Trivial Extensions of Abelian Categories Homological Algebra of Triv
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Robert M. Fossum Phi IIi p A. Griffith ldun Reiten Trivial Extensions of Abelian Categories Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory
Springer-Verlag Berlin· Heidelberg· New York 1975
Authors Dr. Robert M. Fossum Dr. Phillip A Griffith Department of Mathematics University of Illinois Urbana, Illinois 61801 USA
Dr. Idun Reiten Matematisk Institutt Universitetet I Trondheim Norges Laererhøgskole N-7000-Trondheim
Library of Congress Cataloging in Publication Data
Fossum, Robert M Trivial extensions of Abelian categories.
(Lecture notes in mathematics ; 456) Bibliography: p. Includes index. 1. Commutative rings.
2. Associative rings. 3. Abelian categories. I. Griffith, Phillip A., joint author. II. Reiten, Idun, 1942joint author. Title. IV. Series: Lecture notes in mathematics (Berlin) ; 456. QA3.L28 no. 456 [QA2151.3] 510'.8s [512'.55] 75-12984
III.
AMS Subject Classifications (1970): 13A20, 13C15, 13D05, 13H10,
16A48, 16A49, 16A50, 16A52, 16A56, 18A05, 18A25,18 E10, 18GXX ISBN 3-540-07159-8 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07159-8 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 1975. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
v
Introduction
Section 0:
Preliminaries
1
Section 1:
Generalities
3
Section 2:
Coherence
24
Section 3:
Duality and the Gorenstein property
35
Section 4:
Homological dimension in
52
Section 5:
Gorenstein modules
~ ~
F
87
Section
6:
Dominant dimension of finite algebras
104
Section
7:
Representation dimension of finite algebras
113
References
117
Introduction
The notion of the trivial or split extension of a ring by a bimodule has played an important role in various parts of algebra. In most cases, however, it is introduced ab initio and then used with a particular purpose in mind. With no intention of being comprehensive, we mention some important applications of this construction. But first we must describe the construction. Suppose R is a ring (with identity) and M is an R-bimodule. The set R x M, with componentwise addition and multiplication given, elementwise, by (r,m)(r',m') = (rr',mr' + rm'), becomes a ring, which we denote by R ~ M. It has an ideal (0 x M) which has square zero. And there is a ring homomorphism R --> R ~ M and an augmentation rr:R ~ M --> R. Hochshild, in studying the cohomology of R with coefficients in M, notices that R ~ M is the extension of R by M corresponding to the zero element in the 2nd coho
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