##
**Unramified Brauer group and its applications.**
*(English)*
Zbl 1423.14002

Translations of Mathematical Monographs 246. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4072-5/hbk; 978-1-4704-4857-8/ebook). xvii, 179 p. (2018).

From publisher’s description: “This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.”

The topic of this book is the unramified Brauer group and its application to rationality problems and Hasse principles. The theory is developed not as in most other textbooks, but mainly through exercises with many intermediate steps and hints making them very accessible e.g. for graduate students or researchers looking for a quick introduction to the field. Difficult theorems like Merkurjev-Suslin are cited. I think the strategy of this concise book allows one to quickly learn the subject if you like the approach through exercises in the style of Serre’s books. The book has a good balance between modern geometric language and explicit equations. You should not take it as a reference containing full proofs. At the end of most chapters, further reading recommendations are given. The Russian original, extended notes of a 2011 reading seminar at the Steklov Mathematical Institute, can be found at [arXiv:1512.00874].

Other books covering the theory of the Brauer group and the Brauer-Manin obstruction are [A. Skorobogatov, Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015); B. Poonen, Rational points on varieties. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1387.14004)]. These books contain more proofs. [Zbl 0972.14015] covers more applications of Brauer groups and torsors, but does not contain the basics as the book under review and [Zbl 1387.14004]. The applications to rationality problems (Part III) are unique to this book.

Part I “Preliminaries on Galois cohomology” is an introduction to group and Galois cohomology.

In Part II (“Brauer group”) treats the special case of the Brauer group of a field and its relation to Severi-Brauer varieties in Chapter 3 and then in Chapter 4 the (Azumaya algebra) Brauer group of a scheme first defined by A. Grothendieck [in: Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 46–66 (1968; Zbl 0193.21503); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 67–87 (1968); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 88–188 (1968; Zbl 0198.25901)] and the unramified Brauer group \(\mathrm{Br}^{nr} (K)\) (cf. the first part of the title of the book) [J. L. Colliot-Thélène, Proc. Symp. Pure Math. 58, 1–64 (1995; Zbl 0834.14009)] of a function field \(K\) finitely generated over a constant field \(k\) of characteristic 0 defined as the intersection of the kernels of the residue maps \(\mathrm{res}_v:\mathrm{Br}(K)\to\mathrm{Br}(K_v)\to\mathrm{Hom}(G_{\kappa_v},\mathbb Q/\mathbb Z)\) for all discrete valuations \(v\) of \(K\) trivial on \(k^\ast\) with residue field \(\kappa_v\), also with their geometric meaning and the unramified Brauer group of a normal irreducible variety over \(k\). The unramified Brauer group of a smooth proper variety is a stably birational invariant. The book restricts to finitely generated function fields of characteristic 0 because they have perfect residue fields.

Part III “Applications to rationality problems” is the second part of the title and the heart of the book: Chapter 5 produces an example of a quotient of a variety over an al-gebraically closed field of characteristic 0 by a finite group and a proof that is not stably rational using the non-triviality of the unramified Brauer group. Chapter 6 “Arithmetic of Two-dimensional Quadrics” is on the discriminant and the Clifford invariant of quadrics and their geometric meaning. Chapter 7 “Non-rational Double Covers of \(\mathbb P^3\)” proves using the unramified Brauer group that certain unirational threefolds are not rational. In Chapter 8 the Weil restriction of scalars and algebraic tori are introduced and applied to rationality problems. Chapter 9 is on an example [A. Beauville et al., Ann. Math. (2) 121, 283–318 (1985; Zbl 0589.14042)] of a variety over a perfect field of characteristic \(\neq 2\) which is non-rational, but stably rational, hence unirational.

In Part IV (“The Hasse principle and its failure”) the Hasse-Minkowski theorem on quadratic forms over global fields is proved by a geometric reduction to quadrics of dimension 1 and the Hasse principle for the Brauer group (Chapter 11). In Chapter 12 the Brauer-Manin obstruction introduced by Yu. I. Manin [in: Actes Congr. internat. Math. 1970, No. 1, 401–411 (1971; Zbl 0239.14010)] is defined and applied to explaining the failure of the Hasse principle for the Lind-Reichardt equation \(2y^2=x^4-17\).

Appendix A contains a short introduction to etale cohomology without proofs or exercises, but references to the standard source [J. S. Milne, Étale cohomology. Princeton Mathematical Series. 33. Princeton, New Jersey: Princeton University Press. (1980; Zbl 0433.14012)].

