Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
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Jinzhong Xu
Flat Covers of Modules
~ Springer
Author Jinzhong Xu University of Kentucky Department of Mathematics Lexington, Kentucky, 40506-0027 USA
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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. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10479845 46/3142-543210 - Printed on acid-free paper
Dedicated to
Professor Edgar E. Enochs
My Mother Guizhen Zhu and My Father Youcheng Xu
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 encouragement 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 especially did. 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.
Contents
2
Introduction
1
Envelopes and C o v e r s
5
1.1
Preliminaries
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
E n v e l o p e s a n d covers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3
F l a t covers a n d t o r s i o n free coverings . . . . . . . . . . . . . . . . . . .
16
1.4
D i r e c t s u m s of covers a n d envelopes . . . . . . . . . . . . . . . . . . . .
20
Fundamental Theorems
27
2.1
Wakamutsu's Lemmas
. . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4
Injective covers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.5
F l a t envelopes a n d p r e e n v e l o p e s . . . . . . . . . . . . . . . . . . . . . .
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
3.2
M o d u l e s of finite injective d i m e n s i o n
. . . . . . . . . . . . . . . . . . . . .
51
3.3
Cotorsion modules
3.4 3.5
E x t e n s i o n s of p u r e injective m o d u l e s
. . . . . . . . . . . . . . . . . . .
73
3.6
Relative homological theory
. . . . . . . . . . . . . . . . . . . . . . . .
75
. . . . . . . . . . . . . . . . . . .
58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
C o t o r s i o n envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Flat C o v e r s o v e r Commutative Rings
81
4.1
C o t o r s i o n flat m o d u l e s
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
4.3
F l a t covers of c o t o r s i o n m o d u l e s . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . ............
89 93
4.4
F l a t covers of Matlis reflexive m o d u l e s
. . . . . . . . . . . . . . . . . .
98
4.5
A t h e o r e m on A r t i n i a n rings . . . . . . . . . . . . . . . . . . . . . . . .
103
Applications in Commutative Rings 5.1
T h e Bass n u m b e r s of fiat m o d u l e s
107 . . . . . . . . . . . . . . . . . . . .
108
5.2
T h e d u a l Bass n u m b e r s
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
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
5.4
Strongly cotorsion modules
...............
. . . . . . . . . . . . . . . . . . . . . . . .
5.5
Foxby duality
5.6
G o r e n s t e i n projective,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7
G o r e n s t e i n flat m o d u l e s a n d covers
injective m o d u l e s
.................
....................
117 124 129 137 141 145
Bibliography
153
Index
158
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 instance, 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 coverings 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 CohenMacaulay 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
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 categorical 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 extension sequences. With this treatment, the existence of injective envelopes, projective covers and pure injective envelopes can be proved by just specifying the class of modules.
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 envelopes 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 preliminaries 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.e., pure injective fiat modules). In particular, the structure of cotorsion fiat modules and the
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 resolutions, then use them to describe modules over Gorenstein rings. As with injective envelopes and minimal injective resolutions, Cohen-Macaulay rings and Gorenstein rings can be characterized in terms of flat covers and minimal flat resolutions. Using a vanishing property of the dual Bass numbers, we introduce the strongly cotorsion modules. These modules have nice homological properties. At the end of this chapter we introduce the Foxby classes [39] of modules over a Cohen-Macaulay ring admitting a dualizing module and show the existence of Gorenstein injective envelopes and Gorenstein flat covers for modules in these classes. This will demonstrate the nice homologieal properties of the Foxby classes. This monograph is suitable as a reference for researchers who have interest in general theory of covers and envelopes and in the theory of rings and modules and homological methods in commutative algebra. It also can be used by graduate students who have a special interest in homological algebra and commutative algebra.
Chapter 1 Envelopes and Covers In this chapter we define envelopes and covers for a given class of modules and study their basic properties. These notions were directly motivated by injective envelopes. In this general setting, all the well known envelopes and covers, such as injecti.ve envelopes, pure injective envelopes, projective covers and torsion free coverings (which were defined and investigated separately in [25, 41, 71, 8, 27, 28]), can be formulated. Most of the work in t h i s chapter is due to Enochs [27, 30]. References should also be made to Auslander and Buchweitz [5], and to Auslander and Reiten [6]. Section 1 contains a minimal set of concepts, notation and results in the theory of modules and rings which we need to get started. We will give further notions and notation when they become necessary.
At the beginning of Section 2 we give the
definitions of envelopes and covers and their elementary properties. Then we revisit the existing envelopes and covers such as injective envelopes and projective covers, and show the consistency between the original notions and the current descriptions. Section 3 moves to our main point and starts our investigation of fiat covers. In order to have an example of ring over which every module has a fiat cover, we first prove that every module over an integral domain has a torsion free covering. From this we got t h a t every module over a Priifer domain admits a fiat cover agreeing with its torsion free covering. Section 4 is concerned with direct sums of envelopes and covers. The preservation of envelopes and covers under direct sums is relative to a sort of T-nilpotent property.
1.1
Preliminaries
Throughout all rings R are associative with identities and all modules are unitary. If for an R-module M there is no particular side mentioned, it is assumed to be a left R-module. All the concepts and results in this section are standard, and can be found in any algebra text, We take Anderson-Fuller's book [1] as a major reference. For any two modules M and N, a map f : M ~
N is called a linear m a p or
homomorphism if submodule of M,
f(ax + by) = af(x) + bf(y) for all a , b E R and z , y E M. The {x C M I f(x) = 0}, is called the kernel of f , denoted ker(f)
The submodule of N, { f ( x ) I x C M}, or simply
is called the image of f , denoted
ira(f)
f(M). The quotient N / f (M) is called the cokernel, denoted coker(f)
f is said to be injective if ker(f) = 0; surjective (onto) if coker(f) = 0. f is called an isomorphism if f is both injective and surjective. In particular, f is called an automorphism of M if it is an isomorphism of M to itself. When we say that a module M is isomomorphic to a module N, we mean that they are in the same side and there is an isomomorphism from M to N, denoted by M ~ N. For any two R-modules M and N with the same side, HomR(M, N) denotes the set of all R-linear maps from M to N. This set forms an Abelian group naturally. When we say t h a t M is a direct s u m m a n d of M, we mean that there is a submodule L C N such t h a t N ~ M | L. A sequence of modules associated with linear maps d~+l : X~+I --+ X~ 9" - - + X~+I ~ X~ ~ X~_~ ~ . . . is called exact if im(d~+l) = ker(dn) for all n. We do not give a formal definition of commutative diagrams although we use them very often. Roughly, a diagram consists of vertices which are modules and oriented edges which are linear maps between these modules. A diagram is said to be commutative if for any two modules in the diagram all possible routes from one to the other determine the same map. In a diagram, solid arrows mean that the maps are given; and dotted arrows mean that the maps can be determined in one way or other. D e f i n i t i o n 1.1.1 An R-module P is called projective if one of the following statements holds: (1) If f : M --+ N --+ 0 is exact and g : P --+ N is a linear map, then g can be lifted to M (or factored through f ) , i.e., the following diagram can be completed to a commutative one
P
-"Nlg
.0
(2) HomR(P, *) leaves every exact sequence 0 --~ M -+ N -+ L -+ 0 exact, i.e., 0 --~ HomR(P, M) --+ HomR(P, N) ~ HomR(P, L) --+ 0 is exact ; (3) P is a direct s u m m a n d of a free R-module. Here a module F is called free if it is isomorphic to
R (x)--{(rx) ] r x E R , x E X ,
rx=0except
for a finite number of x C X }
which is an R-module with the obvious module structure.
