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

Steffen Sagave

Algebraic & Geometric Topology 16 (2016) 1203–1251

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 spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E 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.

E-infinity space, symmetric spectrum, group completion, units of ring spectra, Gamma-space
Mathematical Subject Classification 2010
Primary: 55P43
Secondary: 55P48
Received: 26 June 2015
Revised: 11 July 2015
Accepted: 13 July 2015
Published: 26 April 2016
Steffen Sagave
Institute for Mathematics, Astrophysics and Particle Physics
Radboud University Nijmegen
PO Box 9010
6500 GL Nijmegen
The Netherlands