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Regulators. (Régulateurs.) (French) Zbl 0617.14008
Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. 644, Astérisque 133/134, 237-253 (1986).
The classical regulator of Dirichlet is defined to be the real number, \(R_ F\), which is associated to a number field \(F\) as the volume of the lattice which is the image of the logarithmic map \(0^*_ F\oplus {\mathbb Z}\to {\mathbb R}^{r_ 1}\oplus {\mathbb R}^{r_ 2}.\) Here \(r_ 1\) (respectively \(2r_ 2)\) is the number of real (respectively complex) embeddings of \(F\). - Borel defined analogous higher regulators associated to higher-dimensional algebraic \(K\)-groups of \(0_ F\), the ring of integers of \(F\).
In this survey the author describes the construction of Bloch and Beilinson of higher regulators of the form \(K_ m(v)\to H^ k_{{\mathcal D}}(V_{{\mathbb R}};{\mathbb R}(n))\quad (m+k=2n)\) where \(V\) is a projective and smooth variety over a number field. Here \(H^*_{{\mathcal D}}\) is Deligne-Beilinson cohomology. With classical results from number theory as motivation the author proceeds to describe the current status of the Beilinson conjectures concerning these regulators, illustrating them with numerous examples.
For the entire collection see [Zbl 0577.00004].
Reviewer: V.P.Snaith

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11R70 \(K\)-theory of global fields
14F40 de Rham cohomology and algebraic geometry