Separable Algebras Over Commutative Rings
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Frank DeMeyer Colorado State University, Fort Collins, CO / USA
Edward Ingraham Michigan State University, East Lansing, MI / USA
Separable Algebras Over Commutative Rings
Springer-Verlag Berlin· Heidelberg· NewYork 1971
ISBN 3-540-05371-9 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-05371-9 Springer-Verlag New York· Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 70-151404 Printed in Germany. Offsetdruck:Julius Beltz, Weinheim/Bergstr.
PREFACE This manuscript has its origin in courses given by the authors at Purdue University and Michigan state University.
We were first intro-
duced to this material by D. K. Harrison and we hope his influence is apparent.
We also had access to notes prepared by G. J. Janusz for a
course he gave at the University of Chicago. We have laid the foundation for the study of the Brauer group of a commutative ring but have omitted the "spectral sequences" and the "method of descent" in the hope this will make the material more accessible.
Cross-references are made as follows:
Theorem 2.1 of Chapter 3
is referred to as Theorem 2.1 if the reference is in Chapter 3 and as Theorem 3.2.1 otherwise.
We would like to thank the many students who
have influenced and encouraged this publication, and Mrs. Glendora Milligan, who carefully typed the final form for reproduction. September 1970
Frank DeMeyer Colorado State University Edward Ingraham Michigan State University
CONTENTS CHAPTER I: Preliminaries . . . . 1• 2.
3. 4. 5.
Progenerator Modules . Categories and Functors of Modules The Morita Theorems . . . . . . . Local Rings; Rings and Modules of Quotients; Localization. ..•... The Projective Class Group Exercises . . . . . . . . . .
CHAPTER II: Central Separable Algebras and the Brauer Group 1•
2.
3. 4. 5. 6.
7.
Definitions, Examples, and Basic Properties. Separable Algebras over Fields Central Separable Algebras . . . . . . • . The Commutator Theorems . . . . . . . . . . . The Brauer Group . . . . . . . . . . . . . Automorphisms of Central Separable Algebras Two Criteria for Separability Exercises . . . .
CHAPTER III: Galois Theory. 12.
3. 4.
The Fundamental Theorem The Imbedding Theorem • •• • The Separable Closure and Infinite Galois Theory Separable Polynomials Exercises . . .
CHAPTER IV: The SixTerm Exact Sequence 12.
Definitions and Statement of the Theorem The Proof of Theorem 1.1 Exercise . • . . . . . •
CHAPTER V: Applications and Remarks 1. Structure Theory • . 2. The Brauer Group of a Dedekind Domain 3. Remar
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