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 $E_{\infty }$-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 $\mathbb{R}$. 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 $E_{\infty }$-refinement and prove a uniqueness result for these refinements.

final version, 40 pages

References in corpus (4)

• Higher Topos Theory
• Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories
• Quillen adjunctions induce adjunctions of quasicategories
• Dualizing cartesian and cocartesian fibrations

Cited by in corpus (4)

• Mixed Hodge structures and formality of symmetric monoidal functors
• Mahler's measure and elliptic curves with potential complex multiplication
• Modules over the de Rham cohomology spectrum
• Borel's rank theorem for Artin $L$-functions