Noncommutative Rings PDF
Preview Noncommutative Rings
Mathematical Sciences Research Institute 24 Publications Editors S.S. Chern 1. Kaplansky C.C. Moore 1.M. Singer Mathematical Sciences Research Institute Publications Volume 1 Freed and Uhlenbeck: Instantons and Four-Manifolds Second Edition Volume 2 Chern (ed.): Seminar on Nonlinear Partial Differential Equations Volume 3 Lepowsky, Mandelstam, and Singer (eds.): Vertex Operators in Mathematics and Physics Volume 4 Kac (ed.): Infmite Dimensional Groups with Applications Volume 5 Blackadar: K-Theory for Operator Algebras Volume 6 Moore (ed.): Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics Volume 7 Chorin and Majda (eds.): Wave Motion: Theory, Modelling, and Computation Volume 8 Gersten (ed.): Essays in Group Theory Volume 9 Moore and Schochet: Global Analysis on Foliated Spaces Volume 10 Drasin, Earle, Gehring, Kra, and Marden (eds.): Holomorphic Functions and Moduli I Volume 11 Drasin, Earle, Gehring, Kra, and Marden (eds.): Holomorphic Functions and Moduli II Volume 12 Ni, Peletier, and Serrin (eds.): Nonlinear Diffusion Equations and their Equilibrium States I Volume 13 Ni, Peletier, and Serrin (eds.): Nonlinear Diffusion Equations and their Equilibrium States II Volume 14 Goodman, de la Harpe, and Jones: Coxeter Graphs and Towers of Algebras Volume 15 Hochster, Huneke, and Sally (eds.): Commutative Algebra Volume 16 Ihara, Ribet, and Serre (eds.): Galois Groups over Q Volume 17 Concus, Finn, Hoffman (eds): Geometric Analysis and Computer Graphics Volume 18 Bryant, Chern, Gardner, Goldschmidt, Griffiths: Exterior Differential Systems Volume 19 Alperin (ed.): Arboreal Group Therapy Volume 20 Dazord and Weinstein (eds.): Symplectic Geometry, Groupoids, and Integrable Systems Volume 21 Moschovakis.(ed.): Logic from Computer Science Volume 22 Ratiu (ed.): The Geometry of Hamiltonian Systems Volume 23 Baumslag and Miller (eds.): Algorithms and Classification in Combinatorial Group Theory Volume 24 Montgomery and Small (eds.): Noncommutative Rings S. Montgomery L. Small Editors Noncommutative Rings Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Susan Montgomery Lance Small Department of Mathematics Department of Mathematics University of Southern California University of California at San Diego Los Angeles, CA 90089 La Jolla, CA 92093 USA USA Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720 USA The Mathematical Sciences Research Institute wishes to acknowledge support by the National Science Foundation. Mathematics Subject Classifications: 16-00, 16A33, 16A27 Library of Congress Cataloging-in-Publication Data Noncommutative rings / Susan Montgomery, Lance Small, editors. p. cm. - (Mathematical Sciences ResearcH Institute publications; 24) Lectures delivered at the Microprogram on Noncommutative Rings, held at MSRI, July 10-21,1989. Includes bibliographical references. ISBN-13:978-1-4613-9738-0 I. Noncommutative rings-Congresses. I. Montgomery, Susan. II. Small, Lance W., 1941- . III. Mathematical Sciences Research Institute (Berkeley, Calif.) IV. Microprogram on Noncommutative Rings (1989 : Mathematical Sciences Research Institute) V. Series. QA251.4.N66 1991 512'.4-dc20 91-32480 Printed on acid-free paper. © 1992 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1992 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue" New York, NY, 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Photocomposed copy prepared by the Mathematical Sciences Research Institute using TEX. 987654321 ISBN-13:978-1-4613-9738-0 e-ISBN-13 :978-1-4613-9736-6 DOl: 10.1007/978-1-4613-9736-6 This volume is dedicated to the memory of Robert B. Warfield Preface This volume collects some of the survey lectures delivered at the Micro program on Noncommutative Rings held at MSRI, July 10-21, 1989. While the program was concerned with recent advances in ring theory, it also had as an important component lectures on related areas of mathematics where ring theory might be expected to have an impact. Thus, there are lectures of S. P. Smith on quantum groups and Marc