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
spaces introduced in joint work with Christian Schlichtkrull.
We construct a group completion model structure for graded
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.
Keywords
E-infinity space, symmetric spectrum, group completion,
units of ring spectra, Gamma-space
