Flat Covers of Modules - Lecture Notes in Mathematics 1634 by Jinzhong Xu

Lecture Notes in Mathematics 1634 Editors: A. Takens, Groningen www.com S rin er BPerling Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo www.com Jinzhong Xu Flat Covers of Modules ~ Springer www.com Author Jinzhong Xu University of Kentucky Department of Mathematics Lexington, Kentucky, 40506-0027 USA Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - C I P - E i n h e i t s a u f n a h m e Xu, Jinzhong: Flat covers of modules / J i n z h o n g Xu. - Berlin ; Heidelberg ; New York ; B a r c c l o n a ; Budapest ; H o n g Kong ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996 (Lecture notes in mathenaatics ; 1634) ISBN 3-540-61640-3 NE: GT Mathematics Subject Classification (1991): 13C 11, 13C 15, 13E05, 13D05, 13H10, 16A50, 16A52, 16A62, 18A30, 18G05, 18G15, 18G25, ISSN 0075-8434 ISBN 3-540-61640-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks.

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Typesetting: Camera-ready TEX output by the author SPIN: 10479845 46/3142-543210 - Printed on acid-free paper www.com Dedicated to Professor Edgar E. Enochs My Mother Guizhen Zhu and My Father Youcheng Xu www.com Acknowledgement Professor Enochs introduced me to the subject of fiat covers five years ago. He also encouraged me to write this monograph two years ago when we made substantial progress in the theory of flat covers. I sincerely thank him for his constant encourage- ment and adivce.

In fact, I have just reorganized and rewritten some of his work in many parts of this monograph. I am grateful to the referees for their careful reading and useful suggestions. Using their reports and advice, I have been able to improve the original draft and remove some flaws. Now it is the time for me to thank all the people who helped me both academically and non-academicaIIy for so many years.

In particular, I would like to thank Mrs. Louise Enochs for her kind care and help. Each time she invited me and other students to have dinner with their family, we just felt like family members. I thank Professor Foxby who gave me his encouragement when I asked him questions about Gorenstein modules and Gorenstein rings.

I would also like to thank Professor Vasconcelos who has encouraged so many young people like me. I am indebted to Professors Belshoff and Professor Jenda for our pleasant cooperation. I also thank Professors Coleman and Sathaye for their help. Finally, I would like to express my gratitude to the Department of M a t h e m a t - ics at the University of Kentucky.

Some part of this book was written when I was awarded the President Dissertation Fellowship. W i t h o u t this support, it would have been impossible for me to start and complete this project. Last, but not least, I thank my wife Wei Cai and my twin sons Siyao, Siyuan for their full support for such a long time.com Contents Introduction 1 Envelopes and C o v e r s 5 1.3 F l a t covers a n d t o r s i o n free coverings .4 D i r e c t s u m s of covers a n d envelopes. 48 Flat Covers and Cotorsion Envelopes 51 3.1 F l a t covers in an e x a c t s e q u e n c e .2 M o d u l e s of finite injective d i m e n s i o n .6 Relative homological theory.

75 Flat C o v e r s o v e r Commutative Rings 81 4.2 M i n i m a l p u r e injective r e s o l u t i o n s of fiat m o d u l e s .4 F l a t covers of Matlis reflexive m o d u l e s. 103 Applications in Commutative Rings 107 5.1 T h e Bass n u m b e r s of fiat m o d u l e s .3 M i n i m a l flat r e s o l u t i o n s of injective m o d u l e s .4 Strongly cotorsion modules. 145 Bibliography 153 Index 158 www.com Introduction Ever since Eckmann and Schopf proved the existence of injective envelopes for modules over any associative ring R and Matlis gave the structure theorem of injective modules over Noetherian rings ([25, 54]), the notion of injective modules and injective envelopes (hulls) has played an important role in the theory of modules and rings, and has had a great impact on homological algebra and commutative algebra [21, 37, 56, 65]. In an attempt to dualize injective envelopes, Bass in [8] successfully studied projective covers of modules, and initiated the study of left perfect rings.

These rings possess nice theoretical and homological properties. The harmony between the global characterizations and the internal descriptions of these left perfect rings exhibits the beauty and the nature of structures in algebra. Motivated by injective envelopes and projective covers, many other varied notions of envelopes and covers have been defined and investigated in various settings. For in- stance, Fuchs in [41] and Warfield in [71] defined and studied pure injective envelopes, and used them to describe algebraically compact Abelian groups and modules.

