Maximal Orders

Hardcover | January 15, 2003

byIrving Reiner

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This is a reissue of a classic text, which includes the author's own corrections and provides a very accessible, self contained introduction to the classical theory of orders and maximal orders over a Dedekind ring. It starts with a long chapter that provides the algebraic prerequisites forthis theory, covering basic material on Dedekind domains, localizations and completions, as well as semisimple rings and separable algebras. This is followed by an introduction to the basic tools in studying orders, such as reduced norms and traces, discriminants, and localization of orders. Thetheory of maximal orders is then developed in the local case, first in a complete setting, and then over any discrete valuation ring. This paves the way to a chapter on the ideal theory in global maximal orders, with detailed expositions on ideal classes, the Jordan-Zassenhaus Theorem, and genera.This is followed by a chapter on Brauer groups and crossed product algebras, where Hasse's theory of cyclic algebras over local fields is presented in a clear and self-contained fashion. Assuming a couple of facts from class field theory, the book goes on to present the theory of simple algebras over global fields, covering in particular Eichler's Theorem on the ideal classes in a maximal order, as well as various results on the KO group and Picard group of orders. The rest of thebook is devoted to a discussion of non-maximal orders, with particular emphasis on hereditary orders and group rings. The ideas collected in this book have found important applications in the smooth representation theory of reductive p-adic groups. This text provides a useful introduction to this wide range of topics. It is written at a level suitable for beginning postgraduate students, is highly suited to classteaching and provides a wealth of exercises.

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This is a reissue of a classic text, which includes the author's own corrections and provides a very accessible, self contained introduction to the classical theory of orders and maximal orders over a Dedekind ring. It starts with a long chapter that provides the algebraic prerequisites forthis theory, covering basic material on Dedeki...

Professor Irving Reiner (1924-1986), was one of the world's leading experts in representation theory. During his life he published more than 80 research papers, four books (including the original issue of Maximal Orders published by Academic Press in 1975) and many research survey articles on topics related to those contained in this ...
Format:HardcoverDimensions:410 pages, 9.21 × 6.14 × 1.02 inPublished:January 15, 2003Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198526733

ISBN - 13:9780198526735

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Table of Contents

PrefacePermanent Notation1. Algebraic preliminaries2. Orders3. Maximal orders in skewfields (local case)4. Morita equivilence5. Maximal orders over discrete valuation rings6. Maximal orders over Dedekind domains7. Crossed-product algebras8. Simple algebras over global fields9. Hereditary ordersAuthors corrections to textReferencesIndex

Editorial Reviews

`This book is unique in its role of providing a self-contained and easily accessible introduction to the theory of orders and maximal orders in both the local and the global setting. Readers of the book will also find it valuable as a guide to many basic algebraic notions such aslocalizations, completions, the Jacobson radical, and Morita theory. The text is well-written and complete, and well-chosen exercises are offered at the end of each chapter. The book is well-suited as a text for graduate courses in representation theory, and for many years has filled a gap in theliterature in the area of orders and maximal orders.'Professor Tsit-Yuen Lam, University of California at Berkeley.