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

TL;DR

This paper investigates the duality properties of Lubin-Tate spectra in the $K(n)$-local category, revealing a twisted self-duality involving a sphere with a non-trivial group action, and constructs a dualizing module for compact $p$-adic groups.

Contribution

It introduces a dualizing module $I_{ ext{G}}$ for any compact $p$-adic analytic group $ ext{G}$, generalizing duality in chromatic homotopy theory and providing computational tools.

Findings

Identifies the $ ext{G}_n$-equivariant dual of $E_n$ as $E_n$ twisted by a sphere with $ ext{G}_n$ action.

Constructs and studies the dualizing module $I_{ ext{G}}$ for compact $p$-adic groups.

Provides explicit computations of $K(n)$-local Spanier-Whitehead duals for certain spectra.

Abstract

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 Morava stabilizer group $\mathbb{G}_n$, or even its subgroup of automorphisms of the formal group in question. In this paper we find that the $\mathbb{G}_n$-equivariant dual of $E_n$ is in fact $E_n$ twisted by a sphere with a non-trivial (when $n>1$) action by $\mathbb{G}_n$. This sphere is a dualizing module for the group $\mathbb{G}_n$, and we construct and study such an object $I_{\mathcal{G}}$ for any compact $p$-adic analytic group $\mathcal{G}$. If we restrict the action of $\mathcal{G}$ on $I_{\mathcal{G}}$ to certain type of small subgroups, we identify $I_{\mathcal{G}}$ with a specific representation sphere coming from the Lie algebra of $\mathcal{G}$. 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 $E_n^{hH}$ for select choices of $p$ and $n$ and finite subgroups $H$ of $\mathbb{G}_n$.