Ri effel on algebraic aspects of quantum field theory. Martin Lorenz and Don ald Passman consider in their lectures various aspects of crossed products: homological and K-theoretic to group actions. Kenneth Brown presents the "modern" theory of Noetherian rings and localization. These contributions as well as the others not presented here show that ring theory remains a vigorous and useful area. The planning and organization of the program were done by the under signed and the late Robert Warfield. His illness prevented his attendance at the meeting. It is to him we dedicate this volume. The organizers wish to extend their thanks to Irving Kaplansky, Director of MSRI, and the staff for all of their efforts in making this conference such a success. Susan Montgomery Lance Small vii NONCOMMUTATIVE RINGS TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . .V ll . . K. A. Brown THE REPRESENTATION THEORY OF NOETHERIAN RINGS 1 A. Joseph SOME RING THEORETIC TECHNIQUES AND OPEN PROBLEMS IN ENVELOPING ALGEBRAS. . . . . . . . . . . . . 2.7 M. Lorenz CROSSED PRODUCTS: CHARACTERS, CYCLIC HOMOLOGY, AND GROTHENDIECK GROUPS . . . . . . . . . . . . . . . . . . .6 9 . . D. S. Passman PRIME IDEALS IN CROSSED PRODUCTS 99 M. A. Rieffel CATEGORY THEORY AND QUANTUM FIELD THEORY 115 S. P. Smith QUANTUM GROUPS: AN INTRODUCTION AND SURVEY FOR RING THEORISTS . . . . . . . . . . . . . . . . . . . .. . . . 1. 3.1 . ix The Representation Theory of Noetherian Rings KENNETH A. BROWN Dedicated to my friend and colleague Bob Warfield Introduction The purpose of this article is to give a survey of part of what is known about the structure of the indecomposable injective modules over a Noethe rian ring R, to indicate how this structure depends on the nature of certain bimodules within the ring which afford "links" between the prime ideals of R, and to suggest some directions for future research in these areas. The subject began with Matlis' analysis of the commutative case in 1958 [31]. During the sixties and beyond many ring theorists studied the Artin Rees property in various classes of noncommutative Noetherian rings, at least partly in an attempt to extend Matlis' theory to such rings. The im possibility of this endeavour was already becoming apparent by the early seventies, notably through Jategaonkar's papers [20, 21] which first high lighted the connections between bimodules, localization at prime ideals and the structure of injective modules. His insights stimulated much subsequent research, focussed largely on the search for a useful and workable theory of localization "near" a prime ideal. But inevitably- in view of the connec tions mentioned above-this work also shed light on representation theory, and in recent years the representation theoretic point of view has assumed a more central position. Since this entire paper is in effect no more than an extended introduc tion, (in which those proofs included are designed to be "skippable"), I won't discuss the contents in detail here. In outline, they are as follows. Noetherian bimodules and their connection with representation theory via Jategaonkar's "Main Lemma" form the subject of §l. In §2 I discuss various open questions, and a few results, concerning those properties which can be passed from one side of a Noetherian bimodule to the other. With §3 we begin the study of indecomposable injective modules over a Noetherian ring. The emphasis in this section is on rings with the right strong sec ond layer condition (Definition 5); in §4 I describe recent wprk which aims to encompass all Noetherian rings within the framework developed in §3. Finally, we specialise to Noetherian PI-rings in §5 . 