This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer-Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications.

Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong's proof of Gabber's theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer-Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples.

Over half a century after Grothendieck's foundational seminars on the topic, The Brauer-Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.

Autorentext
Jean-Louis Colliot-Thélène works in arithmetic algebraic geometry. He contributed to the study of rational points and of zero-cycles on rationally connected varieties. This involved the use of torsors and the BrauerManin obstruction. He applied results from algebraic K-theory (unramified cohomology) to rationality problems, also in complex algebraic geometry. He is the author of some 150 research papers, many written with various collaborators. Jean-Louis Colliot-Thélène received the Fermat prize and a Grand Prix de l'Académie des Sciences de Paris. Alexei Skorobogatov works in arithmetic algebraic geometry with focus on rational points on algebraic varieties, the Brauer group and the BrauerManin obstruction, K3 surfaces and abelian varieties. He is the author of the book Torsors and Rational Points and over 75 research papers. Alexei Skorobogatov is the recipient of a Whitehead prize of the London Mathematical Society.

Inhalt
1 Galois Cohomology.- 2 Étale Cohomology.- 3 Brauer Groups of Schemes.- 4 Comparison of the Two Brauer Groups, II.- 5 Varieties Over a Field.- 6 Birational Invariance.- 7 SeveriBrauer Varieties and Hypersurfaces.- 8 Singular Schemes and Varieties.- 9 Varieties with a Group Action.- 10 Schemes Over Local Rings and Fields.- 11 Families of Varieties.- 12 Rationality in a Family.- 13 The BrauerManin Set and the Formal Lemma.- 14 Are Rational Points Dense in the BrauerManin Set?.- 15 The BrauerManin Obstruction for Zero-Cycles.- 16 Tate Conjecture, Abelian Varieties and K3 Surfaces.- Bibliography.- Index.