Selected Works of MAURICE AUSLANDER 0821806793, 0821809989

Table of contents :
Selected Works of MAURICE AUSLANDER, Part 1
Front Matter
MAURICE AUSLANDER -- August 3, 1926-November 18, 1994 (Photograph courtesy of Gordana Todorov.)
Contents
Foreword
Maurice Auslander 1926-1994
Curriculum Vitae
Publication list for Maurice Auslander
Chapter I. Homological dimension and local rings
[1] On the dimension of modules and algebras (III), Global dimension, Nagoya Math. J. 9 (1955) 67-77.
[2] (with R. C. Lyndon) Commutator subgroups of free groups, Amer. J. Math. 77 (1955) 929-931.
[4] On the dimension of modules and algebras (VI), Comparison of global and algebra dimension, Nagoya Math. J. 11 (1957) 61-65.
[5] On regular group rings, Proc. Amer. Math. Soc. 8 (1957) 658-665.
[6] (with D. Buchsbaum) Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957) 390-405.
[7] (with D. Buchsbaum) Homological dimension in noetherian rings II, Trans. Amer. Math. Soc. 88 (1958) 194-206.
[10] (with D. Buchsbaum) Codimension and multiplicity, Ann. of Math. 68 (1958) 625-657. Correction, Ann. of Math. (2) 70 (1959) 395-397.
[12] (with D. Buchsbaum) Unique factorization in regular local rings, Proc. Nat. Acad. Sci. USA 45 (1959) 733-734.
[19] A remark on a paper of M. Hironaka, Amer. J. Math. 84 (1962} 8-10.
Chapter II. Ramification theory
[13] (with D. Buchsbaum) On ramification theory in noetherian rings, Amer. J. Math. 81 (1959) 749-765.
[14] (with O. Goldman) Maximal orders, Trans. Amer. Math. Soc. 97 (1960) 1-24.
[15] (with O. Goldman) The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960) 367-409.
[16] Modules over unramified regular local rings, Illinois J. Math. 5 (1961) 631-647.
[18] On the purity of the branch locus, Amer. J. Math. 84 (1962) 116-125.
[20] (with D. S. Rim) Ramification index and multiplicity, Illinois J. Math. 7 (1963) 566-581.
[21] Modules over unramified regular local rings, Proc. International Congr. Mathematicians, Stockholm (1962) 230-233, Inst. Mittag-Leffler, Djursholm, 1963.
[26] (with A. Brumer) Brauer groups of discrete valuation rings, Nederl. Akad. Wetensch. Proc. Ser. A, 71, no. 3 (1968) 286-296.
[88] (with I. Reiten and S. O. Smalø) Galois actions on rings and finite Galois coverings, Math. Scand. 65 (1989) No. 1, 5-32.
Chapter III. Functors
[22] Coherent functors, Proc. Conf. Categorial Algebra (La Jolla, Calif. 1965) 189-231, Springer, New York, 1966.
[31] (with I. Reiten) Stable equivalence of artin algebras, Proc. Conf. on Orders, Group rings and related topics (Ohio 1972) 8-71, Lecture Notes in Math. 353, Springer, Berlin 1973.
[35] (with I. Reiten) Stable equivalence of dualizing R-varieties, Adv. in Math. 12 (1974) No. 3, 306-366.
[67] A functorial approach to representation theory, Representations of algebras. (ICRA III, Puebla 1980) 105-179. Lecture Notes in Math. 944, Springer, Berlin New York, 1982.
[105] (with M. Kleiner) Adjoint functors and an extension of the Green correspondance for group representations, Adv. in Math. 104, (1994) No. 2, 297-314.
[107] (with I. Reiten) D Tr-periodic modules and functors, Representation theory of algebras (ICRA VII, Cocoyoc 1994) 39-50, CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996.
Chapter IV. Almost split sequences and artin algebras
[28] Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London (1971).
[29] (with K. W. Roggenkamp) A characterization of orders of finite lattice type, Invent. Math. 17 (1972) 79-84.
[33] Representation theory of artin algebras I, Comm. Algebra 1 (1974) 177-268.
[34] Representation theory of artin algebras II, Comm. in Algebra 1 (1974) 269-310.
[36] (with I. Reiten) Representation theory of artin algebras III: Almost split sequences, Comm. Algebra 3 (1975) 239-294.
[47] Large modules over artin algebras, Algebra, Topology and Category theory, (A collection of papers in honor of Samuel Eilenberg) 1-17 Academic Press, New York, 1976.
[49] (with I. Reiten) Representation theory of artin algebras IV: Invariants given by almost split sequences, Comm. Algebra 5 (1977) 443-518.
[50] (with I. Reiten) Representation theory of artin algebras V: Methods for computing almost split sequences and irreducible morphisms, Comm. Algebra 5 (1977) No. 5, 519-554.
[52] (with I. Reiten) Representation theory of artin algebras VI: A functorial approach to almost split sequences, Comm. Algebra 6 (1978) No. 3, 257-300.
[54] (with M. I. Platzeck) Representation theory of hereditary artin algebras, Representation theory of algebras, (Proc. Conf. Temple Univ., 1976) 389-424. Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978.
[58] (with R. Bautista, M. I. Platzeck, I. Reiten and S. O. Smalø) Almost split sequences whose middle term has at most two indecomposable summands, Canad. J. Math. 31, (1979) No 5, 942-960.
[68] Relations for Grothendieck groups of artin algebras, Proc. Amer. Math. Soc. 91 (1984) No. 3, 336-340.
Chapter V. Some topics in representation theory
[41] (with I. Reiten) On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975) 69-74.
[45] (with E. L. Green and I. Reiten) Modules with waists, Illinois J. Math. 19 (1975) 467-477.
[69] (with I. Reiten) Modules determined by their composition factors, Illinois J. Math. 29 (1985) No. 2, 280-301.
[72] (with J. F. Carlson) Almost-split sequences and group rings, J. Algebra 103 (1986) No. 1, 122-140.
[94] (with I. Reiten) On a theorem of E. Green on the dual of the transpose, Representations of finite-dimensional algebras (Tsukuba 1990) 53-65, CMS Conf. Proc., 11, Amer. Math. Soc., Providence RI (1991) 53-65.
Acknowledgments
Selected Works of MAURICE AUSLANDER, Part 2
Front Matter
MAURICE AUSLANDER -- August 3, 1926-November 18, 1994 (Photograph courtesy of Gordana Todorov.)
Contents
Foreword
Publication list for Maurice Auslander
Chapter VI. Lattices over general orders
[55] Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf. Temple Univ., 1976), 1-244. Lecture Notes in Pure Appl. Math. 37, Dekker, New York 1978.
[56] Applications of morphisms determined by modules, Representation theory of algebras (Proc. Conf. Temple Univ. 1976), 245-327. Lecture Notes in Pure Appl. Math. 37, Dekker, New York 1978.
[76] A survey of existence theorems for almost split sequences, Representations of algebras (Durham, 1985), 81-89, London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge-New York, 1986.
Chapter VII. Tilting theory and homologically finite subcategories
[57] (with M. I. Platzeck and I. Reiten) Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46.
[62] (with S. O. Smalø) Preprojective modules over artin algebras, J. Algebra 66, (1980) No. 1, 61-122.
[63] (with S. O. Smalø) Almost split sequences in subcategories, J. Algebra 69, (1981) No. 2, 426-454, with Addendum J. Algebra, 71 (1981) No. 2 ,592-594.
[92] (with I. Reiten) Applications of contravariantly finite subcategories, Adv. in Math. 86, No. 1 (1991) 111-152.
[96] (with M. I. Platzeck and G. Todorov) Homological theory of idempotent ideals, Trans. Amer. Math. Soc. 332 (1992) No. 2, 667-692.
Chapter VIII. Almost split sequences and commutative rings
[71] Isolated singularities and existence of almost split sequences, Representation theory II (ICRA IV, Ottawa, 1984) 194-241, Lecture Notes in Math. 1178, Springer, Berlin-New York, 1986.
[74] Rational singularties and almost split sequences, Trans. Amer. Math. Soc. 293 (1986) No. 2, 511-531.
[78] (with I. Reiten) Almost split sequences for rational double points, Trans. Amer. Math. Soc. 302 (1987) No. 1, 87-97.
[79] (with I. Reiten) The Cohen-Macaulay type of Cohen-Macaulay rings, Adv. in Math., 73 (1989), No. 1, 1-23.
[83] (with I. Reiten) Almost split sequences for Cohen-Macaulay modules, Math. Ann. 277 (1987) No. 2, 345-349.
[84] The what, where and why of almost split sequences, Proceedings of the International Congress of Mathematicians, vol. 1-2, (Berkeley, CA, 1986) 338-345, Amer. Math. Soc., Providence, RI, 1987.
[85] (with I. Reiten) Cohen-Macaulay modules for graded Cohen-Macaulay rings and their completions, Commutative Algebra, (Berkeley, CA, 1987) 21-31, Math. Sci. Res. Inst. Publ. 15 Springer, Berlin-New York, 1989.
[90] (with I. Reiten) Graded modules and their completions, Topics in Algebra, Part 1 (Warsaw 1989) 181-192, Banach Center Publ. 26, Part 1, PWN, Warsaw 1990.
Chapter IX. Grothendieck groups and Cohen-Macaulay approximations
[73] (with I. Reiten) Grothendieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986) No. 1-2, 1-51.
[86] (with I. Reiten) Grothendieck groups of algebras with nilpotent annihilators, Proc. Amer. Math. Soc. 103 (1988), No. 4, 1022-1024.
[89] (with R. O. Buchweitz) The homological theory of maximal Cohen-Macaulay approximations, Colloque en l'honneur de Pierre Samuel (Orsay 1987), Mém. Soc. Math. France, (N.S.) No. 38 (1989) 5-37.
[97] (with S. Ding and Ø. Solberg) Liftings and weak liftings of modules, J. Algebra 156 (1993) No. 2, 273-317.
Chapter X. Relative theory and syzygy modules
[98] (with Ø. Solberg) Relative homology and representation theory I, Relative homology and homologically finite subcategories, Comm. Algebra 21 (9) (1993) 2995-3031.
[99] (with Ø. Solberg) Relative homology and representation theory II, Relative cotilting theory, Comm. Algebra 21 (9) (1993) 3033-3079.
[102] (with I. Reiten) k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994) 1-27.
[108] (with I. Reiten) Syzygy modules for noetherian rings, J. Algebra 183 (1996) No. 1, 167-185.
Acknowledgments
Not Selected Works of Maurice Auslander
[3] (with D. Buchsbaum) Homological dimension in noetherian rings, Proc. Nat. Acac. Sci. 42 (1956) 36-38.
[8] (with L. Auslander) Solvable Lie groups and locally Euclidean Riemann spaces, Proc. Amer. Math. Soc. 9 (1958) 933-941.
[9] (with A. Rosenberg) Dimension of ideals in polynomial rings, Can. J. Math. 10 (1958) 287-293.
[11] Remark on automorphisms of groups, Proc. Amer. Math. Soc. 9 (1958) 229-230.
[17] (with D. Buchsbaum) Invariant factors and two criteria for projectivity of modules, Trans. Amer. Math. Soc. 104 (1962) 516-522.
[23] Remarks on a theorem of Bourbaki, Nagoya Math. J. 27 (1966) 361-369.
[25] Comments on the functor Ext^1(C, ), Topology 8 (1969) 151-166.
[37] (with I. Reiten) Stable equivalence of dualizing R-varieties II Hereditary dualizing R-varieties, Adv. in Math. 17, (1975) No. 2, 93-121.
[38] (with I. Reiten) Stable equivalence of dualizing R-varieties III Dualizing R-varieties stably equivalent to hereditary dualizing R-varieties, Adv. in Math. 17 (1975) No. 2, 122-142.
[39] (with I. Reiten) Stable equivalence of dualizing R-varieties IV Higher global dimension, Adv. in Math. 17, (1975) No. 2, 143-166.
[40] (with I. Reiten) Stable equivalence of dualizing R-varieties V Artin algebras stably equivalent to hereditary algebras, Adv. in Math. 17, (1975) No. 2, 167-195.
[40.1] Idun Reiten, Stable equivalence of dualizing R-varieties VI. Nakayama Dualizing R-Varieties, Adv. in Math. 17, (1975) No. 2, 196-211.
[42] Almost split sequences I, Proc. Int. Conf. on representations of algebras, Ottawa 1974, Lecture Notes in Math. 488 Springer-Verlag (1975) 1-8.
[43] (with I. Reiten) Almost split sequences II, Proc. Int. Conf. on representations of algebras, Carleton Univ., (Ottawa 1974), Lecture Notes in Math. 488, Springer Verlag (1975) 9-19.
[44] (with E. L. Green and I. Reiten) Modules having waists, Proc. Int. Conf. on representations of algebras, Ottawa 1974, Lecture Notes in Math. 488, Springer Verlag (1975) 20-28.
[48] (with I. Reiten) On the representation type of triangular matrix rings, J. London Math. Soc. (2), 12 (1976) No. 3, 371-382.
[53] (with M. I. Platzeck and I. Reiten) Periodic modules over weakly symmetric algebras, J. Pure Appl. Algebra 11 (1977) 279-291.
[60] (with S. O. Smalø) Preprojective modules An introduction and some applications, Representation theory II (Proc. ICRA II, Carleton Univ., Ottawa 1979) 48-73. Lecture Notes in Math. 832, Springer, Berlin, 1980.
[61] (with I. Reiten) Uniserial functors, Representation theory II (Proc. ICRA II, Carleton Univ., Ottawa 1979) 1-47, Lecture Notes in Math. 832, Springer, Berlin, 1980.
[64] (with S. O. Smalø) Preprojective lattices over classical orders. Integral representations and applications. (Oberwolfach 1980) 326-345. Lecture Notes in Math. 882, Springer, Berlin-New York 1981.
[65] Representation theory of finite-dimensional algebras, Algebraists' homage papers in ring theory and related topics (New Haven, Conn,. 1981), 27-39, Contemp. Math. 13, Amer. Math. Soc., Providence, R.I., 1982.
[66] (with S. O. Smalø) Lattices over orders Finitely presented functors and preprojective partitions, Trans. Amer. Math. Soc. 273 (1982) No. 2, 433-445.
[70] Finite type implies isolated singularity, Orders and their applications (Oberwolfach, 1984), 1-4, Lecture Notes in Math. 1142, Springer, Berlin-New York, 1985.
[75] (with I. Reiten) McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1986) No. 1, 293-301.
[77] Almost split sequences and algebraic geometry, Representations of algebras (Durham, 1985), 165-179, London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge-New York, 1986.
[80] (with I. Reiten) Almost split sequences in dimension two, Adv. in Math. 66 (1987) No. 1, 88-118.
[81] (with I. Reiten) Almost split sequences for Z-graded rings, Singularities, representation of algebras and vector bundles (Lambrecht 1985), 232-243, Lecture Notes in Math. 1273, Springer, Berlin-New York(1987).
[82] (with I. Reiten) Almost split sequences for abelian group graded rings, J. Algebra, 114 (1988) No. 1, 29-39.
[91] (with E. L. Green) Modules over endomorphism rings, Comm. Algebra 20 (5) (1992) 1259-1278.
[93] (with I. Reiten) Cohen-Macaulay and Gorenstein artin algebras, Representation theory of finite groups and finite-dimensional algebras, (Bielefeld 1991) 221-245, Progr. Math. 95, Birkhauser, Basel (1991).
[95] (with I. Reiten) Homologically finite subcategories, Representations of algebras and related topics (ICRA V, Tsukuba 1990). London Math. Soc. Lecture Note Ser. 168, 1-42 (199).
[100] (with Ø. Solberg) Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. Algebra 21 (9) (1993) 3081-3097.
[101] (with Ø. Solberg) Gorenstein algebras and algebras with dominant dimension at least 2, Comm. Algebra 21 (11) (1993) 3897-3934.
[103] (with M. Kleiner) The Green correspondence for adjoint functors, Representations of algebras (ICRA VI, Ottawa 1992) 69-81, CMS Conf. Proc. 14, Amer. Math. Soc., Providence, RI, 1993.
[104] (with Ø. Solberg) Relative homology, Finite-dimensional algebras and related topics (ICRA VI, Ottawa 1992) Kluwer Academic Publishers, Nato Adv. Sci. Inst. Ser. C Math. Phys. Sci. 424 (1994) 347-359.

Citation preview

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

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Contents Foreword

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Maurice Auslander 1926-1994

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Curriculum Vitae

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Publication list for Maurice Auslander

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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 On ramification theory in noetherian rings Maximal orders The Brauer group of a commutative ring Modules over unramified regular local rings On the purity of the branch locus Ramification index and multiplicity Modules over unramified regular local rings Brauer groups of discrete valuation rings Galois actions on rings and finite Galois coverings

105 107 125 149 193 211 221 237 241 253

Chapter III. Functors Coherent functors Stable equivalence of artin algebras Stable equivalence of dualizing R-varieties A functorial approach to representation theory Adjoint functors and an extension of the Green correspondence for group representations D Tr-periodic modules and functors

281 283 327 365 427 471 489

Chapter IV.

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Almost split sequences and artin algebras MAURICE AUSLANDER

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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 647 673 Large modules 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 On a generalized version ofthe Nakayama conjecture Modules with waists Modules determined by their composition factors Almost split sequences and group rings On a theorem of E. Green on the dual of the transpose Acknowledgments

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SELECTED WORKS

815 817 823 835

857 877 891

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 mathematical 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 Trondheim, 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 accomplishments also deals with aspects of his personality. Even though Auslander worked in many different areas, there are characteristic features 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 synthesis 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 Auslander'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

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FOREWORD

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We are grateful to several colleagues who offered their opinion in the selection process, 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

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Sverre O. Smal0

0yvind Solberg

Maurice Auslander 1926-1994 Maurice Auslander died of cancer on November 18, 1994, in Trondheim, Norway. Auslander has made fundamental contributions in many central parts of algebra. It would be senseless to try to describe his work within a given specialty or to present it under a particular title. Quite the contrary, he liked to attack problems by surprise, from apparently nowhere. This approach resulted in many original theorems in commutative and noncommutative ring theory, for orders and Brauer groups, and in the representation theory of Artin algebras as well as in the theory of singularities. Among the main characteristics of his work one will particularly remember the extreme elegance of the methods he liked to introduce and develop and also his ability to present and explain the crucial points. These qualities were also typical of his personality. The importance of his scientific influence is clearly established for many years to come. Maurice Auslander was born on August 3, 1926, in Brooklyn, New York. He received his B.S. in 1949 and his Ph.D. in 1954, both from Columbia University. His thesis was within group theory and was written under the direction of Robert L. Taylor. He spent the years 1953-57 at the University of Chicago, the University of Michigan and at the Institute for Advanced Study in Princeton. He joined Brandeis University in 1957, where he was chairman 1960-61 and 1976-78. He held visiting positions in Paris, Urbana, London, Trondheim, Austin and Blacksburg. He had fellowships from Sloan, Guggenheim and Fulbright, and a few weeks before his death he was awarded a Senior Humboldt Research Prize. Auslander was a Fellow of the American Academy of Arts and Sciences and a member of the Royal Norwegian Society of Sciences and Letters. In commutative ring theory, Auslander was immediately attracted by homological methods. The title of his first joint work with Buchsbaum was "Homological dimension in noetherian rings" (1956). It soon became "Homological dimension in noetherian rings I" for all algebraists, since their collaboration on this topic was long and fruitful. Progressively, they organized and formalized what became "Homological commutative algebra with a view toward algebraic geometry". The characterization of regular rings as the rings with finite homological dimension (Auslander-Buchsbaum-Serre) as well as the factoriality of regular local rings (Auslander-Buchsbaum) are now classic results. The interest of Auslander in this theme culminated with "Modules over unramified regular local rings" (1961). This was also the title of his talk at the World Congress of 1962. In this work he started a research program for thirty years to come. The list of contributors to this program includes the names of his students Peskine and Szpiro and also of Hochster and Fulton (via the local Riemann-Roch theorem). This program was essentially completed with the answers of Roberts to the "Homological Conjectures". Ramification theory was, during the same years, Auslander's other favorite topic. He collaborated there with Buchsbaum, Goldman and Rim. The new methods were first developed for noetherian rings. Later on a general theory of separable algebras over a commutative ring was proposed. The success of this theory and particularly of the homological different of

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an algebra over a ring is now well established. In 1962, Auslander gave a new proof of the "the purity of the branch locus". It was very much in his style, completely surprising for the specialists and strikingly beautiful. First he reduced the question to proving that if R is a regular local ring and M a reflexive finitely generated R-module such that Hom(M, M) is isomorphic to a direct sum of copies of M, then M is free. His proof of this last statement is really exhilarating. With his work on coherent functors in the mid sixties, Auslander started to develop systematically a functorial point of view, in the spirit of his favorite book by CartanEilenberg. The monograph on stable module theory (with Bridger) is rather typical of this evolution. A central role is played by the transpose of a module, one of Auslander's favorite tools. Although this work is still influenced by algebraic geometry, through dualities, the representation theory of Artin algebras is a developing theme. In this monograph are also the roots of Auslander's successful version of noncommutative Gorenstein rings, giving rise to what is now known as Auslander-Gorenstein rings and Auslander regular rings. The new phase for Auslander took a clearer form around 1970, during his visits to Urbana and London, with a decided interest in the module theory of Artin algebras. Modular group representation theory was a first source of inspiration. Higman had much earlier described the group algebras of finite (representation) type in terms of p-Sylow subgroups. Also the Brauer-Thrall conjectures had their origin from group algebras, the first one having been proven by Roiter for algebras over a field. Auslander started by homologically characterizing finite type in terms of endomorphism algebras, now called Auslander algebras. Then he proved the first Brauer-Thrall conjecture for left Artin rings and with an elegant application of functors he characterized infinite type in terms of existence of indecomposable modules which are not finitely generated. This work, together with the work of Roiter and Gabriel's work on representations of quivers, belongs to the start of the modern phase of representation theory of Artin algebras, where Auslander was a main contributor until the end of his life. Starting also in the early seventies, Auslander developed with Reiten the theory of almost split sequences, also called Auslander-Reiten sequences, and the related irreducible maps. The transpose of a module plays a major role in this work. Almost split sequences and the associated Auslander-Reiten quivers are now central in representation theory. They are important for classification theorems, provide a useful combinatorial invariant for an algebra and criteria for finite type. They also play an important role in group representation theory, in particular through the work of Webb and Erdmann. By interpreting work of Bernstein-Gelfand-Ponomarev, Auslander (with Platzeck and Reiten) gave the first module theoretic construction of what is now, through the work of Brenner-Butler and Happel-Ringel, known as tilting theory. A few years ago Auslander and Reiten had also established a close connection between tilting and the theory of contravariantly finite subcategories, as developed by Auslander-Smal0 around 1980. This connection has beautiful applications to quasihereditary algebras, as discovered by Ringel, and inspired applications of tilting to algebraic groups, by Donkin. The contravariantly finite subcategories also were important in Auslander's recent work (with Solberg) on relative homological algebra in representation theory. Even though the representation theory of Artin algebras, including the interplay with group representation theory, was Auslander's main field of research during the last 20-25 years, he always strived to carry his ideas and proofs to their utmost generality. Through these efforts he obtained interesting applications to the theory of commutative and noncommutative noetherian rings. In the mid-seventies, before the theory of almost split xii

