The Brauer Group of Commutative Rings
Author | : Morris Orzech |
Publisher | : Marcel Dekker |
Total Pages | : 204 |
Release | : 1975 |
Genre | : Mathematics |
ISBN | : |
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Author | : Morris Orzech |
Publisher | : Marcel Dekker |
Total Pages | : 204 |
Release | : 1975 |
Genre | : Mathematics |
ISBN | : |
Author | : Stefaan Caenepeel |
Publisher | : CRC Press |
Total Pages | : 280 |
Release | : 2020-08-26 |
Genre | : Mathematics |
ISBN | : 1000103781 |
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.
Author | : Stefaan Caenepeel |
Publisher | : Springer Science & Business Media |
Total Pages | : 516 |
Release | : 2002-03-31 |
Genre | : Mathematics |
ISBN | : 9781402003462 |
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.
Author | : Frank De Meyer |
Publisher | : Springer |
Total Pages | : 162 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540364846 |
These lecture notes were prepared by the authors for use in graduate courses and seminars, based on the work of many earlier mathematicians. In addition to very elementary results, presented for the convenience of the reader, Chapter I contains the Morita theorems and the definition of the projective class group of a commutative ring. Chapter II addresses the Brauer group of a commutative ring, and automorphisms of separable algebras. Chapter III surveys the principal theorems of the Galois theory for commutative rings. In Chapter IV the authors present a direct derivation of the first six terms of the seven-term exact sequence for Galois cohomology. In the fifth and final chapter the authors illustrate the preceding material with applications to the structure of central simple algebras and the Brauer group of a Dedekind domain, and they pose problems for further investigation. Exercises are included at the end of each chapter.
Author | : Jean-Louis Colliot-Thélène |
Publisher | : Springer Nature |
Total Pages | : 450 |
Release | : 2021-07-30 |
Genre | : Mathematics |
ISBN | : 3030742482 |
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.
Author | : F. van Oystaeyen |
Publisher | : Springer |
Total Pages | : 312 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 354039057X |
Author | : Stefaan Caenepeel |
Publisher | : CRC Press |
Total Pages | : 352 |
Release | : 2020-09-29 |
Genre | : Mathematics |
ISBN | : 1000153282 |
"Based on papers presented at a recent international conference on algebra and algebraic geometry held jointly in Antwerp and Brussels, Belgium. Presents both survey and research articles featuring new results from the intersection of algebra and geometry. "
Author | : D. Zelinsky |
Publisher | : Springer |
Total Pages | : 193 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 3540379789 |
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Author | : Carl Faith |
Publisher | : Springer Science & Business Media |
Total Pages | : 589 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642806341 |
VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.
Author | : Charles Small |
Publisher | : |
Total Pages | : 410 |
Release | : 1970 |
Genre | : |
ISBN | : |