Suppose [Cscr ] is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in [Cscr ], and the Picard group Pic([Cscr ]) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with [Cscr ].

There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category [Sscr ]. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere Sn [8, 19]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE. The best-known case is E=K(n), the nth Morava K-theory, considered in [8].

In this paper we study the case E=E(n), where E(n) is the Johnson–Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by [Lscr ]. Our main theorem is the following result.