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sequences was generally accepted in representation theory of Artin algebras, he generalized the basic existence theorem to a setting including lattices over orders and (maximal) Cohen-Macaulay modules over complete local Cohen-Macaulay rings. Auslander several years later discovered connections with singularity theory, thus opening up a new exciting area of research. He showed that his methods were particularly well suited in dimension two, in fact in some sense better than for Artin algebras. He saw that the Auslander-Reiten quivers for two-dimensional invariant rings are isomorphic to the corresponding McKay quivers, which for the special linear group are given by (extended) Dynkin diagrams. This provided a surprising link to the resolution graph of the isolated singularity and was a main topic of his invited address at the 1986 International Congress in Berkeley. Several people were involved in identifying the hypersurfaces of finite (representation) type as the simple isolated singularities in the sense of Arnold, and Auslander found (with Reiten) further examples of finite (representation) type for Cohen-Macaulay rings which are not hypersurfaces. Last but not least was the theory of Cohen-Macaulay approximations which he developed in the late eighties in collaboration with Buchweitz, with strong ties to the theory of contravariantly finite subcategories and the work with Bridger. Maurice Auslander had a warm and sensitive personality. His door was always open to his friends. He enjoyed discussions, often provocative ones, about mathematics and its philosophy in particular. He had considerable administrative talent, doubling the size of the department during his first chairmanship at Brandeis. His interests outside mathematics included art, poetry and music, and he enjoyed playing the violin. He was himself "un homme libre", free of all influences, and wanted others to be the same. He would pin down cliches on the spot. In mathematics he had a sense for beauty; he truly enjoyed some results and their proofs. He had a special interest and concern for young researchers, including his thirty or so doctoral students, who loved and admired him. He would frequently visit their offices or wake them up by phone calls in the early mornings, inquiring about the latest progress. One of his students once complained that Maurice did the easy work. When things were getting harder, he would leave the subject. He answered goodheartedly that the student was welcome to do the same. Maurice loved to travel. On his travels he attracted students and collaborators for himself, and visitors to Brandeis. Through his extensive travels and collaborations, he had a positive influence on the development of algebra in many foreign countries. In the sixties he had French students in Paris whom he invited to work near him at Brandeis. Through collaboration he became a frequent visitor to Trondheim during the past twenty years, and in recent years he also had a formal affiliation. Half of his research publications, including a book on the representation theory of algebras which appeared shortly after his death, have coauthors from Trondheim. With his visit to Mexico in 1975 and the subsequent research stays of Mexicans at Brandeis, he was responsible for the start of the successful Mexican research group on the representation theory of Artin algebras. Through his visit in China in 1986 he helped, as the first western visitor, the Chinese group in representation theory get off to a good start. He also had strong impact in many other countries, in particular Germany, and he had longer research visits in Brazil, Israel, Switzerland and Uruguay. Despite declining health, Maurice managed to continue with his favorite occupations during the last year of his life. He revisited China, seeing the results of his influence and the changes in the society. He enjoyed the spectacular fjords and glaciers of Norway and put the finishing touch to the manuscript for his last book. He attended a conference in Utrecht and saw the impact of the theory of Cohen-Macaulay approximations. He MAURICE AUSLANDER

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gave his final public lecture, pushing homological methods in representation theory, at ICRA VII in Mexico. With cancer in bones, liver and lungs he planned his last nostalgic tour, putting his faith in what money and willpower could do. He enjoyed the company of old friends, wandered through the streets and gardens of Paris; appreciated for the last time his favorite painting, a self portrait of Rembrandt, in London; and enjoyed the Munch museum in Oslo. Shortly after arriving in Trondheim he was hospitalized. He died a week and a half later, amongst close friends and colleagues, in the middle of the European meeting on "Invariants and Representations of Algebras" which he had looked forward to attending. He died the way he lived and worked - elegantly. -

Christian Peskine, University of Paris VI Idun Reiten, University of Trondheim, AVH

Editor's note: During my brief and naive tenure as chair of the Department of Mathematics at Brandeis, Maurice was my close friend and mentor. These were disturbing times (1968-70), and Maurice was ever the steadying influence. For him, the difference between pragmatics and theory was only theoretical: his complete comprehension of complex situations, and only that, guided his actions. There never was any suspicion of compromise with principle: in his life, his politics, his mathematics. He always delivered his message with charm, elegance and humor; because of this he was surprisingly effective . . . I never heard Maurice play the violin . . . I wish I had. - Hugo Rossi NOTICES OF THE

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AMS, April 1995, 450-453

Curriculum Vitae Date of Birth: August 3, 1926 Place of Birth: Brooklyn, New York, USA Education: 1949 B.S., Columbia University 1954 Ph.D., Columbia University Career: 1953-54 Instructor, University of Chicago 1954-56 Instructor, University of Michigan 1956-57 NSF Postdoctoral Fellow, Institute for Advanced Study, Princeton 1957--60 Assistant Professor, Brandeis University 1960-63 Associate Professor, Brandeis University 1960-61 Chairman of the Mathematics Department, Brandeis University 1961-62 NSF Senior Postdoctoral F~llow, University of Paris 1963-94 Professor, Brandeis University 1963--64 Sloan Foundation Fellow 1965 Fulbright Fellow, University of Uruguay, Montevideo (Summer) 1965-66 Visiting Professor, University of Paris 1970 Visiting Professor, University of Illinois, Urbana (Fall) 1971 Senior Research Fellow, Queen Mary College, University of London (Spring) 1976-78 Chairman of the Mathematics Department, Brandeis University 1978-79 Guggenheim Fellow (University of Trondheim, spring 1979) 1981-82 Visiting Professor, University of Texas, Austin 1984 'Visiting Professor, University of Bielefeld (Summer) 1985 Visiting Professor, University of Bielefeld (Summer) 1986-87 Visiting Professor, Virginia Polytechnic Institute and State University, Blacksburg 1988 Visiting Professor, University of Paderborn (Summer) 1989-90 Norwegian Research Council guest researcher, University of Trondheim 1990 Visiting Professor, University of Paderborn (Summer) 1991 Norwegian Research Council guest researcher, University of Trondheim (Winter) 1992-94 Professor II (Adjunct Professor), University of Trondheim 1994 Awarded Humboldt Senior Research Fellowship Special invited lectures: International Congress of Mathematicians, Stockholm, 1962 International Congress of Mathematicians, Berkeley, 1986 Membership in professional associations: Phi Beta Kappa American Mathematical Society

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Honorary memberships: American Academy of Arts and Sciences The Royal Norwegian Society of Sciences and Letters Editorial work: Representations of Algebras (with E. Lluis), Proceedings ICRA III, Puebla, Mexico 1980, LNM 903, Springer 1982 Representations of Algebras (with E. Lluis), Proceedings ICRA III (workshop), Puebla, Mexico 1980, LNM 944, Springer 1982 Communications in Algebra Proceedings of AMS Papers and conferences in his honor: 1987 60th birthday volume, Communications in Algebra, vol. 15 (1 & 2) 1991 65 th birthday conference, University of Utah, Salt Lake City (E. L. Green, B. Huisgen-Zimmermann) 1995 Maurice Auslander Memorial Conference, Brandeis University (K. Igusa, A. Martsinkovsky, G. Todorov), Proceedings published in: Representation theory .and algebraic geometry (ed. A. Martsinkovsky, G. Todorov), Cambridge University Press (1997), LMS, Lecture Notes Series 238. 1996 Representation theory of algebras, ICRA VII (1994), CMS Conf. Proceedings, vol. 18 (ed. R. Bautista, R. Martinez-Villa, J. de la Pena), (dedicated to his memory), contains the article: Maurice Auslander 1926-1994, 1-15 (D. Buchsbaum, C. M. Ringel, I. Reiten) 1998 Algebras and modules I, ICRA VIII (1996), CMS Conf. Proceedings, vol. 23 (ed. Idun Reiten, Sverre 0. Smal~, and 0yvind Solberg), (dedicated to his memory). Maurice Auslander died of cancer on November 18, 1994, at the local hospital in Trondheim, Norway.

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Publication list for Maurice Auslander I* I

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[1] On the dimension of modules and algebras (III), Global dimension, Nagoya Math. J. 9 (1955) 67-77. [2] (with R. C. Lyndon) Commutator subgroups of free groups, Amer. J. Math. 77 (1955) 929-931. [3] (with D. Buchsbaum) Homological dimension in noetherian rings, Proc. Nat. Acac. Sci. 42 (1956) 36-38. [4] On the dimension of modules and algebras (VI), Comparison of global and algebra dimension, Nagoya Math. J. 11 (1957) 61-65. [5] On regular group rings, Proc. Amer. Math. Soc. 8 (1957) 658-665. [6] (with D. Buchsbaum) Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957) 390--405. [7] (with D. Buchsbaum) Homological dimension in noetherian rings II, Trans. Amer. Math. Soc. 88 (1958) 194-206. [8] (with L. Auslander) Solvable Lie groups and locally Euclidean Riemann spaces, Proc. Amer. Math. Soc. 9 (1958) 933-941. [9] (with A. Rosenberg) Dimension of ideals in polynomial rings, Can. J. Math. 10 (1958) 287-293. [10] (with D. Buchsbaum) Codimension and multiplicity, Ann. of Math. 68 (1958) 625657. Correction, Ann. of Math. (2) 70 (1959) 395-397. [11] Remark on automorphisms of groups, Proc. Amer. Math. Soc. 9 (1958) 229-230. [12] (with D. Buchsbaum) Unique factorization in regular local rings, Proc. Nat. Acad. Sci. USA 45 (1959) 733-734. [13] (with D. Buchsbaum) On ramification theory in noetherian rings, Amer. J. Math. 81 (1959) 749-765. [14] (with 0. Goldman) Maximal orders, Trans. Amer. Math. Soc. 97 (1960) 1-24. [15] (with 0. Goldman) The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960) 367----409. [16] Modules over unramified regular local rings, Illinois J. Math. 5 (1961) 631-647. [17] (with D. Buchsbaum) Invariant Jae.tors and two criteria for projectivity of modules, Trans. Amer. Math. Soc. 104 (1962) 516-522. [18] On the purity of the branch locus, Amer. J. Math. 84 (1962) 116-125. [19] A remark on a paper of M. Hironaka, Amer. J. Math. 84 (1962} 8-10. [20] (with D. S. Rim) Ramification index and multiplicity, Illinois J. Math. 7 (1963) 566-581. [21] Modules over unramified regular local rings, Proc. International Congr. Mathematicians, Stockholm (1962) 230-233, Inst. Mittag-Leffler, Djursholm, 1963. [22] Coherent functors, Proc. Conf. Categorial Algebra (La Jolla, Calif.. 1965) 189-231, Springer, New York, 1966. *The Roman numerals indicate the chapter in which the paper appears.

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[23] Remarks on a theorem of Bourbaki, Nagoya Math. J. 27 (1966) 361-369. [24] Anneaux de Gorenstein, et torsion en algebre commutative, Seminar Pierre Samuel, 1966/67 (Notes by M. Mangeney, C. Peskine and L. Szpiro), Ecole Normale Superieure de Jeunes Filles Secretariat mathematique, Paris 1967, 69 pp. [25] Comments on the functor Ext 1 (C, ), Topology 8 (1969) 151-166. [26] (with A. Brumer) Brauer groups of discrete valuation rings, Nederl. Akad. Wetensch. Proc. Ser. A, 71, no. 3 (1968) 286-296. [27] (with M. Bridger) Stable module theory, Memoirs of AMS. 94 (1969), American Mathematical Society, Providence, R.I. 1969, 146 pp. [28] Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London (1971). [29] (with K. W. Roggenkamp) A characterization of orders of finite lattice type, Invent. Math. 17 (1972) 79-84. [30] Representation theory of artin algebras, Brandeis Univ. 1972 (Notes by I. Reiten). [31] (with I. Reiten) Stable equivalence of artin algebras, Proc. Conf. on Orders, Group rings and related topics (Ohio 1972) 8-71, Lecture Notes in Math. 353, Springer, Berlin 1973. [32] (with D. Buchsbaum) Groups, Rings, Modules, Harper and Row. (1974) 470 pp. [33] Representation theory of artin algebras I, Comm. Algebra 1 (197 4) 177-268. [34] Representation theory of artin algebras II, Comm. in Algebra 1 (1974) 269-310. [35] (with I. Reiten) Stable equivalence of dualizing R-varieties, Adv. in Math. 12 (1974) No. 3, 306-366. [36] (with I. Reiten) Representation theory of artin algebras III: Almost split sequences, Comm. Algebra 3 (1975) 239-294. [37] (with I. Reiten) Stable equivalence of dualizing R-varieties II: Hereditary dualizing R-varieties, Adv. in Math. 17, (1975) No. 2, 93-121. [38] (with I. Reiten) Stable equivalence of dualizing R-varieties III: Dualizing Rvarieties stably equivalent to hereditary dualizing R-varieties, Adv. in Math. 17 (1975) No. 2, 122-142. [39] (with I. Reiten) Stable equivalence of dualizing R-varieties IV: Higher global dimension, Adv. in Math. 17, (1975) No. 2, 143-166. [40] (with I. Reiten) Stable equivalence of dualizing R-varieties V: Artin algebras stably equivalent to hereditary algebras, Adv. in Math. 17, (1975) No. 2, 167-195. [41] (with I. Reiten) On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (1975) 69-74. [42] Almost split sequences I, Proc. Int. Conf. on representations of algebras, Ottawa 1974, Lecture Notes in Math. 488 Springer-Verlag (1975) 1-8. [43] (with I. Reiten) Almost split sequences II, Proc. Int. Conf. on representations of algebras, Carleton Univ., (Ottawa 1974), Lecture Notes in Math. 488, Springer Verlag (1975) 9-19. [44] (with E. L. Green and I. Reiten) Modules having waists, Proc. Int. Conf. on representations of algebras, Ottawa 1974, Lecture Notes in Math. 488, Springer Verlag (1975) 20-28. [45] (with E. L. Green and I. Reiten) Modules with waists, Illinois J. Math. 19 (1975) 467-477. [46] Categorical methods in the representation theory of artin rings, Univ. of Trondheim 1975, (Notes by S. 0. Smal0).

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IV [47] Large modules over artin algebras, Algebra, Topology and Category theory, (A collection of papers in honor of Samuel Eilenberg) 1-17 Academic Press, New York, 1976. [48] (with I. Reiten) On the representation type of triangular matrix rings, J. London Math. Soc. (2), 12 (1976) No. 3, 371-382. IV [49] (with I. Reiten) Representation theory of artin algebras IV: Invariants given by almost split sequences, Comm. Algebra 5 (1977) 443-518. IV [50] (with I. Reiten) Representation theory of artin algebras V: Methods for computing almost split sequences and irreducible morphisms, Comm. Algebra 5 (1977) No. 5, 519-554. [51] Existence theorems for almost split sequences, Ring theory II (Proc. Second Conf., Univ. Oklahoma, 1975) 1-44. Lecture Notes in Pure Appl. Math. 26, Dekker, New York, 1977. IV [52] (with I. Reiten) Representation theory of artin algebras VI: A functorial approach to almost split sequences, Comm. Algebra 6 (1978) No. 3, 257-300. [53] (with M. I. Platzeck and I. Reiten) Periodic modules over weakly symmetric algebras, J. Pure Appl. Algebra 11 (1977) 279-291. IV [54] (with M. I. Platzeck) Representation theory of hereditary artin algebras, Representation theory of algebras, (Proc. Conf. Temple Univ., 1976) 389:-424. Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978. VI [55] Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf. Temple Univ., 1976), 1-244. Lecture Notes in Pure Appl. Math. 37, Dekker, New York 1978. VI [56] Applications of morphisms determined by modules, Representation theory of algebras (Proc. Conf. Temple Univ. 1976), 245-327. Lecture Notes in Pure Appl. Math. 37, Dekker, New York 1978. VII [57] (with M. I. Platzeck and I. Reiten) Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46. IV [58] (with R. Bautista, M. I. Platzeck, I. Reiten and S. 0. Smal¢) Almost split sequences whose middle term has at most two indecomposable summands, Canad. J. Math. 31, (1979) No 5, 942-960. [59] Preprojective modules over artin algebras, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Antwerp, 1978) 361-384. Lecture Notes in Pure appl. Math. 51, Dekker, New York 1979. [60] (with S. 0. Smal¢) Preprojective modules: An introduction and some applications, Representation theory II (Proc. ICRA II, Carleton Univ., Ottawa 1979) 48-73. Lecture Notes in Math. 832, Springer, Berlin, 1980. [61] (with I. Reiten) Uniserial functors, Representation theory II (Proc. ICRA II, Carleton Univ., Ottawa 1979) 1-47, Lecture Notes in Math. 832, Springer, Berlin 1980. VII [62] (with S. 0. Smal¢) Preprojective modules over artin algebras, J. Algebra 66, (1980) No. 1, 61-122. VII [63] (with S. 0. Smal¢) Almost split sequences in subcategories, J. Algebra 69, (1981) No. 2, 426-454, with Addendum J. Algebra, 71 (1981) No. 2 ,592-594. [64] (with S. 0. Smal¢) Preprojective lattices over classical orders. Integral representations and applications. (Oberwolfach 1980) 326-345. Lecture Notes in Math. 882, Springer, Berlin-New York 1981.

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[65] Representation theory of finite-dimensional algebras, Algebraists' homage: papers in ring theory and related topics (New Haven, Conn,. 1981), 27-39, Contemp. Math. 13, Amer. Math. Soc., Providence, R.I., 1982. [66] (with S .. 0. Smal0) Lattices over orders: Finitely presented functors and preprojective partitions, Trans. Amer. Math. Soc. 273 (1982) No. 2, 433-445. [67] A functorial approach to representation theory, Representations of algebras. (ICRA III, Puebla 1980) 105-179. Lecture Notes in Math. 944, Springer, BerlinNew York, 1982. [68] Relations for Grothendieck groups of artin algebras, Proc. Amer. Math. Soc. 91 (1984) No. 3, 336-340. [69] (with I. Reiten) Modules determined by their composition factors, Illinois J. Math. 29 (1985) No. 2, 280-301. [70] Finite type implies isolated singularity, Orders and their applications (Oberwolfach, 1984), 1-4, Lecture Notes in Math. 1142, Springer, Berlin-New York, 19~5. [71] Isolated singularities and existence of almost split sequences, Representation theory II (ICRA IV, Ottawa, 1984) 194-241, Lecture Notes in Math. 1178, Springer, Berlin-New York, 1986. [72] ·. (with J. F. Carlson) Almost split sequences and group rings, J. Algebra 103 (1986) No. 1, 122-140. [73] (with I. Reiten) GrotherJ,dieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986) No. 1-2, 1-51. [74] Rational singularties and almost split sequences, Trans. Amer. Math. Soc. 293 (1986) No. 2, 511-531. [75] (with I. Reiten) McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1986) No. 1, 293-301. [76] A survey of existence theorems for almost split sequences, Representations of algebras (Durham, 1985), 81-89, London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge-New York, 1986. [77] Almost split sequences and algebraic geometry, Representations of algebras (Durham, 1985), 165-179, London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge-New York, 1986. [78] (with I. Reiten) Almpst .split sequences for rational double points, Trans. Amer. Math. Soc. 302 (1987) No. 1, 87-97. [79] (with I. Reiten) The Cohen-Macaulay type of Cohen-Macaulay rings, Adv. in Math., 73 (1989), No. 1, 1-23. [80] (with I. Reiten) Almost split sequences in dimension two, Adv. in Math. 66 (1987) No. 1, 88-118. [81] (with I. Reiten) Almost split sequences for Z-graded rings, Singularities, representation of algebras and vector bundles (Lambrecht 1985), 232-243, Lecture Notes in Math. 1273, Springer, Berlin-New York(1987). [82] (with.I. Reiten) Almost split sequences for abelian group graded rings, J. Algebra,114 (1988) No. 1, 29-39. · [83] (with I. Reiten) Almost split sequences for Cohen-Macaulay modules, Math. Ann. 277 (19.87) No. 2, 345-349. [84] The what, where and why of almost split sequences, Proce(;ldings of the International Congress of Mathematicians, vol. 1-2, (Berkeley, CA, 1986) 338-345, Amer. Math. Soc., Providence, RI, 1987.

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VIII [85) (with I. Reiten) Cohen-Macaulay modules for graded Cohen-Macaulay rings and their completions, Commutative Algebra, (Berkeley, CA, 1987) 21-31, Math. Sci. Res. Inst. Puhl. 15 Springer, Berlin-New York, 1989. IX [86) (with I. Reiten) Grothendieck groups of algebras with nilpotent annihilators, Proc. Amer. Math. Soc. 103 (1988), No. 4, 1022-1024. [87) (with E. L. Green) Trace quotient modules, Illinois J. Math. 32 (1988) No. 3, 534-556. II [88] (with I. Reiten and S. 0. Smal¢) Galois actions on rings and finite Galois coverings, Math. Scand. 65 (1989) No. 1, 5-32. IX [89] (with R. 0. Buchweitz) The homological theory of maximal Cohen-Macaulay approximations, Collogue en l'honneur de Pierre Samuel (Orsay 1987), Mem. Soc. Math. France, (N.S.) No. 38 (1989) 5-37. VIII [90] (with I. Reiten) Graded modules and their completions, Topics in Algebra, Part 1 (Warsaw 1989) 181-192, Banach Center Puhl. 26, Part 1, PWN, Warsaw 1990. [91) (with E. L. Green) Modules over endomorphism rings, Comm. Algebra 20 (5) (1992) 1259-1278. VII [92] (with I. Reiten) Applications of contravariantly finite subcategories, Adv. in Math. 86, No. 1 (1991) 111-152. [93] (with I. Reiten) Cohen-Macaulay and Gorenstein artin algebras, Representation theory of finite groups and finite-dimensional algebras, (Bielefeld 1991) 221-245, Progr. Math. 95, Birkhauser, Basel (1991). V [94) (with I. Reiten) On a theorem of E. Green on the dual of the transpose, Representations of finite-dimensional algebras (Tsukuba 1990) 53-65, CMS Conf. Proc., 11, Amer. Math. Soc., Providence RI (1991) 53-65. [95) (with I. Reiten) Homologically finite subcategories, Representations of algebras and related topics (ICRA V, Tsukuba 1990). London Math. Soc. Lecture Note Ser. 168, 1-42 (199). VII [96] (with M. I. Platzeck and G. Todorov) Homological theory of idempotent ideals, Trans. Amer. Math. Soc. 332 (1992) No. 2, 667-692. IX [97] (with S. Ding and 0. Solberg) Liftings and weak liftings of modules, J. Algebra 156 (1993) No. 2, 273-317. X [98] (with 0. Solberg) Relative homology and representation theory I, Relative homology and homologically finite subcategories, Comm. Algebra 21 (9) (1993) 29953031. X [99] (with 0. Solberg) Relative homology and representation theory II, Relative cotilting theory, Comm. Algebra 21 (9) (1993) 3033-3079. [100] (with 0. Solberg) Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. Algebra 21 (9) (1993) 3081-3097. [101] (with 0. Solberg) Gorenstein algebras and algebras with dominant dimension at · least 2, Comm. Algebra 21 (11) (1993) 3897-3934. X [102] (with I. Reiten) k-Gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (1994) 1-27. [103] (with M. Kleiner) The Green correspondence for adjoint functors, Representations of algebras (ICRA VI, Ottawa 1992) 69-81, CMS Conf. Proc. 14, Amer. Math. Soc., Providence, RI, 1993. [104) (with 0. Solberg) Relative homology, Finite-dimensional algebras and related topics (ICRA VI, Ottawa 1992) Kluwer Academic Publishers, Nato Adv. Sci. Inst. Ser. C Math. Phys. Sci. 424 (1994) 347-359.