Dually an R-module E is called injective if for every exact sequence of R-modules
0 --+ Nf-~M, any linear map g : N --+ E can be extended (or factored through f) to M, i.e., the diagram
0
.N f'M
can be completed by a linear map h such that g = hr. Note that any direct sum of projective modules is projective, and any direct product of injective modules is injective. Any exact sequence of R-modules 0 ~ Mf--~N~,L 0 is called an extension of M by L. This extension is said to be trivial if it is split, or equivalently there is a linear map # : N -+ M such that # f = 1M, or there is a linear map v : L -+ N such that gv = 1 L . As standard we use E x t , ( M , N) to denote the derived homological groups by the functor Horn, and Ext with certain parameters for the corresponding homological functors. See Rotman [66] for the detailed description. T h e o r e m 1.1.1 For any two R-modules M and N the following statements are equiv-
alent: (1) Every extension of M by N is trivial, i.e., every exact sequence 0 -~ M -+ X -+ N -~ 0 is split; (2) E x t , ( N , M) = 0 . Let {M~, ~ji} be a direct (inductive) system of R-modules with the directed index set I. Then the the direct limit, denoted lim Mi, exists. It is isomorphic to ~ M i / S --+ where S is the submodule generated by all elements {Aj~j~(a~) - A~(a~)} where A~ : Mi --+ ~ M i is the canonical injection. Since direct limit arguments will often be used in this monograph, it is appropriate to state the following result (see Rotman [66, Thm2.17]). P r o p o s i t i o n 1.1.2 With the notation as above, the direct limit of a direct system
{ Mi, ~gji} has the following properties (1) l i m M i consists of all .~(ai) + S; (2))~i(a~) + S = 0 if and only if ~ji(a~) = 0 for some j > i. As standard we use M | N for tensor product and
Tor~(M, N) for the derived
homological groups. D e f i n i t i o n 1.1.2 A left R-module F is called fiat if the tensor functor - |
F leaves
every exact sequence of right R-modules 0 -~ M -+ N exact, that is, 0 -+ M | N |
F is exact.
F -+
Note that all projective modules are flat, but the converse is not true in general. The following result, due to Lazard [53], is useful for our purpose.
An R-module M is
called finitely generated if there are finite many elements x l , . . . ,xn E M such that M = RXl + Rx2 + . . . + Rxn. T h e o r e m 1.1.3 Every fiat module is a direct limit of finitely generated projective modules. Any direct limit of fiat modules is fiat. T h e o r e m 1.1.4 The following statements about a left R-module F are equivalent : (1) F is fiat; (2) For each (finitely generated) right ideal I, the Z-linear map #I : I |
F --+ I F
with #i(r | x) = rx, r C I, x c F, is injective. The proof of the above theorem can be found in Anderson and Fuller
[1, 19.17].
We need the following result for our future use. The proof also can be found in the Anderson and Fuller's book. T h e o r e m 1.1.5 Let F be a fiat left R-module. Suppose we have an exact sequence O-~ K ~ F--+ G--+ O of left R-modules. Then G is fiat if and only if I K = I F N K for each (finitely generated) right ideal I. Note that this is equivalent to saying that K is a pure submodule of F. For completeness we recall the definition of pure submodule here. We will consider purity and related topics in the next chapter. D e f i n i t i o n 1.1.3 An exact of sequence of left R-modules O----~M----~N---~L~O is pure exact if, for every right R-module A, we have exactness of O~ A|
A|
A|
We say that M is a pure submodule of N in this case. Let Q be the rational numbers, Z C Q. It is well known that for a left R-module M, M is flat if and only M* = Homz(M, Q / Z ) is injective. Here the module structure on M* is defined naturally. Let R and S be two rings, and let RNs an R-S-bimodule.
For any right R-
module M and any right S-module E, there is a canonical isomorphism (see Caftan and Eilenberg [15] or Glaz [44, Thm.l.l.8]): P0: HomR(M, Homs(N, E))
> H o m s ( M @R N, E)
f --~ Po(f), po(f)(m | n) = f(m)(n) Furthermore, if E is an injective R-module, then the naturally induced maps p~: E x t , ( M , Horns(N, E)) -+ Homs(TorR(M, N), E) are isomorphisms for all integers n _> 0. An R-module M is called finitely presented if there is an exact sequence R (m) ~ R (n) ~ M --+ 0 D e f i n i t i o n 1.1.4 A ring R is called right coherent provided that every finitely generated right ideal is finitely presented. R is called right Noetherian if every finitely generated right R-module is finitely presented. For later use we state some useful characterizations of coherent rings and Noetherian rings. The first theorem is due to Chase [16] and the second is due to Matlis [54]. Both proofs can be found in Anderson and Fuller's book [1, 19.20, 25.6]. T h e o r e m 1.1.6 For a ring R the following are equivalent:
(1) R is right coherent; (2) Any product of fiat left R-modules is .fiat. T h e o r e m 1.1.7 For a ring R the following are equivalent:
(1) (2) (3) (4) (5)
R is right Noetherian; R has ascending chain condition (ACC) on right ideals; An arbitrary direct sum of injective right R-modules is injective; An arbitrary direct limit of injective right R-modules is injective; Any injective right R-module has an indecomposable decomposition.
Note that in (5) the representatives, up to isomorphism, of indecomposable injective modules form a set. We will use this fact in Chapter 2 when we study injective covers. Since pullback and pushout diagrams are very useful in our arguments, we briefly discuss them. Let M, N and L be R-modules. For any linear maps f : M --+ L and g : N --+ L, there is a completed commutative diagram, the so-called pullback of f and g:
pu.M
1I Ng,L such that for every pair of linear maps u' : X --+ M and v' : X ~
N satisfying
fu' = gv' there is a unique linear map h : X -~ P satisfying u' = hu and v' = hv. Actually the module P can be chosen as the submodule {(x, y) E M @ N I f(x) = g(y)} Moreover if both f and g are surjective, then we have the full commutative diagram with exact rows and columns:
10 0
0
K=K
where
0
.L
0
.L
K = k e r ( f ) and L = ker(g).
.P
ii
I
.M
1
.N
.L
0
0
.0 .0
Dually we have the pushout diagram for every
pair of linear maps f : L --+ M and g : L -+ N. There are the dual description and properties (see Stenstr6m [14] for the details). We use R M to stand for all the left R-modules, or from time to time we call RA/[ the category of all left R-modules. A left R-module C is called a cogenerator in R M provided t h a t for any nonzero module M there is a nonzero linear m a p f : M -+ C. Furthermore C is called an injective cogenerator if it is injective and a eogenerator in RA4. It is well known that there is an injective cogenerator in R M (see Anderson and Fuller [1, Cor.18.19]). We need the next result for our future use. Theorem
1.1.8 Let C be an injective cogenerator. Then the the following hold :
(1) 0 --+ M --+ N ~ L --+ 0 is exact if and only if 0 ~ HomR(L, C) ~ HomR(N, C) -+ HomR(M, C) --+ 0
is exact;
(2) Every R-module M can be embedded into a product C I for some set I. As usual we use p r o j . d i m R ( M ) to denote the projective dimension of M, and we use i n j . d i m R ( M ) to denote the injective dimension, and f . d i m R ( M ) the fiat dimension of M . l.gl.dim(R) stands for the left global dimension of R and w.gl.dim(R) for the weak global dimension of R. For the notions and notation in homological algebra, we take R o t m a n [66] as a major reference.
1.2
Envelopes and covers Let X be a class of left R-modules. We assume that X is closed under isomor-
phisms, i.e., if M E X and N = M, then N E X. We also assume t h a t X is closed under taking finite direct sums, and direct summands, i.e, if M 1 , . . . , Mt E X, then
M1 0 ' " @ M r E X; i f M = N ~ L
E X, then N , L E X .
D e f i n i t i o n 1.2.1 For a left R-module M, a module X E X is called an X-envelope of M if there is a linear map ~ : M --+ X such that the following hold: (1) for any linear m a p ~' : M --+ X ' with X ' E X, there is a linear m a p f : X -+ X ' with ~' = f ~ . In other words, Homn(X, X ' ) --+ HomR(M, X ' ) --+ 0 is exact for any X ' E X;
11 (2) If an endomorphism f : X --4 X is such that ~ = fg), then f must be an automorphism. If (1) holds (and perhaps not (2)), we call ~a : M --+ X an X-preenvelope . For convenience we sometimes call X or the map ~ an X-envelope (preenvelope) of M. This definition was first introduced by Enochs in [30] where particular attention was paid to the class 2( of all the injective left R-modules, or all the flat left R-modules. Auslander and Reiten in [6] given the essentially same notion for modules over an Artinian algebra, but they called an X- envelope a minimal left X - a p p r o x i m a t i o n generalizing the maximal Cohen-Macaulay approximations investigated in [5, 19]. One of our main problems is the existence of X-envelopes for a given class X. This is highly dependent on the structure of the given class. Before we start the study of the existence problem, let us establish some elementary properties. We first note that if M -+ X is an X-preenvelope and if S C M is a direct s u m m a n d of M, then S -+ M -+ X is an X-preenvelope of S. Proposition
1.2.1 I f ~1 : M ~ X1 and ~2 : M --+ X2 are two different X-envelope
of M , then X1 ~- Xu. Proof:
Since both X r and X2 are X-envelopes of M, there exist linear maps fl :
X2 --~ X1 and f2 : X, --~ X2 such that the following diagrams are commutative:
M~X1
M ~2, .X2
t h a t is, ~2 = f2pl and ~1 = flp2. Then easily we have ~a = f l f 2 T l and ~2 = f2fl~2. By the hypothesis (2) in the definition, both f~f2 and f 2 f l are automorphisms. This implies t h a t both fl and f2 are isomorphisms. [] Proposition X-preenvelope.