2 KA. . Brown Throughout, I have tried to highlight some of the many open questions in this area; in fact I've listed 15 of these at appropriate points in the text. Most of the results in this paper have appeared elsewhere, the chief ex ception being Theorem 10, a result of K.R. Goodearl on bimodules over simple rings, which answers a question I raised in my lectures at the MSRI conference. I have tried to include full references to the literature, espe cially where I have not included the full proof of a result. For the reader who wishes to take the subject further, the relevant books are [18], [25], and [32]. With such a reader in mind I have included references to some papers which, though related to the subject at hand, are not mentioned in the text; but I am afraid the list of references should not be regarded as exhaustive in this respect. And the text itself is certainly very far from exhaustive. In particular, to escape drowning in a sea of technicalities, I have not discussed what extra information is available for specific classes ofrings, with the (brief) exception of PI-rings in §5. The most noteworthy absentee in this respect is the class of enveloping algebras of solvable Lie algebras, for which detailed results can be found in [11, 12]. ACKNOWLEDGEMENTS: Some of the work for this paper was done while I was visiting the University of Washington (Seattle) in the summer of 1989, with financial support from the N.S.F. and the Carnegie Trust for the Universities of Scotland. §1. Bimodules DEFINITION 1: (i) A bond B between prime Noetherian rings Sand T is a non-zero S-T-bimodule which is finitely generated and torsion-free both as a left S-module and as a right T-module. (ii) Let R be a Noetherian ring with prime ideals P and Q. There is an internal bond, or ideal link, from Q to P if there exist ideals A and B of R with A c Band B/A a torsion free left R/Q-module and a torsion-free right R/ P-module. In this case we write Q ~ P. (iii) If Q ~ P as in (ii) with QP ~ A, we say that Q is (right second layer) linked to P, or that there is a (second layer) link from Q to P, and write Q P. f'Vf'V-t REMARK 2: (i) If Q P then it is easy to see that there is an internal f'Vf'V-t bond of the form Q n P / A'; thus second layer links are internal bonds occurring 'as high as possible' in R. Representation Theory of Noetherian Rings 3 (ii) The example R = (~ ~), Q = (~~), P = (~ ~) shows that the definitions l(ii) are not symmetric. (iii) It is a consequence of the definitions that Q :::::J P if Q rvrv~ P. The converse is false: In the ring R = (~ ~ ~ ), Q = (~~ ~) :::::J (~ ~ ~) = ooe ooe 000 P via the ideal link (~ ~ ~ ), but P n Q = Q P . Nevertheless, for rings with 00 0 the second layer condition (see Definition 5 below) rvrv~ and:::::J generate the same equivalence relation on Spec(R). (For this, see [25, Theorem 8.2.4], or consult Theorem 26 below.) This is no longer true for rings without the s.l.c. [39, Theorem 4.1j. (iv) The importance of internal bonds for representation theory was first pointed out in [20j. Second layer links and the notation Q rvrv~ P were introduced (for fully bounded Noetherian rings) in [33j. The notation Q :::::J P was first used in [28j. The first connections between the ideal structure of a Noetherian ring and its representation theory are achieved through what has come to be called Jategaonkar's Main Lemma. To state it we need one more set of definitions. r r DEFINITION 3: Let be a subset of Spec(R). An ideal I of R is semiprime if and only if I is a finite intersection of primes in r. An R module M is r -semiprime if and only if the annihilator of every finitely gen erated submodule of M is r -semiprime. If r is finite and S = n{ P: PEr}, we shall also say that M is S -semiprime. If r = {S}, we say that M is S-prime. LEMMA 4. (i) (Jategaonkar [25, 6.1.2, 6.1.3]) Let P and Q be prime ideals of R. Let o (1) ---+ U ---+ M ---+ V ---+ 0 be an exact sequence of R-modules, with U, M and V all finitely generated, non':'zero and uniform. Suppose that (a)U is P-prime, (,6)V is Q-prime and (-r) for all submodules M' of M not contained in U, Ann(M') = Ann(M) = A. Then either (a) Q rvrv~ P via the second layer link Q n PIA, or (b) A = Q ~ P. If (a) holds and U is (R/P)-torsion-free, (so that U is a right ideal of R/ P), then V is (R/Q)-torsion-free, (so that V is a right ideal of R/Q). If (b) holds, then V is (R/Q)-torsion.
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