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III [105] (with M. Kleiner) Adjoint functors and an extension of the Green correspondance for group representations, Adv. in Math. 104, (1994) No. 2, 297-314. [106] (with I. Reiten and S. 0. Smal¢) Representation theory of artin algebras, Cambridge Univ. Press, Cambridge 1995. III [107] (with I. Reiten) D Tr-periodic modules and functors, Representation theory of algebras (ICRA VII, Cocoyoc 1994) 39-50, CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996. X [108] (with I. Reiten) Syzygy modules for noetherian rings, J. Algebra 183 (1996) No. 1, 167-185.

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CHAPTER I

Homological dimension and local rings [1] On the dimension of modules and algebras (III), Global dimension (1955). [2] Commutator subgroups of free groups (with R. C. Lyndon, 1955). [4] On the dimension of modules and algebras (VI), Comparison of global and algebra dimension ( 195 7) . [5] On regular group rings (1957). [6] Homological dimension in local rings (with D. Buchsbaum, 1957). [7] Homological dimension in noetherian rings II (with D. Buchsbaum, 1958). [10] Codimension and multiplicity (with D. Buchsbaum, 1959). [12] Unique factorization in regular local rings (with D. Buchsbaum, 1959). [19] A remark on a paper of M. Hironaka (1962). Many of Auslander's early papers deal with first applications of homological algebra to commutative ring theory. The book by Cartan-Eilenberg on homological algebra served as general background. The inspiration for several results came from algebraic geometry, and many developments gave important feedback to geometry. A large portion of the work in this chapter has become textbook material. In the mid fifties there was a series of papers on the dimension of modules and algebras in the Nagoya Mathematics Journal. Several distinguished mathematicians were the authors of these papers, and Auslander was responsible for two of them [1, 4]. Here we find the by now standard result that the global dimension for rings can be computed by considering only the cyclic modules. For noetherian rings the weak global dimension is shown to coincide with the left and right global dimension. In [5] the (von Neumann) regular rings are shown to be those having weak global dimension zero. This result is applied to obtain information about when a group ring over a commutative ring is regular. There are later improvements in the literature, covering also skew group rings. In the papers [6, 7] basic foundations are laid for the homological theory of commutative noetherian rings. Regular sequences and codimension are treated here. Codimension, which is one of the most important tools in commutative algebra, appears independently by other authors, and is also known as depth or grade. The finitistic dimension is introduced and is shown to be the codimension, for a local noetherian ring, as a consequence of the famous codimension formula established here. Besides being a basic formula in commutative ring theory, this formula has also inspired generalizations, for example for G-dimension and virtual projective dimension. The difference between the dimension and the codimension for a finitely generated module over a regular local ring is investigated in [19]. Another famous result from these papers is the Auslander-Buchsbaum-Serre theorem, which characterizes the regular local rings as the local rings of finite global dimension. Indeed, this homological description is now often taken as a definition. An important

MAURICE AUSLANDER

1

CHAPTER I

2

consequence, which was of interest in algebraic geometry, was that localizations of regular local rings are regular local. There is still no nonhomological proof of this fact. In [10] the Koszul complex is used to obtain simpler proofs of some of the results on codimension. This method was motivated by the fact that Serre had shown connections between the Koszul complex and the local theory of multiplicities, which plays a central role in algebraic geometry. It is also shown in [10] that a local ring is Cohen-Macaulay if and only if the multiplicity of a system of parameters is equal to its length. The celebrated Auslander-Buchsbaum theorem that a regular local ring is a unique factorization domain appears in [12]. This was viewed as a big achievement at the time, and served to establish homological algebra as an important tool in commutative ring theory. Even though Auslander's doctoral thesis was in group theory, very little of his later production is within this area. One exception is the paper [2]. Auslander was very fond of this paper, which is still being referred to in the literature.

2

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ON THE DIMENSION OF MODULES AND ALGEBRAS (III) GLOBAL DIMENS10N 11 MAUR~CE AUSLANDER Let A be a ring with unit. If A is a left A-module, the dimension of, A (notation: I. dim" A) is defined to be the least integer n for which there exists an exact sequence

where the left A-modules Xo, •.. , Xn are projective. exists for any n, then I. dim" A

= oo.

If no such sequence

The left global dimension of A is

I. gl. dim A = sup I. dim" A where A ranges over all left .A-modules. The condition I. dim" A< n is equivalent with Ext!(A, C)=O for all left .A-modules C. is equivalent with Ext!= 0.

The condition l.gl.dim.A 0), then I. gl. dim A : : : 1 + sup I. dimA L

(c)

l

The equivalence of (a) and (b) is obvious. From [l; I, 4.2] we deduce that

I.t follows from this and [1 ; IV, 2.3], which we state below without proof as Proposition 2, that (b) implies (c). A is semi-simple if and only if A/ I is projective for all left ideals / of A.

PROPOSITION 2. If O~ A ' ~ A ~ A"~ 0 is an exact sequence of left Amodules with A projective and A" not projective, then I. dimA A":::::: 1 + I. dimA A'. Therefore in order to prove Theorem 1, it suffices to prove statement (a) of Theorem 1. This proof is based on PROPSITION 3. Let A be a left A-module, I a non-empty well-ordered set and (A;), e A;~Aj.

1

a family of submodules of A such that if i, j E I and i

!§ia

j, then

IfUA;=A and 1.dimA(A;/A~)!!een for all iElwhere A}::::::UAi, iEI

then I. dimA A

~

n.

Proof. The proof is by induction on n. If n : : : 0, then for all i E I we have 1. dim (A;/ A}) !§ia 0. Therefore each A;/ A~ is projective. This implies that each of the exact sequences

splits.

Thus there exist submodules C; of A; such that

(i) A;::::::A}+C; (direct sum), (ii) each C; is isomorphic to A;/ A} and therefore is projective.

From (i) and the hypothesis that A:::::: U A;, it follows that A:::::: ~C; (direct iE l

sum).

4

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iE l

69

ON THE DIMENSION OF MODULES AND ALGEBRAS (III)

From (ii) we have that A is projective, since by [1; I, 2.1] the direct sum of projective modules is projective. Therefore 1. dimA A= 0 and the proposition is established in the case n = 0. Suppose n

>0

and the proposition has been established for n - I.

Also,

suppose I. dimA (A;/ A}) s n for all i E I. Let F be the free A-module generated by the elements of A and F; (respectively Fi) the free A-module generated by the elements of A; (respectively AD. Further, let R = Ker (F~ A) and define

R; = F;

n R, Ri = P; n R.

From the relations A; g A}, F; g P;, R;

;;;?

R} and the exact sequences

0 ~ R;~ F;~ A;~O 0 ~ R}~F}~A,~O,

it follows that the sequences

are exact for all i E I. Each F;/ F} is a free A-module and therefore projective, since each F} is generated by a subset of a basis for F;. position 2 we have I. dimA (R;/ R}) !2i n - I.

It can easily be established that the

family (R;);ez has the properties that i, j E / and i

R = U R; and R} = U Ri. iEl

Therefore by Pro-

§

j implies that R; ~ Ri,

Thus by the induction hypothesis 1. dim R s n - 1.

jO. Therefore, by Proposition 2, we deduce from the exact sequence O~N~A~T~O

+ 1. dim" N. If N = 0, then A ] .dim" 0 = -1, gl. dim A= l + 1. dim" N.

that gl. dim A= 1. dim" T = l gl. dim A= 0. Since (d). Since

r

is semi-simple, i.e.,

is semi-simple, T:::::. ::SC;, finite direct sum of simple left A-

modules, where the C; have the property that if C is a simple left A-module, Therefore sup I. dim" C = I. dim" T = gl. dim.A, where C , C

then C :::::. C; for some i.

ranges over all simple left A-modules. (e). This is proved in an analogous fashion to (d). COROLLARY 12. If A is a semi-primary ring, then the following are equivalent: (a) gl. dim A

0,

ON THE DIMENSION OF MODULES AND ALGEBRAS

(III)

75

then there is an indecomposable left ideal J in A such that

J2 = O

and

gl. dim A= 1 + I. dimAJ.

Proof. Let J be a left ideal contained in the radical N, minimal with respect to the property that 1. dimA J = I. dimA N ( ideals of this type exist since N is such an ideal). l. dim., B).

If

J =A+ B ( direct

sum), then 1. dim AJ = sup (1. dimA A,

By the minimal property of ], either A or B must be the trivial

ideal. Thus J is indecomposable. Suppose ] 2 ¾ 0. Then there is an element ,1* E J such that Jl* ¾O. Consider the exact sequence J

(*) 0----+K-+J-+Jl*-+O

wheref0)=,1,1* and K=Ker f.

Since J is nilpotent, Jl*¾] and f is not a

monomorphism. Thus O¾ K ¾ ]. Therefore J,1* and K are proper ideals in J. Consequently we have sup (1. dimAJ,1*, 1. dim" K) the exact sequence(*) and [l; l. dimA K).

< I. dimAJ.

But in view of

VI, 2.3] we have l.dimJ;easup(l.dimAJ,1*,

This contradiction proves that

J2 = 0.

Since by definition 1. dimA] ==

1. dimA N, we have by Corollary 11 (c) gl. dim A = 1 + I. dimA ]. Q.E.D. § 4.

Applications

PROPOSITION 14. Let A be a semi-primary ring such that each simple left A-module is isomorphic to a left ideal in A, then gl. dim A = 0,

Proof.

Suppose gl. dim A= n, 0 < n

0,

~

I,

A/ I is

not projective. Therefore Proposition 2 applied to the exact sequence

0----+l-+A-+A/J-+O gives I. dim A/ I= 1 + 1. dim I= 1 + n. However, L dimA A/ I ;ea gl. dim A = n.

This

contradiction proves the proposition. PROPOSITION 15. The hypothesis of Proposition 14 is satisfied in each of the following cases : (a) A is a direct sum of a finite number of primary rings (a semi-primary ring A is primary if I'= A/N is a simple ring). (b) A is a semi-primary commutative ring. ( c) A satisfies both the left and right minimum conditions and every two·

MAURICE AUSLANDER

11

MAURICE AUSLANDER

76

sided ideal in A is a principal right ideal and a principal left ideal. (d) A is a quasi-Frobenius ring.

Proof. (a). Suppose A is a primary ring. If N

= 0, we are

finished. Assume

N ~ 0. Let k be the maximum index such that Nk ~ 0. Since NNk = 0, Nk is a Fmodule. Thus Nk is semi-simple and therefore contains at least one simple left ideal of A. But

r

is a simple ring.

class of simple left A-modules.

Thus there is only one isomorphism

Therefore each simple left A-module is iso-

morphic to a simple ideal in A. The rest of (a) is obvious. (b). Since N is nilpotent, every set of orthogonal idempotents in "lifted" to an orthogonal set of idempotents in A.

r can be

From this and the com-

mutativity of A, it follows that A is a finite direct sum of primary rings. Thus (d) is reduced to (a). (c ). By [3; Chapter 4, Theorem 37] we have that A is a direct sum of primary rings. Therefore (c) is also reduced to (a). (d). This is an immediate c9nsequence of the definition of a quasi-Frobenius ring as given in [ 4]. § 5.

Tensor products of semi-primary algebras

THEOREM 16. If A1 and A2 are algebras over a field K, then w. gl. dim (A1 ® A2) i.1. w. gl. dim A1 + w. gl. dim A2. If further A1 and A2 are semi-primary algebras and I'1 ® I'2 is semi-simple, then

A1 ® A2 is a semi-primary algebra and

Proof. By [l ; XI, 3.1] we have hTor#'(C 1, A1)®Tor~ 1 (C2, A2):::::Tor~•®A'(C1®C2, A1®A2) p+q=n

for all n a11, O, where C 1 and C2 are right A1 and A2·modules and A1 and & are left A1 and A2-modules. Since K is a field, Tort'(C1, A1) ~ 0 and Tor~2 (C2, A2) ,i,c 0 implies that Tort!rA•(C1 ® C2, A1 ® A2) ~o. Thus

From the exact sequence j

N1 ® A2+A1 ®N2~A1 ® A2~I'1 ® I'2~0 we deduce that the Ker / is nilpotent.

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If we assume that T1 ® I'2 is semi-

77

ON THE DIMENSION OF MODULES AND ALGEBRAS (III)

~imple, we have that A1 ® A2 is semi-primary with radical the Ker /.

Now we

have ~Tor~'(I'i, T1)®Tor~'(T2, T2)~Tor~•®A2(T1®T2, T1®T2) JHq=n

for all n ~ 0. Since K is a field, we deduce from Corollary 12 and these isomorphisms that gl. dim (A1 ® A2)

= gl. dim A1 + gl. dim A2.

REFERENCES

Ill H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1955. [2] S. Eilenberg, Algebras of cohomologically finite dimension, Comment. Math. Helv.,

28

( 1954 ), 310-319. [3] N. Jacobson, Theory of Rings, Amer. Math. Soc., 1943. [4] T. Nakayama, On Frobeniusean Algebras II, Ann. of Math., 42 (1941), 1-21.

University of Michigan

MAURICE AUSLANDER

13

COMMUTATOR SUBGROUPS OF FREE GROUPS.* 1 By MAURICE AUSLANDER and R. C. LYNDON.

Let F be a non-abelian,free group, Ra non-trivial normal subgroup of F; and G = F JR. Then F and R are free groups of ranks at least two. Let [R, R] be the commutator subgroup of R, and let F 0 =FJ[R, R] and R 0 =RJ[R,R]. Then R 0 is a free abelian group of rank at least two. Moreover, the inner automorphisms of F induce a natural operation of G on R 0 which we denote by g · ro. THEOREM 1. The operation of G on R 0 is effective ; that is, only the unit element of G leaves all elements of R 0 fixed. Proof. We may suppose G #= l. · Let g "F 1 in G generate a cyclic group G', and let F' be the inverse image of G' under the canonical map of F onto G. Then G' =F'JR, and. the operation of G' on R 0 agrees with that of G on R 0 • We must show that g, and hence that G', does not operate trivially on R 0 • Since G' = F'JR where F' is a free group, the cup product reduction theorem [2, Theorem 10.1] asserts that for any G'-module A

(1)

H 3 (G',A) °"H1(G', Hom(R 0 ,A) ),

where the operation of G' on Hom ( R 0, A) is defined by

(g · 6) (r0 ) = g · [O(g- 1 • ro) ], 6t: Hom((Ro, A),

r 0 t: Ro.

We take A to be isomorphic with the cyclic group G' = Z,., n > O, with G' operating trivially on A. Suppose that G' operates trivially on R 0 as well. Then all operations in ( 1) are trivial. Since G' is cyclic, H 3 ( G', A) is cyclic or trivial [2, 16.2]. On the other hand H 1 (G',Hom(R 0,A)) is isomorphic to Hom(Z,.,Hom(R 0 ,Z,.)) and hence to Hom(Z,.@R 0 ,Z,.) where Z,.©R 0 denotes the tensor product of Z,. and R 0 • Since R 0 is a free abelian group of rank at least two, Z,. 0 R 0 is a direct product of at least two copies of Z,.. * Received February 25, 1955; revised May 16, 1955. ' Part of the work presented in this paper was done while the first named author was at the University of Chicago. Substantial simplification of the proofs, and a strengthening of Theorem 3, are due to R. L. Taylor.

929 MAURICE AUSLANDER

15

MAURICE AUSLANDER AND R. C. LYNDON.

930

But Zn# 1. Therefore, Hom (Zn® R 0 , Zn) is neither cyclic nor trivial. This contradicts ( 1). Therefore G' does not operate trivially on Ro. For the following corollaries, we asume that Ry= F. COROLLARY

1. 1.

[ R, R] is

ll

proper subgroup of [F, R].

Proof. Let f £ F, f ~ R. If [f, r] = frf- 1 r- 1 were in [ R, R] for all r in R, then, setting g = fR in G, we should have g · r 0 = ro for all ro = r[R, R] in R 0 •

This contradicts Theorem 1.

COROLLARY

1. 2.

[ R, R] is a proper subgroup of [ F, F].

COROLLARY

1. 3.

The center C0 of F 0 is a proper subgroup of Ro.

Proof. We first show that 0 0 C R 0 • Suppose fo is in Co. Then f 0 r 0 = r 0 f O for all r 0 in Ro. This implies that the image g = f oR of f o in G operates trivially on R0 • By Theorem 1, this implies that g = 1. Hence

f o is

in R 0 • Suppose that C0 =R 0 • Then foro=rofo for all fo in Fo and ro in Ro. Hence all g in G operate trivially on R0 • By Theorem 1, this is possible only if G = l, which contradicts the hypothesis that R # F. COROLLARY 1. 4. The center C0 of F 0 consists precisely of those elements of R 0 that are left fixed under the operation of G. THEOREM

2. The center C0 of F 0 is trivial if and only if G is infinite.

Proof. Let G be of finite order n, with elements g 1 = f ,R, · · ·, g,. = f nR. Suppose C0 = 1. For any r in R, and corresponding To= T[ R, R] in R 0 , n

the element

IT g, · T0 is fixed under G, hence in Co. Therefore IT gi ·To= 1. i=l

This implies that ITfiTf,- 1 [R,R] C [F,F]. Since fiTf1-1=rmodulo[F,F], this implies that TnE[F,F], where n>l. Since F/[F,F] is torsion free, r is in [F, F]. Thus C0 = 1 implies RC [F, F]. But this would imply that the free abelian group F /[F, F] were a homomorphic image of the finite group G = F JR, a contradiction. Therefore 0 0 # 1. Let G be infinite, with elements g0 = 1, g 1 , g2 , • • •• Let ZG be the integer group ring of G. Its elements are of the form y = ~ ni% where only a finite number of the integers n, are not zero. Let G operate on ZG by left multiplication. If y were fixed under G, then g;-1 · y = y for all i. This implies that all rzi = n 0 , and hence that all n, are zero. Then y = 0. It follows that in any free G-module only the zero element is fixed under G. Since [ cf. 1, Ch. XIV, Ex. 11] R 0 is a submodule o:f a free G-module., R 0 is without non-trivial fixed elements. Therefore, by Corollary 1. 4, C0 is trivial.

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COMMU'L'A'L'OR SUBGROUPS OF FREE GROUPS.

931

~--,or any group G, let G2 = [ G, GJ, Ga= [ G2, GJ. If F is a non-abelian free group, and R a norrnal subgroup such that F 2 CR and Pa C R2, then F = R. LEMMA.

Proof. From Fa C R2 it follows that F2/R 2 C C0 • If R # F, then R 2 ¥= F 2 by Corollary 1. 2, whence C0 # 1, and, by Theorem 2, G = F / R is finite. Thus, in any case, G is finite.

The commutator operation determines a homomorphism K: F/F2®F/F 2 ~F2/Fa.

This induces a homomorphism p.: G ® R 0 ~ R 0 ; explicitly, p.(g o r 0 ) = g · r 0 ~ r 0 • Since G is finite, G 0 R 0 is a torsion group. But R 0 is a free abelian group. It follows that ,u = 0. This means that R 0 C C0 • By Corollary 1.3, this implies that R = F. THEOREM 3. If S and R are norrnal subgroups of a non-abelian free group, and [ S, SJ C [ R, RJ, then S C R.

Proof. Let F = RS. We may suppose that F is a non-abelian free group. We have

F2 = [F, FJ = [RS,RSJ = [R, R][R, S][S, SJ c R 2 (R n S), Fa=[F2,FJ C [R2(RnS),RSJ cR2[R,R][S,S]=R 2 • The lemma now gives R = F, whence S

C

R.

UNIVERSITY OF MICHIGAN.

REFERENCES.

(1] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1955.

[2] S. Eilenberg and S. MacLane, "Cohomology theory in abstract gmups. I," Annals

of M atkematics, vol. 48 ( 1947), pp. 51-78.

MAURICE AUSLANDER

17

ON THE DIMENSION OF MODULES AND ALGEBRAS, VI COMPARISON OF GLOBAL AND ALGEBRA DIMENSION MAURICE AUSLANDER Throughout this paper all rings are assumed to have unit elements. A ring A is said to be semi-primary if its Jacobson radical N is nilpotent and I'= A/ N

satisfies the minimum condition. The main objective of this paper is THEOREM

I. Let A be a semi-primary algebra over a field K. Let N be the

radical of A and T = A/ N. If dim A


K(G). (c) K(G), considered as a two-sided K(G')-module, contains K(G') as a two-sided direct summand. Thus if we set R=K(G') and S=K(G), the results of Proposition 2 remain valid.

PROOF. (a) It is easily seen that any system of representatives for the left cosets of G' in G, is a basis for K(G) considered as a left K(G')-module. (b) Analogous argument to that used in (a). (c) We may assume that F=G-G' is not empty. Since G'F=FG' = F, the K-submodule generated by Fin K(G), which we will denote by K(F), is a two-sided K(G')-submodule of K(G). It is clear that K(F)r\K(G') =0. Thus K(G') is a two-sided direct summand of K(G). We define the ring epimorphism E: K(G)-tK by E(Lkigi) = Lki. This homomorphism is called the unit augmentation. From now on we consider Kasa two-sided K(G)-module as follows: xk = E(x)k and kx=kE(x) for all kin Kand x in K(G). PROPOSITION 4. Let K be a left K(G)-module. Then we have w. gl. dim K ~ w. gl. dim K(G) ~ w. l. dimK(G) K

+ w. gl. dim K.

PROOF. The first inequality follows from Proposition 3 by setting G' = {1}. Thus in proving the second inequality, we need only consider the case where both w. dimKcG> K =rand w. gl. dim K =s are finite. Suppose A and Bare right and left K(G)-modules respectively. Let O-+X.-+X._1-+ · · · -+Xo-+A-+O be an exact sequence of right K(G)-modules where the X, are K(G)projective for i=O, · · ·, s-1. Then we have by [1, Chapter V, 7.2] that (1)

K(G)

K(G)

TorI>+• (A, B) ""' Torp

(X., B)

for all P>O. SinceK(G) isaprojectiveK-module, each of theX,, i= 1, · · ·, s-1,

MAURICE AUSLANDER

27

1 957)

ON REGULAR GROUP RINGS

661

is a projective module when considered as K-modules. Applying [1, Chapter V, 7.2] again we have IC

Tors,+, (A, B)

~

IC

Torp (X,, B)

for all p>O. Since w. gl. dim K=s, we have Tor: (X,, B) =O for all p>O. Therefore it follows from [1, Chapter XV, 7.6a] that (2)

K(G) Torp (X, ®KB, K)

~

IC(G) Torp (X,, B)

for all p>O, where X,®KB is the right K(G)-module defined by (x®b)g= (xg®g- 1b) for all x in X,, bin B, gin G. Since w. dimK K =r, we have Tor~r. Thus it follows from (1) and (2) that Tor:s+r. Therefore w. gl. dim K(G) ~s+r. PROPOSITION 5. Let G be a group and (Ga) a directed family of subgroups of G such that G is the direct limit of the Ga, If A is a left K(G)module, we have w. 1. dimKco> A

= sup w. l. dimK A. a.

From this it follows that w. gl. dim K(G)

= sup w. gl. dim K(Ga), Q

PROOF, Since G is the direct limit of the Ga, it follows that K(G) is the direct limit of the K(Ga). By [1, Chapter VI, Exercise 17] we have Tor~ (C, A) = lim Tor~(G,.l (C, A) (direct limit) --+

for all n and all right K(G)-modules C and all left K(G)-modules A. Therefore we see that w. I. dimK A ;a;supoa w. l. dimKcGa> A. On the other hand, we know by Proposition 3 that for each Ga, w. 1. dimK ~ w. 1. dimKco..>A..

This establishes the first equality. A similar argument proves the second equality. PROPOSITION 6. Let G be a locally finite group. Then the following statements are equivalent: (a) w. l. dimKcG> K=O; (b) w. l. dimK K < oo ; (c) K is uniq_uely divisible by the order of each element in G.

