Central Simple Algebras And Galois Cohomology
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Central Simple Algebras and Galois Cohomology
Author | : Philippe Gille,Tamás Szamuely |
Publsiher | : Cambridge University Press |
Total Pages | : 431 |
Release | : 2017-08-10 |
Genre | : Mathematics |
ISBN | : 9781107156371 |
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The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.
Central Simple Algebras and Galois Cohomology
Author | : Philippe Gille,Tamás Szamuely |
Publsiher | : Cambridge University Press |
Total Pages | : 432 |
Release | : 2017-08-10 |
Genre | : Mathematics |
ISBN | : 9781108293679 |
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The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.
Central Simple Algebras and Galois Cohomology ICM Edition
Author | : Gille,Tamás Szamuely |
Publsiher | : Unknown |
Total Pages | : 135 |
Release | : 2014-12-01 |
Genre | : Electronic Book |
ISBN | : 0521168910 |
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The Brauer Grothendieck Group
Author | : Jean-Louis Colliot-Thélène,Alexei N. Skorobogatov |
Publsiher | : Springer Nature |
Total Pages | : 450 |
Release | : 2021-07-30 |
Genre | : Mathematics |
ISBN | : 9783030742485 |
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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.
An Introduction to Galois Cohomology and its Applications
Author | : Grégory Berhuy |
Publsiher | : Cambridge University Press |
Total Pages | : 328 |
Release | : 2010-09-09 |
Genre | : Mathematics |
ISBN | : 9781139490887 |
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This is the first detailed elementary introduction to Galois cohomology and its applications. The introductory section is self-contained and provides the basic results of the theory. Assuming only a minimal background in algebra, the main purpose of this book is to prepare graduate students and researchers for more advanced study.
Cohomological Invariants in Galois Cohomology
Author | : Skip Garibaldi,Alexander Merkurjev,Jean-Pierre Serre |
Publsiher | : American Mathematical Soc. |
Total Pages | : 168 |
Release | : 2003 |
Genre | : Mathematics |
ISBN | : 9780821832875 |
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This volume is concerned with algebraic invariants, such as the Stiefel-Whitney classes of quadratic forms (with values in Galois cohomology mod 2) and the trace form of etale algebras (with values in the Witt ring). The invariants are analogues for Galois cohomology of the characteristic classes of topology. Historically, one of the first examples of cohomological invariants of the type considered here was the Hasse-Witt invariant of quadratic forms. The first part classifies such invariants in several cases. A principal tool is the notion of versal torsor, which is an analogue of the universal bundle in topology. The second part gives Rost's determination of the invariants of $G$-torsors with values in $H^3(\mathbb{Q}/\mathbb{Z}(2))$, when $G$ is a semisimple, simply connected, linear group. This part gives detailed proofs of the existence and basic properties of the Rost invariant. This is the first time that most of this material appears in print.
A Gentle Course in Local Class Field Theory
Author | : Pierre Guillot |
Publsiher | : Cambridge University Press |
Total Pages | : 309 |
Release | : 2018-11 |
Genre | : Mathematics |
ISBN | : 9781108421775 |
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A self-contained exposition of local class field theory for students in advanced algebra.
Quadratic Forms Linear Algebraic Groups and Cohomology
Author | : Skip Garibaldi,R. Sujatha,Venapally Suresh |
Publsiher | : Springer Science & Business Media |
Total Pages | : 344 |
Release | : 2010-07-16 |
Genre | : Mathematics |
ISBN | : 9781441962119 |
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Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.