# zbMATH — the first resource for mathematics

The regulators of Beilinson and Borel. (English) Zbl 0994.19003
CRM Monograph Series. 15. Providence, RI: American Mathematical Society (AMS). xi, 104 p. (2002).
If $$K$$ is a number field with $$r_{1}$$ real embeddings and $$2r_{2}$$ complex ones the Dirichlet regulator, which is very important in number theory (because of its central role in the conjectures of Brumer-Stark, Gross, Tate and others) is a map of the form $$\rho : K_{1}({\mathcal O}_{K}) \cong {\mathcal O}_{K}^{*} \rightarrow {\mathbb R}^{r_{1}+r_{2}-1}$$. The image of $$\rho$$ is a lattice of rank $$r_{1}+r_{2}-1$$ whose covolume features in the analytic class number formula. Here $${\mathcal O}_{K}$$ denotes the algebraic integers of $$K$$.
In higher-dimensional K-theory there are two similar regulators of the form $r_{Bo}, r_{Be} : K_{2p-1}({\mathcal O}_{K}) \rightarrow {\mathbb R}^{d_{p}}$ where $$d_{p}$$ equals $$r_{1}+r_{2}$$ if $$p$$ is odd and $$r_{2}$$ otherwise. Each of these regulators has some very important properties. The Borel regulator $$r_{Bo}$$, constructed by A. Borel [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 613-636 (1977; Zbl 0382.57027)], features in the Lichtenbaum conjectures on the leading terms of the Taylor’s series of the zeta function of $$K$$ at negative integers. A. A. Beilinson’s regulator $$r_{Be}$$ [J. Sov. Math. 30, 2036-2070 (1985; Zbl 0588.14013)], is given by the Chern class from K-theory to Deligne cohomology and has been related to the polylogarithms by Beilinson. For these and other reasons it was important to determine precisely the relation between $$r_{Bo}$$ and $$r_{Be}$$.
In this book the author carefully traces through a number of canonical isomorphisms which translate the regulators so that both land in Lie algebra cohomology and there he shows that $$2 r_{Be} = r_{Bo}$$. Beilinson’s sketch of the relationship between the regulators was very brief most of whose details were fleshed out by M. Rapoport [Perspect. Math. 4, 169-192 (1988; Zbl 0667.14005)] and this book synthesises the features of both sources to give a complete proof of the correct relationship for the first time. This synthesis requires a lot of background in simplicial techniques, Hopf algebras, rational homotopy theory, cohomology and continuous cohomology of groups, de Rham cohomology and Lie algebra cohomology together with their inter-relationships. The first six chapters of the book are devoted to this, making an excellent background source for graduate students.

##### MSC:
 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
##### Citations:
Zbl 0382.57027; Zbl 0588.14013; Zbl 0667.14005