28

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662

MAURICE AUSLANDER

[August

PROOF. (a)=>(b). Obvious. (b)=>(c). Let G' be a cyclic subgroup of G of order n. Since the cohomology groups of a group depend only on the group and not on the ground ring K (see [I, X, 3.1]), we have by [I, XII, 7] that Torf« K =0. Combining Propositions 4 and 6 we have THEOREM 7. If G is a locally finite group and w. I. dimxca> K then w. gl. dim K(G) = w. gl. dim K.

< oo,

In particular, K(G) is regular if and only if K is regular and is uniquely divisible by the order of each element in G. LEMMA 8. If G is an infinite cyclic group, then w. dimxca> K = 1. Further, if K is a regular ring, then w. gl. dim K(G) = 1. PROOF. Let e: K(G)-.K be the unit augmentation. If g is a generator for G, it is easy to show that I(G) =Ker Eis a free left K(G)module with basis (g-1). Thus the sequence f 0-. K(G) -. K(G) -. K-. 0

where f(x) = (1-g)x, is a projective resolution of K. It follows that w. I. dimxca> K ~ 1. Since Torf K -.K(G)®xca> K)=K~O, we see that w. I. dimxca> K=l. The rest of the lemma follows from Proposition 4. THEOREM 9. If K(G) is a regular ring, then G is a torsion group and

K is uniquely divisible by the order of each element in G. PRO@F. If G' is an infinite cyclic subgroup of a group G, we have by Proposition 3 and Lemma 7 that w. gl. dim K(G) ~w. gl. dim K(G') ~w. I. dimKO. HenceRs®RE isan exact functor of E. Thus, if E' is a submodule

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M. AUSLANDER AND D. A. BUCHSBAUM

[July

of E, then Rs© RE' is a submodule of Rs© RE. Suppose now that Risa noetherian ring and that Eis a finitely generated R-module. We recall that for the submodules of E there exists a primary decomposition theory analogous to that for ideals (see [8]). A prime ideal\) in R is said to belong to a submodule E' of E if \) belon_gs to the annihilator in R of E/E'. In the following lemma, which we state without proof, we summarize the part of this theory that we shall need. LEMMA 1.1. Let S be a multiplicatively closed subset of R not containing 0, and let E' be a submodule in the R-module E. Then \) 1 , • • • , \),., the prime ideals in R belonging to E' in E, have the following properties: (a) An element x in R is in one of the \Ji if and only if xis a zero divisor for E/E'; (b) An ideal f in R is contained in one of the \); if and only if there is an element e in E which is not in E' such that fe is contained in E'; (c) Let \)1, • • • , \Jr be those ideals which do not meet S. Then E' © RRs is a proper submodule of the Rs-module E©RRs if and only if r>O. If r>O, then Rs\)1, · · · , Rs\)r are the prime ideals in Rs belonging to E' © RRs in E © RRs. LEMMA .1.2. Let E be an R-module and let ))" be a prime ideal in R which belongs to (O) in E. If xis an element in R such that the ideal (\l", x) is :;c.R, then E:;c.xE. Further, if xis not a zero divisor of E, and\) is a proper prime ideal containing ()) 11 , x), then there is a prime ideal \J' belonging to (0) in E/xE such that \J:Jll':J(ll'', x).

Proof. Since (ll", x) is a proper ideal of R, there exists a proper prime ideal \) in R which contains (\l", x). From the exact sequence of R-modules f E---tE---tE/xE---tO we deduce the exact sequence of Rp-modules (1)

g

E ©R Rp---tE ©R Rp---t (E/xE) ©R Rp---t O,

where f is multiplication by x in R and g is multiplication by the image of x in R'!J which we shall denote also by x. Since \J'' C\J, and \) 11 is a prime belonging to (0) in E, we have by 1.1 that E©RRp,:6.(0). Since xis in Rpp, the maximal ideal of the local ring Rp, we know by [3, VIII, Proposition 5.1'] that g is not an epimorphism. Therefore (E/xE) ©RRp:;c-0, which means that E/xE:;c-0. Suppose x in R is not a zero divisor for E. Then it can easily be seen that x in Rp is not a zero divisor for E@RRu. Since p" belongs to (0) in E, we have by 1.1 that Rp\) 11 belongs to (O) in E©RRp. Therefore we know that there is a prime ideal p in the local ring Rp such that p belongs to (0) in (E©RRp)/x(E©RRp) and Rpp:)p:)(Rp\) 11 , x) =Rp(p", x) (see [5, 135 no. 8]; also [10, Lemma 1]). From the exact sequence (1) we deduce that

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(E ®R Rp)/x(E ®R Rp)

~

(E/xE) ®R Rp.

Since (E/xE)®RRp~O, we have by 1.1 that ~=Rp\) 1 , where \) 1 is a prime belonging to (0) in E/xE which is contained in \). Thus by contraction to R we have)):::)))':::)(\)'', x). Since\) was any proper prime ideal containing (\l", x), the lemma has been established. Unless stated otherwise, we assume throughout the rest of this section that Risa local ring with maximal ideal m, and F=R/m. PROPOSITION 1.3. Let \:)1, • • • , \:),. be the prime ideals in R belonging to (0) in E. If xi, · · · , x. is an E-sequence, then (xi, · · · , x.) is contained in m and s ~ dim R/\:);for all i. Thus we have that codimRE ~ dim R < oo.

Proof. Suppose X; is not in m for some i. Then X; is a unit in R and thus x;E = E. Since (x1, · · · , x.)E contains x;E = E, we have that E/ (x1, · · · , x.)E =0, which contradicts the definition of an E-sequence. Let\:) be one of the\:);. We will show by induction that s~dim R/\:). For s = 0, it is obvious. Assume the statement true for s-1. By 1.2, we know there is a prime \l' belonging to (0) in E/x 1E with j:)':::)()), x1). Now X2, • • • , x. is an E/x1E-sequence. Therefore we have by our induction hypothesis that s-1 ~dim R/\:)'. Since X1 is not contained in the prime ideal \:), we have that dim Rh'~ dim R/\:)-1. From this it follows that s~dim R/\:). The rest of the proposition follows from the definition of codimension and the fact that dim R/f ~ dim R for any ideal f in R. PROPOSITION

1.4. If x1, ; · · , x. is an E-sequence, then hdRE/ (x1 ,

• • · ,

x,)E

=s+hdRE.

Proof. Suppose an element x in m is not a zero divisor for E. From the exact sequence f o-E-E-E/xE-o

where f is multiplication by x, we deduce the exact sequence ···-

f'

Torn(E, F) - Tor,.(E, F) -

Torn(E/ xE, F) -

f'

Tor,,_1(E, F) -

···

where f' is multiplication by x. Since xF=O, the homomorphisms f' are the zero homomorphisms for all n. Hence we have, for every n, the exact sequences (1)

0 - Torn(E, F) - Tor,.(E/ xE, F)

-t

Torn-1(E, F) - 0.

If hdRE = oo, then we know by [3, VIII, 6.1'] that Torn(E, F) ~O for all n. From (1) we deduce that Torn(E/xE, F) ~O for all n and thus hdRE/xE = oo. If hdRE = p < oo, then we have Torp(E, F) ~O and Torn(E, F) = 0 for all n > p.

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[July

We therefore deduce from (1) that Torp+1(E/xE, F) ~O, and Torn(E/xE, F) =0 for all n>P+l, which yields the desired result for s=l. The proof for all s follows by induction. LEMMA 1.5. Let R, T be arbitrary rings with identity element (i.e., not necessarily commutative or noetherian), and f: R__;,T a ring homomorphism. If E is an arbitrary (left) R-module (i.e., not necessarily finitely generated), and Tor!(T, E) =0 for all n>O, then hdRE~hdT(T®RE). Furthermore, if there exists an R-module homomorphism g: T--tR such that gf is the identity on R, then the above inequality is an equality.

Proof. The first statement is essentially contained in the proof of [3, VIII, 3.1]. Hence we need only prove the reverse inequality. For any R-module C, consider the identifications

HomT(T ®RE, T ®RC)

~

HomR(E, HomT(T, T ®RC))

~

HomR(E, T ®RC).

If we replace Eby an R-projective resolution X of E, and pass to homology, we obtain H(HomT(T ®RX, T ®RC)) ~ H(HomR(X, T ®RC)). Since Tor!(T, E) =0 for all n>O, we have that T®RX is a T-projective resolution of T®RE. Therefore we deduce that

ExtT(T ®RE, T ®RC)

~

ExtR(E, T ®RC).

Since gf is the identity, R is a direct summand of T as an R-module, which implies that C is a direct summand of T®RC as an R-module. Therefore, ExtR(E, C) is a direct summand of ExtR(E, T®RC), which proves the reverse inequality. As an application of 1.5 we prove PROPOSITION 1.6. Let R be an arbitrary commutative ring with identity element (not necessarily noetherian), S a multiplicatively closed subset of R not containing 0, and E an arbitrary R-module (not necessarily finitely generated). Then hdRE~hdR8 (Rs®RE). Further, we have that gl. dim R~gl. dim Rs.

Proof. By [9, no. 48] we know that for an arbitrary R-module E. Tor!(Rs, E) =0 for all n>O. Thus we have by 1.5 that hdRE~hdR 8 (Rs®RE). Let f be an ideal in Rs. It is well known that Rs/f~Rs®R(R/(fnR)). Since gl. dim R~hd&R/(fnR) ~hdR8 Rs/f for all ideals f in Rs, it follows from [ 1, Theorem 1] that gl. dim R ~ gl. dim Rs. THEOREM

1. 7. If R is a local ring, then f. gl. dim R = codim R.

Proof. If x1, · · · ,

Xn

is an R-sequence, then

hd&R/(x1, · · · , Xn)R = n

+ hd&R =

n.

Thus codim R ~ f. gl. dim R.

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395

Given any R-module E, we can construct an exact sequence

(2) with X free and K = Ker gCmX. For let e1, · · · , e, be .a minimal generating system for E. Let X be the free R-module with basis xi, · · · , x,. and g the homomorphism which sends x; into e;. Now I:r;x; is in K if and only if I:r ;e; = 0. Suppose r1 is not in m. Then r1 is a unit. Thus e1 = -r1 1( L•>I r ,e;), which contradicts the minimality of e1, · · · , e,. Hence r1 and similarly all the other r, must be in m. Therefore K is contained in mX. Assume that codim R=O and that f. gl. dim R>O. Then there exist modules E with OO. Thus by 1.5 we have that hdR'K' = hdR,(R' 0R K) ~ hdRK < oo. Since codim R' ~ n -1, it follows from the induction assumption that hdR,K'

~

n - 1.

We now use the inequality hdRK'

~

hdR,K'

+ hdRR'

[3, XVI, Exercise 5 ]. Substituting previous results in this inequality we obtain that hdRE~n. Thus f. gl. dim R~codim R. The followin12: result is due to Serre [10, Theorem 41:

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[July

1.8. gl. dim R?;, (m/m 2 : R/m).

A well known theorem of Krull [7, 3.5, Theorem 7] tells us that dim R ~ (m/m2 : R/m). Summarizing the results of this section we obtain THEOREM

1.9. If R is a local ring, then

f. gl. dim R = codim R

~

dim R

~

(m/m 2 :R/m)

~

gl. dim R.

THEOREM 1.10. A necessary and sufficient condition that a local ring R be regular is that gl. dim R < oo •

Proof. If gl. dim R < oo, then gl. dim R = f. gl. dim R. Hence, by Theorem 1'.8, we h~ve that dim R= (m/m 2 : R/m), so that R .is regular. · ' Now suppose that R is regular. Then it is known that m = (u1 , • • • , Un) where (u1, · · ·, Ui) is a prime ideal for 1 ~i~n, -[7, p. 72]. Hence u 1, • · ·, u,. is an R-sequence and gl. dim R=hd&R/m=n. Combining 1.10 and 1.6 we have THEOREM 1. 11. If R is a regular local ring and )) is a prime ideal in R, then Rp is a regular local rin,g.

2. Regular local rings. In this section, all rings will be regular local rings unless otherwise specified. PROPOSITION 2.1. Let R be an arbitrary local ring (not necessarily regular) and let Ebe an R-module. The E-sequence xi, · · · , x, is maximal (i.e., there is no x,+1ER such that xi, · · · , x,+1 is an E-sequence) if and only if m belongs to (0) in E/(xi, · · · , x,)E.

Proof. The E-sequence Xi, • • • , x, is maximal if and only if every element of mis a zero divisor of E/(xi, · · · , x,)E. By 1.1 that is precisely the condition form to be contained in some prime belonging to (0) in E/(x1, • • • , x,)E, Since mis maximal, this means that m belongs to (0) in E/(x 1, • • • , x,)E. PROPOSITION

2.2. hdRE = gl. dim R if and only if m belongs to. (0) in E.

Proof. Suppose m does ncit belong to (0) in E. Then there is an x in m which is not a zero divisor of E. Therefore, by 1.4, we have hd&E/xE = 1 +hd&E. Since gl. dim R is finite, this shows that hd&E s there is an x,+i in lJ, such that xi, · · · , x,+i is an R-sequence. By 1.2 we know that there is a prime ll" belonging to (xi, · · · , X,+1) such that lJ =:)lJ''=:) (ll', x,+i) =:)lJ' (ll', x,+i~ll'). Therefore lJ=lJ". Since (xi,···, X,+1) is unmixed and of rank s+l, we have rank l)=s+l =l+rank lJ'. The following corollaries are immediate consequences of 2.8 and 1.9. COROLLARY 2.9. If lJ is a prime ideal in R, then dim Rh+rank lJ =dim R. COROLLARY 2.10. If Risa factor ring of a regular local ring, and lJ=:)lJ' are two prime ideals of R, then all saturated chains of prime ideals between lJ and p' have the same length, namely dim Rh' -dim Rh. LEMMA 2.11. Let R be a commutative noetherian ring having the property that if lJ=:)lJ' are prime ideals of R, then any two saturated chains of prime ideals between lJ and lJ' ha·ve the same length (we will say that such a ring R satisfies the "saturated chain condition"). If f is any ideal of Rand xis a nonunit in R, then rank (f, x) ~ 1 +rank f.

Proof. Let rank f =s, and let lJ be a prime of ranks belonging to f. Then in the ring Rh, the ideal (ll, x)h has rank at most one. It follows from the saturated chain condition in R that the ideal (ll, x) has rank at most s+l. Since (ll, x) contains (f, x), we have rank (f, x) ~ 1 +s. We can now prove the following slight refinement of the Cohen-Macaulay Theorem: THEOREM 2.12. Let f = (xi, · · · , x,) be an ideal of rank s. Then Xi, · · · , x, is an R-sequence and thus f is unmixed.

Proof. By induction on s. If s = 1, then f = (xi) and thus by 2.6 we have that the theorem is true. Consider the ideal (xi, · · · , x,-i), s > 1. Since rank f==s, it follows from 2.11 that rank (xi, · · · , x,-i) ~s-1. But clearly the reverse inequality holds, since Xi, · · · , x,-i is generated by s -1 elements. Applying the induction hypothesis, we have that xi, · · · , x,-i is an Rsequence and thus Xi, · · · , x,-i is unmixed. Since f has ranks, x, cannot be contained in any of the primes belonging to (xi, · · · , x,-i). Therefore xi, · · · , x, is an R-sequence, and consequently unmixed. COROLLARY 2.13. Let xi, · · · , x, be an R-sequence and let q be a permutation of 1, · , · , s. Then Xu(i), • • • , Xu(s) is an R-sequence. THEOREM 2.14. A regular local ring of dimension two is a unique factorization domain.

Proof. Since a regular local ring is integrally closed in its field of quotients, it suffices to prove that every minimal prime of R, i.e., every prime of rank one, is principal. If lJ is a prime ideal of rank one, then m does not belong to

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399

\) so that hdRR/ll "': codim R. Now suppose m does belong to (0) in E. Then, passing to completion, we have belongs to (0) in E. Let R..= T/t where Tis a regular local ring. If we denote by m the maximal ideal of T, then mbelongs to (0) in E, so that hdTE=dim T. From the relationship

m

hdTE ~ hdTR.. + hdi.E, we obtain dim T - hdTR.. = codim

R~

hdii:E.

But, by 3.2, hdRE=hdRE, and codim R=codim R, so that codim R~hdRE. But the reverse inequality is given by 1.6, so the proof is complete. PROPOSITION 3.6. Let o-E'-E-E"-o be an exact sequence of R-modules. If codimRE" 0, and rank R 11 f ~rank fin R 11 , R 11f:;=O. Now, R11P belongs to {O) in R 11 , so codim R 11 =0. Therefore, by 1.6, we have that hdRIIR11/ R 11f = oo, since R 11 / Rpf is not projective. But then we have hd&R/f = oo, which is a contradiction. 4. Regular rings. In this section, we assume that R is a commutative noetherian ring with identity element. DEFINITION. An ideal fin R is said to be regular if R 11 is a regular local ring for each prime p belonging to f. R is said to be regular if every ideal of R' is regular. PROPOSITION

4.1. R is regular if and only if each maximal ideal in R is

regular. Proof. The necessity follows from the definition. To prove sufficiency, we remark that it suffices to prove that R11 is regular for every prime p. Therefore, let p be a prime, and p' a maximal ideal containing p. Since R11 = (Rp,) Rp'lh and since R 11 , is regular by hypothesis, the result follows from 2.14. PROPOSITION

4.2. A regular integral domain R is integrally closed.

Proof. Since R = n11 Rp where p runs through all maximal ideals, and since each Rp, being regular, is integrally closed, R is the intersection of integrally closed rings. But then R is integrally closed. PROPOSITION 4.3. R is the direct sum of a finite number of integral domains if and only if Rp is an integral domain for every prime p.

Proof. Suppose R 11 is an integral domain for every prime p. Let (0) =q1r'I . · · r'lq,. be a normal decomposition of (0), and let Pi, · · · , p-,. be the primes belonging to Q1, • • • , q,. respectively. We will show that q,=p, for all i, and hence each p, is minimal, due to the irredundancy of the decomposition. Suppose q,:;=p,. Then R 11 , contains zero divisors, contrary to assumption. Thus q, = p, for all i. Ifn = 1, we are finished. Suppose, then, that n> 1. If (p,, Pi) :;=R (i:;=j), let p be a proper prime ideal containing (p,, P;). Then in R 11 we have zero divisors, again contrary to assumption. Thus, for all pairs (i, j), i:;=j, we have (p,, P;) = R. Applying the Chinese Remainder Theorem, we have R..,, R/Pi +Rfp,. (direct sum). Now let R=R1 + · · · +R,. (direct sum) with each R, an integral domain. Let e, denote the ith natural base elements and let p, be the ideal generated

+ ···

MAURICE AUSLANDER

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HOMOLOGICAL DIMENSION IN LOCAL RINGS

by all e;, j r5-i. It is easy to see that each lJ; is a prime. Let lJ be a proper prime in R. Then since I:ei= 1, lJ does not contain ei for some i. If lJ does not contain lJ;, then there is an ei, j r5-i, such that e;Ef+J. But this is impossible since e;e;=O for jr5-i. Thus every prime contains some lJ;. Furthermore, if lJ:)lJ;, then lJ=lJ;+lJ/\R; (direct sum). Therefore Rp=(R;)pnR, and hence Rp is an integral domain for each lJ. COROLLARY 4.4. A regular ring is the direct sum of regular integral domains. PROPOSITION 4.5. A ring R of finite global dimension is regular.

Proof. For each prime ll, Rp has finite global dimension, hence is regular. LEMMA 4.6. If E is an R-module, then hdRE = sup (hdRp(Rp ®RE)) where lJ runs through all maximal ideals of R. Hence gl. dim R=sup. (gl. dim Rp), where lJ runs through all maximal ideals of R.

Proof. The first statement is found in [3, VII, Exercise 11]. To prove the second, we again make use of [1, Theorem 1], and restrict our attention to the modules R/f, where f is any ideal of R. Thus we have -gl. dim R =sup (hdRR/f) =sup (sup (hdRp(Rp®RR/f))) ~sup (gl. dim Rp). On the other hand gl. dim R~sup (hdRR/ll) =sup (gl. dim Rp), and so we are done. THEOREM 4.7. In a regular ring R, gl. dim R=dim R.

Proof. In a regular ring, gl. dim Rp=dim Rp. Since dim R=sup (dim Rp), 4.6 gives us the result. COROLLARY 4.8. A ring of finite dimension is regular if and only if it has finite global dimension. PROPOSITION 4.9. Let R be the quotient of a regular ring. Then R satisfies the saturated chain condition (see 2.11).

Proof. Let R= T /f, where Tis regular ring. Let lJ 1 ClJ 2 be two primes in R, and suppose that +12 is the image of ll' in T. Then Rp 2 = Tp, /Tp,f. Hence Rp 2 is the quotient of a regular local ring. Since there is a 1-1 correspondence between primes of R contained in lJ 2 and primes in Rp 2 , the result follows from 2.10. PROPOSITION 4.10. If f is a regular ideal in R, then hdRR/f ~rank ll, where lJ is any prime belonging to l. Thus, if hdRR/f =rank f, then f is unmixed.

Proof. We know that if lJ is a prime belonging to f, then hdRR/f ~hdRP(Rp®RR/f) =hdRPRp/Rpf. Since Rpp belongs to Rpf in Rp, we have hdRR/f ~ gl. dim Rp = dim Rp = rank lJ. PROPOSITION 4.11. Let R be a regular ring, and f = (x1 , • of rank r. Then xi, · · ·, x, is an R-sequence. Hence hdRR/(x 1, f is unmixed.

46

SELECTED WORKS

• • ,

x,} an ideal x,) =r, and

• • • ,

404

M. AUSLANDER AND D. A. BUCHSBAUM

[July

Proof. Since R is a regular ring, it satisfies the saturated chain condition by 4.9. Hence, applying 2.11, we see that (x1, · · ·, x,-1) has rank r-1. Suppose the proposition true for r-1. Then, as in 2.12, we have that X1, · · ·, x, is an R-sequence. Since rank (x1, · · · , x,) =r, we knowthathdRR/(x1, · · · , x,) ~r. Hence we need only show that hdRR/(x1, · · · , x,) ;;£r. However, this follows from the general fact that if Eis an R-module, and xis an E-sequence, then hdRE/xE ;;£ 1 +hdRE, For from the exact sequence

f 0-tE-tE-tE/xE-tO where f is multiplication by x, we obtain the exact sequence n

• · · -t

ExtR(E, C)

n+l

-t

ExtR (E/ xE, C)

n+l

-t

ExtR (E, C)

-t · · ·

for every R-module C. Hence if Ext~(E, C) = 0 for all p > s, then Ext~+ 1(E/xE, C) =0 for all p >s. Now we must prove the proposition for r = 1. Let k = (x) be of rank 1. If x were a zero divisor, it would be contained in some prime belonging to (0). But O=hdRR/(0) =sup rank lJ where j) runs through all primes belonging to (0). Hence all the primes belonging to (0) are of rank zero, and therefore cannot contain an ideal of rank one. Since x is not a zero divisor, hdRR/(x) ;;£ 1. But hdRR/(x) ~sup rank lJ where lJ belongs to x. Hence hdRR/(x) ~ 1, thus hdRR/(x) = 1, and rank lJ = 1 for all j) belonging to (x). Thus our proposition is established for r=l, hence for all r. PROPOSITION 4.12. In a regular integral domain R of dimension less than or equal to two, all minimal primes are invertible.

