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The Brauer-Grothendieck group. (English) Zbl 1490.14001

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 71. Cham: Springer (ISBN 978-3-030-74247-8/hbk; 978-3-030-74248-5/ebook). xv, 453 p. (2021).
For fields \(k\), the Brauer group \(\operatorname{Br}(k)\) goes back to work of Brauer, Albert and Noether. Its elements are equivalence classes of central simple algebras \(A\) of finite degree. Addition and inverses come from tensor product and opposite algebras, and the matrix algebras \(\operatorname{Mat}_n(k)\) are made trivial by the equivalence relation. These algebras \(A\) can also be characterized as twisted forms of matrix rings, and in this way generalize to algebras over arbitrary rings \(R\), or sheaves of algebras over structure sheaves \(\mathscr{O}_X\). In this more general context, they are called Azumaya algebras.
Brauer groups of fields, rings, or ringed spaces are truly foundational invariants that have striking applications in various fields of mathematics. To mention a few: In group theory, they measure the obstruction to pass from a projective representation to a linear representation. In algebraic number theory, they provide an elegant formulation of class field theory. In algebraic geometry, they are directly related to twisted forms of projective \(n\)-spaces, and frequently contain obstructions against existence of tautological objects for moduli problems. In complex geometry, they describe the relation between algebraic and transcendental cycles. In arithmetic geometry, they can be used to explain why certain schemes over numbers fields may or may not contain rational points.
One of Grothendieck’s many insights was that each Azumaya \(\mathscr{O}_X\)-algebra \(\mathscr{A}\) induces, via non-abelian cohomology, a class in \(H^2(X,\mathbb{G}_m)\), which embeds the Brauer group into the cohomology group. This is analogous to the interpretation of the Picard group, with two crucial differences: For the Brauer group, one gets in general only an inclusion rather than an equality, and one has to work with the étale topology. This point of view, with its numerous ramifications and applications, was developed by Grothendieck in a series of three highly influential papers [A. Grothendieck, Adv. Stud. Pure Math. 3, 88–188 (1968; Zbl 0198.25901)]. Since then, several books appeared that treated certain aspects of the theory, for example [J. S. Milne, Étale cohomology. Princeton, NJ: Princeton University Press (1980; Zbl 0433.14012)] and [P. Gille and T. Szamuely, Central simple algebras and Galois cohomology. Cambridge: Cambridge University Press (2006; Zbl 1137.12001)] and [S. Gorchinskiy and C. Shramov, Unramified Brauer group and its applications. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1423.14002)].
The monograph of Colliot-Thelene and Skorobogatov is the first book entirely devoted to the subject ob Brauer groups. It has a length about 450 pages and contains 16 chapters. The first three chapters provide the necessary prerequisites from Galois cohomology and étale cohomology, and introduce the Brauer group for schemes. The authors use the somewhat idiosyncratic notation \(\operatorname{Br}(X)=H^2(X,\mathbb{G}_m)\), and call it the Brauer-Grothendieck group, and write \(\operatorname{Br}_{\text{Az}}(X)\subset H^2(X,\mathbb{G}_m)\) for the group of classes of Azumaya algebras, and here is referred to as the Brauer-Azumaya group.
Section 4 discusses methods from stack theory, which in the last two decades found some striking applications to Brauer groups. Namely, one may view \(\alpha\in H^2(X,\mathbb{G}_m)\) as a gerbe \(\mathfrak{X}\) on \(X\), interpret it as an Artin stack, and then study coherent sheaves \(\mathscr{E}\) on this Artin stack. From this viewpoint, the Azumaya algebras on \(X\) representing \(\alpha\) are nothing but certain locally free sheaves on \(\mathfrak{X}\). Using this approach, de Jong gave a proof of a result attributed to Gabber, that the Brauer-Azumaya group is the torsion part in the Brauer-Grothendieck group, provided that \(X\) is quasi-compact and admits an ample invertible sheaf. The book gives the first published version of de Jong’s arguments.
In Section 5, the authors concentrate on separated schemes \(X\) of finite type over a ground field \(k\), which are called varieties, and examine the interplay between the geometric Brauer group \(\operatorname{Br}(X^s)\) obtained after passing to a separable closure, the kernel \(\operatorname{Br}_1(X)\) for the base-change map \(\operatorname{Br}(X)\rightarrow\operatorname{Br}(X^s)\), and the image \(\operatorname{Br}_0(X)\) for \(\operatorname{Br}(k)\rightarrow \operatorname{Br}(X)\). The topic of Section 6 is birational invariance of Brauer groups, and its relation to the ramified Brauer group \(\operatorname{Br}_{\text{nr}}(K/k)\bigcap\operatorname{Br}(A)\), where \(A\) runs through all discrete valuation rings with \(K=\operatorname{Frac}(A)\) and \(k\subset A\). In Section 7 the relation to the Severi–Brauer varieties is discussed, which are twisted forms of some \(\mathbb{P}^n\), whereas Section 8 discusses the contribution of singularities to the Brauer groups. Section 9 discusses the results of Bogomolov and Saltman about the unramified Brauer group of certain fields of invariants.
Section 10 contains various computations concerning schemes over local noetherian rings that are henselian. In Section 11, the authors study more generally the Brauer groups arising in connection with dominant morphisms \(f:X\rightarrow Y\) between integral schemes, and give a discussion of the examples of M. Artin and D. Mumford [Proc. Lond. Math. Soc. (3) 25, 75–95 (1972; Zbl 0244.14017)] for unirational schemes that are not rational. Such rationality questions are examined in more detail in Section 12, with a discussion of the examples of B. Hassett et al. [Acta Math. 220, No. 2, 341–365 (2018; Zbl 1420.14115)] for families of fourfolds, where some members are rational, while others are not stably rational.
The sections 13–15 are devoted to a comprehensive presentation of the Brauer-Manin obstruction for algebraic varieties \(X\) over number fields \(k\). They discuss in detail the set of adelic points \(X(\boldsymbol{A}_k)\) and the ensuing Brauer-Manin set \(X(\boldsymbol{A}_k)^{\operatorname{Br}}\), which contains the closure of \(X(k)\). If the scheme \(X\) has empty Brauer-Manin set but admits adelic points one says that there is a Brauer-Manin obstruction to the Hasse principle for \(X\). Density properties of \(X(k)\subset X(\boldsymbol{A}_k)^{\operatorname{Br}}\) lead to various notions of weak approximation and strong approximations, coming with related obstructions, all of which is discussed in neat detail. The role of Schinzel’s Hypothesis is explained, and applications to rationally connected varieties and zero-cycles are studied. The final Section 16 is devoted to the Tate Conjecture, mainly in the context of K3 surfaces and abelian varieties. The volume closes with a comprehensive list of references, containing about 470 entries, and a short list of symbols. There are no exercises.
The monograph is directed to researchers in algebraic and arithmetic geometry who use Brauer groups in one form or another, and also to graduate students who want to learn about the topic and its applications. The book gives a comprehensive, clear, up-to date presentation of the theory, including most proofs. A particular strength is that it nicely collects many results, examples and counterexamples from various areas of algebraic and arithmetic geometry that are otherwise somewhat scattered in the literature. Summing up, the book fills a wide gap and is a most welcome addition to the literature.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14F22 Brauer groups of schemes

Keywords:

Brauer groups
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