The Brauer–Grothendieck Group, 1st ed. 2021 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Series, Vol. 71

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.

Alexei Skorobogatov works in arithmetic algebraic geometry with focus on rational points on algebraic varieties, the Brauer group and the Brauer–Manin 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.

Provides a self-contained introduction to the Brauer group of schemes

Presents recent applications to rational points on varieties and to rationality problems in algebraic geometry

Offers a detailed guide to the computation and finiteness of the Brauer group for various classes of varieties