##
**Cohomology of number fields.**
*(English)*
Zbl 0948.11001

Grundlehren der Mathematischen Wissenschaften. 323. Berlin: Springer. xv, 699 p. (2000).

This book is so rich in substance that its title is an understatement. Of course, the promise to present “cohomology of number fields”, that is, cohomology theory of arithmetical modules over absolute Galois groups of a number field, is amply fulfilled; but a lot more is to be found. The book covers a vast field which one might call “Advanced Algebraic Number Theory”; this also encompasses lots of homological algebra and more than a bit of algebraic geometry.

With some over-simplification, one might say: The basic part of algebraic number theory deals with number fields per se and finite (Galois) extensions of them. (By all means one should recommend here Neukirch’s book on that subject, to which the book under review is a kind of sequel.) “Advanced Algebraic Number Theory”, at least as far as the book under review is concerned, now considers infinite Galois extensions, so the top field is only a union of number fields and not itself a number field. One basic case is the maximal algebraic extension, in other words: the algebraic closure, of a given number field or local field; in the number field case, the resulting Galois group is still not completely understood. There is a whole class of very important modifications of this case: the maximal extensions unramified outside a given set \(S\) of places. In another direction, so-called Iwasawa theory, one considers certain abelian but still infinite extensions of a base field, and looks at the behavior of arithmetic invariants (class numbers in particular) in the resulting towers. The unifying techniques are commutative algebra and homological algebra, with a particular emphasis on the cohomology theory of groups.

The book is organized as follows: In the first half “Algebraic Theory”, the purely algebraic and group-theoretical foundations are presented. In the second half “Arithmetic Theory”, all this is brought to bear on arithmetic, that is, the theory of number fields and related objects. It is easy, and advisable, to skip back and forth to some extent between these two halves of the book. We will now discuss the contents in a little more detail. The reader might equally well consult the very well-written introduction of the book itself at this stage!

Chapter I presents the general cohomology theory of profinite groups. As the authors point out, this has been treated before, but the presentation here is remarkably efficient and complete. The same goes for Chapter II on homological algebra: here the reader gets a real chance to quickly obtain a good understanding of spectral sequences. Chapter III is on duality properties; it contains an abstract form of class field theory, the reciprocity homomorphism. It also treats Demuškin (pronounced “Dyómushkin”) groups, which will reappear later as absolute Galois groups of local fields. Similarly, the contents of Chapter IV on free products is intended for later applications to Galois groups. Finally, Chapter V gives the module-theoretic foundations for Iwasawa theory, that is: the theory of towers \(K_0 \subset K_1 \subset K_2\ldots\), where \(K_i\) is a cyclic Galois extension of \(K_0\) of degree \(p^i\) for all \(i\) and \(p\) is a fixed prime. The profinite group in question is here \(\Gamma\), a copy of the additive group of \(p\)-adic integers, and the main issue is a classification of Iwasawa modules, that is, modules over the so-called Iwasawa algebra \(\Lambda = {\mathbb Z}_p[[\Gamma]]\). A particularly valuable topic is Jannsen’s classification of Iwasawa modules using homotopy, which is finer than the usual one which is only up to quasi-isomorphism. That latter classification is likewise presented in a very elegant way, avoiding matrix computations.

Chapters VI to XII are concerned with arithmetical theory in the proper sense. Chapter VI calculates various cohomology groups attached to absolute Galois groups and the standard arithmetical modules, mainly the additive group and the multiplicative group of the top field on which the group acts. Applications include the Brauer group and Milnor K-theory. This still concerns general fields. The next chapter (VII) now considers local fields, that is, locally compact fields complete with respect to a discrete valuation. Here we arrive at one of the highlights of the theory: the description of the absolute Galois group \(G_k\) of a \(p\)-adic local field \(k\) via generators and relations. For the maximal pro-\(p\)-quotient \(G_k(p)\) full details are given; for the entire group \(G_k\) the results of Jannsen-Wingberg and Diekert are explained, but for proofs, the reader is directed to the literature. The basic idea is the following: For a pro-\(p\)-group \(G\), the ranks of the cohomology groups H\(^1(G,{\mathbb Z}/p)\) and H\(^2(G,{\mathbb Z}/p\)) provide the minimal number of generators, and minimal number of relations for \(G\) respectively. The first cohomology group is tackled directly, and for the second one, duality theorems are used. These duality theorems (treated in the first half of the book) provide a link between second and zeroth cohomology of appropriate modules. In Chapter VIII which deals with global fields (thus in particular with number fields) the approach is in principle similar, but the corresponding duality theorem of Poitou and Tate is much harder to prove. This chapter culminates in the calculation of generator and relation rank of the group \(G_S(p)\), where \(G_S\) is the Galois group of the maximal algebraic extension \(k_S\) which is unramified outside the set \(S\). Most results in this section suppose that \(S\) is finite, so the case of the algebraic closure \(\bar k\) (i.e. \(S\) is the set of all primes) is not captured yet. This is one topic of Chapter IX, another topic being so-called local-global theorems, also known as Hasse principles. While the structure of \(G_k=G(\bar k/k)\) is not known in general, important results are available, and proved in this chapter. We first mention Iwasawa’s theorem which states that the maximal prosolvable quotient of \(G_{k(\mu)}\) is a free-prosolvable group. In particular every finite solvable group occurs as a Galois group over \(k(\mu)\), but better still: every finite solvable group can be realized as a Galois group over \(k\) already. This is Šafarevič’s theorem, which is proved in the last section of Chapter IX; the way in which this difficult theorem is presented is particularly beautiful and illuminating.

