2020/10/19 by Beaudry, Agnès, Goerss, Paul G., Hopkins, Michael J. +1
#20E18 #22E41 #55P92 #55R25 #55R50 #55U30 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.09518
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 Morava stabilizer group \mathbbGn, or even its subgroup of automorphisms of the formal group in question. In this paper we find that the \mathbbGn-equivariant dual of En is in fact En twisted by a sphere with a non-trivial (when n>1) action by \mathbbGn. This sphere is a dualizing module for the group \mathbbGn, 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 \mathbbGn.