Picard groups of higher real $K$-theory spectra at height $p-1$

TL;DR

This paper computes the Picard groups of higher real $K$-theory spectra at height $p-1$ using descent spectral sequences, revealing they are cyclic and generated by suspension.

Contribution

It provides the first explicit computation of Picard groups for these spectra at height $p-1$, extending understanding of their structure.

Findings

Picard groups are cyclic for the spectra considered.

The Picard group is generated by the suspension.

Results apply to homotopy fixed points spectra of Lubin-Tate $E$-theory.

Abstract

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$-theory spectra of Hopkins and Miller at height $n=p-1$, for $p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra $E_n^{hG}$, where $E_n$ is Lubin-Tate $E$-theory at the prime $p$ and height $n=p-1$, and $G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.