For the remaining chapters, we will be more brief. Chapter X is still concerned with groups \(G_S\) as in Chapter VIII, proving structural results. In particular, the case \(S=\emptyset\) which is the famous Class Field Tower Problem is carefully discussed. A shift of viewpoint happens in Chapter XI which treats Iwasawa theory of number fields, that is, roughly speaking: the theory of certain modules over the Iwasawa algebra \(\Lambda\) (see above), and as an application, a law for the growth of the \(p\)-part of the class group of \(k_n\), where \(k_n\) runs through a tower of cyclic degree \(p^n\) extensions of the base field \(k\). The Main Conjecture in Iwasawa Theory expresses characteristic power series of Iwasawa modules via \(L\)-functions. Although the authors very reasonably refer to the literature for proofs for the main conjecture, the section on this topic is most instructive, containing motivation and the most important applications, in particular to algebraic \(K\)-theory. Finally, Chapter XII presents the Neukirch-Uchida theorem which says, roughly speaking, that a number field \(k\) is characterized by its absolute Galois group (the actual statement is much sharper); the book ends with hypothetical extensions of this idea to algebraic geometry, so-called ”anabelian” geometry.

This outline does not give the book its proper due; many topics have been omitted or barely touched in this review. The efforts of the authors (in the German preface Schmidt and Wingberg very modestly write: “Wir haben uns alle Mühe gegeben”, i.e. “we really tried hard”) have resulted, in the reviewer’s opinion, in an excellently written book, which is pleasurable to read even during technical stretches. (In some rare places the reviewer found the presentation just a bit terse.) The author’s maintain an errata page, see http://www.mathi.uni-heidelberg.de/ag-wingberg/papers/cohen.html. These errata (only one of which was spotted by the reviewer) are few in number, and minor.

The book brings together a remarkable wealth of topics, many of them not easily available before, some of them quite new, and it is organized in a very systematic and coherent way. Therefore this most useful monograph can be warmly recommended, and it should very soon turn into a standard reference in its field.

With some over-simplification, one might say: The basic part of algebraic number theory deals with number fields per se and finite (Galois) extensions of them. (By all means one should recommend here Neukirch’s book on that subject, to which the book under review is a kind of sequel.) “Advanced Algebraic Number Theory”, at least as far as the book under review is concerned, now considers infinite Galois extensions, so the top field is only a union of number fields and not itself a number field. One basic case is the maximal algebraic extension, in other words: the algebraic closure, of a given number field or local field; in the number field case, the resulting Galois group is still not completely understood. There is a whole class of very important modifications of this case: the maximal extensions unramified outside a given set \(S\) of places. In another direction, so-called Iwasawa theory, one considers certain abelian but still infinite extensions of a base field, and looks at the behavior of arithmetic invariants (class numbers in particular) in the resulting towers. The unifying techniques are commutative algebra and homological algebra, with a particular emphasis on the cohomology theory of groups.

The book is organized as follows: In the first half “Algebraic Theory”, the purely algebraic and group-theoretical foundations are presented. In the second half “Arithmetic Theory”, all this is brought to bear on arithmetic, that is, the theory of number fields and related objects. It is easy, and advisable, to skip back and forth to some extent between these two halves of the book. We will now discuss the contents in a little more detail. The reader might equally well consult the very well-written introduction of the book itself at this stage!

Chapter I presents the general cohomology theory of profinite groups. As the authors point out, this has been treated before, but the presentation here is remarkably efficient and complete. The same goes for Chapter II on homological algebra: here the reader gets a real chance to quickly obtain a good understanding of spectral sequences. Chapter III is on duality properties; it contains an abstract form of class field theory, the reciprocity homomorphism. It also treats Demuškin (pronounced “Dyómushkin”) groups, which will reappear later as absolute Galois groups of local fields. Similarly, the contents of Chapter IV on free products is intended for later applications to Galois groups. Finally, Chapter V gives the module-theoretic foundations for Iwasawa theory, that is: the theory of towers \(K_0 \subset K_1 \subset K_2\ldots\), where \(K_i\) is a cyclic Galois extension of \(K_0\) of degree \(p^i\) for all \(i\) and \(p\) is a fixed prime. The profinite group in question is here \(\Gamma\), a copy of the additive group of \(p\)-adic integers, and the main issue is a classification of Iwasawa modules, that is, modules over the so-called Iwasawa algebra \(\Lambda = {\mathbb Z}_p[[\Gamma]]\). A particularly valuable topic is Jannsen’s classification of Iwasawa modules using homotopy, which is finer than the usual one which is only up to quasi-isomorphism. That latter classification is likewise presented in a very elegant way, avoiding matrix computations.

