The Brauer-Grothendieck group / Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov.

Format:

Book

Author/Creator:

Colliot-Thélène, J.-L. (Jean-Louis), author.

Skorobogatov, Alexei, 1961- author.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 71.

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 ; Volume 71

Language:

English

Subjects (All):

Grothendieck groups.

Brauer groups.

Schemes (Algebraic geometry).

Cohomology operations.

Arithmetical algebraic geometry.

Physical Description:

xv, 453 pages ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer Nature Switzerland AG, [2021]

Summary:

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. -- Publisher's website

Contents:

Galois Cohomology

Étale Cohomology

Brauer Groups of Schemes

Comparison of the Two Brauer Groups, II

Varieties Over a Field

Birational Invariance

Severi-Brauer Varieties and hypersurfaces

Singular Schemes and Varieties

Varieties with a Group Action

Schemes Over Local Rings and Fields

Families of Varieties

Rationality in a Family

The Brauer-Manin Set and the formal lemma

Rational Points in the Brauer-Manin Set

The Brauer-Manin Obstruction for Zero-Cycles

The Tate Conjecture, Abelian Varieties and K3 Surfaces.

Notes:

Includes bibliographical references (pages 427-448) and indexes.

Current copyright fee: GBP19.00 42\0.

Other Format:

Online version: Colliot-Thélène, J.-L. (Jean-Louis). Brauer-Grothendieck group.

ISBN:

9783030742478

3030742474

OCLC:

1242465146