Enochs in [27, 28] defined torsion free coverings and proved the existence of torsion free cov- erings over any integral domain. And then Teply [45, 68] generalized these to certain torsion theories. Concerning envelopes and covers, there are two primary problems: (1) How can we define envelopes or covers in a general setting? (2) How can we prove the existence of the defined envelopes and covers? Considering all the envelopes and covers mentioned above, we found that the processes were totally different. To reveal the consistency of various kind of envelopes and covers, Enochs first in [30] noticed the categorical version of injective envelopes, and then made a general definition of envelopes and covers by diagrams for a given class of modules.

In this setting, all the existing envelopes and covers can be recovered by specializing the class of modules. The essentially same notion was also studied by Auslander and Buchweitz in terms of maximal Cohen- Macaulay approximation for modules over a Cohen-Macaulay ring, and Auslander and Reiten in terms of minimal left (or right) approximation for modules over Artinian algebras (see [5, 6]). Now the notion of flat covers can be easily stated by taking the class of flat modules. For a left R-module M, a linear map ~ : F -+ M with F flat is called a flat cover of M if every linear map p' : F ' --+ M from any flat module F ' can be factored through p; and if ~ itself is factored through g) by an endomorphism f of F, then f must be an automorphism.

Enochs conjectured that every module over any associative ring admits a flat cover. One of the reasons to believe this is true is because many properties of flat modules are highly dualized counterparts of those for injective modules. This monograph is mainly devoted to (1) giving an introduction to envelopes and covers under the general setting and providing a uniform treatment to deal with the www.com existence problem; (2) showing that Enochs' conjecture is true for a quite large class of rings, for instance, all right coherent rings of finite weak global dimension and all commutative Noetherian rings of finite KrulI dimension (which include all coordinate rings of algebraic varieties over any field); (3) applying fiat covers and minimal fiat resolutions to study commutative Noetherian rings as it is done with injective envelopes and minimal injective resolutions. In particular, we characterize Cohen-Macaulay rings and Gorenstein rings by the dual Bass numbers.

By doing so, we can better see the dual relation between injective modules and fiat modules. Chapter 1 gives an introduction to envelopes and covers and presents the basic properties. Although we can phrase most of concepts and statements in purely cate- gorical fashion, we choose to use the terminology of module theory and ring theory. In order to provide the first class of rings (Priifer domains) over which every module has a fiat cover, in this chapter we prove the existence of torsion free coverings over any integral domain.

Direct sums of envelopes and covers are discussed in the last section. Chapter 2 establishes the fundamental results on envelopes and covers. With the assumption that a certain class of modules is closed under direct limits, we develop a general technique to solve the existence problem by manipulating generators of exten- sion sequences. With this treatment, the existence of injective envelopes, projective covers and pure injective envelopes can be proved by just specifying the class of mod- ules.

As a nontrivial application, we show the existence of injective covers of left modules over a ring R is equivalent to R being left Noetherian. We show that the existence of nonzero injective covers of every nonzero module implies that _R must be Artinian. The main results in Chapter 3 are the existence of fiat covers and cotorsion en- velopes over a right coherent ring of finite weak global dimension. Cotorsion modules were studied by many authors with different interests (for instance, Fuchs in [41] and Harrison in [47]).

The consistency of the existence of fiat covers and cotorsion envelopes is ensured by the special properties of these modules. The interesting relations among the classes of injective modules, pure injective modules and cotorsion modules will also be explored in this Chapter. We show that in general the class of pure injective modules is not closed under extensions although both injective modules and cotorsion modules are. Assuming the existence of fiat covers, a relative homological theory can be developed by using fiat resolutions.

This will be briefly discussed in the last section. In Chapter 4 it is shown that every module over a commutative Noetherian ring of finite Krull dimension has a fiat cover. This makes it possible to apply fiat covers to study commutative Noetherian rings. In order to prove this result, necessary prelim- inaries on modules over commutative Noetherian rings are needed.

For instance, the completion of a free R-module is useful in describing cotorsion fiat modules (i., pure injective fiat modules). In particular, the structure of cotorsion fiat modules and the www.com minimal pure injective resolutions of flat modules are very important in the procedure of the proof. Chapter 3 and 4 contain the main ideas and techniques in the study of flat covers and related problems. In Chapter 5 , as an application of the theory of fiat covers developed in Chapter 1 through Chapter 4, we define the dual Bass numbers by using minimal flat reso- lutions, then use them to describe modules over Gorenstein rings.