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

For a stable homotopy category [6], M. Hopkins introduced a Picard group (cf. [11], [4]) as a category consisting of isomorphism classes of invertible objects. For the stable homotopy category of Ln-local spectra, M. Hovey and H. Sadofsky [7] showed that the Picard group is actually a group containing the group of integers as a direct summand. We constructed an injection in [8] from the other summand of the Picard group to the direct sum of the Er-terms E r,r−1 r over r ≥ 2 of the Adams-Novikov spectral sequence converging to the homotopy groups of the Ln-localized sphere spectrum. In this paper, we show in a classical way that the injection is a bijection under a condition.