Proof. Let lJ' be a minimal prime, i.e., a prime of rank 1. Then hdRR/lJ' = sup (hdRpRp/ Rpp') where lJ runs through all maximal ideals containing lJ'. Since all maximal primes have rank;;£ 2, Rp/ Rpj) 1 has codimension 1 as an Rp-module, hence hdRpRp/ Rpj)' = 1. Thus hdRR/ll' = 1, hdRp' = 0, lJ' is projective and hence invertible [3, VII, 3.2 ]. PROPOSITION 4.13. A ring is hereditary if and only if it is the direct sum of a finite number of Dedekind rings.

Proof. A ring R is hereditary if and only if gl. dim R ;;£ 1. Thus the result follows from 4.4, and the fact that if R = R1 + · · · + Rn (direct sum), then gl. dim R = sup gl. dim R;. APPENDIX (added in proof). Some recent work of the authors (to appear) has shown that 3.4 can be proved without using the structure theorems of Cohen. These techniques also yield the result of 2.13 for an arbitrary local ring.

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It can also be shown that 3.9 holds without the hypothesis that rank

f>O. BIBLIOGRAPHY

1. M. Auslander, On the dimension of modules and algebras, III. Global dimension, Nagoya Math. J. vol. 9 (1956). 2. M. Auslander and D. A. Buchsbaum, Homological dimension in noetherian rings. Proc. Nat. Acad. Sci. U.S.A. vol. 42 (1956). 3. H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956. 4. I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. vol. 59 (1946) pp. 54-106. 5. W. Grobner, Moderne algebraische Geometrie, Berlin, Springer, 1949. 6. W. Krull, Dimensiontheorie in Stellenringen, J. Crelle vol. 179 (1938). 7. D. G. Northcott, Ideal theory, Cambridge University Press, 1953. 8. P. Samuel, Commutative algebra (Notes by D. Hertzig), Cornell University, 1953. 9. J.-P. Serre, Faisceaux algebriques coherents, Ann. of Math. vol. 61 (1955). 10. - - - , Sur la dimension des anneaux et des modules noethtriens, Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955, Science Council of Japan, Tokyo, 1956. UNIVERSITY OF MICHIGAN, ANN ARBOR, MICH. UNIVERSITY OF CHICAGO, CHICAGO, ILL.

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MAURICE AUSLANDER AND DAVID A. BUCHSBAUM

Introduction. Throughout this paper it is assumed that all rings are commutative, noetherian rings with unit element and all modules are unitary. The major purpose of this paper is to extend to arbitrary noetherian rings the homological invariants which were introduced in [2] for local rings. In §1 we study the codimension of modules, and prove, in particular, that if E is any finitely generated R-module, then

codimR E = sup codimRm Em m

where m runs through all maximal ideals of R, Rm is the local ring of quotients of R with respect tom, and Em=Rm®RE, We also show that the result which we obtained in [2], codim R = sup hdR E E

where E runs through all finitely generated R-modules of finite homological dimension, holds when R is any noetherian ring. Another result obtained in this section is that if R[[X1, · · ·, Xn]] =Sis the ring of formal power series over R, then codims S = n

+ codimR R,

gl. dim S = n

+ gl. dim R.

In §2 we introduce the weak homological dimension of an R-module E (w. hdR E). It is defined as the smallest integer n (or + oo) such that Tor! (E, C) =0 for all P>n, and all R-modules C. The finitistic weak global dimension of R is defined by f.w. gl. dim R = sup w. hdR E E

where E runs over all modules of finite weak homological dimension. We then show that f. w. gl. dim R = sup codimR~ R~ ~

~

dim R

Received by the editors October 31, 1956. ( 1) This work was done while one of the authors was on a National Science Foundation Fellowship.

194 MAURICE AUSLANDER

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HOMOLOGICAL DIMENSION IN NOETHERIAN RINGS. II

where lJ runs through all prime ideals in R. For polynomial rings R R, we obtain the equalities

195

[x] over

f.w. gl. dim R[x] = codimRcxJ R[x] = 1 + f.w. gl. dim R. An example is given to show that codimR!zJ R[x] can be arbitrarily larger than codimR R. §3 deals with the question of unique factorization in local rings. It has been communicated to the authors that some of the results we establish in this section have been proven (but not yet published) by Mori. However, the methods used in our proofs are homological and, we feel, are of sufficient interest to be included in this paper. The main result of the section is that a local integral domain Risa unique factorization domain if and only if hdR R/(x, y) ~ 2 for every pair of elements x, yER. From this it follows that R is a unique factorization domain if its completion, R, is. We thereby conclude that every nonramified regular local ring is a unique factorization domain. The terminology used throughout is the same as that used in [2]. The notation used is standard except for the following situation: If Sis a multiplicatively closed subset of the ring R, not containing zero, and Eis an R-module, we denote by Es the Rs-module Rs®RE. However, if S=R-p, where lJ is a prime, we write E'9 instead of Es. 1. Codimension. Throughout this section we assume that all modules are finitely generated. An element x in the ring R is said to be a zero divisor for the R-module E if there is a nonzero element e in E such that xe=O. A finite sequence Xi, · · · , x, of elements in R is called an E-sequence if Xi is not a zero divisor for the module E/(xi, · · · , x,-i)E and E/(x1, • • • , x,)E-,,fO. Given an ideal f in R, an E-sequence is said to be a maximal E-sequence in f if f contains (xi, · · · , x,) and given any y in f, the sequence xi, · · · , x,, y is not an Esequence. It is clear that if Xi, · · · , x. is an E-sequence then (xi, · · · , Xi) -,,f (xi, · · · , Xi+i) for any i. Since R is a noetherian ring, it follows that each E-sequence inf can be imbedded in a maximal E-sequence inf of finite length. If xi, · · · , x, is a maximal E-sequence in f =R, we shall simply say that xi, · · · , x. is a maximal E-sequence. The least upper bound of lengths of E-sequences (finite or + oo) is called the codimension of E (notation: codimR E). LEMMA 1.1. Let E be an R-module and lJ a prime ideal containing the annihilator of E. If xi, · · · , x, is an E-sequence contained in p, then lJ contains the annihilator of E/(xi, · · · , x,)E.

Proof. The proof proceeds by induction on s. If s = 0, there is nothing to prove. Suppose lemma true for s=n~O and let s=n+L By the induction hypothesis lJ contains f, the annihilator of E/(xi, · · · , Xn)E. Thus lJ contains

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some prime deal ))' belonging to f. Since Xn+i in lJ is not a zero-divisor for E/(x1, · · ·, Xn)E, it follows from [2, 1.2] that ll contains a prime ideal belonging to the annihilator of E/(x1, · · · , Xn+1)E. Therefore lJ contains the annihilator of E/(x1, · · · , Xn+1)E. PROPOSITION 1.2. Let Ebe an R-module, ma maximal ideal in R containing the annihilator of E, and xi, · · · , x, an E-sequence contained ln m. Then the following statements are equivalent: (a) x 1, • • • , x, is a maximal E-sequence in m. (b) m belongs to (0) in E/(x1, · · · , x,)E. (c) The sequence X1, • • • , x., considered as elements in Rm, is a maximal Em-sequence. (d) codimRm Em=s.

Proof. (a)=?(b). Suppose m does not belong to (0) in E/(x1, · · · , x,)E. Since mis a maximal ideal in R, we have that mis not contained in any prime ideal belonging to (0) in E/(x1, · · · , x,)E. Thus there is a y in m which is not a zero-divisor for E/(x1, · · · , x,)E. Since m contains x1 , • • • , x, and the annihilator of E, it follows from 1.1 that m contains the annihilator of E/(x1, · · · , x,)E. Consequently, m contains a prime ideal lJ belonging to (O) in E/(x1, · · · , x,)E. Therefore m contains (ll, y) and thus by [2, 1.2] we have that E/(xi, · · · , x., y)E~O. This contradicts the maximal nature of X1, · · · , Xs.

(b)=}(c). From the exact sequence of R-modules 0-

f E/(x1, · · · , X;-1)E - E/(x1, · · · , X;-1)E -

E/(x1, · · · , x;)E - 0

we deduce the exact sequence of Rm-modules

where f is multiplication by X; and g is multiplication by the image of x, in Rm which we shall denote also by Thus is not a zero-divisor for Em/(x1, · · · , x,-1)Em. Since m belongs to (0) in E/(x1, · · · , x,)E, we have by the general theory of noetherian modules [see 2, 1.1] thatEm/(x1 , • • • , x,)Em ~O and that the maximal ideal mRm of the local ring Rm belongs to (0) in Em/(x1, · · · , x,)Em. It follows therefore that x 1, • • • , x, in Rm is a maximal Em-sequence. (c)=?(d). This follows immediately from the fact that in a local ring all maximal sequences for a module have the same length [see 2, 3.4]. (d)=?(a). Suppose x 1 , • • • , x, is not a maximal E-sequence in m. Then there is an E-sequence xi, · · · , x,, x,+1, · · · , Xn which is a maximal Esequence in m. Applying the implication (a)=?(d) we have codimRm Em =n>s, which is a contradiction.

x,.

x,

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PROPOSITION 1.3. Let E be an R-module and let the ideal t in R be the annihilator of E. If xi, · · · , x. in R is an E-sequence, then the ideal (f, Xi, · · · , x.) r5-R.

Proof. Suppose (f, xi, · · · , x.) ~ R. Then there exist x in f and y in (xi, · · · , x.) such that x+y = 1. Therefore ye= e for all e in E which means that E/(xi, · · · , x.)E=O. This contradicts the fact that Xi, · · · , x. is an E-sequence. THEOREM

1.4. Let E be an R-module. Then

codimR E = sup codimRm Em m

where m runs through all maximal ideals in R.

Proof. Let m be a maximal ideal in R. If m does not contain the annihilator of E, then Em=O and thus codimR E~codimRm Em, If m does contain the annihilator of E, then by 1.2 we have that all maximal E-sequences in m have the same length, namely codimRm Em. Thus we have codimR E

~

supm codimRm Em.

Suppose xi, · · · , x. in R is a. maximal E-sequence. Then by 1.3 we know there is a maximal ideal m containing Xi, · · · , x, and the annihiiator of E. Since xi, · · · , x. is a maximal E-sequence it is maximal in m. We then have by 1.2 that codimRm Em=s. Therefore we have codimR E~supm codimRm Em. COROLLARY

1.5. Let E be an R-module. Then

codimR E

~

dim R.

By [2, 1.3] we know that codimRm Em~ dim Rm~ dim R. Therefore it follows from 1.4 that codimR E~dim R. 1.6. For a ring R we have (a) codimR R~dim R, (b) codimR R=supm codimRm Rm, (c) codimR R=supE hdR E, where m runs through all maximal ideals in R and E runs through all finitely generated R-modules such that hdR E < oo. THEOREM

Proof. The first two relations are obtained by substituting R for E in 1.5 and 1.4 respectively. Suppose E is an R-module such that hdR E < oo. Then we have by [3, VII, Exer. 11] that hdR E=supm hdRm Em, where m runs through all maximal ideals in R. Since for each m we have that Rm is a local ring and hdRm Em< oo, it follows from [2, 1.7] that hdRm Em~codim Rm for all m. From this we conclude that hdR E~supm codimRm Rm=codim R. On the other hand if

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x1, • • • , x. is an R-sequence, then hdR R/(x1, · · · , x.) =s (see [3, VIII, 4.2]). Thus the last equality is established. COROLLARY

1. 7. If gl. dim R < 00 ' then

codimR R

= gl. dim R == dim R.

By [1, Theorem 1] we know that gl. dim R=sup8 hdR E where E runs through all finitely generated R-modules. It follows therefore from 1.6 that codimR R=gl. dim R. The second equality was proven in [2, 4.7]. PROPOSITION

pose that hdR S < have

1.8. Let S be the factor ring R/t, where t is an ideal in R. Sup00 and that E is an S-module such that hds E < 00. Then we

and thus codims S

~

codima R.

If we assume in addition that rank f>O, then we have hds E

< hda E
O, then f contains at least one nonzero divisor. Consequently we know that km;;-60 ([see 3, VII, Exercise 9]). Since Rm is a local ring and Sm=Rm/km, we have that hdRm Sm >O. The third and fourth inequalities now follow immediately from (*). ( 1) We have recently shown that if Eis an R-module which has a finite resolution and if f, the annihilator of E, is not zero, then f contains a nonzero divisor. Thus, in this proposition, if S has a free finite R-resolution (e.g. if Risa local ring), and f.eO, then it follows that rank f >0. (It should be observed that in [1, Appendix] one has to assume that f has a finite free resolution.)

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PROPOSITION 1.9. Let S be the factor ring R/t of R. Suppose that hdR S < and that E is an S-module such that hds E < oo • Then we have

hdR E

~

hdRS

+

hds E

~

oo

2hdR E.

Proof. The first inequality is given us by [3, XVI, Exercise 5]. We know that there are maximal ideals m and m' in R such that hds E = hdsO and all R-modules C. Also R[[X]] contains Ras a direct summand. Thus we have that hdR E=hdR[[XlJ R[[X]] ®RE for all R-modules E. Therefore if gl. dim R = oo, then gl. dim R [ [X]] = oo. Assume gl. dim R is finite. Let f: R[[X]]--tR be the ring epimorphism f("'E, 1: 0 a.Xi)=ao, Since Xis in the Jacobson radical of R[[X]], all the maximal ideals in R[[X]] contain X. Therefore there is a one-one correspondence between the maximal ideals in R[[X]] and those in R given by m--tf(m). Given a maximal ideal m in R[[X]], the map f induces a ring epimorphismfm: R[[X]]m--tR1(mJ. Since Ker fm= (X), we have hdR[[XJJm R1cm> = 1. Also the fact that gl. dim R i?; gl. dim Rf(m) [2, 1.6], means that gl. dim R1(mJ < oo. Combining this information with [2, 3.8] gives us that gl. dim R [ [X]]m = 1 +gl. dim R1(m)• Since gl. dim R[[X]] = supm gl. dim R[[X]]m and gl. dim R = SUP/Cm) gl. dim Rf(m) where m runs through all maximal ideals in R [[X]], we conclude that gl. dim R[[X]]=l+gl. dim R. 2. Finitistic homological dimension. The modules considered in this section need not be finitely generated. We define the weak homological dimension of the R-module E as follows: -l~w.hdRE~oo where w. hdR E 0). For each i we define the complex Rx 1 as follows : (Rx ! )p = R for p = 0, 1 and is O otherwise; d1 : (Rx)i (Rx)o is multiplication by x 1 • The complex Rx, 1, ••• ,, = Rx l @ · · · @ Rxs is called the Koszul complex generated over R by Xi, • • • , x,. If s = 0, we define Rx, 1, ... ,, = R. It should be noted that this complex is the exterior algebra over R generated by Xi, • • • , x,. If C is a complex, we define Cx, 1, ... ,, to be C@Rx, 1, .... , . Simple computation shows that if a= (xi,· · ·, x,), and C is an R-module, then H 0 (Cx,i, ... ,,) = C/aC, the module H,(Cx, 1, ... ,,) = 0: a (i.e., the set of all c in C such that ac = 0), and HiCx, 1, ... ,,) = 0 for p < 0 and p > s. If x is in R, we have the natural inclusion map of R--+ Rx of degree zero. This induces an exact sequence DEFINITION.

(1)

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where R is the factor complex R,;f R. If C is any complex, we have the exact sequence

o-c-c,,-c-o

(2)

obtained by tensoring the exact sequence (1) with C. Since (C)p and (d)p = dp-i we obtain

= Cp-i

PROPOSITION 1.1. If C is a complex, and xis in R, the sequence (2) yields the exact homology sequence

(3)

where a,,. is multiplication by (-l)"x. The only fact that must be proved is that a,,.= (-l)"x. But this can be demonstrated trivially by direct computation. PROOF.

1.2. If E is an R-module, and Xu • • • , x. are in R such that x1 is not a zero divisor for E/(xi, • • • , X1-1)E, then Hp(E,,, 1, ...,.) = 0 for all p =I= 0 (see [4, VIII, 4.3]). COROLLARY

We proceed by induction on s. If s = 0, there is nothing to prove. Suppose s = t + 1 where t ;;;;; 0. Then x1 is not a zero divisor for E/(xi, • • • , x. _1)E, i = 1, · · · , t. Thus by the induction hypothesis we have that Hp(E,,, 1 ,.,.,t) = 0 for all p > 0. By (1.1) we have an exact sequence PROOF.

• • • -H,,.+l(E,,,1, ... ,1+1)-H,.{E,,,1,... ,t)

-

Hn(E,,,1, .... , ) - Hn(E,,,1, ... ,t+1)- · · ·

from which it follows that Hp(E,,, 1, ••• ,t+i)

= 0 for p > 1, and

is- exact. Since Xt+i is not a zero divisor for E/(xi, · · ·, x,)E = Ho(E:i:,1 .... ,,), we have that Hi(E,,,1, ... ,t+1) = 0. 1.3. Let C be a complex and x be in R. If f: C,, C,, (8) C,, is the natural, inclusion map of degree zero, then there is a map g: C,, (8) c,,C,, such that gf: C,, G_,, is the identity. LEMMA

Since C,, = R,, (8) C, it suffices to prove the lemma in the case It that case we have the commutative diagram

PROOF.

C

= R.

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...

-

···...

0-----tO---t

R ~R-O

l

l

1

l

l

lu

11

d1

llfo

0 ---t R---tR+R- R---t 0

liuo

1

o-o-R~R-o where d1(ri, r 2) = x(r1 + r2), fi(r) = (r, 0), gi(ri, r 2) = r1 + r2. it follows that gtf, = identity for all i, hence gf = identity. ~

LEMMA 1.4. Let C and x be as above. Then xHn( C,,)

From this

= 0 f 0 where a(E) is the. annihilator of E. Then s + q = n where q is the largest integer, such that Hq(Ey, 1,... ,n) * 0. Further, if'b cc+ a(E), then Hg{Ey, 1, ... ,n) = (x1 , • • · , x,)E: c/(x1 , • • • , x,)E. PROOF. Since a(E)H(Ey, 1 , ••• ,n) = 0 = cH(Ey, 1, ... ,n), we have that b"'H(Ey, 1 , ••• ,n) = 0 and, therefore, every element of ois a zero divisor for each HiEy, 1,... , Now we proceed with the proof by induction on s. Ifs= 0, then every element of o, hence of c, is a zero divisor for E. Therefore Hn(Ey, 1,.... n) = (0): c * (0). Now suppose the theorem true for s = t ~ 0, and let s = t + 1. Clearly x 2 , • • • , x, is a maximal E/x1Esequence in o. Thus by our induction hypothesis, the theorem is true for E/x1E. From the exact sequence 11 ) .

X1

o - E - E - E / x1E-O we obtain the exact sequence X1

o - Hq+i((E/x1E\,1, ... ,n)- Hq{E11,1, ... ,n)-Hq(Ey:1,.,.,n) • Since x 1 is in o, we know that x 1 is a zero divisor for Hq(Ey, 1, ••• ,n), and therefore multiplication by x 1 is not a monomorphism. Hence

Hq+i((E/x1E)y,1, ... ,n) * 0,

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while Hg{(E/x 1E)y, 1,... ,n) = 0 for p > q + 1. Therefore we have that s + q = t + (q+ 1) = n. If ois contained inc+ a(E), then x 1Hg{Ev, 1.... ,n) = 0 so that Hq+i((E/x1E)v, 1,... ,n) ~ Hg(Ev, 1,... ,n). Again applying the induction hypothesis, we see that Hq(Ey:J, ... ,n)

~

Hq+i((E/x1E)y:J, ... ,n)

~

((x2, • • • , x,)E/x1E: c)/(x2 , • • • , x,)E/x1E ((xi, • · · , x,)E: c)/(x1, • • • , x,)E .

~

As an immediate result of 1.7, we have the following result due to Rees [14]. COROLLARY 1.8. Let b be an ideal in R and E an R-rrwdule. Then all maximal E-sequences in b have the same length. lf c is an ideal contained in b such that bis contained inc+ a(E), then (xi, · · · , xk)E: c/(xi, · · • , xk,)E ~ (x;, • • · , x~)E: c/(x;, · · • , x~)E where

Xi, • • • ,

xk and x~, • • • , x~ are any two maximal E-sequences in

o.

DEFINITION. Let Ebe an R-module and a an ideal in R (not necessarily a proper ideal). The a-codimension of E (notation : a-codim E) is the sup of the lengths of E-sequences in a (finite or + oo ). We observe that if aE =:/= E, then a-codim E < oo and all maximal Esequences in a have the same length. LEMMA 1.9. Let G be a complex over Rand let a be an ideal in R such that aHiG) = 0. Then there exists an integer h ;;=;: 0 such thatfor all integers n > h the natural map

is a rrwnomorphism.

PROOF. Let zp = Ker (Gp Gp-1) and BP= Im(Gp+I Gp). Since Gp is a noetherian R-module, we know by [12] that there is an h such that Zp n anGP = an-h(Zp n ahGP) for all n ;;=:_;;: h. From the fact that aHiG) = 0, we see that aZP C BP and thus zp n anGP = an-h(Zp n ahGP) C BP for all n > h. Since Ker (Hp(G)- HiG/anG)) = ((Zp n anGP) + Bp)/Bp, we have that Ker (HiG)-HP(G/anG)) = 0 for all n > h. PROPOSITION 1.10. Let E be an R-module, a an ideal in R such that aE =:/= E. If b is an ideal contained in a, then there exists an interger h ;;=;: 0 such that for all integers n > h, we have a-codim E ;;=;: a-codim E/onE. PROOF. Let a= (Y1, · • • , Yt). Then for all n we have (E/onE)y,1, ... ,t = (R/on 0 E) 0 Rv,1, ... ,t = R/on@ (E 0 Ry,1, .... e) = Ev,1 ... ,tfo''(Ev,1, ... , t). By

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(1.9), we know there is an integer h ~ 0, such that for all n > h, the natural maps HP(Ev, 1 , ••• , t ) - HP((E/b"E)v, 1 , ... ,t) are monomorphisms for all p. Let q be the highest integer, such that Hq(Ev, 1, ... ,t) 0. Then for all n > h we have that Hq((E/b"E)v, 1, ••• ,t) 0. Hence, by (1.7) it follows that a-codim E ~ a-codim E/b" E.

*

*

2. Graded and semi-local rings

We now assume that R, in addition to being noetherian, is also graded, i.e., R = Ro + R1 + · · · , R,R1 c Ri+J• and that all R-modules E are also graded R-modules, i.e., E = E 0 + E 1 + · · · , R,E1 c E;+J· All maps will be homogeneous maps, and all elements and ideals explicitly mentioned will be homogeneous, and contained in the "graded radical" m = m0 + R1 + Rz + · · · of R where m0 is the Jacobson radical of Ro. In particular, all E-sequences will consist of homogeneous elements in m. It should also be noted that, if C is a complex of which each component is a graded, noetherian R-module and' the boundary operator of which is homogeneous, then C,,, 1, ... ,, is again such a complex and, consequently, its homology modules are graded, noetherian R-modules. 2.1. If E = E~=o E, is a gra 0) that there is a system of parameters for R that is an E-sequence. This will clearly prove the theorem. Since (0) in E is unmixed and of rank zero, any element x in m which is not in any prime ideal of rank zero in R, is not a zero divisor for E, and can be extended to a system of parameters for R. Therefore, in the case dim R = 1, this completes the proof. Suppose the theorem established if dim R = s ~ 1, and let dim R =:= s + 1. Now, by the above argument, we know there is an element x in m which is not a zero divisor for E and such that dim R/(x) = s = dimE/xE. We will show that if Yu···, y.,. in Rare such that rank (0) in E/(x, yi, • • • , y,.)E in R/(x) is k, then it is unmixed in R/(x). Let v be a prime ideal in R containing x. Then x in Rp can be extended to a system of parameters, since x is not in any of the primes of rank zero in Rp. Thus rank v in R = 1 + rank v/(x) in R/(x). Next, observe that all the primes in R belonging to (0) in E/(x, Yi, • • • , y,.)E contain x. It is clear that if e e (E/(x, y1, • • · , y,.)E), then a(e) = v (a prime ideal) in R if and only if a(e) = p/(x) in R/(x). Hence p c R belongs to (0) in E/(x, y1, • · · , y,.)E if and only if v/(x) c R/(x) belongs to (0) in E/(x, Y1, · · • , y,.)E (see appendix A.4). Consequently we see that rank of (0) in E/(x, yi, • • • , yk)E considered as an R-module is k + 1. Thus all the primes in R belonging to (0) in E/(x, Yi, • • • , Y1c)E have rank k + 1, so that all the primes in R/(x) belonging to (0) in E/(x, y1, · · • , Y1c)E have rank k. Therefore, E/xE and R/(x) satisfy the induction hypothesis, and thus we can find a system of parameters x, ~. · · · , x,+ 1 for R which is an E-sequence. Hence codim E ~ dim R, and so by (5.3), codim E = dim R. PROPOSITION 5.7. Let Ebe an R-module (dimR > 0). Then the foll 0. (c) If Xi, · • • , Xa is a system of parameters for R, then Xi,···, Xa is an E-seqiience. (d) There exists a system of paramete1·s for R which is an E-sequence.

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PROOF. By (1.7), we know that (a) implies (b). By (2.8), (b) implies (c). Clearly (c) implies (d) and (d) implies (a). LEMMA 5.8. Let E be an R-rrwdule, and x1 , • • • , xd (d > 0) a system of parameters for R. ·11 eE(q) = L(E/qE) where q = (x1, · · • , xd), then (a) ((xu • • · , Xd-1)E) : xd = (x11 · • • , Xd-1)E, (b) eE((xf1, • • • , xM) = L(E/(xf1, • • • , x:a)E), (c) codim E > 0. PROOF.

eE(q)

By (4.3) we know that

= L(E/qE) -

L((x11

• • • ,

Xa- 1)E: xd/(x1, · • · , Xa-1)E)

- .[: e(q,._1E):x,./q1r,_1Ef_q/q1r,) . Since eE(q) = L(E/qE), we have that L(((x11 • · ·, Xa-1)E): Xa/(x 11 • • · , Xa- 1 )E) = 0 and that x11 • · • , Xa is a reducing system of parameters with respect to E. Hence ((x1, · • · , Xa-,1)E) : xd/(x1, · • · , xd-1)E = 0, and (x1, • · ·, Xd-1, x:) is a reducing system of parameters with respect to E for all n. Moreover, ((x11 • • • , xd-1)E) : x: = (x11 •·· • , Xa- 1)E for all n so that eE( (x1, · • · , x:)) = L(E/(x1, • • • , x:)E) for all n. Since d could have been replaced by any other index, we have (a) and (b). Now from (a) and (b) we know that codim E/(xf, • • • , x:)E > 0 for all n so that (0): m c (xr, • • ·, x:)E for all n. Since (xf, · • ·, x:):) (x1, • • ·, xa)", we have (0): m c n (x1, • · • , xa)"E = 0, i.e., codim E > 0. PROPOSITION 5.9. Let E be an R-module (dim R > 0), and q an ideal generated by a system of parameters for R. Then E is a Macaulay module if and only if eiq) = L(E/qE).

If Eis a Macaulay modtlle, then by (5.7b) we have HlE.,, 1 , ••• ,a) (i > 0) where x1, • · · , Xa is a system of parameters generating q. Since x(H(E.,,1, ... ,a)) = L(Ho(E.,, 1,... ,a)) = L(E/qE), we have eE(q) = L(E/qE). Now suppose eiq) = L(E/qE). If dim R = 1 then q = (x) and eE(q) = L(E/xE) - L((O) : x). Thus (0) : x = 0 so that xis not a zero divisor for E which shows that Eis Macaulay. Suppose dim R = s + 1 (s ~ 1) and that the proposition is true if dim R = s. By (5.8), we have that codim E > 0. Thus none of the primes belonging to (0) in E contain m and hence they do not contain q. Therefore we can find a system of parameters x 11 • • • , x.+ 1 which generate q and such that x1 is not a zero divisor for E. Then by (4.2) PROOF.

=0

eE(q)

= eE/.,1E(q/(x1)) = L(E/qE) = L((E/x1E)/(q/(x1))E/x1E) •

Applying the induction hypothesis, we obtain the result.

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We summarize and extend the preceding results in the following theorem. THEOREM 5.10. Let R be a semi-local ring (dim R > 0) with radical m, let E be an R-module, and let .Xh, • • • , Xa be a system of parameters!or R. Then the following statements are equivalent : (a) Eis a Macaulay module. (b) The system of parameters Xu • • • , Xa is an E-sequence. (c) If (Y1, • · · , Ya) = (x11 • • • , Xa) then codim E/(Yu • • • , Ya-1)E > 0. (d) If q = (x1, • • • , Xa), then e.s(q) = L(E/qE). Moreover, if Eis a Macaulay module, then any system of parameters for R satisfies conditions (b), (c), (d). 6 PROOF. That (a) implies (b) has been shown in (5.7). By (5.7) we also know that if x 11 • • • , Xa is an E-sequence, then Yi, • ·, , Ya is an Esequence so that (b) implies (c). Let y1, • · • , Ya be a reducing systE)m of parameters with respect to E which generates q. Then es(q)

= L(E/qE) -

L(((yi, • • • , Ya-1)E): Yaf(yi, • • • , Ya-a1)E) .

Since codimE/(yi,• • · ,Ya-1)E>O even when considered as an R/(Yu • • • ,Ya-1)module we have that 1

= dim Rl(y

11 • • • ,

Ya-1) ~ codim E/(Yi. • • • , Ya-1)E > 0 .

Therefore E/(y1, • • • , Ya-1)E is a Macaulay module over R/(yi, • · • , Ya-1). Since the image of Ya in Rf(Yi, • • • , Ya-i) is system of parameters, we see by (5.7) that Ya is not a zero divisor for E/(Yi. • • • , Ya-i)E. Therefore eB(q) = L(E/qE). Thus (c) implies {d). That (d) implies (a) follows from (5.9). The remainder of the theorem follows from (5.7) or (5.9). 7

a

6. Some generalizations

Throughout this section R is an arbitrary noetherian ring. If Xis a free R-module we denote by r(X) the number of elements in a base of X. It is well known that any two bases of a free module over a commutative ring have the same number of elements. As a consequence we have We point out here that if R is a local ring andµ a prime ideal of R, then dim R - codim R ~ dim Rµ - codim Rµ . Hence, if R is Macaulay, so is R:i, for every prime ideal µ. 7 If E is an R-module, and R is a local ring, one can introduce the notion of a system of parameters xi.,···, xa for E. In that case, x(H E,,,1, ... ,a)) = BB(X1, · · ·, x,) where the multiplicity is taken over R/(nE). Thus one can obtain analogues of the theorem in this section if one studies modules E such that dim E = codim E. One need only consider JJ; as a Macaulay R/n(E)-module. 6

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LEMMA 6.1. If R has the property that every direct summand, of a free module is free (e.g., a local ring) and, 0--+ X,.--+ • • ·--+ X1.Xo--+ 0 is an ex 0, then H;(E.,, 1, ...• ,) = 0 for all i > n, and so codim E < s - n. PROOF. If E is an R-module, we denote the completion of E by E. It is shown in [3] that if; = R ® E and is an exact functor of E. It follows that if E has finite length, then E has finite length and L(E) = L(E). It also follows that H;(E.,, 1 , ...• ,) = fi.i(E.,, 1 , .•. ,,) for all i. Thus it suffices to prove the theorem in the case that R is complete. Since we are in the equal characteristic case, R contains a coefficient field which we will denote also by k. Then there exists a ring homomorphism k[ [Xu···, X,] ] - R such that X 1 - x1 • Therefore it follows from [7, Theorem 8] that R is a finitely generated k[[X1 , • • • , X,J ]-

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module. Since X, operates on Ethe same as x" we have that H.CEx, 1, ...•• ) = Hi{E,,, 1, •••••). From the fact that X 1, • • • , X, is a k[ [Xx, • • • , X,] ]-sequence we have by (2.8) that k[ [X1, • • • , X,] ]x, 1, ..... is acyclic and therefore a free resolution of k. Thus H.CEx, 1, ...••) = Tor:ux1,···,x,1J(k, E) for all i. Now if an R-module has finite length, then it also has finite length as a k[ [Xx, • • • , X,] ]-module and these length are the same. Thus

E:_,. {-l)'L(Tort[[xl'... ,x,u (k, E)) = 0. Since k[ [Xx, • • • , X,]] is a regular local ring, E (considered as a k[ [X1 , • • • , X,] ]-module) has finite homological dimension. Thus by (6.4) we have that Tort[[xl' ... ,x,JJ (k, E) = 0 for all i ~ n. Therefore H,(E,,, 1, .•••• ) = 0 for all i ~ n. The rest of the theorem follows from (1. 7). The above result gives another proof of the fact that if R is a local ring with the same characteristic as its residue field and q is generated by a system of parameters in R, then eFJ(q) = L(E/qE) if and only if codim E = dim R. We do not know if the above result can be generalized to arbitrary local rings. .f\ppendix

Throughout this appendix R is a noetherian ring and all R-modules are finitely generated. We begin with a brief review of the noether primary decomposition theory for modules as can be found in [3]. Let E' be a proper submodule of the module E. The set of all x in R such that x"E c E' for some n > 0 is an ideal in R called the railical of E' (notation : t{E')). It is clear that t{E) = t{a(E/E')), where a(E/E') is considered a submodule of R. E' is said to be a primary submodule of E if every zero divisor of E/E' is in the radical of E'. · If E' is a primary submodule of E, then t{E') is a prime ideal in R which we call the prime ideal belonging to E'. Now every proper submodule E' of E can be represented as the intersection of primary submodules Qi, • · · , Q,. of E having the following properties: t{Q,) = t{Q1) implies that i = j, and Q1 t> Q,. Such a representation is called a reduced primary decomposition of E'. It is clear that Q1 n · · · n Q,. = E' is a reduced primary decomposition of E' if and only if nQ.JE' = (0) is a reduced primary decomposition of (0) in E/E'. Thus there is no loss of generality if we assume from now on that E' ={0). We summarize the properties of a reduced primary decomposition that we shall need in the following theorem.

n,,,.J

THEOREM A.1. Let Ebe a non-trivial R-mo s - n ''. 2. A non-spectral sequence proof of Theorem 3.6

THEOREM. Let E be a finitely generated module over the (arbitrary) noetherian ring R, and let b = (x 11 • • · , Xs) be an ideal in R such that E /bE has finite length. Then X(H(E,,, 1 , ... ,s)) = AsX(n; E, b) for n sufficiently large. PROOF. We let C = E,,, 1 , ... ,s and C = G(E; b);, 1 , ... ,s and we still see that H 1(C) and H;(C) have finite length for all i ~ 0. Since H;(C) = 395

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Ek Hik), there is an integer n 0 such that Hf,.,) = 0 for all k

~ n0

and all i.

Let c(I,) be the complex:

with the differentiation induced by that of C. Then CCk) is a subcomplex of C and we have the exact sequence 0-+ C(k) -+ C-+ Cf C(k)-+ 0 . The aim of the remaining argument is to show that HtCC(lc)) = 0 for all i ~ 0 and fork sufficiently large. If we show this, then we will have H 1(C) ,;;::; Ht{C 1cu,,)) so that L (- l)i£(Ht(C)) = E (- l) 1L(H;(C /C) 1 =Ctfbk+s-ict has finite length for each i, we will then have E (- 1)1 L(H;(C /C 0. Now if R is a regular local ring, then it is well known that the hd M codim M = dim R for any :finitely generated non-zero R-module 11!. Therefore d(M) =dim (M)- (dimR-hdllf) =hdM-rankM where rank M = rank(a(M)) by definition.

+

PROPOSITION 1. Let R be a regular local ring, M a non-zero finitely generated R-module and t an integer greater than or equal to 0. Then d ( M) > t if and only if there exist integers i > 0 and h > t such that Ext•R(M, R) # 0 and Ext•+hR(M, R) # 0.

Proof. We first recall a result of D. Rees (see [3]). If S is a noetherian ring, 11! a :finitely generated S-module and i the smallest integer such that Ext•s(llf, S) # O, then i is the length of the largest S-sequence contained in the a ( M). If S is a regular local ring, then we have that i=rank(a(ll{)) (see [1, Proposition 2. 7]). Now we return to the proof of the proposition. Suppose that d(llf) > t. Then we have that the hd 11! > rank(llf) + t. By the previous remark we

* Received April 10, 1961. 8

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9

A REMARK ON A PAPER.

know that if we set i=rank(M), then Exti(M,R) #0, But it is well known that the hd M is the largest integer k such that Extk(M, R) # 0 (see [2; VI, Exercise 9]). Therefore setting h = k - i > t we have the desired integers i and h. On the other hand suppose that ExtiR(M, R) # 0 and Exti+hR(M, R) # 0 with h > t. Then it follows from what has been observed before that the rank (M) < i and that the hd M > i h. Therefore

+

d(M) = hd M -ranklY[> h > t.

We say that a noetherian ring R is regular if Rp is a regular local ring for each prime ideal .IJ in R and dim R is finite. THEOREM 2. Let R be a regular ring, M a finitely generated, non-zero R-module. For each integer t > 0, there exists an ideal Ct ::J a(M) such that a prime ideal 1J ::J Ct if and only if d (Mp) > t.

Proof. Let a;=a(ExtiR(M,R)) for all i>O. It is clear that each q; ::J a(M) and only a finite number of a;#R since only a finite number of ExtiR(M,R) # 0. For each pair of integers i > 0 and h > 0 define Ii (i, h) = ( ai, ai+n). It should be observed that only a finite number of the

li(i,h) #Rand that each li(i,h) ::J a(M). For each integer t > 0 define is trivial.

Since

n Ii (i, h)

Ct

to be

n Ii (i, h).

The fact that

Ct ::J

a(M)

h?::.t

is really a finite intersection, we know that a

h?::.t

prime ideal 1J in R- contains Ct if and only if .IJ ::J Ii ( i, b) for some i and some h>t. But .IJ ::J li(i,h) if and only if Rp®EstiR(M,R) #0 and Rp ® Exti+hR(M, R) # 0. Since RP is R-flat, it follows from [2; VI, Exercise 11] thatRp®ExtiR(M,R) =ExtiRp(.ilI1i,Rp) for all j. Therefore applying Proposition 1, we have that .IJ ::J c1 if and only if d(Mp) > t. The following module theoretic version of Theorem O of [ 4] is an easy consequence of Theorem 2. COROLLARY 3. Let S be a noetherian ring which is a factor ring of a regular ring and let M be a finitely generated non-zero S-module. ;I'hen for each integer t > O, the set of all prime ideals 1J E Spec(S) such that d(Mp) > t forms a closed set.

Proof. Let f: R---,) S be the canonical ring epimorphism. Let 1J be a prime ideal in S and ~ the prime ideal f- 1 ( 1J) in R. Then it is well known that dim (M$) = dim (MlJ.) and codim (111,;ll) = codim (1111-1). Therefore we have that d(M$) = d(M1i),

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Now given an integer t > 0 we have by Theorem 2 that there exists an ideal Ct in R which contains a ( M), and therefore the Ker f, such that a prime ideal ~ in R contains Ct if and only if d(M\ll) > t. It therefore follows by standard arguments that a prime ideal .p in S contains f (Ct) if and only if d(Mp) > t. BRANDEIS UNIVERSITY.

REFERENCES.

[l] M. Auslander and D. Buchsbaum, "Homological dimension in local rings," Transactions of the American Mathematical Society, vol. 85 ( 1957), pp. 390-405. [2] H. Cartan and S. Eilenberg, Homological Algebra, Princeton, 1956. [3] D. Rees, "The grade of an ideal or module," Proceedings of the Cambridge Philosophical Society, vol. 52 ( 1957), pp. 28-42. [ 4] H. Hironaka, "A generalized theorem of Krull Seidenberg on parameterized algebras of finite type," American Journal of Mathematics, vol. 82 (1960), pp. 831-850.

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CHAPTER II

Ramification theory [13] [14] [15] [16] [18] [20] [21] [26] [88]

On ramification theory in noetherian rings (with D. Buchsbaum, 1959). Maximal orders (with 0. Goldman, 1960). The Brauer group of a commutative ring (with 0. Goldman, 1960). Modules over unramified regular local rings (1961). On the purity of the branch locus (1962). Ramification index and multiplicity (with D. S. Rim, 1963). Modules over unramified regular local rings (1962). Brauer groups of discrete valuation rings (with A. Brumer, 1968). Galois actions on rings and finite Galois coverings (with I. Reiten, S. 0. Smal¢, 1989).

This chapter contains the second part of Auslander's early work. Concepts like ramification and separability play a main role, and homological methods are applied in these settings. The focus is still on regular local rings, and now the modules over these rings become more important. These papers contain conjectures and basic theories which still are of importance today. Auslander's work on unramified regular local rings has been extremely influential. A basic question in [16, 21] is what it means for two R-modules A and B that A@ B is torsionfree, for an unramified regular local ring R. It is shown that it is necessary that A and B are torsionfree, and that if the n-fold tensor product of A is torsionfree, where n is the dimension of R, then A must be free. These results were obtained as consequences of the following: If A and Bare two modules and Tori(A, B) = 0 for some i, then Torj(A,B) = 0 for all j ~ i. Auslander asked if this last statement is true for arbitrary regular local rings, and this was proved later by S. Lichtenbaum. He also asked which modules A over a noetherian local ring are rigid, that is, the above property of Tor holds with respect to all finitely generated modules B. This work was motivated by results on the Koszul complex from [10]. A general theme at the end of the sixties and in the early seventies was to try to extend properties of modules over regular local rings to modules of finite projective dimension over arbitrary local rings. The above problem about Tor for a module A of finite projective dimension (and arbitrary B) was known as Auslander's Tor conjecture, or rigidity conjecture. An example that it fails was given by R. C. Heitmann in the early nineties. The problem remains open if it is assumed that also B has finite projective dimension. Another important problem which has its roots in [16], where it was shown to be a consequence of rigidity, is what was known as Auslander's zero divisor conjecture: If M is a finitely generated module of finite projective dimension over a local noetherian ring R and x in Risa nonzero divisor on M, then xis a nonzero divisor on R. This conjecture stimulated a lot of further research. It was proved by C. Peskine and L. Szpiro in the late sixties, for equicharacteristic local rings of characteristic p, a setting in which they at the same time proved other related conjectures. They also obtained information on the

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zero divisor conjecture in characteristic zero and later M. Hochster gave a new approach and succeeded in proving the conjecture for equicharacteristic local rings of characteristic zero. The zero divisor conjecture was finally proved by P. Roberts in mixed characteristic. Both [14] and [15] are foundational papers which even today are standard references for maximal orders and for Brauer groups. The starting point of [14] is the classical theory of orders over a complete valuation ring, as treated in "Algebren" by M. Deuring. The work splits into two parts: commutative algebras and central algebras. The paper [15] lays the foundations for a general theory of separable algebras over a commutative ring, and was, for example, the starting point for Grothendieck's work on Brauer groups. The purely commutative theory is treated in [13], where homological techniques are used, and for example the homological different is introduced. In [26] it is proved that the Brauer group of the function field in one variable over an algebraically closed field is zero, giving a converse of C. C. Tsen's theorem. In [19] a beautiful proof is given of the theorem on the purity of the branch locus, and [17] contains an axiomatic treatment of ramification index. The last paper, [88], included in this chapter was written much later, but in spirit it belongs here, following up some of the ideas from [14, 15]. It was motivated by the covering theory developed in the representation theory of algebras.

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ON RAMIFICATION THEORY IN NOETHERIAN RINGS.* 1 By M. AUSLANDER and D. A. BUCHSBAUM.

IntroduS 6 ®RT-'>S@RT-'>0 is exact and splits. But S 0 ®RT= (S ®R T)®T(S ®RT)= (S ®R T) 6 as a T-algebra. Hence S ®RT is (S ®R T) 6 -projective. Thus, by 2. 2, it follows that S ®T T is an unramified T-algebra. If, in addition, T is an unramified R-algebra, then it easily follows from the fact that S ®RT is an unramified T-algebra, that S ®RT is an unramified R-algebra.

3. The homological different and different. Throughout this section, R will be an integral domain with field of quotients K, L a finite-dimensional K-algebra, and S a subring of L containing R such that S ®R K = L. We shall denote HomR(S,R) by S* and Homx(L,K)by L*. We define the map r: S®RS-'>HomR(S*,S) to be r(x®y) (f) =xf(y) for fin S*. By [3; VI, 5. 2] if S is a projective and finitely generated Rmodule, then r is an isomorphism. In particular, a: L ®x L-'> Homx(L*, L) which is similarly defined, is an isomorphism. Since S ®R K = L, every element of S* is uniquely extendable to an element of L*. If S* generates all of L* over K, then we have a natural map p: HomR(S*,8)-'>Homx(L*,L). We therefore obtain the following diagram which is easily shown to be commutative:

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M. AUSLANDER AND D. ,\. BUCHSBAUM.

Hom 8 (S*,S) -HomL(L*,L)

l

T

l

p

S®Rs-HomR(S*,S)-HomK(L*,L)-L®KL

¥1

L.

If f is in HomL ( L*, L), it can be seen by standard techniques of linear algebra that cf>'u- 1 (f) = f (Tr), where Tr: L ""-7 K is the trace map.

An explicit description of p and of cf>'cr- 1 p can be given as follows: let v1, · · ·, vn be elements of S* which form a basis for L* over K. Then for fin HomR(S*,S) we have p(f) defined by p(f) (vi) =f(vi). If we let v1 , · · · , Vn be the basis of L over K dual to v1, · · ·, vn and set vi= ui/r 0 with ui in S, r 0 in R (using the fact that S®RK =L), we can easily see that every element s of S can be written s = :S rivi with r; in R and that Tr ( r 0 S) is contained in R. Thus if we let T': S ""-7 R be the restriction of r 0 Tr to S, we have for fin Homs(S*,S) that 4>'u- 1 p(f) = (l/r0 )f(T'). If Tr(S) were contained in R (e.g. if S were integral over R) then cf>'cr- 1 p(f) = f (Tr') where Tr' is the restriction of Tr to S. We now define the complementary module, Cis;R, and the different, ':ns;R, as follows: ·Cis;R={x in L/Tr(xS) is contained in R} ':ns;R = {x in L/xCis;R is contained in S}.

From the above remarks, we can se,e that cf>'cr- 1 p (Hom 8 (S*, S)) is contained in ':ns;R- For suppose x is in Cis;R and f is in Hom 8 (S*, S). Then x[ (1/r0 )f(T')] = x(p(f) (Tr)) =p(f) (Trox), where Tr ox: L""-7 K 1s defined by Tr ox (y) = Tr ( xy) for y in L. Since Tr ox restricted to S maps S into R, p(f) (Trox) is in S. Therefore x·4>'u- 1 p(f) is in S for all x in fiis/R• Now let us make HomR ( S*, S) an 8 6 -module by defining ( x 0 y )f (g) =X·f(goy) for x, yins, fin HomR(S*,S), gin S*. Then Tis an se_ homomorphism. Furthermore, Hom 8 (S*, S) is equal to the set of all f in HomR(S*, S) such that jf = 0. Thus, since "h, = annihilator of j in se, we have r('n) is contained in Hom 8 (S*,S). We can go even a little further. Let

9t = Ker(S®RS--?L®KL = S ®R S®RK), and let

a in 8° be

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the annihilator of

j /'>t (9t

is obviously contained in

j).

