The Chromatic Behavior of the Spherical Units
Abstract: To a commutative ring spectrum R, one can associate a connective spectrum of units, which generalizes the groups of units of ordinary commutative rings. While the underlying spaces of R and of its spectrum of units have the same identity components, they are very different from each other as spectra; one is based on the additive structure of R and the other on its multiplicative structure.
In this talk, I will discuss the spectrum of units of the (p-completed) sphere spectrum from the perspective of chromatic homotopy theory. The chromatic tower of this spectrum is not well understood beyond its monochromatic layers. However, I will show that a suitable ''chromatic co-localization'' of this spectrum is concentrated in height 1. Among other things, this fact excludes the existence of certain higher-height analogs of the classical J-homomorphism.