1.2.2 Suppose that M admits an X-envelope and ~a : M ---+ X is an Then X = X* @ K for some submodules X* and K such that the
composition M ~ X -+ X* gives rise to an X - e n v e l o p e . Proof." Let ~ : M --+ X0 be an X-envelope of M. Then we have the commutative diagram:
12 such that ~ = f r and r = g~. Hence, r = g f r
It follows that g f is an automorphism
of X0, and X =im(S)@kerg. Obviously X* = i m ( f ) ~ X0 is an 2(-envelope of M. [] C o r o l l a r y 1.2.3 Suppose M has an 2(-envelope. Let cp : M -+ X be an 2(-preenvelope. Then it is an envelope if and only if there no direct sum decomposition X = X~ 9 K with K 7t 0 and im(9)) c X1. Proof:
Suppose ~ : M --+ X is an envelope, and that there is a decomposition
X = X I @ K with im(~) C X1 and K -r 0. We construct a linear m a p S : X I @ K ~ X which agrees with the the projection onto X > It then is easy to verify t h a t g) = f ~ holds. But then by the second condition of the definition, ] must be an automorphism. This is impossible unless K = 0. The other direction easily follows from the previous proposition. [] Proposition
1.2.4 Suppose that the class 2( is closed under arbitrary direct sum .
I f f o r each i, ~i : ]Vii --+ X i is an 2(-preenvelope, then O~i : OMi --4 O X i is an 2(- preenvelope. P r o o f : Let ~' : @Mi --+ X ' be any linear map. If qi : Mi --4 @Mi is the canonical injection, then since Mi --+ Xi is a preenvelope we have a linear m a p fi : Xi --+ X ' such t h a t ~p' o qi = fi o ~oi. Then if f : O X i --+ X ' is the unique linear map such t h a t f I X i = f/, then ~' = f o ( G ~ ) . [] Note t h a t in general O~i : @Mi --+ @Xi may fail to be an 2(-envelope even though each ~i : M, --+ Xi is an envelope. We will see an example later. But with a finite number of terms, we do get an envelope. We only need to look at the case of two terms. Theorem
1.2.5 Let qoi : Mi --+ Xi, i = 1,2,
be 2(-envelopes.
Then ~1 @ g)2 :
M1 @ M2 --4 X1 G X2 is an Xx-envelope. P r o o f : By the preceding result we know that it is an 2(-preenvelope (2( is closed under finite direct sums). Now suppose that there is an endomorphism S of X1 (9X2 such that ~1@~2 = f (~1G~2). We want to show t h a t f is an automorphism. Let qi : X i --+ X~G X2 , i = 1, 2, be the canonical injections, and let Pi : X1 @ X2 --+ Xi, i = 1, 2, be the canonical projections. For convenience we express the elements in X1 @ X2 as columns ( XLl )e t C nf = ~ p l f q lE' X r 1I 6' x221E6X2 2 " Then f can be expressed as a matrix
For a E M~, b C M2, we have the following equations:
13 and then Pl (a) = r (a) + r ~2(b) -- r (a) + r (b). Therefore c21 = r 0 = r ~22 = r 0 = r This implies that r is an automorphism of X1. Consider the matrix multiplication -r Note that r
r
r
1
r
=
0
-r162162
+ r
= 0. Hence ~2 = ( - r 1 6 2 1 6 2
This shows that (-r162162
+r
+ r
is an automorphism of X2 by the second condition
of envelopes. Now by a standard matrix argument we see that the last matrix above is invertible. So the matrix corresponding to f is invertible. [] Dually we have the following definition and properties for X-covers. We just state them and omit most of the proofs. D e f i n i t i o n 1.2.2 With the same assumption as in the Definition 1.2.1 on the class X, for an R-module M, X C X is called an X-cover of M if there is a linear map : X --+ M such that the following hold: (1) For any linear map ~' : X ' --+ M with X ' C X , there exists a linear map f : X ' -+ X with ~' = ~ f , or equivalently HomR(X', X ) ~ HomR(X', M ) ~ 0 is exact for any X ' C X. (2) If f is an endomorphism of X with ~ = ~ f , then f must be an automorphism. If (1) holds (and perhaps not (2)), ~ : X ~ M is called an X-precover. Note that an X-cover (precover) is not necessarily surjective. Note also that if X --+ M is an X-precover of M and if M ~ S is the projection of M onto a direct summand S of M, then X --+ M --+ S is an X-precover of S. One of our main interests is to determine for which classes X, X-covers exist. T h e o r e m 1.2.6 Let M be an R-module. I f ~i : X~ --+M, i = 1, 2, are two different X-covers, then X1 ~ X2.
T h e o r e m 1.2.7 Suppose M admits an X-cover, and ~ : X ~ M is an X -precover. Then X = X1 O K for submodules X1 and K such that the restriction qo Ix1:X1 --+ M gives rise to an X - c o v e r of M and K C ker(~).
C o r o l l a r y 1.2.8 Suppose M admits an X-cover. Then an X-precover ~ : X --+ M is a cover if and only if there is no nonzero direct summand K of X contained in ker(~).
14 Proof." By the theorem above the condition is sufficient. For necessity, let X
=
X 1~K
with KC ker(q@ Define f : X -+ X by sending Xl + k --+ xl. Easily ~ f = ~2. Now we note that f is not an automomorphism of X unless K -- 0. [] T h e o r e m 1.2.9 Suppose X is closed under an arbitrary direct product, and f o r each i, ~i : X i -+ Mi is an X-precover. Then the natural product If ~i : [I Xi -+ [I Mi is an X-precover.
Note that even when each ~ : X~ --~ Mi is an X-cover, the product [I ~i : [I Xi -+ I-[ Mi may fail to be a cover. One counterexample will be given in the next section.
T h e o r e m 1.2.10 I f ~i : Xi -+ Mi is an X-cover f o r i = 1 , . . . ,n, 9 Mi is an X-cover.
then @Pi : O X i -+
So far we have discussed envelopes and covers in general, we now review some well known envelopes and covers by specifying the class X. First let C be the all injective left R-modules. Recall that an injective module E is called an injective envelope of M if M can be essentially embedded into E, i.e., there is an injection p : M -+ E such that i m ( ~ ) N K = 0 for any submodule K of E only i f K = 0. Eckmann and Schopf [25] proved that over any ring every module M has an injective envelope, denoted E ( M ) . This result together with the Matlis' structure theorem [54] for injective modules has played an important role in homological algebra and its application in commutative algebra (see [37, 21, 56]). The following show the consistency between the notion of injective envelope and the notion of g-envelope. T h e o r e m 1.2.11 Let M be a left R-module, and let E E $. Then the following are equivalent. (1) ~ : M -+ E is an g-envelope; (2) ~ : M --+ E is an injective envelope in the Eckmann-Schopf's sense.
P r o o f : (1) ~
(2) By Theorem 1.1.8, M can be embedded into an injective module
E'. Hence there is an injection ~' : M -+ E'. By the first condition of g-envelopes there is a linear map f : E -+ E ' such that 9~' = f ~ . This shows that ~ is an injection. Suppose that ~(M) is not essential in E. So there exists a nonzero submodule K C E with ~(M) NE---- 0. Since K + ~ ( M )
= K~(M),
p : KO~(M)
= ~(m),k E K,m
--+ E w i t h p ( k + ~ a ( m ) )
g : E --+ E , we have the commutative diagram
M --~K G ~(M) z-~ E P
.,-) P
E
we can define a l i n e a r map E M.
E x t e n d i n g p to
15 Note that ~ = g~,. By the second condition of t;-envelopes, g must be an automorphism of E. This is impossible because g ( K ) = O. (2) ==~ (1) By the definition of injective modules it is obvious t h a t ~ : M --+ E is an C-preenvelope of M. Now suppose there is an endomorphism f of E with ~ = f ~ . We shall claim t h a t f is an automorphism, f is injective since T is essential. Hence E = f(E)~K
for a s u b m o d u l e K
C E. I f K ~ 0 ,
takinganon-zerox
C K C E,
there exists an element r C R such that r x :~ 0 and r x = y E ~ ( M ) . But then
y=~(m)
= f~(m) e f(E)AK=O
.
