Brauer Groups

Download or read book The Brauer–Grothendieck Group written by Jean-Louis Colliot-Thélène and published by Springer Nature. This book was released on 2021-07-30 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Richard Brauer - Collected Papers written by Richard Brauer and published by MIT Press (MA). This book was released on 2003-02-01 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: Richard Brauer (1901-1977) was one of the leading algebraists of this century. Although he contributed to a number of mathematical fields, Brauer devoted the major share of his efforts to the study of finite groups, a subject of considerable abstract interest and one that underlies many of the more recent advances in combinatorics and finite geometries.

Download or read book Brauer Groups, Hopf Algebras and Galois Theory written by Stefaan Caenepeel and published by Springer Science & Business Media. This book was released on 2002-03-31 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and étale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included. The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph. Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.

Download or read book Brauer Groups and the Cohomology of Graded Rings written by Stefaan Caenepeel and published by CRC Press. This book was released on 2020-08-26 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces various notions defined in graded terms extending the notions most frequently used as basic ingredients in the theory of Azumaya algebras: separability and Galois extensions of commutative rings, crossed products and Galois cohomology, Picard groups, and the Brauer group.

Download or read book Central Simple Algebras and Galois Cohomology written by Philippe Gille and published by Cambridge University Press. This book was released on 2017-08-10 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.

Download or read book Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties written by Jorg Jahnel and published by American Mathematical Soc.. This book was released on 2014-12-02 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.

Download or read book The Brauer Group of Commutative Rings written by Morris Orzech and published by Marcel Dekker. This book was released on 1975 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: