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**Cohomology of number fields.
2nd ed.**
*(English)*
Zbl 1136.11001

Grundlehren der Mathematischen Wissenschaften 323. Berlin: Springer (ISBN 978-3-540-37888-4/hbk). xv, 825 p. (2008).

The second edition of this important book differs from the first in the following respects: many minor slips and errors have been corrected (cf. the online errata page to the first edition), material has been added, and the exposition was rearranged in some places, splitting for instance one section into two. As a minor side effect, the numbering of results in the new edition is not strictly compatible with the old one, but the digits seldom differ by more than one or two. The numbers and headings of the chapters did not change at all. The changes resulted in an increase of over a hundred pages, but the general bipartite outline of the book (algebraic part, and arithmetical part) and its clear style remain unaffected. The reviewer would like to refer to his review of the first edition [D. Hilbert, The theory of algebraic number fields. Berlin: Springer (1998; Zbl 0984.11001)] for a description of the contents.

The second edition will continue to serve as a very helpful and up-to-date reference in cohomology of profinite groups and algebraic number theory, and all the additions are interesting and useful. For example, the chapter on spectral sequences now has a section on filtered complexes, and some results on the derived functors of the projective limit functor that were only quoted in the first edition are now given with a proof. We learn in Proposition 2.7.4 that the second and all higher such derived functors vanish for categories of modules. I only quote this to illustrate how the authors succeed in presenting important results that are not too well known, or of difficult access (as for instance the deep theorem of Shafarevich on solvable Galois groups). The reviewer thinks it would be tedious to list all changes, or even only the non-minor ones, with respect to the first edition in this review (see also the preface to the new edition), but perhaps the most remarkable among the major additions to the book is a section on pro-2-extensions of number fields in considerable generality, which presents recent work of the second author.

Just a few comments: Corollary 5.2.20 is actually a special case of a general result which says that projective modules over local rings are free, and the proof of the authors, even though written out for group rings, is in fact the general proof for finitely generated modules. As the authors mention themselves in the new errata page, it was forgotten to transfer the mention of Voevodsky’s proof of the Milnor conjecture from the old errata page to the second edition. In reviews of books, it is all too easy to mention topics that might have been included, in this case perhaps étale \(K\)-theory of number fields (only Milnor \(K\)-theory is treated), or equivariant versions of the main conjecture, but the book is fine as it is: systematic, very comprehensive, and well-organised. This second edition will be a standard reference from the outset, continuing the success of the first one.

The second edition will continue to serve as a very helpful and up-to-date reference in cohomology of profinite groups and algebraic number theory, and all the additions are interesting and useful. For example, the chapter on spectral sequences now has a section on filtered complexes, and some results on the derived functors of the projective limit functor that were only quoted in the first edition are now given with a proof. We learn in Proposition 2.7.4 that the second and all higher such derived functors vanish for categories of modules. I only quote this to illustrate how the authors succeed in presenting important results that are not too well known, or of difficult access (as for instance the deep theorem of Shafarevich on solvable Galois groups). The reviewer thinks it would be tedious to list all changes, or even only the non-minor ones, with respect to the first edition in this review (see also the preface to the new edition), but perhaps the most remarkable among the major additions to the book is a section on pro-2-extensions of number fields in considerable generality, which presents recent work of the second author.

Just a few comments: Corollary 5.2.20 is actually a special case of a general result which says that projective modules over local rings are free, and the proof of the authors, even though written out for group rings, is in fact the general proof for finitely generated modules. As the authors mention themselves in the new errata page, it was forgotten to transfer the mention of Voevodsky’s proof of the Milnor conjecture from the old errata page to the second edition. In reviews of books, it is all too easy to mention topics that might have been included, in this case perhaps étale \(K\)-theory of number fields (only Milnor \(K\)-theory is treated), or equivariant versions of the main conjecture, but the book is fine as it is: systematic, very comprehensive, and well-organised. This second edition will be a standard reference from the outset, continuing the success of the first one.

Reviewer: Cornelius Greither (Neubiberg)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R34 | Galois cohomology |

11R23 | Iwasawa theory |

11S25 | Galois cohomology |

18G10 | Resolutions; derived functors (category-theoretic aspects) |

18G20 | Homological dimension (category-theoretic aspects) |

20J05 | Homological methods in group theory |

11R37 | Class field theory |