Spectra of units for periodic ring spectra and group completion of graded $E_{\infty}$ spaces

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 $E_{\infty }$ spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded $E_{\infty }$ spaces and use it to exhibit our spectrum of units functor as a 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.