RAMIFICATION THEORY IN NOETHERIAN RINGS.

757

Since ')t is the torsion submodule of S 0 ( as an R-module), and since S is torsion-free as an R-module, T(w) =0 for all win '>t. Thus, if a is in a, we have a J is contained in ')t and O = T(a = T(a) Thus by the remark above T(a) is in Hom 8 (S*,S) which implies that T(a) is contained in Hom 8 (S*, S). Combining all the above remarks, and resorting to the commutative diagram above, we have shown

J)

J.

PROPOSITION 3. 1. Let R be an integral domain with field of quotients K, L a finite-dimensional K-algebra, and S a subring of L containing R such that S®RK=L and S*®RK=L*. Then if a is the annihilator of J!'>t, cf, (a) is contained in '1) 81 R. In particular, /?Js;R is contained in 'l)S!R·

3. 2. Let R, K, L and S be as in 3. 1 and in addition assume L is a field. Then L is a separable extension of K if and only if /?Js;R =I= 0 (i.e. ·91, is not contained in PROPOSITION

J).

Proof. If L is not separable, the trace map is identically zero, so that for all fin Hom 8 (S*,S), p(f) (Tr) =0. Since /?Js;R=cf:,('n) =cf:,'u- 1 pT('n) and T('n) is contained in Hom8 (S*,S), we have /?Js;R=O. If L is separable, the exact sequence

o-

J'-L®KL_L_O

splits. However, L = S ®R K, L ®KL= 8° ®R K, and J' = J®R K. Thus the annihilator 'n' of J' is 'l1 ®RK (by 2. 6), and 'n' is not contained in J'· Therefore 91, is not contained in J and /?Js;R =I= 0. PROPOSITION 3. 3. Let R be an integrally closed integral domain with field of quotients K, L a separable K-algebra, and S a subring of L containing R which is integral over R and such that S ®R K = L. Then Homs (S*, S) is isomorphic to '1) 81 R under the map f - f(Tr). If Sis a projective, finitely generated R-module, then /?Js;R = 'l)S/R·

Proof. Since L is a separable K-algebra, the map L - L* given by x - Tr O x is an isomorphism. Under this isomorphism, CEs;R is mapped onto S*. Thus Homs(S*,S):::::::Hom 8 (CEs;R,S) and this latter module is isomorphic to 'Ils;R, The composite isomorphism is the one described above, namely f-f(Tr) (Tr is here. restricted to S). Furthermore, by standard arguments, it is easy to see that S* ®R K = L* so that all our previous discussion (including commutative diagram) holds. Moreover, if S is finitely generated over R and R-projective, then T is an isomorphism, ')t = 0, and T('n) =Homs(S*,S). Thus in this case /?Js;R='Ils;R,

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Throughout the rest of this section we shall denote the R-module HornR ( E, R) by E*, where E is an arbitrary R-module. 3. 4. Let R be a noetherian domain such that every proper principal ideal is unmixed (e.g. R is integrally closed). Let A be a finitely PROPOSITION

generated R-module such that A= A**, and Ba finitely generated torsion-free R-module containing A such that B / A is a non-trivial torsion module. Then a(B/A) is unmixed of rank one (a(B/A) is the annihilator of B/A in R). 3 Proof. Let b1 , • · · , bt be generators of B. Then a(B/A) = {r in R/rb, is in A for i=l,· · ·,t}=nai, where ai={r in R/rbi is in A}. Thus if each ai is unmixed of rank one, then so is a(B/A). We may therefore suppose that B=A +Rb, b not in A, and B/A is a torsion module. Observe next that if h is in A*, then h can be extended to a map h: A®RK~K. Moreover, if x in A®RK is such th2-t h(x) =0 for all h in A*, then x=O. As a result, we have that r is in a(B/A) if and only if rh ( b) = h ( rb) is in R for all h in A*. For if r is in a ( B /A), then rb is in A so that h(rb) =h(rb) is in R. Conversely, if h(rb) is in R for all h in A*, then the map A*~ R given by h ~ h ( rb) is an element of A**= A so that h(rb) =h(a 0 ) for some a0 in A and all h in A*. Thus h(rb-a0 ) =0 for all h in A* and by the above remarks, rb=a 0 i.e. r is in a(B/A). Since A* is finitely generated, say by h 1 , · • · , hn, we see that a ( B /A) ={r in R/hi(rb) is in R for i=l,· · ·,n}. Let hi(b) =u;/v. Then r is in a(B/A) if and only if r is inn (v): ui i.e. a(B/A) = n (v): ui. Now by assumption on R, ( v) is an unmixed ideal of rank one so that ( v) : 11i is also. Thus a(B/A) is unmixed of rank one. CoROLLAHY 3. 5. Let R be an integrally closed noetherian integral domain with field of quotients K, L a separable field extension of K, and S the integral closure of R in L. Then if 'lJs;R # S, 'lJs;R must be of rank one.

This follows from 3. 4 by letting R = A = S and B = that 'lJs;R=a(Cis;R/S).

fis;R

and observing

PROPOSITION 3. 6. With R, K, L and S as above, we have 'lJs;R = S if and only if every minimal prime ideal of S is unramified.

Proof. By 2. 7, every minimal prime ideal of S is unramified if and only if rank S')s;R is greater than one. Since 'lJs;R contains S;Js;R, we have that if every minimal prime is unramified, then 'lJs;R has rank greater than one. Thus, by 3. 5, 'lJs;R = S. 3

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We would like to thank 0. Goldman for suggesting this proposition to us.

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759

Conversely, let SDs;R = S, $ be a minimal prime of S, and l) = $ n R. Then ,l.) is a minimal prime of R and Rp is a regular local ring of dimension one. Since Sp is a finitely generated, torsion-free Rp-module, Sp is Rp-free. We have by 3. 3 that "5sp/Rp = SDsp/Rp· But it is easily seen that "5sp/Rp = "5s;R © 8 Sil and SDsp/Rp = SDs;B ©s Sil. Therefore "5sll/Bp = Sil and thus$ it unramified. COROLLARY 3. 7. Let R, K, S and L be as in 3. 5 and assume further that S is R-projective. Then S is unramified if and only if every minimal prime ideal of S is unramified.

Proof. Since S is R-projective and finitely generated, we have "5s;R = SDs;R· Furthermore, B is unramified if and only if "5s;R = S (by 2. 5): Thus 3. 6 implies 3. 7. The following theorem has also been obtained independently by Serre. 'fHEOREM 3. 8. Let R be a regular local ring of dimension less than or equal to two, and let K, S, and L be as above. Then S is unramified if and o,nly if ,every minimal prime ideal of S is unramified.

Proof. We need only show that S is R-projective, for then we may apply 3. 7. However, by [2, 2.10] it is sufficient to show that Sis a Macaulay ring ( S is semi-local). Since dim S < 2, and since S is integrally closed, we have that every principal ideal of S is unmixed and that every ideal of rank two that is generated by two elements is unmixed. Hence S is Macaulay and therefore R-projective.

4. On being free. Throughout this section, R will be an integrally closed local domain with maximal ideal m and field of quotients K. The residue class field R/m will be denoted by F. PROPOSITION 4.1. Let L be a separable K-algebra and S an integral extension of R in L such that S ®BK= Land is unramified. Then SDs;R = S.

Proof. Since S is unramified, "5s;R = S. tained in SDs;R, we have SDs;R = S.

However, since "5s;B is con-

LEMMA 4. 2. Let S be an R-algebra containing R which is torsion-free 01Jer R and such that

a)

B is finitely generated over R,

b)

there is an element t in B/mS such that B/mS = F [ t].

If 0 in B is

S'UCh

that () ~ t under the natural map S ~s;ms, then

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8=R[0] and {1,0,· · n= [S/mS: F]).

·,en-1 } is a free basis for S over R (where

Proof. Since S/mS = F[ t], we have that {1, t, · · ·, tn-1 } is a basis for S /mS over F. Since R is a local ring, this implies that {1, 0, · · · , 0n-1 } generates S over R and is a minimal generating set for S over R. Also, since R is integrally closed, we know that the minimal polynomial f in K[ x] for 0 has its coefficients in R. We will show that degree f = n, hence that {1, 0, · · ·, 0n-1 } is a basis for K[ 6] over K. This will imply that {l, 0, · · ·, 0n-1 } is a free basis for S over R. Let Jin F[X] be the corresponding polynomial off. Then l(t) =0

so that deg J> n.

On the other hand, since

en

n-1

is in S, we have

ri in R. Therefore deg f < n and we are done.

en=~ ri0i, i=O

PROPOSITION 4. 3. Let S be a torsion-free R-algebra containing R which is unramified and finitely generated over R. Then S is a free R-module on n generators (where n= [S/mS: F]). Moreover, if L is the full ring of quotients of S, [ L: K] = n and S is integrally closed in L.

Proof. Let us assume first that F is an infinite field. Then it is well known [ 4] that S/mS = F [ t]. Thus, by 4. 2, we have that S is R-free with basis {1, 0, . .. 'en-l }.

Now suppose F is finite. Let X be an indeterminate, and consider the local domain R[X]m•=R', where m* is the extension of m to R[X]. The maximal ideal m' of R' is R'm* and R'/m'=F'=F(X). R' is integrally closed since R [ X] is and rings of quotients of integrally closed rings are integrally closed. We now have R' contained in S', where S' = S [ X] m•, S' a finitely generated torsion-free R'-module, and [S'/m'S': F'] =n. If we show that S' is unramified over R', and use the fact that F' is infinite, we will have that S' is a free R'-module on n generators. This will imply that S is R-free on n generators for if s1 , · • · , Sm is a minimal generating set for S over R, it is also one for S' over R', hence a free basis for S' over R' and therefore a free basis for S over R, with m = n. Since S is unramified, S is Se-projective so that the exact sequence 0-')

j -') se-') S-') 0

splits.

Therefore, the exact sequence

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splits. Since S' = S[X] @Rcxi R' = S @R (R[X] @Rcxi R') = S@RR' and ( S') e= S' ®R'S'= se @RR', we see that S' is ( S') e-projective and therefore S' is unrami:fied over R'. This then shows that S is R-free on n generators. The rest of the proposition follows from standard arguments [4]. THEOREM 4. 4. Let R be a noetherian integral domain ( not necessarily local) with field of quotients K, and L a separable K-algebra. If S is an unramified integral extension of R in L such that S@RK =L, then S is R-projective.

Proof. By standard localization arguments, this result follows from 4. 3. 4. 5. Let S be a local ring containing R which is unramified and finitely generated over R. Then S is R-free. PROPOSITION

Proof. Since S is unrami:fied, S@RK is a separable K-algebra. Let S' = Im (S ~ S @R K). Then S' is torsion-free and :finitely generated over R, and we have the exact sequence

(E)

where t(S) is the R-torsion submodule of S, and is :finitely generated over R. If we can show that t(S)/mt(S) = 0, we will have t(S) = 0. Therefore S ;:::; S' and so S will be torsion-free, hence free (by 4. 3). Since S is unrami:fied, S /mS is a field and therefore the map S /mS ~ S'/mS', being an epimorphism, must be an isomorphism. Moreover, S/mS is a separable extension of F so that S' /mS' is also. Renee S' is unramified (by 2. 5 it is sufficient to test ramification of S' by its unique maximal ideal) and by 4. 3 is free over R. Therefore the sequence (E) splits over R so that the sequence

o~ t(S)/mt(S) ~ S/mS ~ S'/mS' ~ 0 is exact. Since S/mS;::::;S'/mS', we have t(S)/mt(S) =0, hence t(S) =0 and S;:::; S'. PROPOSITION 4. 6. Let R be analytically normal ( i. e. R, the completion of R, is also an integrally closed local domain) and let S be a ring containing R which is unramified and finitely generated over R. Then S is R-free.

Proof. Sm is the radical of S, so that S contains R and "S is a :finitely S,. ( direct sum), where each S, generated .R-module. Now S = S 1 is a local ring which is an R-algebra. In fact, each Si contains a copy of ii. It can also be easily seen that each Si is an unrami:fied .R-algebra ( since S is unrami:fied over R). Therefore, by 4. 5, each Si is free over R, which implies

+ · · ·+

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that 8 is R-free. Since S is finitely generated over R, 8 being R-free implies that S is R-free [1, Theorem 3. 2]. Since an integrally closed geometric local ring is analytically normal, we see that an unramified integral, finitely generated extension ring S of an integrally closed geometric local ring R is R-free. Hence if R is a normal affine ring ( not necessarily local), S is R-projective. LEMMA 4. 7. Let R be a noetherian ring (not necessarily an integrally closed local domain) and let S be a ring containing R which is finitely generated as an R-module. If S is R-projective, then R is a direct summand of S as an R-module.

Proof.

From the exact sequence

it is clearly sufficient to prove that S/R is R-projective. ideal of R. Then the exact sequence

Let m be a maximal

splits since Sm is a projective (hence free) Rm-module, and 1 is part of a free basis for Sm over Rm. Therefore (S/R)m is free for every maximal ideal m and so by [3, VII, Exercise 11] S/R is R-projective. PROPOSITION 4. 8. Let R C S C T be noetherian rings with T a finitely generated projective unramified R-algebra. Then S is unramified over R if and only if T is S-projective.

Proof. diagram

Suppose T is S-projective.

Then we have the commutative

s®Rs~T®RT

l

l

S---T.

Since Tis S-projective, T ®RT is S ®RS-projective, and S is a direct summand of T as an S- hence also as an S ®RS-module. But T is T ®RT-projective since T is an unramified R-algebra. Hence S is S 0R S-projective. By [3, IX, Proposition 2. 3] (letting A= r = S, A= S, :S = R, B = T) we have that since S is S ®RS-projective (being unramified) and T is Rprojective, then T is S-projective.

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5. Ramification and homology. 5. 1. Let S and T be R-algebras such that S is R-projective

PROPOSITION

and Sis Se-projective. If Eis an S®RT-module, then hds® 8 TE=hdpE and thus gl. dim S ®RT< gl. dim T. Further, if S is R-free, then gl. dim S ®RT= gl. dim T. Proof. By [3; XVI, sec. 4) we have the spectral sequence HP(S, Ex~T(E, 0))

~

Extns® 8 T(E, 0),

p

where O is an S ®RT-module. Since S is Se-projective, this spectral sequence collapses to H 0 (S,Extnp(E,O)) ;:::::Extns®BT(E,O). From the fact that 0 is an arbitrary S ®RT-module, it follows that hd 8 ® 8 TE < hdT E. But considering S ®RT as a T-algebra, ,we have by [3; XVI, Exercise 5] that hdpE < hdTS ®RT+ hds® 8 TE. Since S is R-projective, it follows that S ®RT is T-projective. Therefore hdpS ®RT= 0 and thus hdT E < hd8 ® 8 p E, which gives the desired equality. From the fact that hds® 8 pE=hdTE for arbitrary S®RT-modules E it follows that gl. dim S ®RT< gl. dim T. Further, if S is R-free and A is a T-module, then hdpA =hdpS®RA since S®RE is a direct sum of copies of A. But hdpS®RA =hds®RTS®RA by the previous arguments. Thus hdTA = hdTS ®RA, which means that gl. dim T < gI. dim S ®RT.

Let S and T be noether-ian R-algebras such that S is imramified and R-projective and ;} is a finitely generated ideal in S 0 • If T i.s a regular ring of finite (Krull) dimension and S ®RT is noetherian, then S®RT is a regular ring of (Krull) dimension less than or equal to that of T. COROLLARY

5. 2.

Further, if S is R-free (e.g. R a local ring) then the dimensions of S ®RT and T are equal. Proof. By 2. 5 we have that S is S 6 -projective. Since T is a regular ring of finite dimension, we have by [1, Corollary 4. 8] that gl. dim T < oo. Therefore it follows from 5. 1 that gl. dim S ®RT< gI. dim T, which means that S ®RT is a regular ring ef. dimension less than or equal to that of T. The rest of the corollary follows from the fact that if gl. dim 8 ©RT = gl. dim T, then the dimensions of S ®RT and T are equal. PROPOSITION

5. 3. Let S be an R-algebra, where R is an integral domain

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with field of quotients K such that Se is noetherian and O < [S ® R K: K] Then hd 8 @R 8 S = 0 or oo.

< oo.

Proof. First we observe that hd 8 ,S > hdL,L where L = 8 ® K and Le= (S®RK) ®K (8®RK). Since [L: K] (E* ® E)*--'> Hom(E, E**),

where Hom(E, E)-Hom(E, E) ** is the usual imbedding of a torsion-free module into its second dual, and u* is the map defined in Proposition 4.1. Since by hypothesis Hom(E, E) = Hom(E, E)**, while the other maps are isomorphisms by Proposition 4.1, it follows that y;* is an isomorphism. This completes the proof of Proposition 4.6, as well as the proof of the implication (c)=?(b) of Theorem 4.4. We are now left with the implication (b)=>(a) of Theorem 4.4. Before presenting the details of its proof, we first review some definitions and results from the theory of local rings. For a fuller discussion, the reader is referred to [2]. If Risa local ring and Eis a nonzero finitely generated R-module, then the codimension of E (denoted by codim E) is defined to be the length of the longest sequence xi, X2, · · · , x, of nonunits of R such that Xi is not a zero divisor in E/(x1, · · · , X;-i)E. Then, codim Eis at most equal to dim Rand is thus always finite. If R is a regular local ring, then the relation codim E +hd E=dim R always holds. Thus a module E over a regular local ring R is free if, and only if, codim E=dim R. PROPOSITION 4. 7. Let R be a local ring, A and B finitely generated R-modules such that Hom(A, B)-,c-0. If codim B "?;,i,jori= 1, 2, then codim Hom(A, B) "?;,i.

Proof. Suppose that the nonunit x is not a zero divisor in B. From the exact sequence o-B-"'B-B/xB--'>0 we deduce the exact sequence 0---'>Hom(A, B)--'>"'Hom(A, B)--'>Hom(A, B/xB), which shows that xis not a zero divisor in Hom(A, B) and therefore that codim Hom(A, B) '?;, 1. If y is not a zero divisor in B/xB_, then y is not a zero divisor in Hom(A, B/xB). But Hom(A, B)/x Hom(A, B) is a nonzero submodule of Hom(A, B/xB). Thus, if codim B '?;, 2, then also codim Hom(A, B) '?;, 2, which proves the proposition. COROLLARY.

If R is a regular local ring of dimension at most two, then

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Hom(E, R) is projective for every finitely generated R-module E( 3). Thus, if E=E**, then Eis projective.

Proof. Since codim R = dim R for a regular local ring R, the previous proposition shows that codim Hom(E, R) = dim R if dim R ~ 2 and Hom(E, R) ,;,!O. Thus Hom(E, R) is always projective. In view of this corollary, we see that Theorem 4.4 has been established for dim R~2. LEMMA 4.8. Let R be a regular local ring of dimension at least three and let B be a finitely generated R-module of codimension at least two. If A is a finitely generated R-module such that Hom(A, B) is projective and Ext 1 (A, B) ,;,!O, then codim Ext 1 (A, B) >O.

Proof. Let 0---tB---t"'B---tB/xB---tO be exact with x a nonunit in R. Then we have the exact sequence 0---tHom(A, B)---t""Hom(A, B)---tHom(A, B/xB) ---tExt 1 (A, B)---t"'Ext 1 (A, B). Suppose that codim Ext1(A, B) =0. Then C=ker(Ext 1 (A, B)---t"'Ext 1 (A, B)) is not zero and also has codimension zero. Thus we have an exact sequence 0---t Hom(A, B)/xHom(A, B) ---t Hom(A, B/xB) ---t C---t 0. Since hdR C=dim R~3, and hdR(Hom(A, B)/x Hom(A, B)) ~ 1, it follows that hdRHom(A, B/xB) =dim R. But this means that Hom(A, B/xB) ,;,!O and has codimension 0. This is impossible, since codim B ~ 2. (See Proposition 4.7.) As an immediate consequence of this lemma we have: PROPOSITION 4.9. If R is a regular local ring, E a finitely generated Rmodule such that E=E** and Hom(E, E) is R-free, then Ext 1 (E, E) =0.

Proof. If dim R ~ 2, the proposition is trivially true since by the corollary to Proposition 4. 7, E is projective. Suppose that dim R = k > 3 and the proposition is true for rings of dimension less than k. Let lJ be a prime ideal of R other than the maximal ideal m. Then Rp is a regular local ring of dimension less thank, while E©Rp is equal to its second dual (with respect to Rµ) and HomR-p(E©Rµ, E©Rp) =Rp©HomR(E, E) is Rp-projective. Thus, Rp©Ext1(E, E) = Ext1p(E©Rµ, E©Rµ) is 0. Therefore m 1 Ext 1 (E, E) =0 for some integer t, and consequently were Ext 1 (E, E) ,;,!O, we would have c.odim Ext 1 (E, E) = 0. In view of Lemma 4.8, we must have Ext 1 (E, E) = 0. PROPOSITION 4.10. Let R be a local ring, Ea finitely generated R-module of finite homological dimension n. If A is any nonzero finitely generated R-module, then Extjt{E, A) ,;,!O. ( 3 ) It can be shown that this property actually characterizes local rings which are reg-ular and of dimension not greater than two,

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Proof. Let o-xn-Xn-1- · · · -xo-E-o be a minimal resolution of E in which each Xi is R-free and Im (X H1-X i) C mX i for i = 0, 1, · · · , n -1. Then we have the exact sequence Hom(Xn-1, A) -

Hom(Xn, A) - Extn (E, A) - 0.

If Extn(E, A) =0, then every fin Hom(Xn, A) is the restriction to Xn of some gEHom(Xn-i, A). Since Xn is contained in mXn-1, this would mean that f(Xn) CmA for every fin Hom(Xn, A). Since Xn is a nonzero free R-module, we find that A = mA. This is impossible for a nonzero finitely generated Rmodule. Thus Extn(E, A) ~O. Suppose now that Risa regular local ring of dimension three. If E=E** and HomR(E, E) is R-free, we have several facts. Because codim R=3, it follows from Proposition 4. 7 that codim E '?;, 2, or that hd E ~ 1. At the same time, because of Proposition 4.9 we know that Ext1(E, E) = 0. Therefore, because of Proposition 4.10 we cannot have hd E= 1, so that hd E must be 0, and hence Eis free. This shows that the implication (b)=>(a) of Theorem 4.4 is proved in case dim R ~ 3. We now complete the proof of the implication (b)=>(a) of Theorem 4.4 by induction on dim R. We suppose that dim R=k'?;,4, and that the result is valid for rings of dimension less thank. Assuming that E=E** and that HomR(E, E) is R-free, we shall prove that E is R-free. Let x be in m, not in m 2 • Then xis not a zero divisor in E = E**. From the exact sequence o-E-"'E-E/xE-o, we deduce the exact sequence o-Hom(E, E)-"Hom(E, E)-Hom(E, E/xE)-Ext 1 (E, E). By Proposition 4.9, Ext 1 (E, E) =0, so that HomR(E, E/xE)"-'Hom(E, E)/x Hom(E, E). If we put R=R/xR, we have therefore that HomR(E, E/xE) is a free R.-module. It is easily seen that HomR(E, E/xE) = Hom;/E, E), with E=E/xE. Hence, by Lemma 4.5 we know that Eis a torsion-free R.-module because Hom~(E, E) is a torsion-free R.-module being R.-free. Denoting HomR(E, R.) by E*, we know by Proposition 4.1 that HomR(E, E)**"-'HomR(E*, E*) since R is regular and therefore integrally closed. Thus HomR(E*, E*) is R.-free and E* = (E*) **. Therefore, by the induction hypothesis, we have E* is R.-free. By a standard change of rings argument, we have Extf(E, R.)"-'Extk(E, R.) for i '?;, 0. Since dim R '?;, 4, we have dim R '?;, 3, so that by Lemma 4.8, codim Ext1(E, R.) > 0 if Extl(E, R) ~0. Thus, if Ext1(E, R.) ~O, then codim Ext1(E, R) > 0. From the exact sequence o-R-"R_R_O we deduce the exact sequence X

0-

E* - E* -

HomR (E, R) - Ext 1 (E, R) X - Ext 1 (E, R) -

-

Ext 1 (E, R).

