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arXiv · 1509.05667

The Beilinson regulator is a map of ring spectra

Ulrich Bunke·Thomas Nikolaus·Georg Tamme
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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.

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Ulrich Bunke, Thomas Nikolaus, Georg Tamme. 2015-09-18. The Beilinson regulator is a map of ring spectra. https://doi.org/10.1016/j.aim.2018.05.027

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