Value Functions on Simple Algebras, and Associated Graded Rings

Value Functions on Simple Algebras, and Associated Graded Rings

Author: Jean-Pierre Tignol

Publisher: Springer

Published: 2015-04-03

Total Pages: 652

ISBN-13: 3319163604

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This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of non crossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.


Book Synopsis Value Functions on Simple Algebras, and Associated Graded Rings by : Jean-Pierre Tignol

Download or read book Value Functions on Simple Algebras, and Associated Graded Rings written by Jean-Pierre Tignol and published by Springer. This book was released on 2015-04-03 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of non crossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.

Graded Rings and Graded Grothendieck Groups

Graded Rings and Graded Grothendieck Groups

Author: Roozbeh Hazrat

Publisher: Cambridge University Press

Published: 2016-05-26

Total Pages: 244

ISBN-13: 1316619583

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This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.


Book Synopsis Graded Rings and Graded Grothendieck Groups by : Roozbeh Hazrat

Download or read book Graded Rings and Graded Grothendieck Groups written by Roozbeh Hazrat and published by Cambridge University Press. This book was released on 2016-05-26 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.

Central Simple Algebras and Galois Cohomology

Central Simple Algebras and Galois Cohomology

Author: Philippe Gille

Publisher: Cambridge University Press

Published: 2017-08-10

Total Pages: 431

ISBN-13: 1107156378

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The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.


Book Synopsis Central Simple Algebras and Galois Cohomology by : Philippe Gille

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.

Algebraic Groups and Number Theory

Algebraic Groups and Number Theory

Author: Vladimir Platonov

Publisher: Cambridge University Press

Published: 2023-08-31

Total Pages: 379

ISBN-13: 052111361X

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The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.


Book Synopsis Algebraic Groups and Number Theory by : Vladimir Platonov

Download or read book Algebraic Groups and Number Theory written by Vladimir Platonov and published by Cambridge University Press. This book was released on 2023-08-31 with total page 379 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.

Algebraic Groups and Number Theory: Volume 1

Algebraic Groups and Number Theory: Volume 1

Author: Vladimir Platonov

Publisher: Cambridge University Press

Published: 2023-08-31

Total Pages: 380

ISBN-13: 1009380656

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The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.


Book Synopsis Algebraic Groups and Number Theory: Volume 1 by : Vladimir Platonov

Download or read book Algebraic Groups and Number Theory: Volume 1 written by Vladimir Platonov and published by Cambridge University Press. This book was released on 2023-08-31 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.

Amitsur Centennial Symposium

Amitsur Centennial Symposium

Author: Avinoam Mann

Publisher: American Mathematical Society, Bar-Ilan University

Published: 2024-05-14

Total Pages: 322

ISBN-13: 1470475553

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This volume contains the proceedings of the Amitsur Centennial Symposium, held from November 1–4, 2021, virtually and at the Israel Institute for Advanced Studies (IIAS), The Hebrew University of Jerusalem, Jerusalem, Israel. Shimshon Amitsur was a pioneer in several branches of algebra, the leading algebraist in Israel for several decades who contributed major theorems, inspiring results, useful observations, and enlightening tricks to many areas of the field. The fifteen papers included in the volume represent the broad impact of Amitsur's work on such areas as the theory of finite simple groups, algebraic groups, PI-algebras and growth of rings, quadratic forms and division algebras, torsors and Severi-Brauer surfaces, Hopf algebras and braces, invariants, automorphisms and derivations.


Book Synopsis Amitsur Centennial Symposium by : Avinoam Mann

Download or read book Amitsur Centennial Symposium written by Avinoam Mann and published by American Mathematical Society, Bar-Ilan University. This book was released on 2024-05-14 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Amitsur Centennial Symposium, held from November 1–4, 2021, virtually and at the Israel Institute for Advanced Studies (IIAS), The Hebrew University of Jerusalem, Jerusalem, Israel. Shimshon Amitsur was a pioneer in several branches of algebra, the leading algebraist in Israel for several decades who contributed major theorems, inspiring results, useful observations, and enlightening tricks to many areas of the field. The fifteen papers included in the volume represent the broad impact of Amitsur's work on such areas as the theory of finite simple groups, algebraic groups, PI-algebras and growth of rings, quadratic forms and division algebras, torsors and Severi-Brauer surfaces, Hopf algebras and braces, invariants, automorphisms and derivations.

Abelian Varieties and Number Theory

Abelian Varieties and Number Theory

Author: Moshe Jarden

Publisher: American Mathematical Soc.

Published: 2021-05-03

Total Pages: 200

ISBN-13: 1470452073

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This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc. The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.


Book Synopsis Abelian Varieties and Number Theory by : Moshe Jarden

Download or read book Abelian Varieties and Number Theory written by Moshe Jarden and published by American Mathematical Soc.. This book was released on 2021-05-03 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of articles on Abelian varieties and number theory dedicated to Gerhard Frey's 75th birthday. It contains original articles by experts in the area of arithmetic and algebraic geometry. The articles cover topics on Abelian varieties and finitely generated Galois groups, ranks of Abelian varieties and Mordell-Lang conjecture, Tate-Shafarevich group and isogeny volcanoes, endomorphisms of superelliptic Jacobians, obstructions to local-global principles over semi-global fields, Drinfeld modular varieties, representations of etale fundamental groups and specialization of algebraic cycles, Deuring's theory of constant reductions, etc. The book will be a valuable resource to graduate students and experts working on Abelian varieties and related areas.