This contradiction implies t h a t f is also surjective. [] Example
We give an example which shows t h a t injective envelopes is not closed
under taking direct products. Let p be a prime and Zp~ = { ~ + Z I n C Z , t > 0}. As Z-modules, E ( Z / ( p ) ) = Zp~. Consider the product of a countable number of copies of Z/(p). It is easy to see that 0 --+ [I Z / ( p ) --+ 11 Zpo~ is an injective preenvelope. Note t h a t 11 Z / ( p ) is torsion (i.e., every element can be annihilated by an nonzero integer.), but 11 Zp~ is not, because we can choose an element x = ( ~ + Z , . . . , ~ + Z , . . . )
e I-[ Zpo~
and x can not be annihilated by any nonzero integer. If l-] Zp~ is an injective envelope of l-I Z/(p), then l l Zp~ must be torsion. This gives a contradiction. []. Bass [8] defined projective covers as the dual of injective envelopes. Surprisingly the existence of projective covers is not so common, and it forces the ring to be a left perfect ring. Let us recall some notions. For a left R-module M, a submodule S C M is said to be small or superfluous if for any submodule L C M, S + L = M implies L = M. This is denoted by S < < M. Let P be a projective R-module. A surjective linear m a p ~ : P -+ M is called a projective cover if ker(~) < < P . Let P_ be the class of all projective R-modules. Then we have the consistency between the notion of projective cover and the notion of P-cover for a module M.
Theorem
1.2.12 For a left R-module M and a linear map ~ : P --+ M with P E P ,
the following statements are equivalent: (1) ~ : P --+ M is a P-cover; (2) ~ : P -+ M is a projective cover.
Proofi
(1) ==~ (2) First we see that p is surjective. This is ensured by the fact
t h a t any left R-module is an image of a projective module and the first condition of P-covers. Now let k = ker(p) and K + L
= P for a s u b m o d u l e L C P. We claim
t h a t L must be P itself. Note t h a t the restriction TL : L --~ M is onto. Then by the definition of projective module ~ can be factored through ~L, i.e., there is a linear m a p f : P ~ L with ~
=
(flLf.
Hence we have a commutative diagram:
16 P
P Easily ~ = ~ f .
By the second hypothesis of P-covers it follows that f must be an
automorphism of P.
This is impossible unless L = P.
We have shown t h a t K is
superfluous (small) in P , and then ~o : P --+ M is a projective cover of M. (2) ==~ (1) Clearly ~ : P -+ M is a P - precover. Suppose there is an endomorphism f of P satisfying ~ = ~ f . We have to show that f is an automorphism. First note t h a t such an f must be surjective because P = ker(~) + f ( P ) and K = ker(~a) < < P is superfluous. But since P is projective, 0 I-+ ker(f) --+ P ~ P -+ 0 is split. There exists a linear map g : P ~ P such that fg = 1p and then g is injective with P = g(P)+ ker(f). By the equation p = ~ f , ker(f) C ker(~) < < g . This implies that g(P) = P and g is an automorphism, and hence so is f . [] Next we state Bass' theorem P [8] on the existence of projective covers. A ring R is said to be left perfect provided every left R-module has a projective cover. We use
J(R) or simply J to stand for the Jacobson radical of R. We will reconsider projective covers and left perfect rings in the next chapter from a different point of view. Here we give the definition of T-nilpotence which will be used to describe specific rings. For other terminology used in Bass' theorem, see his paper [8] or Anderson and Fuller's book [1]. D e f i n i t i o n 1.2.3 Let R be a ring. A subset S C R is called left T-nilpotent if for any countable sequence {ai ~ S I i > 1}, there is an integer n such t h a t ala2 . . . . . . a,~ = O.
T h e o r e m 1.2.13 The following are equivalent for an associative ring R: (1) R is left perfect; (2) R / J is semisimple and J is left T-nilpotent; (3) R / J is semisimple and every nonzero left R-module has a maximal submodule; (4) Every fiat left R-module is projective;
(5) R satisfies the descending chain condition (DCC) for principal right ideals; (7) Any direct limit of projective left R-modules is projective.
1.3
Flat covers a n d t o r s i o n free c o v e r i n g s
In this section we will switch our attention to fiat covers. Let _~ be the class of all fiat left R-modules. For an R-module M, an ~-cover (precover) of M is called a
17
fiat cover (precover). Our main goal is to investigate the existence of fiat covers. It is appropriate to state the following conjecture which was initiated in [30]: E n o c h s ' C o n j e c t u r e : Over any associative ring R every R-module has a fiat cover. In some sense this is the dual of the existence of injective envelopes. For many reasons it is believed that the duality between flat modules and injective modules is better than that between projective modules and injective modules.
One of these
reasons is that the structure of projective modules is relatively simple but that of injective modules and fiat modules is more mysterious. Actually there are very few injective modules and flat modules (not projective) that can be described explicitly. So far Enochs' conjecture is still open. In Chapter 3 and 4 we will prove that the conjecture is true for quite a large class of rings, including all local rings. But at this moment we at least know that the conjecture is true for left perfect rings. P r o p o s i t i o n 1.3.1 Every left module over a left perfect ring R has a fiat cover which is the same as its projective cover. Note that if ~ : F --+ M is a flat precover of M, ~ must be surjectivc. Before we give a nonperfect ring over which the Enochs' conjecture holds, we consider torsion free coverings. To do so, we temporarily assume R to be an integral domain, that is, R is commutative and ab ~ 0 for any a ~ 0, b ~ 0 E R. Recall that an R-module M is said to be torsion free if for a E R, x E M , ax = 0 implies that a = 0 or x = 0. Let be the class of all torsion free R-modules. A Tf-cover (precover) of M is called a torsion free covering (precovering). Note that if W : F -+ M is a torsion free precover, 7~ is always surjective. Enochs proved in [27, Theorem 1] that every module over an integral domain R has a torsion free covering. Teply generalized this result to the torsion theory setting in [45, 68]. Here we give the proof of Enochs' result. T h e o r e m 1.3.2 Every module over an integral domain has a torsion free covering. The proof is going to be split into several lemmas. L e m m a 1.3.3 If ~ : F -+ M is a torsion free preeovering and N C M is a submodule, then the restriction ~1 : ~ - t ( N ) -+ N is a torsion free precovering of N .
Here
7~-1(N) = {x e F I 7)(x ) G N}. P r o o f : First note that ~ - I ( N ) C Tf. Now for any linear map f : G --+ N with G E Tf, since 7~ : F -+ M is a torsion free precovering, there is a linear map g : G -+ F such that f = ~zg. Easily g(G) c ~ - I ( N ) . This shows that f can be factored through ~ . [] L e m m a 1.3.4 I f E is injective, then ~ : F -+ E is a torsion free precoverin9 if and only if for any linear map ~' : F' -+ E with F' torsion free and injective can be factored through ~.
18 P r o o f i The condition is clearly necessary. If pl : F1 -4 E is a linear map with F1 E Tf, then the injective envelope of ['1, E(F~) is torsion free and injective. Note that ~a can be extended to E(F~) since E is injective. By the assumption this extension can be factored through p. In other words we have a commutative diagram:
gi ',,. Here ~' is the extension such t h a t ~' = qog. Hence ~o'oe= ~oga. It follows that ~o~ can be factored through ~. [] Lemma
1.3.5 Every R-module M has a torsion free precovering.
Proof." By Lemma 1.3.3 and the fact that every module is a submodule of an injective module, it suffices to assume that M is injective. Now we only need to find a linear m a p ~o : F -4 M with F torsion free such that any linear map ~' : F ' -4 M with F ' torsion free and injective can be factored through qo. Let H = HomR(K, M), where K is the fraction field of R. Set F = K (H), Define qo : F - 4
M with ~(kh) = ~ h ( k h ) for (kh) E F. Then for any F ' t o r s i o n free and
injective, F ' = •K,,, # C I for some index set I. Here K u ~ K for each # C I. Now by the construction of F it is easy to see t h a t any linear m a p ~' : F ' -4 M can be factored through ~o, and so ~ : F -4 M is a torsion free precovering of M. [] Lemma
1.3.6 Let r : G -4 M be a torsion free precovering. Then from this we can
derive a torsion free preeovering qo : F -4 M such that there is no nontrivial submodule S C ker(~) with F / S torsion free. Proof." Let ~ be the set of all submodules S C ker(r
with G / S torsion free. Then
the union of any chain of elements of ~ is still in ~ . Consequently by Zorn's L e m m a (see [1, 0.9. p.5.]) there is a maximal element S E E . Consider the diagram
G ~O.M
al
.."~
here ~ is the induced m a p since S C ker(r
" Therefore if F = G / S , ~o : F -9 M is a
torsion free precovering since ~b : G -4 M is one. It is easy to see t h a t this precovering satisfies the desired property. [] Lemma
1.3.7 If ~ : F -4 M is a torsion free precovering of M with no nontrivial
submodule S C ker(~) such that F / S is torsion free, then this precovering actually is a torsion free covering of M.
