The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum En, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that En is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group Gn of the formal group in question. In this paper we find that the Gn-equivariant dual of En is in fact En twisted by a sphere with a non-trivial (when n> 1) action by Gn. This sphere is a dualizing module for the group Gn, and we construct and study such an object IG for any compact p-adic analytic group G. If we restrict the action of G on IG to certain type of small subgroups, we identify IG with a specific representation sphere coming from the Lie algebra of G. This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of EnhH for select choices of p and n and finite subgroups H of Gn.
ASJC Scopus subject areas