Applying the induction hypothesis, if pis any prime ideal in Rother than m, then Rp0Ext 1 (E, R) = Ext1p(E0Rp, Rp) =0. Therefore Ext 1(E, R) has finite

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length. Assume for the moment that Ext 1 (E, R) ~O. Then, Ext 1 (E, R)/xExt 1 (E, R) has finite nonzero length and is contained in Ext 1 (E, R), so that codim Ext 1 (E, R) = 0. This contradicts the conclusion arrived at above, so that we must have Ext 1 (E, R) = 0. Consequently, the sequence o-E*-"'E* -Hom;lE, R)-o is exact. Since Hom.il(E, R) =E*/xE* is R-free, we conclude that E* is R-free. Since E = E**, we find that E is R-free and the proof of Theorem 4.4 is complete. It should be observed that Theorem 4.4 gives another proof that a regular local ring is a unique factorization domain. For, if pis a minimal prime ideal in R, then Hom(p, p) = R since R is integrally closed, while p** = (p-1) - 1 = p because p is minimal. Hence by Theorem 4.4 p is a free R-module and is therefore principal, which means that R is a unique factorization domain. It should also be observed that Theorems 4.3 and 4.4 show that if R is a regular domain of Krull dimension at most two, then every maximal order in a full matrix algebra is a projective R-module. However, this statement is not true in higher dimension. For, suppose that R has dimension greater than two; then there is a nonprojective finitely generated R-module E such that E=E**. Then A= HomR(E,E) is a maximal order in a full matrix algebra, and is not projective because of Theorem 4.4. It is an open question whether every central simple algebra over K, the quotient field of a regular local ring of dimension at least three, contains a maximal order which is a projective R-module. Appendix. For the convenience of the reader, we present in this appendix some facts about projective modules and their endomorphism rings which are used in the body of the paper. Since a number of these results have already appeared in the literature (see for example Morita [7] and Curtis [5]) we shall omit the details of some of the proofs. If r is a ring and Eis a left r-module, there are a number of additional rings and modules associated with this pair. For example, Homr(E, E) is the endomorphism ring of E. Denoting Homr(E, E) by D, we will consider E a left D-module through the operation wx=w(x), where wED, xEE. The group Homr(E, r) is both a right r and right D-module by means of the following operations. IfJEHomr(E, r), 'YEr, xEE, then we define {!'Y}(x)=f(x)'Y; and for wED, {Jw}(x)=f(wx). The operations of r and Q commute on Homr(E, r). We define the mapµ: Homr(E, r)©rE-Uas follows: µ(f©x)(y)=f(y)x. Because the operations of Q and r commute both on E and on Homr(E, r), the tensor product Homr(E, r) © rE is a two-sided D-module. It is clear that µ is a two-sided D-mapping. PROPOSITION

144

A, 1. A necessary and sufficient condition that E be a finitely

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generated projective I'-module is that the map µ: Homr(E, I') ©rE-Q be an epimorphism. Ifµ is an epimorphism, then µ is an isomorphism.

Proof. The proposition is a reformulation of the statement that E is a finitely generated projective I'-module, if and only if, there are elements xi, · · · , xnEE and Ji, · · · , fnEHomr(E, I') such that L,f;(y)x;=y, all yEE. (See [4, p.132].) The pairing of Homr(E, I') with E tor defines a map r: Homr(E, r) ©oE - r by r(f©x) =f(x), which is easily seen to be a two-sided I'-map. The image of r is thus a two-sided ideal in r which will be called the trace ideal of E in rand will be denoted by ;;rr(E). It will shortly be shown that ;;rr(E) is connected with the properties of E as an 0-module. The fact that r and Q commute on E gives rise to a ring homomorphism i: r-Homo(E, E), whose kernel a(E) is the annihilator of E in r. Finally we have a map y;: Homr(E, r)-Homo(E, 0) as follows: y;(j)(x) =µ(f©x). After these preparatory remarks, we have: A.2. (a) The diagram:

THEOREM

1/1 © 1 Homr (E, r) © o E - - - Homo (E, Q) © o E

rl

r

--i

! µ' Homo (E, E)

is commutative. (b) a(E);;rr(E) =0. If a(E)+;;rr(E)=I', then: (c) µ' is an epimorphism, so that E is a finitely generated projective 0module. (d) i is an epimorphism and splits as a left I'-map so that a(E) is a left direct summand of I'. If ::tr(E) =I', then: (e) a(E) = 0. (f) y; is a monomorphism. (g) All the maps in the diagram are isomorphisms. (h) Homr(E, r) is a finitely generated projective SJ-module.

Proof. (a) and (b). The statements follow directly from the definitions of the various objects. (c) and (d). Since E is a left Homo(E, E)-module with operations of Hom 0 (E, E) and Q commuting on E, it follows that Homr(E, r) © oE is also a left Hom 0 (E, E)-module. The operations of r on Homr(E, I') ©oE given by the mapping i is the same as that given by r on E. The hypothesis a(E) +::tr(E) =I' implies that there is an element zEHomr(E, r) ©oE with

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ir(z) = 1. Define a: Hom 0 (E, E)-Homr(E, r) ®oE by a(x) =xz. It is easily checked ttat a is a left r-map. Since ira=identity, we have that i is an epimorphism and splits as a left r-map. We also have that µ'(,t,®l)a=identity, so thatµ' is an epimorphism and consequently Eis a finitely generated projective U-module. (e) and (f). If '.tr(E) =r, then (b) shows that a(E) =0. From this it follows readily that ,t, is a monomorphism. (g). Because E is a projective U-module, the fact that ,t, is a monomorphism assures that ,t,®1 is also a monomorphism. Combining this with the fact that i andµ' are isomorphisms and that Tis an epimorphism, it follows that the maps i, r, µ', ,t, ® 1 are isomorphisms. (h) We shall apply Proposition A.1 to Homr(E, r) as a (right) U-module. To do so, we first define a map {3:E-Homo(Homr(E, r), Q) by means of: {3(x)(f) =µ(f®x). This leads to the diagram:

l®tl Homr (E, r) ® o E - - Homr (E, r) ® o Homo (Homr (E, r), 1"2)

d

r

-i'

!µ" Homo (Homr (E, r), Homr (E, r))

in which the map i' is defined analogously to i. One checks readily that the diagram commutes. Since T is an epimorphism, the image of i'r contains 1, so that 1 is in the image ofµ". It follows by Proposition A.1 that Homr(E,r) is a finitely generated projective n-module. PROPOSITION A.3. If E is a finitely generated projective r -module and n = Homr(E, E), then '.tr(E)E = E and '.to(E) = n. If in addition r is commutative, then a(E)+'.tr(E)=r, and consequently, E is a finitely generated projective U-module.

Proof. The statements follow from the definition of the-trace ideal combined with the remark immediately following the statement of Proposition A.1. REMARK. The part of Proposition A.3 concerned with the case in which r is commutative is not in general true without that assumption. That is, there exist finitely generated faithful projective r-modules E with '.tr(E) ;:er and E not n-projective. · If E is a left r-module and n=Homr(E, E), we denote by :m(r) the category of right r-modules and by :m(U) the category of left U-modules. If ME:m(n) define ff(M) =Homr(E, r)®oM. Then, ff(M) is a right r-module because Homr(E, r) is a right r-module and the operations of r .and n commute on Homr(E, r). Thus, ff(M)E:m(r). If M, M'Emt(U) and cf>: M-4M' is an n-mapping, we define ff(): ff(M)-ff(M') by ff()= 1 ®. These definitions serve to show that ff is a covariant additive functor from :m(U) to :m(r). Similarly, if NE:m(r), we set g(N)=N®rE. Here g(N) is a

146

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MAXIMAL ORDERS

Z3

left Q-module because E is a left Q-module, and the operations of Q and r commute on E. If cf>: N---,,,N' is a I'-map, then S(cf>): g(N)---,,,g(N') is defined to be cf>©l. With these definitions, g is a covariant additive functor from grr(r) to grr(Q). PROPOSITION A.4. 5' and g are covariant additive functors; for MEgrr(Q), we have that g5'(M)=(Homr(E, I')©rE)©oM, while for NEgrr(r), we have 5'g(N)=N©r(Homr(E, I')©oE). If Eis a finitely generated projective I'module, then g is exact and g5' is the identity. If ~r(E) = r, then 5' is exact and 5'9 is the identity.

Proof. The statements result from combining Proposition A.1 and Theorem A.2 with the definitions of 5' and S. There are several special cases of Proposition A.4 of sufficient importance to be stated separately: THEOREM A.5. If ~r(E) = r then N © rE = 0 for a right I'-module N implies N = 0. If E is a finitely generated projective I'-module and ~r(E) = r then 5' and g establish a one-to-one correspondence between the two-sided ideals in Q and the two-sided ideals in r which preserves inclusion and preserves projectivity (left Q and right r).

Proof. The proof follows directly from Proposition A.4. It might be of interest to have an explicit description of the correspondence between the ideals of Q and of r. If l is a two-sided ideal of Q, we form 5'(l©oE) = Homr(E, r) ©ol©oE. The monomorphism Q---,,,{---,,,Q induces a monomorphism Homr(E, r) ©ol©oE---,,,Homr(E, r) ©oE=I', so that Homr(E, r) © ol ® oE is naturally isomorphic to a two-sided ideal of r. Suppose now that E is a finitely generated projective I'-module and ~r(E) =I'. If Q=Homr(E, E), then we know by Theorem A.2 and Proposition A.3 that Eis symmetric in relation tor and Q, that is, I'=Homo(E, E), and E is a finitely generated projective Q-module with ~o(E) = Q. Also, Theorem A.2 shows that the map y;: HomrlE, r)---,,,Homo(E, Q) is a monomorphism. Using the symmetry, we have also a map y;': Homo(E, Q) ---,,,Homr(E, I') and there is no difficulty in verifying that y; and y;' are inverses · of each other. PROPOSITION A.6. If E is a finitely generated projective I'-module with ~r(E) =I' then if;: Homr(E, r)---,,,Homo(E, Q) is an isomorphism. If Mis a left S1-module, then the mapping p: Hom 0 (E, M) ©rE---,,,M defined by p(f)(x) = f(x) is an isomorphism.

Proof. The first part of the statement follows from the remarks immediately above. For the mapping p we have the following situation. By Proposition A.4, we have M = g5'(M) =5'(M) © rE and 5'(M) = Homr(E, r) © oM '..:::'.Homo(E, Q) © oM. The latter isomorphism is y; © 1. Now it is easily checked

MAURJCE AUSLANDER

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24

M. AlTSLANDER AND 0. GOLDMAN

that Homn(E, Q) © uM ,..,_, M when E is a finitely generated projective Qmodule. Composing these various isomorphisms shows that p is also an isomorphism. REFERENCES

1. (1955) 2. Amer. 3.

M. Auslander, On the dimensions of modules and algebras (III), Nagoya Math. J. vol. 9 pp. 67-77. M. Auslander and D. A. Buchsbaum, Homological dimension in local rings, Trans. Math. Soc. vol. 85 (1957) pp. 390--405. - - - , Ramification theory in noetherian rings, Amer. J. Math. vol. 81 (1959) pp. 749-

764. 4. H. Cartan and S. Eilenberg, Homological algebra, Princeton, 1956. 5. C. W. Curtis, On commuting rings of endomorphisms, Canad. J. Math. vol. 8 (1956) pp. 271-291. 6. M. Deuring, Algebren, Berlin, Springer, 1935. 7. K. Morita, Duality for modules, Science reports of the Tokyo Kyoiku Daigaku, See. A. vol. 6 no. 150 (1958). BRANDEIS UNIVERSITY, WALTHAM, MASSACHUSETTS

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THE BRAUER GROUP OF A COMMUTATIVE RING BY

MAURICE AUSLANDER AND OSCAR GOLDMAN(1 )

Introduction. This paper contains the foundations of a general theory of separable algebras over arbitrary commutative rings. Of the various equivalent conditions for separability in the classical theory of algebras over a field, there is one which is most suitable for generalization; we say that an algebra A over a commutative ring R is separable if A is a projective module over its enveloping algebra A•=A®~ 0 • The basic properties of separable algebras are developed in the first three sections. The results obtained show that a considerable portion of the classical theory is preserved in our generalization. For example, it is proved that separability is maintained under tensor products as well as under the formation of factor rings. Furthermore, an R-algebra A is separable over R if, and only if, A is separable over its center C and C is separable over R. This fact shows that the study of separability can be split into two parts: commutative algebras and central algebras. The purely commutative situation has been studied to some extent by Auslander and Buchsbaum in [1 }. The present investigation is largely concerned with central algebras. In the classical case, an algebra which is separable over a field K, and has K for its center, is simple. One cannot expect this if the center is not a field; however, if A is central separable over R, then the two-sided ideals of A are all generated by ideals of R. In the fourth section we consider a different aspect of the subject, one which is more analogou·s to ramification theory. If A is an algebra over a ring R, the homological different !i;)(A/R) is an ideal in the center C of A which essentially describes the circumstances under which A®RR-p is separable over R-p, when l) is a prime ideal of R. (Suitable finiteness conditions must be imposed in the statement of these theorems.) The general question of ramification in noncommutative algebras is only touched on in the present paper; various arithmetic applications will be treated in another publication. In the classical theory of central simple algebras, the full matrix algebras have a special significance. The proper analogue of the full matrix algebra in the present context is the endomorphism ring HomR(E, E) of a finitely generated projective R-module E. It is easy to show for such a module E that HomR(E, E) is separable Qver R, and central if Eis faithful. By analogy with the classical theory, we introduce an equivalence relation between central separable algebras over a ring R, under which the equivalence classes form Received by the editors March 16, 1960. '.This work was carried out under a grant from the National Science Foundation.

( 1)

367 MAURICE AUSLANDER

149

368

MAURICE AUSLANDER AND OSCAR GOLDMAN

[December

an abelian group, the Brauer group

A® A0

!

! ct>o

! cf>1

! cf>

S-'>

A-->

A

in which (a) = 1, or what amounts to the same thing, if Sis S -projective. The reader is referred to [2] for the basic properties of Sj(S/R) and its connections with ramification theory. Now suppose f E HomR(S, R). Then we define a(f) : Se ---+ S by a(f) (x ® y) = f(x)y. It is clear that a(f) is an S-homomorphism if S ® S is considered an S-module by means of the operation s(x ® y) = x ® sy but not in general an Se-homomorphism. However a simple calculation shows that a(f) I Ct is an S -homomorphism. For an element x; ® Yi is in Ct if and only if (x ® 1)(:I:x; ® y;) = (1 ® x)(Lx; ® Yi) for all x in S. Therefore if x; ® y; Ea, then we have that 6

L

6

L

(x ® y)(Lx; ® y;) =(I® y)((x ® l)(Lx; ® y;)) (1 ® y)((l ® x)Lxi ® y;) = LXi ® yixy.

Therefore a(f)((x ® y)(Lxi ® y,))

= a(f)(Lxi

® y;xy)

= (x

= XYLf(xi)Yi

® y)(a(f)(L Xi® Yi)).

Thus we have a homomorphism a: HomR(S, R) ---+ Hom 8 .( K is the only element of HomK(L, K) with the property that a(t) = ¢KI a @R K (see Proposition 4.1) and t(S) c R (because R is integrally closed), it follows that tis the only element in HomR(S, R) such that a(t) = ¢ I a. It is clear that the image of a(t) : S @ S ---'> S definedbyt(x ® y) = t(x)yist(S)S. Nowbydefinition.p(S/R) is¢( 0 and that A is an R-module. Then

(a) A is a free R-nwdule if and only if the tensor product of A with itself n times is torsion free.

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232

(b) If HomR(A,R)=!=O, then A is a free module if and only if A®RA®R Hom(A,R) is torsion free. (c) If A satisfies the foUowing conditions: (i) hd A=hd Hom(A,R) (ii) A® Hom(A, R) is torsion free (iii) Al> is Rp-free for all non-maximal prime ideals .\) in R, then the hd A = (n - I) /2 or O. In particul,ar if n is even then A is free.

It should be observed that each of the results given in Theorem 4 are the best possible. For if n;;;,,, 3, then there exist modules A such that the tensor product of A with itself (n -1) times is torsion free but A is not free. As far as (c) is concerned, in case n > I and odd, there always exist modules A satisfying the hypothesis of (c) with the hd A =(n-1)/2; which shows that the result (b) is the best possible as well as indicating that there seem to be some differences in the module theory of unramified regular local ring depending on whether the dimension of the ring is odd or even. We now devote the rest of this report to a discussion of some of the information concerning the Ext functor which can be derived from the above discussion of the Tor functor. If A and B are R-modules, there exists a natural homomorphism A be the automorphism given by conjugation with the matrix [~ ~] . 1 1 1 Then G = (g) is a group of order 2. One can show that A is a A[G]-generator, but not a projective A[G]-module, by using Corollary 1.5 and Proposition 1. 7. We now give an example where (A, G) is pregalois, showing that AG is not always a twosided AG-summand of A. EXAMPLE 1.11. Let k be a field and let A be the subring of the lower 4 by 4 matrix ring over k described by

i;

A-I G ~ }~b,,,dek} Now, conjugation by the matrix

0 0 001 1

0 0 1 0 1 0 is obviously of order two, and acts as an automorphism 0) and Pis a projective resolution of A (see [2] for further details). Similarly if Xis a complex in -A 1 -'>-A2-'>- 0 is exact. c) There exists an exact sequence O -'>-Ao -'>-A 1 -'>-A 2 -'>- 0 in CC such that O-'>- (-,Ao)-'>- (-, A1)-'>- (-, A2)-'>- F-'>- 0 is exact.

" which is left exact, d} Exti(F, G) = 0 for i = O,I and any Gin CC i.e. if X-'>- Y -'>-0 is exact in CC, then O-'>- G(Z)-'>- G(Y)-'>- G(X) is exact.

-z

" which is left exact. e) (F, G) = 0 for any G in CC f) (F, (-, A)) = 0 for all A in CC. Proof. a) b). Suppose O -'>-Ao-'>- A1 -'>-A 2 is exact such that 0-'>- (-, A 0 )-'>- (-, A1)-'>- (-, A2)-'>- F-'>- 0 is exact. Then applying v to this exact sequence, we obtain the exact sequence O -'>-Ao -'>-A1 -'>-A 2 -'>-V(F)-'>- 0. Thus v(F) = 0 if and only if O -'>-Ao -'>-A1 -'>-A2-'>- 0 is exact, which gives us our desired result. b) => c) Trivial. c) => d) Follows immediately from Lemma 3.1. d) => e) and e) => f) are trivial. f) => b) Suppose O -'>-Ao -'>-Ai -'>-A 2 is exact such that O-'>- (-, A 0 ) -'>- (-, A1)-'>- (-, A2) -'>-F -'>-0 is exact. Then it follows from Lemma 3.1, that (F, (-,A))= Ker((A 2, A)-'>- (A 1, A)). Thus (F, (-,A))= 0 for all A if and only if O-'>- (A 2, A)-'>- (A 1 , A) is exact for all A, i.e. if and only if A1 -'>-A 2 -'>- 0 is exact. Thus f) => b), which finishes the proof. We now recall the definition of the v: ~-'>- CC. For each Fin~ we chose a fixed exact sequence O -'>-Ao-'>- A 1 -'>-A2 and a fixed map (-,Ai)-'>- F such that O-'>- (-, A 0 )-'>- (-, A1)-'>- (-, A2) -'>-F-'>- 0 is exact, subject only to the condition that if F = (-, X), then we choose Ao= 0 = A1 and A 2 = X with the map(-, A 2)-'>- F the identity. Then v(F) = A3 = Coker(A1-'>- A 2). Now if we let B = Coker(A 0 -'>-A1) = Ker(A2 -'>-A3), then from the exact sequence O -'>-A 0 -'>-A1-'>- B-'>- 0 and O-'>- B-'>-A2 -'>-A3 -'>-0 we deduce the following commutative diagram with exact rows and columns 0

t

0-'>- (-, A 0 )-'>- (-, A1)-'>- (-, B) -'>-F0 -'>- 0 (3.3)

II

II

+

0-'>- (-, A 0 )-'>- (-, A1)-'>- (-, A2)-'>- F-'>- 0

t

(-,A3)

t t

F1 0

MAURICE AUSLANDER

297

204

M.

AUSLANDER

where F 0 and F1 are defined by the exact sequences and are obviously in i 0 • From this it follows that there is a unique sequence of maps

F 0 -'>-F-'>-(-, As)-'>-F1 which makes the above diagram commutative. Further elementary diagram chasing shows that O-'>- F 0 -'>- F-'>- (-, As) -'>-F1-'>- 0 is exact. Since As= v(F), this exact sequence can be rewritten as 0----'>-Fo -'>-F -'>- (-, v(F))-'>- F1-'>- 0. It is not difficult to see that: a) the map F-'>- (-, v(F)) is functorial in F; b) F 0 and F 1 are functorial in F; c) the exact sequence 0-'>-F0 -'>-F-'>- (-, v(F)) -'>-F1 -'>-0 is functorial in F. Let G be a left exact functor in '2. Since Fi is in ~o for i = 0 and 1, we know by proposition 3.2, that Exti (Fi, G) = 0 for j = 0 and 1 and i = 0 and 1. From this it follows by easy direct computations that the map ((-, v(F)), G)-'>- (F, G) is an isomorphism. Thus given any map F-'>- G there exists one and only one map (-, v(F))-'>- G which makes the diagram F-'>-(-, v(F))

t

II

F-'>-G

commutative. As an application of the above observations, we obtain: Proposition 3.4. Let F be in and let O -'>- F' -'>- F -'>- G -'>- F" -'>- 0 be an exact sequence in i with G left exact and F' and F" in io. Then there exists one and only one map of exact sequences

i

0-'>-F0 -'>-F-'>-(-, v(F))-'>-F1 -'>-0

II

t

t

0----'>-F'-'>-F-'>- G

-'>-

t

F" -'>-0

and this is an isomorphism. Proof. The existence and uniqueness of the map has already been shown. A similar argument shows that since F' and F" are in i 0 , then the induced map (G, G') -'>- (F, G') is an isomorphism whenever G' is left exact. Thus we obtain that there is one and only one map of exact sequences 0----'>-F'-'>-F-'>- G

t

II

-'>-

t

F"-'>-0

t

0----'>-Fo -'>-F-'>-(-, v(F))-'>-F1 -'>-0

and this map is easily shown to be the inverse af the previous map. We now briefly recall the definition of the category o, the quotient category of i by i 0 (see [l] for details). The objects of ~/~ 0 are the same as the objects in '2. Given F and Gin we define i/~o(F, G) =

i/i

i

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205

Coherent Functors

Jim i(F', G/G') when F' and G' run through all subobjects of F and G y""J;•

respectively such that F/F' and G' are in ~ 0 • Then it is well known that i/io is an abelian category and that the canonical functor i-+ i/