Chapters VI to XII are concerned with arithmetical theory in the proper sense. Chapter VI calculates various cohomology groups attached to absolute Galois groups and the standard arithmetical modules, mainly the additive group and the multiplicative group of the top field on which the group acts. Applications include the Brauer group and Milnor K-theory. This still concerns general fields. The next chapter (VII) now considers local fields, that is, locally compact fields complete with respect to a discrete valuation. Here we arrive at one of the highlights of the theory: the description of the absolute Galois group \(G_k\) of a \(p\)-adic local field \(k\) via generators and relations. For the maximal pro-\(p\)-quotient \(G_k(p)\) full details are given; for the entire group \(G_k\) the results of Jannsen-Wingberg and Diekert are explained, but for proofs, the reader is directed to the literature. The basic idea is the following: For a pro-\(p\)-group \(G\), the ranks of the cohomology groups H\(^1(G,{\mathbb Z}/p)\) and H\(^2(G,{\mathbb Z}/p\)) provide the minimal number of generators, and minimal number of relations for \(G\) respectively. The first cohomology group is tackled directly, and for the second one, duality theorems are used. These duality theorems (treated in the first half of the book) provide a link between second and zeroth cohomology of appropriate modules. In Chapter VIII which deals with global fields (thus in particular with number fields) the approach is in principle similar, but the corresponding duality theorem of Poitou and Tate is much harder to prove. This chapter culminates in the calculation of generator and relation rank of the group \(G_S(p)\), where \(G_S\) is the Galois group of the maximal algebraic extension \(k_S\) which is unramified outside the set \(S\). Most results in this section suppose that \(S\) is finite, so the case of the algebraic closure \(\bar k\) (i.e. \(S\) is the set of all primes) is not captured yet. This is one topic of Chapter IX, another topic being so-called local-global theorems, also known as Hasse principles. While the structure of \(G_k=G(\bar k/k)\) is not known in general, important results are available, and proved in this chapter. We first mention Iwasawa’s theorem which states that the maximal prosolvable quotient of \(G_{k(\mu)}\) is a free-prosolvable group. In particular every finite solvable group occurs as a Galois group over \(k(\mu)\), but better still: every finite solvable group can be realized as a Galois group over \(k\) already. This is Šafarevič’s theorem, which is proved in the last section of Chapter IX; the way in which this difficult theorem is presented is particularly beautiful and illuminating.

For the remaining chapters, we will be more brief. Chapter X is still concerned with groups \(G_S\) as in Chapter VIII, proving structural results. In particular, the case \(S=\emptyset\) which is the famous Class Field Tower Problem is carefully discussed. A shift of viewpoint happens in Chapter XI which treats Iwasawa theory of number fields, that is, roughly speaking: the theory of certain modules over the Iwasawa algebra \(\Lambda\) (see above), and as an application, a law for the growth of the \(p\)-part of the class group of \(k_n\), where \(k_n\) runs through a tower of cyclic degree \(p^n\) extensions of the base field \(k\). The Main Conjecture in Iwasawa Theory expresses characteristic power series of Iwasawa modules via \(L\)-functions. Although the authors very reasonably refer to the literature for proofs for the main conjecture, the section on this topic is most instructive, containing motivation and the most important applications, in particular to algebraic \(K\)-theory. Finally, Chapter XII presents the Neukirch-Uchida theorem which says, roughly speaking, that a number field \(k\) is characterized by its absolute Galois group (the actual statement is much sharper); the book ends with hypothetical extensions of this idea to algebraic geometry, so-called ”anabelian” geometry.

This outline does not give the book its proper due; many topics have been omitted or barely touched in this review. The efforts of the authors (in the German preface Schmidt and Wingberg very modestly write: “Wir haben uns alle Mühe gegeben”, i.e. “we really tried hard”) have resulted, in the reviewer’s opinion, in an excellently written book, which is pleasurable to read even during technical stretches. (In some rare places the reviewer found the presentation just a bit terse.) The author’s maintain an errata page, see http://www.mathi.uni-heidelberg.de/ag-wingberg/papers/cohen.html. These errata (only one of which was spotted by the reviewer) are few in number, and minor.

The book brings together a remarkable wealth of topics, many of them not easily available before, some of them quite new, and it is organized in a very systematic and coherent way. Therefore this most useful monograph can be warmly recommended, and it should very soon turn into a standard reference in its field.

Reviewer: Cornelius Greither (Neubiberg)

### MSC:

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

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

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11R34 | Galois cohomology |

11S25 | Galois cohomology |

12G05 | Galois cohomology |

11R32 | Galois theory |

12F10 | Separable extensions, Galois theory |

11R23 | Iwasawa theory |

20J06 | Cohomology of groups |

20E18 | Limits, profinite groups |