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**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.

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.

Reviewer: Victor P.Snaith (Southampton)

### 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 |