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

v3: 33 pages; exposition improved, accepted for publication in Algebraic and Geometric Topology

References in corpus (10)

• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• Units of ring spectra and Thom spectra
• An Inverse K-Theory Functor
• Localization sequences for logarithmic topological Hochschild homology
• Group completion and units in I-spaces
• Logarithmic structures on topological K-theory spectra
• Logarithmic topological Hochschild homology of topological K-theory spectra
• Algebraic K-theory of strict ring spectra

Cited by in corpus (9)

• Model Structures on Commutative Monoids in General Model Categories
• Localization sequences for logarithmic topological Hochschild homology
• Twisted differential cohomology
• Logarithmic structures on topological K-theory spectra
• Logarithmic topological Hochschild homology of topological K-theory spectra
• Algebraic K-theory of strict ring spectra
• Virtual vector bundles and graded Thom spectra
• Adjoining roots in homotopy theory
• On Deformation Theory in Higher Logarithmic Geometry