Algebraic & Geometric Topology

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

Steffen Sagave

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1203-1251.

Received: 26 June 2015
Revised: 11 July 2015
Accepted: 13 July 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 55P48: Loop space machines, operads [See also 18D50]

E-infinity space symmetric spectrum group completion units of ring spectra Gamma-space


Sagave, Steffen. Spectra of units for periodic ring spectra and group completion of graded $E_{\infty}$ spaces. Algebr. Geom. Topol. 16 (2016), no. 2, 1203--1251. doi:10.2140/agt.2016.16.1203.

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