19 P r o o f i Let f : F -+ F be an endomorphism with ~o = qof. We claim that f is an automorphism. Suppose we have a linear map ~ : F' ~ M with F ' torsion free such that the following diagram is completed into a commutative one by a linear map f:
F
Then f must be injective. In fact ~ = qo'f implies that ker(f) C ker(~) and F/ker(f) is torsion free. This induces that ker(f) = 0 by the special property of ~ : F --+ M. In particular, f is injective. We will prove that f is also surjective in an indirect way. As usual, Card(Y) stands for the cardinality of a set Y. Let X be a set such that F C X and Card(X) >Card(F). Let ~ = {(F0,~0)}, where 9~0 : F0 --+ M is a torsion free precovering of M without nontrivial submodule S Cker(~0) with Fo/S torsion free and with F0 C X as a subset.
Then ~ is not
empty since (F, ~) E ~ . Partially order ~ by setting (F0, ~o) _< (F1, ~1) if and only if Fo C F1 and the restriction ~1
I o=
Then for any chain {(Fi, ~i)} E ~ , define
F* = U F i a n d ~* : F* ~ M with~*(x) = pi(x) i f x C Fi. It can be verified that (1) F * E T f and F* C X; (2) for a n y G c T f a n d a l i n e a r m a p ~ ' : G - + M ,
thereis
a linear map f ' with ~*f' = ~'; (3) if S C ker(~*) and S C F* with F * / S torsion free, then F * / S = ~''F~+SSis torsion free. Since each F,+SS"~=SnF~F'and SMFi C ker(~i), this implies that S N Fi = 0 for all i. Therefore, S = 0. By all of these observations we get that (F*, 9<) E ~ . Hence by Zorn's lemma there exists a maximal element in E , say (F*, ~*). Consider the commutative diagram
F:
sq; F ~tM Since all elements in ~ have the special property, by an argument similar to that above, fl must be injective. We will show that fl is also surjective. Suppose fl is not surjective. Let Y C X be such that Card(Y) =Card(F\fl(F*)) and such that F*M Y is empty. Such a Y is available because Card(X) > C a r d ( F ) =Card(F*). The latter is true because there is an injection mapping F into F*. Let F0 = F* U Y and let g : F0 -+ F be the bijection such that g IF.= f l , g ( Y ) = F - fl(F*). Then F0 can be made uniquely into an R-module so that g becomes an isomorphism. Let Fo denote this module. We see that F* is a submodule of Fo. Consider the pair (F0, ~ o g). We assert that it is an element in ~ strictly greater than (F*, ~*). Since g is an isomorphism and ~ : F --+ M is a torsion free precovering, ~ o g : Fo -+ M is a
20 torsion free precovering of M. Suppose that there exists a submodule S c ker(~ o g) with Fo/S torsion free. Then g(S) ~ ker(~) and Fo/S ~ g(Fo)/g(S) is torsion free. This implies t h a t S = 0. Finally ~ o g IF.= ~ o f l = ~*, and (F0, ~ o g) _> (F*, ~*). This is a contradiction because F0 strictly contains F*. Therefore Card(Y) = C a r d ( F - ft(F*)) = 0 and fl(F*) = F, i.e., ft is surjective. We now finally are able to show that f is an automorphism if ~ = ~ o f . Using the m a p fl in the argument, we have ~* = qoofl = qoo (f fl). Then by the same argument
f fl must be isomorphism, and so f is surjective. [] As we know, torsion free modules and torsion modules are very useful in describing commutative rings. Recall that a commutative integral domain is called Priifer domain if every finitely generated ideal is projective. Over such a domain a module is flat if and only if it is torsion free (see R o t m a n [66] for the details). Combining this fact with the previous result, we have that 1.3.8 Let R be a Priifer domain. agreeing with its torsion free covering.
Theorem
Example
Then every R-module has a fiat cover
Let (R, m) be a discrete valuation d o m a i n , / ~ be the completion of R with
respect to m-adic topology. Then the natural surjection ~ : R --+ R / m is a torsion free covering of the quotient field k = R/rn (also see Section 4.1). [] By Proposition 1.2.4 torsion free precovers are preserved under taking direct products.
But covers may fail to be preserved through this operation.
We state the
following result without giving the proof (see Enochs [28] for the proof). Theorem
1.3.9 Let R be an integral domain. Then the following are equivalent:
(1) Every torsion free module G 7~ 0 has a simple submodule; (2) If for each i E I (an index set), y) : Fi --+ M~ is a torsion free covering of Mi, then I-[ Fi ~ [[ Mi is a torsion free covering of 1-I Mi ; (3) An R-module M is injective if and only if E x t , ( S , M) = 0 for every simple R-module S. Further, if R is Noetherian (and not a field), then R satisfies one of the above statements if and only if K.dim(R) _< 1.
1.4
Direct
sums of covers and envelopes
Let X be a class of modules as stated in Section 2. A direct sum of X-covers may fail to be a precover, or it may be a precover and still not be a cover, In this section we will give a necessary condition for a direct sum of X-covers (envelopes) to be an X-cover (envelope). This condition is a sort of T-nilpotence. Using this property we c a n construct an example which shows that a direct sum of flat covers is no longer a
flat precover. All these results are taken from Enochs [30].
21 Theorem
1.4.1 I f for each i = 1, 2 , . . . , , Si c Ei is a submodule such that
Ei~Ei/Si
can be completed only by an automorphism of Ei, then the same is true of
(gE/ |
(gE//|
eE~ if and only if each sequence 1 ~ kl < k2 < . . . of positive integers and linear maps f~ : Ek, -+ Ek~+l with im(f~) C Sk~+~ and for each x E Ekl there is an m > 1 such that fm o fro-1 o . . . o f l ( x ) = O.
Proof:
For necessity, let the f ~ s and k~s be as stated.
r : O E i --+ •Ei
Define an endomorphism
so t h a t if i r k~ for all n, then r IE~ is the identity m a p and so t h a t
r tE~ agrees with the map Ekn -+ Ek~ 9 Ekn+l which takes y to ( y , - f , ( y ) ) .
Then,
since fn(Ek~) C Ski+l, r completes the diagram above. Hence r is an automorphism by the hypothesis. For every x G Ekl , x = r calculation we get that x = Xkl = Ykl,
for some (y~) E @E/. By a simple
Yk~ -- fk~(Yk~) = 0, . . . , Yk~+~ -- fk~(Yk~) = O.
Then Yk~+~ = fkt o 9 9 9o fk~ (X) for all t _> 1. But since (y/) E @Ei, there is an integer m such t h a t Ykm+~ = 0. This finishes the proof. For sufficiency, we shall employ matrix notation for endomorphisms of @E~. Suppose r completes the diagram, i.e., ~)~i = r
and the stated properties are satis-
fied. Denote every element x G @El in column vector form, i.e., x = ( X l , . . . , x . . . . . )~. Here each xi E Ei and T means the transpose. Let qi : E~ --+ @Ej be the canonical injection, let p/ : @Ej ~ E~ the canonical projection, and let Cji = PjCq/. W i t h this notation r can be expressed as a matrix:
r r
r r
9 " "
r
" '
'
9 " "
(~)2F~
" '
'
9 " "
r
" '
"
, , ,
r
r
Since G~ai = (@~/)r for x / E E/, 1 < i < n we have ~nCnl(xl) + " " + ~nCn~(X~) = ~ ( X ~ ) , SO ~ O Cjj = 0 for i ~ j; ~ o r
= ~/. By the assumption this shows t h a t r
is an automorphism of Ei for each i and Cj/(E/) C Sj for j ~ i. Also the m a t r i x (r is locally column finite in the sense t h a t for any j, x E Ej, eli(X) = 0 except for finite
22 defines an endomorphism of @El.
number of i. This guarantees that the matrix (r Furthermore any such a matrix (r and im(r
with r
satisfying the conditions: (1) ~i = ~ir
C Sj; (2) (~ij) is locally column finite, we defines an endomorphism r of
OEi with @~i = ( O ~ ) r To argue that r is an automorphism we only need to find a m a t r i x r which is an automorphism of |
having ~
= ( G ~ i ) r and such that r o r or r o r is an
isomorphism. The argument proceeds by showing t h a t r has a triangular decomposition, i.e., it is the product of an upper and lower triangular matrix (corresponding to an automorphism of |
If r is upper triangular, then since its diagonal elements
are automorphisms of the E~, it is a standard argument to show it is invertible, and clearly its inverse satisfies the conditions above, so guaranteeing that it corresponds to an automorphism of @Ei of the desired type. we now construct an upper triangular matrix r of the desired form so t h a t r o r is lower triangular. Then r = r o r o r
will give the desired decomposition.
