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Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real Deligne cohomology?

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To elaborate on my comment, the comparison between the regulators of Beilinson and Borel can be found in the book

  • J.I. Burgos Gil. The regulators of Beilinson and Borel. CRM Monograph Series, 15. Amer. Math. Soc., 2002. (link to book)

The main result is that ${\rm reg}_{\rm Borel}=2{\rm reg}_{\rm Beilinson}$. A short discussion of the history of the comparison and earlier work can be found on p. 3 of the book. (As a side note, let me point out that $H^1(X,\mathbb{Q}(n))=K_{2n-1}(X)$ for $X$ the spectrum of a number field or number ring, so there is no real need to talk about motivic cohomology in this case.)

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