Note

Picard Groups of Quotient Ring Spectra: Real K-Theory and Lifting Obstructions

Author: Wei Yang · Added: September 15, 2026

Abstract

We compute Picard groups for the descent-tower quotients $\mathrm{KO}/\eta^k$ , their ordinary two-completions, their $K(1)$ -local module categories, and specified $\mathbb{E}_2$ double quotients $\mathrm{KO}/(2^m,\eta^n)$ for $m\ge5$ . Ordinary completion preserves the Picard group at every eta stage, whereas local invertibility introduces two-adic families. The calculations use antipodal spherical cochains and the distinction between an invertible local system and one realized by an invertible cochain module. A coherent Bockstein and a connection argument determine the exponent- $32$ input; a finite Postnikov comparison determines the finite-coefficient extensions. We also construct a rational quotient tower with a non-liftable Picard class at any prescribed stage, and, for every prime $p\ge5$ , a height-two generalized Moore algebra whose $K(2)$ -localization has a Picard class of order $p$ modulo classes coming from local spectra.