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?


1 Answer 1


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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.