Value Functions on Simple Algebras, and Associated Graded Rings

Value Functions on Simple Algebras, and Associated Graded Rings
Author :
Publisher : Springer
Total Pages : 652
Release :
ISBN-10 : 9783319163604
ISBN-13 : 3319163604
Rating : 4/5 (04 Downloads)

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

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.

Value Functions on Simple Algebras, and Associated Graded Rings

Value Functions on Simple Algebras, and Associated Graded Rings
Author :
Publisher :
Total Pages :
Release :
ISBN-10 : 3319163612
ISBN-13 : 9783319163611
Rating : 4/5 (12 Downloads)

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

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 noncrossed 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 :
Publisher : Cambridge University Press
Total Pages : 244
Release :
ISBN-10 : 9781316619582
ISBN-13 : 1316619583
Rating : 4/5 (82 Downloads)

Synopsis Graded Rings and Graded Grothendieck Groups by : Roozbeh Hazrat

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 :
Publisher : Cambridge University Press
Total Pages : 431
Release :
ISBN-10 : 9781107156371
ISBN-13 : 1107156378
Rating : 4/5 (71 Downloads)

Synopsis Central Simple Algebras and Galois Cohomology by : Philippe Gille

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 :
Publisher : Cambridge University Press
Total Pages : 379
Release :
ISBN-10 : 9780521113618
ISBN-13 : 052111361X
Rating : 4/5 (18 Downloads)

Synopsis Algebraic Groups and Number Theory by : Vladimir Platonov

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 :
Publisher : Cambridge University Press
Total Pages : 380
Release :
ISBN-10 : 9781009380652
ISBN-13 : 1009380656
Rating : 4/5 (52 Downloads)

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

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 :
Publisher : American Mathematical Society, Bar-Ilan University
Total Pages : 322
Release :
ISBN-10 : 9781470475550
ISBN-13 : 1470475553
Rating : 4/5 (50 Downloads)

Synopsis Amitsur Centennial Symposium by : Avinoam Mann

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 :
Publisher : American Mathematical Soc.
Total Pages : 200
Release :
ISBN-10 : 9781470452070
ISBN-13 : 1470452073
Rating : 4/5 (70 Downloads)

Synopsis Abelian Varieties and Number Theory by : Moshe Jarden

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.

Integral Closure of Ideals, Rings, and Modules

Integral Closure of Ideals, Rings, and Modules
Author :
Publisher : Cambridge University Press
Total Pages : 446
Release :
ISBN-10 : 9780521688604
ISBN-13 : 0521688604
Rating : 4/5 (04 Downloads)

Synopsis Integral Closure of Ideals, Rings, and Modules by : Craig Huneke

Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

Encyclopaedia of Mathematics

Encyclopaedia of Mathematics
Author :
Publisher : Springer Science & Business Media
Total Pages : 543
Release :
ISBN-10 : 9789401512336
ISBN-13 : 9401512337
Rating : 4/5 (36 Downloads)

Synopsis Encyclopaedia of Mathematics by : Michiel Hazewinkel

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.