Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces

TL;DR

This paper introduces a new spectrum of units for commutative symmetric ring spectra that captures periodicity and connects to graded E-infinity spaces, advancing the understanding of ring spectra with graded structures.

Contribution

It constructs a novel spectrum of units detecting periodicity, develops a group completion model for graded E-infinity spaces, and links these to graded logarithmic structures.

Findings

Spectrum of units detects periodicity in ring spectra.

Group completion model structure for graded E-infinity spaces.

Spectrum of units as right adjoint in homotopy categories.

Abstract

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity. Our approach builds on the graded variant of E-infinity spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E-infinity spaces and use it to exhibit our spectrum of units functor as right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.