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.