The Beilinson regulator is a map of ring spectra

Abstract

We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E∞-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E∞-refinement and prove a uniqueness result for these refinements.