Selected Works of MAURICE AUSLANDER

2272 Pages·1999·150.178 MB·English

by  (ed.) Idun Reiten, Sverre O Smalø, Øyvind Solberg

SELECTED WORKS OF
MAURICE
AUSLANDER
Part 1
Idun Reiten
Sverre 0. 'Smal0
0yvind Solberg
Editors
American Mathematical Society
Providence, Rhode Island

Selected Titles in This Series
Volume
10 Maurice Auslander: Selected Works of Maurice Auslander, Parts 1 and 2 (Idun
Reiten, Sverre 0. Smal¢, and 0yvind Solberg, Editors), 1999
9 Lipman Bers: Selected Works of Lipman Bers: Papers on Complex Analysis, Parts
1 and 2 (Irwin Kra and Bernard Maskit, Editors), 1998
8 Walter E. Thirring: Selected Papers of Walter E. Thirring with Commentaries,
1998
7 Robert Steinberg: Robert Steinberg Collected Papers, 1997
6 Julia Robinson: The Collected Works of Julia Robinson (Solomon Feferman,
Editor), 1996
5 Freeman Dyson: Selected Papers of Freeman Dyson with Commentary, 1996
4 Witold Hurewicz: Collected Works of Witold Hurewicz (Krystyna Kuperberg,
Editor), 1995
3.2 A. Adrian Albert: A. Adrian Albert Collected Mathematical Papers:
Nonassociative Algebras and Miscellany, Part 2 (Richard E. Block, Nathan Jacobson,
J. Marshall Osborn, David J. Saltman, and Daniel Zelinsky, Editors), 1993
3.1 A. Adrian Albert: A. Adrian Albert Collected Mathematical Papers: Associative
Algebras and Riemann Matrices, Part 1 (Richard E. Block, Nathan Jacobson,
J. Marshall Osborn, David J. Saltman, and Daniel Zelinsky, Editors), 1993
2 Salomon Bochner: Collected Papers of Salomon Bochner, Parts 1-4 (Robert C.
Gunning, Editor), 1992
1 R. H. Bing: The Collected Papers of R. H. Bing, Parts 1 and 2 (Sukhjit Singh,
Steve Armentrout, and Robert J. Daverman, Editors), 1988
SELECTED WORKS OF
MAURICE
AUSLANDER
(Photograph courtesy ofGordana Todorov.)
MAURICE AUSLANDER
August 3, 1926-November 18, 1994
Editorial Board
Jonathan L. Alperin Elliott H. Lieb
Cathleen S. Morawetz
1991 Mathematics Subject Classification. Primary 13-xx, 14-xx, 16-xx, 18-xx, 19-xx.
Library of Congress Cataloging-in-Publication Data
Auslander, Maurice.
[Essays. Selections]
Selected works of Maurice Auslander / Idun Reiten, Sverre 0. Smal!2:I, 0yvind Solberg, editors.
p. cm.
Includes bibliographical references.
ISBN 0-8218-0679-3 (set). - ISBN 0-8218-0998-9 (pt. 1). - ISBN ()..8218-1000-6 (pt. 2)
1. Algebra. I. Reiten, Idun, 1942- . II. Smal!2:I, Sverre 0. III. Solberg, 0yvind, 1961- .
IV. Title.
QA155.2.A882 1998
512---dc21 98-2926
CIP
© 1999 by the American Mathematical Society. All rights reserved.
Printed in the United States of America.
A complete list of acknowledgments can be found at
the back of this publication.
The American Mathematical Society retains all rights
except those granted to the United States Government.
§ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at URL: http://www. ams. org/
10987654321 04 03 02 01 00 99
Contents
Foreword ix
Maurice Auslander 1926-1994 xi
Curriculum Vitae xv
Publication list for Maurice Auslander xvii
Chapter I. Homological dimension and local rings 1
On the dimension of modules and algebras (III), Global dimension 3
Commutator subgroups of free groups 15
On the dimension of modules and algebras (VI), Comparison of global and algebra
dimension 19
On regular group rings 25
Homological dimension in local rings 33
Homological dimension in noetherian rings II 49
Codimension and multiplicity 63
Unique factorization in regular local rings 99
A remark on a paper of M. Hironaka 101
Chapter II. Ramification theory 105
On ramification theory in noetherian rings 107
Maximal orders 125
The Brauer group of a commutative ring 149
Modules over unramified regular local rings 193
On the purity of the branch locus 211
Ramification index and multiplicity 221
Modules over unramified regular local rings 237
Brauer groups of discrete valuation rings 241
Galois actions on rings and finite Galois coverings 253
Chapter III. Functors 281
Coherent functors 283
Stable equivalence of artin algebras 327
Stable equivalence of dualizing R-varieties 365
A functorial approach to representation theory 427
Adjoint functors and an extension of the Green correspondence for group
representations 471
D Tr-periodic modules and functors 489
Chapter IV. Almost split sequences and artin algebras 501
MAURICE AUSLANDER vii
viii CONTENTS
Representation dimension of artin algebras 505
A characterization of orders of finite lattice type 575
Representation theory of artin algebras I 581
Representation theory of artin algebras II 625
Representation theory of artin algebras III: Almost split sequences 647
Large modules 673
Representation theory of artin algebras IV: Invariants given by almost split