Define
- e l l -1 -r162
....
0
1
9 "'
0
"' 9
0
0
9 ''
1
" 9149
Then the m a t r i x product (r162
~ll-lr
...
turns out to be
1
0
...
0
...
r
r
"'"
r
""
r
r
"'"
r
9
with r 3' satisfying the properties so that (r162 is locally finite and @r = (@r162162 Define r define r
as we defined r ", r
but using the second row of (r162
and then similarly
. It is easy to see that the (i,j) entry of r
.....
or
o ... o r
is
constant for n sufficiently large and so the infinite products converges; and the upper oo
triangular m a t r i x r = 1-I r
gives the desired automorphism so t h a t r o r is lower tri-
i=l
angular with the diagonal entries identities, and locally finite with O~i = ( ~ i ) ( r Now we claim t h a t A = r o r is an automorphism of @Ei. For convenience, we say A = (r
with r
= 1; r
Owheni
< j . For a n y x = ( X l , ' " , x , ~ - - - ) ~ E GEi,
suppose A(x) = O, then 0 = x l , - ' - , 0 = r 1 6 2
. . . . i(Xn-1)+Xn for all n _> 1.
It easily follows that x = 0 and A is injective. Next we prove A is surjective by solving the equation Ay = x for any x = ( x l , ' " , x n , " equation for x = ( 0 , . - - , x ~ , 0 , . . . ) ~ E ~ E i
.)t E ~ E i . We only need to solve the
i > 1. W i t h o u t loss of generality, let us
assume i = 1. Set y = ( y i , . . . , y m . . . ) ~ E @Ei. We hope y can be found such t h a t
23 + " " + r . . . . l(Yn-1) +Yn.
y,~ = 0 for all m sufficiently large. For all n _> 1, x~ = r But then Yl = xl, Y2 = x2 - r
". By continuously substituting, we have
here sgn = + or - , n > k, > k s - l - - " > kt > 1. Suppose there were infinite many terms such that
Ca%' o Ck~k~_l o . . . o Ck~kl o Ckb(Xl) r 0 Look at the sequence {k~l [ t > 1}. Since
(r
is locally finite, Cki,(xl) = 0 for k~
sufficiently large. So we can choose Ck~ol(Xl) 7~ 0 such t h a t it appears infinitely times, say Ck~ol(Xl) = Ck,]l(xl) for infinite many tj. Similarly since r
o Cktjl(Xl) = 0 for
k~~ sufficiently large, we can have k~' such that 0 7~ Ck~,kt] o Cktjl(Xl) appears infinite times. Repeating this procedure, we can find an infinite sequence
. . . > ktm ' > ... >
kt9~ > k~~ such t h a t
r
o...o Ck;'k',' o r
:/= 0
By using our hypothesis and setting f m = Ck~k~_,' we are led to a contradiction. [] Remark
1.4.2 If we only have finite many terms, then the hypothesis is a u t o m a t i -
cally satisfied. So this means that covers and envelopes are closed under taking finite direct sums. The very last step in the proof can be realized by applying the KSnig graph theorem. Also note that our hypothesis was not needed until when we proved t h a t the lower triangular matrix A was a surjective map. Hence we have the following. C o r o l l a r y 1.4.3 If for each i = 1, 2 , . . . , , S~ C Ei is a submodule such that
can be completed only by automorphism of Ei, then the diagram
eE, e~ eerie&
eE, can be completed only by injective endomorphisms. Notice t h a t in the above restricting the index set to be only positive integers is not necessary. Actually, the index set could be any set. Dually, we have similar results for envelopes.
24 Theorem
1.4.4 If for each i = 1, 2 , . . . , : Si C Ei is a submodule such that
~i I ...".'" gi I~'"
can be completed only by automorphism of Ei, then the same is true of
@Vii
..."
if and only if each sequence 1 <_ kl < k2 < "" of positive integers and linear maps f,~ : Ek~ -4 Ek~+l with fn(Sk,) = 0 and for each x E Ekl there is an m > 1 such that fm o fm-~ o...o
A (z) = o.
C o r o l l a r y 1.4.5 If for each i = 1, 2 , . . . , , Si C Ei is a submodule such that
Ei" can be completed only by automorphism of Ei, then the diagram
9~i] ........"""
can be completed only by injective endomorphisms. Theorem
1,4.6 Suppose the the class ,u is closed under countable direct sums. If
for each integer i > 1, ~i : Mi -4 Xi is an X-envelope and if @Mi has an X-envelope, then OMi -+ @Xi is an X-envelope. Proof." By Proposition 1.2.4 ~Mi -4 OXi is an X-preenvelope. Let r : ~Mi -4 X be an X-envelope. Then we have the linear map s : @Mi -4 @Xi and the commutative diagram:
~M, = ~M, - eM~
,
25 Hence , r = (fg)r
@~i = ( g f ) ( |
f g is an automorphism since r is an envelope.
But by the previous corollary, g f must be injective. Hence f is an isomorphism. [] We have the following similar results which are dual to the above: T h e o r e m 1.4.7 I f for each integer i > 1, Xi -4 Mi is an X-cover, and if @Xi --+
@Mi is an X-precover (this means that G X i must be in X ) , then @Xi --9 OMi is an X-cover. C o r o l l a r y 1.4.8 For a left perfect ring R (every left R-module has a projective cover)
if Pi --+ ]Vii is a projective cover for each integeri > 1, then |
--+ @Mi is a projective
cover.
R e m a r k 1.4.9 Now we have an alternative proof of the fact that if R is left perfect, then the Jacobson radical J = J ( R ) is left T-nilpotent. Note that R --9 R / J projective cover, hence so is OP~ --9 o P ~ / J ~ ,
is a
where R~ = R, J~ = J for all n. Now
for any countable set {ri E J I i = 1,. 9 n , . 9.}, define f~ : R~ --9 R~+I to be the right multiplication by r~. Obviously fn(P~) C Jn+l. By Theorem 1.4.1, there is an integer m such that fm o . . . o fl(1) = rl 9r 2 " "rm-lrm = 0. Hence J is left T-nilpotent. [] Example:
Let R -- Zp = Homz(Zpo~, Zpoo) be the p-adic numbers. Then
o
2p
Z/(p) --+ o
yields a fiat cover of Z/(p). Here p is the map which is the multiplication by p. Suppose
0 2 p --9 O Z / ( p ) is still a flat cover. Then by a similar argument we see that p~ = 0 for some positive integer n. This is not true. Namely, in general, flat covers are not preserved under direct sums. [] As a conclusion we state a result, which will be used in Chapter 5, about the locally nilpotent property of injective envelopes over a commutative Noetherian ring R. This result was originally proved by Matlis in [54]. Now it follows from the results we have proved in this section. P r o p o s i t i o n 1.4.10 Let R be a commutative Noetherian ring, let p a prime ideal and
E ( R / p ) be the injective envelope of R i p . Then for any x C E ( R / p ) , r E p, there is an integer n such that r '~ 9x = O. P r o o f i It is easy to see that @Ri/pi ~ |
is still an injective envelope, where
Ri = R, pi = p, E i ( R i / p i ) = E ( R / p ) for all i > 1. Define fi : Ei -+ Ei+l to be the multiplication by r. Note that f i ( R / p ) = 0. By Theorem 1.4.4 there is an integer n such that f~ o . . . o f l ( x ) = r ~ ' x = O. [] This result implies that for any finitely generated submodule S C E ( R / p ) ) , and r c p, S will be annihilated by some power of r.
Chapter 2 Fundamental Theorems In this chapter we continue to study the properties of X-envelopes and X-covers by adding conditions on the class X. One of them is to assume t h a t X is closed under extensions. All the conditions we are going to put on the class are satisfied by the most common classes of modules. Of course our special attention will go to the class of fiat modules, the class of injective modules and the class of projective modules. By using a direct limit argument, we will provide a general process to deal with the existence problem of X-envelopes and X-covers. As applications we can easily prove the existence of injective envelopes, projective covers, pure injective envelopes and injective covers in a uniform way. For flat covers, the fundamental results ensure t h a t the exib~ence of flat precovers implies the existence of flat covers. This fact will be crucial in showing the existence problem of flat covers over certain rings in Chapter 3 and 4. In the last section we briefly consider the notion of fiat envelopes and flat preenvelopes. The'main results presented in this chapter are taken from [30, 22, 24, 12].
