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

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

Label
The Brauer-Grothendieck group
Title
The Brauer-Grothendieck group
Statement of responsibility
Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov
Creator
Contributor
Author
Subject
Language
eng
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. --
Member of
Assigning source
Publisher's website
Cataloging source
YDX
http://library.link/vocab/creatorName
Colliot-Thélène, J.-L.
Index
index present
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/relatedWorkOrContributorDate
1961-
http://library.link/vocab/relatedWorkOrContributorName
Skorobogatov, Alexei
Series statement
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,
Series volume
Volume 71
http://library.link/vocab/subjectName
  • Grothendieck groups
  • Brauer groups
  • Schemes (Algebraic geometry)
  • Cohomology operations
  • Arithmetical algebraic geometry
Label
The Brauer-Grothendieck group, Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov
Instantiates
Publication
Copyright
Bibliography note
Includes bibliographical references (pages 427-448) and indexes
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
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
Control code
1242465146
Dimensions
24 cm
Extent
xv, 453 pages
Isbn
9783030742478
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other control number
10.1007/9783030742485
System control number
(OCoLC)1242465146
Label
The Brauer-Grothendieck group, Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov
Publication
Copyright
Bibliography note
Includes bibliographical references (pages 427-448) and indexes
Carrier category
volume
Carrier category code
  • nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
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
Control code
1242465146
Dimensions
24 cm
Extent
xv, 453 pages
Isbn
9783030742478
Media category
unmediated
Media MARC source
rdamedia
Media type code
  • n
Other control number
10.1007/9783030742485
System control number
(OCoLC)1242465146

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