adams–Novikov spectral sequence converging to the homotopy groups of the $L_n$-localized sphere spectrum. In this paper, we show in a classical way that the injection is a bijection under a condition.