The topic of this book is the unramified Brauer group and its application to rationality problems and Hasse principles. The theory is developed not as in most other textbooks, but mainly through exercises with many intermediate steps and hints making them very accessible e.g. for graduate students or researchers looking for a quick introduction to the field. Difficult theorems like Merkurjev-Suslin are cited. I think the strategy of this concise book allows one to quickly learn the subject if you like the approach through exercises in the style of Serre’s books. The book has a good balance between modern geometric language and explicit equations. You should not take it as a reference containing full proofs. At the end of most chapters, further reading recommendations are given. The Russian original, extended notes of a 2011 reading seminar at the Steklov Mathematical Institute, can be found at [arXiv:1512.00874].

Other books covering the theory of the Brauer group and the Brauer-Manin obstruction are [A. Skorobogatov, Torsors and rational points. Cambridge: Cambridge University Press (2001; Zbl 0972.14015); B. Poonen, Rational points on varieties. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1387.14004)]. These books contain more proofs. [Zbl 0972.14015] covers more applications of Brauer groups and torsors, but does not contain the basics as the book under review and [Zbl 1387.14004]. The applications to rationality problems (Part III) are unique to this book.

Part I “Preliminaries on Galois cohomology” is an introduction to group and Galois cohomology.

In Part II (“Brauer group”) treats the special case of the Brauer group of a field and its relation to Severi-Brauer varieties in Chapter 3 and then in Chapter 4 the (Azumaya algebra) Brauer group of a scheme first defined by A. Grothendieck [in: Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 46–66 (1968; Zbl 0193.21503); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 67–87 (1968); in: Dix Exposes Cohomologie Schemas, Advanced Studies Pure Math. 3, 88–188 (1968; Zbl 0198.25901)] and the unramified Brauer group \(\mathrm{Br}^{nr} (K)\) (cf. the first part of the title of the book) [J. L. Colliot-Thélène, Proc. Symp. Pure Math. 58, 1–64 (1995; Zbl 0834.14009)] of a function field \(K\) finitely generated over a constant field \(k\) of characteristic 0 defined as the intersection of the kernels of the residue maps \(\mathrm{res}_v:\mathrm{Br}(K)\to\mathrm{Br}(K_v)\to\mathrm{Hom}(G_{\kappa_v},\mathbb Q/\mathbb Z)\) for all discrete valuations \(v\) of \(K\) trivial on \(k^\ast\) with residue field \(\kappa_v\), also with their geometric meaning and the unramified Brauer group of a normal irreducible variety over \(k\). The unramified Brauer group of a smooth proper variety is a stably birational invariant. The book restricts to finitely generated function fields of characteristic 0 because they have perfect residue fields.

Part III “Applications to rationality problems” is the second part of the title and the heart of the book: Chapter 5 produces an example of a quotient of a variety over an al-gebraically closed field of characteristic 0 by a finite group and a proof that is not stably rational using the non-triviality of the unramified Brauer group. Chapter 6 “Arithmetic of Two-dimensional Quadrics” is on the discriminant and the Clifford invariant of quadrics and their geometric meaning. Chapter 7 “Non-rational Double Covers of \(\mathbb P^3\)” proves using the unramified Brauer group that certain unirational threefolds are not rational. In Chapter 8 the Weil restriction of scalars and algebraic tori are introduced and applied to rationality problems. Chapter 9 is on an example [A. Beauville et al., Ann. Math. (2) 121, 283–318 (1985; Zbl 0589.14042)] of a variety over a perfect field of characteristic \(\neq 2\) which is non-rational, but stably rational, hence unirational.

In Part IV (“The Hasse principle and its failure”) the Hasse-Minkowski theorem on quadratic forms over global fields is proved by a geometric reduction to quadrics of dimension 1 and the Hasse principle for the Brauer group (Chapter 11). In Chapter 12 the Brauer-Manin obstruction introduced by Yu. I. Manin [in: Actes Congr. internat. Math. 1970, No. 1, 401–411 (1971; Zbl 0239.14010)] is defined and applied to explaining the failure of the Hasse principle for the Lind-Reichardt equation \(2y^2=x^4-17\).

Appendix A contains a short introduction to etale cohomology without proofs or exercises, but references to the standard source [J. S. Milne, Étale cohomology. Princeton Mathematical Series. 33. Princeton, New Jersey: Princeton University Press. (1980; Zbl 0433.14012)].

Reviewer: Timo Keller (Bayreuth)

### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14F22 | Brauer groups of schemes |

14E08 | Rationality questions in algebraic geometry |

14G05 | Rational points |

14M20 | Rational and unirational varieties |

12G05 | Galois cohomology |