2.1
Wakamutsu's Lemmas
Assume the class X is the same as before. We say X is closed under extensions provided that for every exact sequence of R-modules : 0 -+ M -+ N --+ L --+ 0 if both M and L are in X, then N C X. For instance, P , E and 9v are closed under extensions. We call the following two results Wakamutsu's Lemmas because Auslander and Reiten mentioned them in [6] without giving a proof. Since we could not locate the original proof in the literature, for completeness we give the proof here. Lemma 2.1.1
Let ~ : X ~
M be an X-cover of M, and assume that X is closed
under extensions. Set K = ker(~). Then E x t ~ ( X ' , K ) = 0 for any X ' C X .
P r o o f ' . By Theorem 1.1.1 it suffices to show t h a t all extensions of K by X r C X are trivial. Suppose 0 ~ K --+ L ~ X ' ~ 0 is such an extension. Consider the induced exact sequence 0 -+ K --+ X --~ ~ ( X ) -~ 0 with the image ~ ( X ) C M. Then we have the pushout diagram of g : K -+ L and a : K --+ X (the inclusion map):
28
0
0
.
0
9L h P ~ ( X ) ~ O
1
X' = X'
I
l
0
0
N o t e t h a t b o t h X a n d X ' are in 2(, h e n c e so is P . Since ~ : X ~ ~ ( X ) r
M is a n
2(-cover, t h e r e is a l i n e a r m a p g : P ~ X such t h a t ~ o g = ~. H e n c e ~ o g f = a o f = ~. T h e r e f o r e by t h e s e c o n d c o n d i t i o n of covers g f m u s t b e a n a u t o m o r p h i s m of X . N o t e t h a t ~ o ( g f ) - l gh = ~ o gh = c~o h = O. We c a n define a l i n e a r m a p u = a - l ( g f ) - : gh : L --+ K . It is easy t o s e e t h a t u o v = a - : ( g f ) 1K. T h i s implies t h a t 0 ~ Lemma
l g h o v = o~ : ( g f ) - : ( g f ) o o ~ = a - : o o e =
K -+ L --+ X ' --+ 0 is split. []
2 . 1 . 2 Let ~ : M ~ X be an 2(-envdope of M, and assume that 2( is closed
under extensions.
Set D = c o k e r ( ~ ) = X / ~ ( M ) .
Then
Ext~(D,X')
= 0 for all
X'EX. Proof:
T h e p r o o f is t h e d u a l of t h a t of L e m m a 2.1.1 w i t h t h e n e c e s s a r y m o d i f i c a t i o n s .
For a n X ' E 2(, c o n s i d e r a n a r b i t r a r y e x t e n s i o n of X ' by D. Set I = i m ( F ) . T h e n we get t h e p u l l b a c k d i a g r a m of h : L --+ D a n d a : X - ~ D as follows:
0 M
0
X' = X'
0
,I
a
,L
0
,I
i
cr D
I
0 F i r s t P is in 2( b e c a u s e b o t h X ' a n d
,0
1
0
X are in 2(.
2(-envelope, t h e r e is a l i n e a r m a p 9 : X ~
0
Since ~ : M ~
I ~-~ X is a n
P such t h a t a o p -- g o i o ~. A n d t h e n
f o a ~ = (.fg) o i o ~, ~ = ( f g ) o ~. T h i s implies t h a t ( f g ) is a n a u t o m o r p h i s m of X . N o t e t h a t / 3 o g ( f g ) - : o ~ = / 3 o g o ~ = t3 o a ~ = 0. So u : D --+ L b y s e n d i n g a ( x ) to ~ g ( f g ) - : ( x )
afg(fg)-:(x)
for a n y x E X is well defined. B u t t h e n h o ucF(x) = h ~ g ( f g ) - l ( x ) = a ( x ) , h o u = l b . T h e r e f o r e 0 - ~ X ' --~ L --+ D --+ 0 is split. []
=
29 Let s be a class of left R-modules. We have the two associated classes: E • = {X 9 RA4 I E x t ~ ( L , X ) = O, L 9 s •163 { X C R M I E x t l ( X , L ) = 0 , L e s
E L is called the right orthogonal class of E, • 1 6 3is called the left orthogonal class of s Concerning the orthogonal operations above, we are interested in the question of when the following is true:
L=•
•
or
•
As before let P be the class of projective modules and $ the class of injective modules. Then it is obviously true that P • = Rill, and •177 = 7). Similarly, • = RA/[ and (• • --- g. But in general there is no explicit description of the classes • and g• By the previous lemmas we know that • is related to 7)- envelopes and g____~_~is related to S-covers. The latter will be discussed later. For the former we refer to Asensio and Martinez [57]. The orthogonal operations are very useful in describing modules. For instance, recall that a left R-module X is called absolutely pure (FP-injective) if X is a pure submodule of every module containing it as a submodule. It is well known that X is FP-injeetive if and only if E x t , ( M , X) = 0 for all finitely presented modules M. In other words if 5rv stands for all finitely presented left R-modules, and ~ P z stands for all FP-injective modules, then ~ T ' z = .Tv • On the other hand a finitely generated Rmodule is finitely presented if and only E x t , ( M , X) = 0 for all FP-injective modules X (see Enochs [29] or Glaz [44, Thm.2.1.10]). Let us return to X-covers and envelopes. P r o p o s i t i o n 2.1.3 If ~ : X -+ M is surjective with X E X and ker(~) E 2( • X is an X-precover of M. Such a precover is called a special X-precover.
then
P r o o f : Let X ' E X and apply H o m ( X I, - ) to the exact sequence 0 --+ ker(~) -+ X --+ M --4 0 Then Hom(X', X) --+ Hom(X', M) ~ E x t l ( X ', ker(~)) = 0 is exact. [] Dually, we have that P r o p o s i t i o n 2.1.4 If ~ : M -+ X is inject•
witJi X E X and D = coker(~) E I X ,
then X is an X-preenvelope. Such a preenvelope is called a special X-preenvelope of M.
30
2.2
Fundamental theorems
D e f i n i t i o n 2.2.1 Let s be a class of left R-modules, and let M be a left R-module. An extension 0 --+ M -+ G --+ L -+ 0 with L E s is called a generator for s
M ) if
for any extension 0 -+ M --+ G --+ L -+ 0 with L E 12, there is a commutative diagram:
0
,M
,G
,L
.0
0
.M
,G
,L
,0
Furthermore, such a generator is said to be minimal provided t h a t any commutative diagram
0
,M
,G
.L
.0
0
.M
.G
,L
.0
always implies that f is an automorphism (so that g is too). Note t h a t if 0 --+ M --+ G --+ L -+ 0 is a generator for Sxt(s M) and
0
,M
0
,M
II
,G
1'
,G'
,L
l
, L'
,0 , 0
is a commutative diagram with exact rows and L' E s
then 0 ~ M --+ G' --+ L' ~ 0
is also a generator. Example:
For a given an R-module M, any exact sequence 0 --+ M --+ E --+ L --+ 0
with E injective gives a generator for E x t ( n M , M).
Moreover if E is the injective
envelope of M, then this generator is minimal. The most interesting question to us is when there is a generator, and when there is a minimal generator for a given class s and module M. We will see t h a t the existence of a generator is related to the existence of an s177
and the existence of a
minimal generator is related to the existence of an s177 Proposition
2.2.1 Assume that the class s is closed under extensions, and assume
that 0 -+ M -+ K -+ L -+0 is a minimal generator for Ext(s P r o o f : For any L E s
then K C f~•
consider an arbitrary extension of K by L, 0 -+ K -+ N -+
-+ 0. Using a pushout diagram, we have the following commutative diagram with exact rows and columns.
