Hopf Algebras and Galois Theory
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Stephen U. Chase Moss E. Sweedler Cornell University, Ithaca, N.Y.
1969
Hopf Algebras and Galois Theory
Springer-Verlag Berlin· Heidelberg· New York
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich Series: Forschungsinstitut fUr Mathematik, ETH, ZUrich, Adviser: K. Chandrasekharan
97
Stephen U. Chase Moss E. Sweedler Cornell University, Ithaca, N.Y.
1969
Hopf Algebras and Galois Theory
Springer-Verlag Berlin· Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer- Verlag Berlin· Heidelberg 1969 Library of Congress Catalog Card Number 75- 84143 . Printed in Germany. Title No. 3703
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Preface and Acknowledgement consists of three chapters on separate but related topics, followed by several pages of footnotes which were inserted to correct and clarify certain points in the text. Chapters I and III, written by Stephen U. Chase, are based upon a series of lectures given during the fall and winter of 1966-67 at the ETH Forschungsinstitut fur Mathematik in Zurich. The author is indebted to the Forschungsinstitut and its director, Professor Beno Eckmann, for the gracious hospitality which he enjoyed there, and to the Alfred E. Sloan Foundation, ETH, and NSF GP-6484 for providing the necessary financial support. Chapter II, which contains joint work of Stephen U. Chase and Moss E. Sweedler, was written during the fall of 1967 at Cornell University with the partial support of NSF GP-7945. Chapter III was subsequently revised so as to take advantage of some of the techniques of Chapter II. During this academic year the first-named author also received partial support from the Alfred E. Sloan Foundation. The authors are indebted to their colleagues at Cornell and the Forschungsinstitut for many stimulating conversations on matters related to the material of these notes; in particular, to Jon Beck, Bill Lawvere, Fred Linton, Stephen Lichtenbaum, George Rinehart, and Miles Tierney.
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Contents
I. Galois Objects by Stephen U. Chase
O. Introduction and Preliminaries
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1. Elementary Properties of Galois Objects. • • • •
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2. Adjointness and the Functor E. • • • •
• 14
3. Products of Galois Objects and the Functor X 4. Galois Algebras and Coalgebras
• • • •
• • • • • 21
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• • 28
5. Triples, Extensions, and Cohomology.
• 41
II. Bopf Algebras and Galois Theory of Rings by Stephen U. Chase and Moss E. Sweedler
6. Introduction
• 52
7. Definitions, Theorems and Consequences
• 54
• • • • • • • • • • • • • • • •
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9. Elementary properties of Bopf Algebras and Galois Objects.
• 66
8. The Morita Correspondence
10.Subalgebras of Galois Objects.
• • • • • • • •
11.Proof of the Main Theorem. • • • • • • • • • •
• • 71 • • 77
12.Flatness, and the Functor)( • • • • • • • • • • • • • • • • • ••
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III. Galois Objects and Extensions of Bopf Algebras by Stephen U. Chase
13.Introduction
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