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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1$\begingroup$ What about the book of Burgos Gil? icmat.es/miembros/burgos/files/brbr.pdf $\endgroup$– Matthias WendtCommented Jan 11, 2018 at 9:30
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$\begingroup$ probably this helping you to predict the relationship you want :faculty.math.illinois.edu/~dan/KtheoryHandbook-for-authors-only/… $\endgroup$– zeraoulia rafikCommented Jan 11, 2018 at 11:53
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1 Answer
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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.)
answered Jan 11, 2018 at 16:43