sequences 691
Representation theory of artin algebras V: Methods for computing almost split
sequences and irreducible morphisms 727
Representation theory of artin algebras VI: A functorial approach to almost split
sequences 745
Representation theory of hereditary artin algebras 769
Almost split sequences whose middle term has at most two indecomposable
summands 789
Relations for Grothendieck groups of artin algebras 809
Chapter V. Some topics in representation theory 815
On a generalized version ofthe Nakayama conjecture 817
Modules with waists 823
Modules determined by their composition factors 835
Almost split sequences and group rings 857
On a theorem of E. Green on the dual of the transpose 877
Acknowledgments 891
viii SELECTED WORKS
Foreword
In view of Maurice Auslander's important contributions to many parts of algebra,
there is great interest in the gathering of several of his papers into the present volumes:
"Selected Works of Maurice Auslander". In these we are pleased to present to the mathe
matical public the core of Auslander's work. Experts also, acquainted perhaps with only
parts of his scientific production, will enjoy the wide range of his contributions and the
impact of his work on the development of modern algebra.
Because of Auslander's long and close connection with the algebra group in Trond
heim, through numerous visits and joint papers, we found it natural to undertake the
task of editing such volumes. The American Mathematical Society, of which Auslander
was a long time member, was the appropriate choice of publisher.
The selected works start with an article from the Notices of the AMS, written shortly
after his death, which in addition to a short summary of his mathematical accomplish
ments also deals with aspects of his personality.
Even though Auslander worked in many different areas, there are characteristic fea
tures common to most of his work. The use of homological methods, including functor
categories, should especially be mentioned. While his early work is mostly concerned
with commutative rings and the later work mainly with artin algebras, he was always
interested in finding common features and common settings for these situations. A syn
thesis of the topics was given through his theory of general orders, with applications to
singularity theory. Typical for his work is also the large number of papers initiating new
directions, and his name is attached to several concepts in algebra, including algebras,
rings, modules, quivers, sequences, translates, conditions. In addition he is responsible
for influential conjectures.
Despite these interconnections, we feel that it is easier to get an impression of Aus
lander's contribution by grouping related papers together. According to this principle we
have organized the selected papers in ten chapters, of which five appear in each volume.
For each chapter we provide some background material, interrelationship between the
papers, and an indication of further developments. For this we give no specific references.
Even though a large number of mathematicians would have deserved to be cited, we have
limited mentioning names to a minimum.
Space requirements did not allow to include all of Auslander's papers. With the
exception of the Queen Mary College notes which are not readily available, monographs
are not included. A main criterion in the difficult selection process was to reflect the
broad range and impact of his contributions. While the inclusion of some of the papers
was unquestionable, some arbitrariness is involved in other cases.
Due to low quality of typing, several papers have been retyped. To be faithful to
the original, we have made changes only when there were obvious typographical errors or
inconsistencies in notation. We take the full responsibility for any new typos which may
have been introduced.
MAURICE AUSLANDER ix
X FOREWORD
We are grateful to several colleagues who offered their opinion in the selection pro
cess, and suggested improvements in what we have written. We also thank our doctoral
students Aslak Buan, Ole Enge, Inger Heidi Slungard and Stig Venas for their help with
proof reading the retyped manuscripts.
We express our thanks to the American Mathematical Society, and especially to
Chris Thivierge, for fruitful collaboration. We also appreciate financial support from the
Norwegian University of Science and Technology and from the Maurice Auslander Award
of the American-Scandinavian Foundation.
Trondheim, November, 1997.
Idun Reiten Sverre O. Smal0 0yvind Solberg
x SELECTED WORKS