Graded Ring Theory

Graded Ring Theory

Author: C. Nastasescu

Publisher: Elsevier

Published: 2011-08-18

Total Pages: 352

ISBN-13: 0080960162

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This book is aimed to be a ‘technical’ book on graded rings. By ‘technical’ we mean that the book should supply a kit of tools of quite general applicability, enabling the reader to build up his own further study of non-commutative rings graded by an arbitrary group. The body of the book, Chapter A, contains: categorical properties of graded modules, localization of graded rings and modules, Jacobson radicals of graded rings, the structure thedry for simple objects in the graded sense, chain conditions, Krull dimension of graded modules, homogenization, homological dimension, primary decomposition, and more. One of the advantages of the generality of Chapter A is that it allows direct applications of these results to the theory of group rings, twisted and skew group rings and crossed products. With this in mind we have taken care to point out on several occasions how certain techniques may be specified to the case of strongly graded rings. We tried to write Chapter A in such a way that it becomes suitable for an advanced course in ring theory or general algebra, we strove to make it as selfcontained as possible and we included several problems and exercises. Other chapters may be viewed as an attempt to show how the general techniques of Chapter A can be applied in some particular cases, e.g. the case where the gradation is of type Z. In compiling the material for Chapters B and C we have been guided by our own research interests. Chapter 6 deals with commutative graded rings of type 2 and we focus on two main topics: artihmeticallygraded domains, and secondly, local conditions for Noetherian rings. In Chapter C we derive some structural results relating to the graded properties of the rings considered. The following classes of graded rings receive special attention: fully bounded Noetherian rings, birational extensions of commutative rings, rings satisfying polynomial identities, and Von Neumann regular rings. Here the basic idea is to derive results of ungraded nature from graded information. Some of these sections lead naturally to the study of sheaves over the projective spectrum Proj(R) of a positively graded ring, but we did not go into these topics here. We refer to [125] for a noncommutative treatment of projective geometry, i.e. the geometry of graded P.I. algebras.


Book Synopsis Graded Ring Theory by : C. Nastasescu

Download or read book Graded Ring Theory written by C. Nastasescu and published by Elsevier. This book was released on 2011-08-18 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is aimed to be a ‘technical’ book on graded rings. By ‘technical’ we mean that the book should supply a kit of tools of quite general applicability, enabling the reader to build up his own further study of non-commutative rings graded by an arbitrary group. The body of the book, Chapter A, contains: categorical properties of graded modules, localization of graded rings and modules, Jacobson radicals of graded rings, the structure thedry for simple objects in the graded sense, chain conditions, Krull dimension of graded modules, homogenization, homological dimension, primary decomposition, and more. One of the advantages of the generality of Chapter A is that it allows direct applications of these results to the theory of group rings, twisted and skew group rings and crossed products. With this in mind we have taken care to point out on several occasions how certain techniques may be specified to the case of strongly graded rings. We tried to write Chapter A in such a way that it becomes suitable for an advanced course in ring theory or general algebra, we strove to make it as selfcontained as possible and we included several problems and exercises. Other chapters may be viewed as an attempt to show how the general techniques of Chapter A can be applied in some particular cases, e.g. the case where the gradation is of type Z. In compiling the material for Chapters B and C we have been guided by our own research interests. Chapter 6 deals with commutative graded rings of type 2 and we focus on two main topics: artihmeticallygraded domains, and secondly, local conditions for Noetherian rings. In Chapter C we derive some structural results relating to the graded properties of the rings considered. The following classes of graded rings receive special attention: fully bounded Noetherian rings, birational extensions of commutative rings, rings satisfying polynomial identities, and Von Neumann regular rings. Here the basic idea is to derive results of ungraded nature from graded information. Some of these sections lead naturally to the study of sheaves over the projective spectrum Proj(R) of a positively graded ring, but we did not go into these topics here. We refer to [125] for a noncommutative treatment of projective geometry, i.e. the geometry of graded P.I. algebras.

Methods of Graded Rings

Methods of Graded Rings

Author: Constantin Nastasescu

Publisher: Springer Science & Business Media

Published: 2004-02-19

Total Pages: 324

ISBN-13: 9783540207467

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The Category of Graded Rings.- The Category of Graded Modules.- Modules over Stronly Graded Rings.- Graded Clifford Theory.- Internal Homogenization.- External Homogenization.- Smash Products.- Localization of Graded Rings.- Application to Gradability.- Appendix A:Some Category Theory.- Appendix B: Dimensions in an abelian Category.- Bibliography.- Index.-


Book Synopsis Methods of Graded Rings by : Constantin Nastasescu

Download or read book Methods of Graded Rings written by Constantin Nastasescu and published by Springer Science & Business Media. This book was released on 2004-02-19 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Category of Graded Rings.- The Category of Graded Modules.- Modules over Stronly Graded Rings.- Graded Clifford Theory.- Internal Homogenization.- External Homogenization.- Smash Products.- Localization of Graded Rings.- Application to Gradability.- Appendix A:Some Category Theory.- Appendix B: Dimensions in an abelian Category.- Bibliography.- Index.-

Encyclopaedia of Mathematics

Encyclopaedia of Mathematics

Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 543

ISBN-13: 9401512337

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This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fme subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.


Book Synopsis Encyclopaedia of Mathematics by : Michiel Hazewinkel

Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 543 pages. Available in PDF, EPUB and Kindle. Book excerpt: This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fme subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.