Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

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 E_n, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that E_n is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group G_n of the formal group in question. In this paper we find that the G_n-equivariant dual of E_n is in fact E_n twisted by a sphere with a non-trivial (when n> 1) action by G_n. This sphere is a dualizing module for the group G_n, and we construct and study such an object I_G for any compact p-adic analytic group G. If we restrict the action of G on I_G to certain type of small subgroups, we identify I_G 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 G_n.