Central simple algebras and Galois cohomology.

*(English)*Zbl 1137.12001
Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press (ISBN 0-521-86103-9/hbk; 0-511-22315-3/ebook). xi, 343 p. £ 45.00; $ 60.00/e-book (2006).

The main object of the book under review is the celebrated Merkurjev-Suslin theorem on the bijectivity of the Galois symbol map \(K^M_2(k)/mK^M_2(k)\to H^2(k,\mu_m^{\otimes 2})\) for all fields \(k\) and all integers \(m\) invertible in \(k\). The authors view this theorem as the culmination of the classical theory of Brauer groups of fields on the one hand, and as a starting point of motivic cohomology theory on the other hand. Correspondingly, the structure of the book reflects both traditional algebraic approaches and more modern geometric insight. As a result, a motivated reader can profit much from studying this monograph containing rich material in one of rapidly developing areas of mathematics.

More specifically, on their way to the statement of the Merkurjev-Suslin theorem (the first seven chapters of the book), the authors introduce and thoroughly discuss many topics of their own interest. They start with classical material on quaternion algebras (focusing on Witt’s and Albert’s theorems), give the basic theory of central simple algebras (from a modern viewpoint, with emphasis on Galois descent), present some basic notions from the cohomology theory of groups with abelian coefficient modules, apply this theory to the study of the Brauer group, introduce Severi-Brauer varieties, study residue maps in some detail (including Faddeev’s theory and digressions related to applications to class field theory and rationality problems in the theory of invariants), define and study Milnor groups (focusing on tame symbols and norm maps and bringing the reader to the cutting edge of the modern high-tech algebra – the Bloch-Kato conjecture). The eighth chapter is devoted to the proof of the Merkurjev-Suslin theorem. Although the proof is considerably simplified with the respect to the original one, this chapter contains some more technical material, such as Gersten complexes in Milnor \(K\)-theory and \(K\)-cohomology of Severi-Brauer varieties. The last chapter of the book is devoted to symbols in positive characteristic. The authors start with classical results of Teichmüller and Albert and use them as an opportunity to introduce and discuss some more modern tools (Cartier operators, logarithmic differentials, flat \(p\)-connections) to finish with the Bloch-Gabber-Kato theorem.

The presentation of the material is reader-friendly, arguments are clear and concise, exercises at the end of every chapter are original and stimulating, the appendix containing some basic notions from algebra and algebraic geometry is very helpful. To sum up, the book under review can be strongly recommended to everyone interested in the topic.

More specifically, on their way to the statement of the Merkurjev-Suslin theorem (the first seven chapters of the book), the authors introduce and thoroughly discuss many topics of their own interest. They start with classical material on quaternion algebras (focusing on Witt’s and Albert’s theorems), give the basic theory of central simple algebras (from a modern viewpoint, with emphasis on Galois descent), present some basic notions from the cohomology theory of groups with abelian coefficient modules, apply this theory to the study of the Brauer group, introduce Severi-Brauer varieties, study residue maps in some detail (including Faddeev’s theory and digressions related to applications to class field theory and rationality problems in the theory of invariants), define and study Milnor groups (focusing on tame symbols and norm maps and bringing the reader to the cutting edge of the modern high-tech algebra – the Bloch-Kato conjecture). The eighth chapter is devoted to the proof of the Merkurjev-Suslin theorem. Although the proof is considerably simplified with the respect to the original one, this chapter contains some more technical material, such as Gersten complexes in Milnor \(K\)-theory and \(K\)-cohomology of Severi-Brauer varieties. The last chapter of the book is devoted to symbols in positive characteristic. The authors start with classical results of Teichmüller and Albert and use them as an opportunity to introduce and discuss some more modern tools (Cartier operators, logarithmic differentials, flat \(p\)-connections) to finish with the Bloch-Gabber-Kato theorem.

The presentation of the material is reader-friendly, arguments are clear and concise, exercises at the end of every chapter are original and stimulating, the appendix containing some basic notions from algebra and algebraic geometry is very helpful. To sum up, the book under review can be strongly recommended to everyone interested in the topic.

Reviewer: Boris Kunyavskii (Ramat Gan)

##### MSC:

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

12G05 | Galois cohomology |

16K20 | Finite-dimensional division rings |

14F22 | Brauer groups of schemes |

19C30 | \(K_2\) and the Brauer group |