31
0
.M
II 0
,M
Since both L and L a r e in 12, P i s i n
0
0
,K
.L-
fish
g~!l
,N
,P-
~
= ~
0
0
.0 .0
/2. Note t h a t 0 - 4
M -4 K -4 L -4 0 is a
generator, so there are linear maps h, l making the diagram commutative. Now by the minimality of the generator, both (h f ) and (lg) are automorphisms. This means t h a t the middle column is split. Therefore KC12 •
E x t , ( L , K ) = 0 for any L E 12, and so
[]
When we have generators, we hope to find the minimal one. We now state one of our main theorems in this section. This can be considered as a sort of Zorn's lemma [24]. Theorem
2.2.2 Assume the class of left R-modules 12 is closed under direct limits.
Then for a left R-module M , if ~xt(12, M ) has a generator, then there must be a minimal generator. The proof will be completed through several steps. We present them as the following three lemmas. The main ingredients of the proof were initiated by Enochs in [30]. L e m m a 2.2.3 Assume that s is closed under direct limits. For an R-module M , if 0 ---4 M --+ N -4 L ~
0 is a generator for Ext(12, M ) , then there is a generator
0 -4 M -4 N -4 -L -4 0 and the commutative diagram
0
,M
,N
,L
,0
0
.M
.N
.~
.0
II
II
such that for any generator 0 -4 M -+ N* -+ L* -4 0 and for any commutative diagram with exact rows
0
.M
.N
.L
.0
0
.M
"N
.L--~O
0
9M---*N* ~ L * - ~
0
32 we have that ker(g) = ker(hg).
P r o o f : We try to derive a contradiction by assuming that such a generator does not exist. Let (0 ~ M ~ No ~ L0 ~ 0 )
= (0 ~ M - - + N - + L - - + 0 ) . By the assumption
there is a generator 0 ~ M --+ N1 --+ L1 --+ 0 such that in the commutative diagram
0
,M
,N
0
,M~N1
"
.L
90
1 , L1I
, 0
glo is not injective. By the assumption again 0 --+ M -+ N1 -+ L1 -+ 0 does not satisfy the desired property. In other words there is a generator 0 -+ M -+ N2 -+ L2 -+ 0 and a diagram
0
.M
,NI~L1
0
.M~N2
,0
.L2
. 0
such that ker(gl0) g ker(g21910) = ker(g20), here g2o = g21glo. By repeating the same process, we have that for any positive integer n there is a generator 0 --+ M ~ Nn ~ L~ --+ 0 and linear maps g~ for all i < n such that for any triple k < m < n, g~k = gnmg,~k; and ker(gl0) g ker(g2o) g ker(g3o) g -.- g .-- C N. From this we easily see that Card(N) _> Card(Z). We wish to demonstrate that the cardinality of N must be greater than that of any ordinal number/3. This is the way to create our expected contradiction. To do so we have to use the hypothesis that the class s is closed under an arbitrary direct limit. For the first infinite ordinal w we form the exact sequence by taking direct limits ( 0 -+ M ~
limM~ ~
limL~ ~
O) and note that since l i m L . C s
we have a
commutative diagram
0
9M
0
,M
,,
, lira N n - - ~ l i m L~
I
,N~-
]
,L~
9 0
, 0
with the bottom sequence a generator. We let g,on : N,~ -+ N~o be the obvious map. For any triple k < n < l(_< w), gta =- glmgmk. Also ker(gn0) g ker(g~o) for all n. Otherwise ker(gno) = ker(g~o) for some n. Choose x E ker(g~+l,o), but not in ker(g~o). Since g~o = g~,~+lgn+l,o, x C ker(g~oo) = ker(gno). This contradicts the choice of the element X.
Since
33 0
,M
0
,M
.N
9L
[
,,
. 0
1
.N~
,L~
, 0
does not satisfy the conclusion of the lemma, we can find a generator 0 -+ M -+ N~+I --+ L~+I --+ 0 and a commutative diagram
0
, M
,N~
, L~
90
0
.M
,N~+I
.L~+I
, 0
so that ker(g~o) c ker(g~+l o) where g~o+l o is the composition N --+ N~ --+ N~+l,O. Proceeding in this manner, given any ordinal 3, we can find generators 0 --+ M -+ No --+ L~ -+ 0 for all a
_< fl, with No = N and with ga0 : N
~
N o so that for A < # _<
fl, ker(g~0) ~ker(g,0). As a consequence, Card(N) _> Card(fl). Since fl is arbitrary, this leads to a contradiction and finishes our proof. [] L e m m a 2.2.4 A s s u m e the class s is closed u n d e r direct limits. g e n e r a t o r 0 --+ M M --+ N ~
~
N -+ L --+ 0 f o r 8 x t ( s
I f there exists a
t h e n there is a g e n e r a t o r 0 --+ M -+ N * ~
L --~ 0 such that f o r a n y g e n e r a t o r 0
L* -+ 0 a n d a n y
c o m m u t a t i v e diagram
m
0
.M
,N
"L
0
, M---~N*---~L*
,
, 0
Is 90
g m u s t be injective.
P r o o f i By the preceding lemma there is a generator 0 -+ M -+ N1 -+ LI --~ 0 having the property such that in any commutative diagram with exact rows and N*, L* E s
0
,M
,N
.L
,0
0
,M
, N1---~L1---,- 0
0
, M---~N*~L*
,0
34 ker(hg) = ker(g). R e p l a c i n g 0 - 4 M -4 N -4 L -4 0 by 0 ~ M - 4 N~ -4 L1 -4 0, we can find a g e n e r a t o r 0 -4 M --, N2 -4 L2 -4 0 having the same stated property. T h e n , by the s a m e procedure for all n we can find 0 -4 M -4 N~ --+ L~ -4 0 such t h a t for any g e n e r a t o r 0 -4 M -4 N* -4 L* --+ 0 and any d i a g r a m
0
.M----~N~L~
0
*M~N,~+I*L~+I~ 0
0
,M~N*
II
1
90
1
.L*
,0
we have ker(hg~+t,~) = ker(g,~+t,~). Now let
. M
, lim N~lim
L~
ii
, 0
]
9M
,N~
.L~
, 0
be c o m m u t a t i v e w i t h the b o t t o m row a g e n e r a t o r and let g ~ obvious maps.
: N~ -4 N~ be the
We claim t h a t this g e n e r a t o r has the desired property.
Consider a
commutative diagram
0
,M
O~M
II
,N~,~L~
1
1
,N*---~L*
, 0 90
If h is not an injection, there exists an x r 0 E N~ w i t h h(x) = 0. Since x = 9~,~(x~) for some x,~ C Nn, h(x) = hg~,~(x~) = 0. Note t h a t h g ~ = hg~,n+~g,~+l,n. T h e r e f o r e we have ker(hg,,,~+lgn+],,~) = ker(g~+l,~). Hence x~ C ker(g~+l,~), and x = g ~ ( x ~ ) =
9~,,~+lgn+l,n(X,~) = 0. T h i s yields a contradiction. [] Lemma
2 . 2 . 5 Assume the class s is closed under direct limits. I f 0 -4 M -+ N -+
L -4 0 is a generator having the property stated in the previous lemma, then it is a minimal generator Proof:
Suppose this were not true. T h e n for (0 -4 M -4 N1 -4 L1 - 4 O) =
(0 -4 M - 4 N ~ L -4 0), there is a c o m m u t a t i v e d i a g r a m
O----~M
,N
0
,N1---*L1
~M
,L
,0 90
35 such t h a t 910 is injective, but not surjective. Again set (0 -+ M --+ N2 -+ L2 --+ 0 ) = (0 --+ M -+ N -+ L -+ 0). There is a commutative diagram 0
"M---~N1
0
,M~N2
II
.L1
. 0
Ig~'l 1 .L2
90
in which the map g2,1 is injective, but not surjective. Let g20 = g21g10, then g2o(N) 9 g~l(N) C N. In general for all integers n there is a commutative diagram with a linear m a p gn+l,n which is injective, but not surjective. There are also a family of maps g,l for all l < n such that for any triple k < m < n, g,k = gnmgmk. Therefore g~o(N) 9 g~,l(N) ~ g~,2(N) ~ . . . ~ g. . . . 2(N)
C g ....
I(N) C iV for any positive integer n. This
shows t h a t card(N) > n for any integer n, and then C a r d ( N ) _> Card(w), here w is the first infinite ordinal number. We intend to deduce t h a t
C a r d ( N ) > Card(/3) for
any ordinal number/3, which obviously is impossible. For an a r b i t r a r y ordinal number /3, suppose we have built up all those maps g~x with the properties stated previously for all ordinal numbers A < /3. If/3 is not a limit ordinal, then this is easy by using the assumption. Assume /3 to be a limit ordinal number.
First take a direct limit
I
t
(0 --+ M --+ N'~ --~ L z --+ 0) =(0 --+ M --+ limMx~ -+ limL~ --+ 0), and define g~x to be the canonically induced maps for all A < /3. Since (0 -+ M --+ N --+ L --+ 0) t
is a generator , there is a pair of maps (h, p) from (0 --+ M --+ N~ --+ LZ --+ 0 ) to (0 --+ M --+ ?/1 --+ L1 -+ 0) making the associated diagram commutative. Now define (0 -+ M --+ N~ --+ LZ --+ 0) =(0 --+ M --+ N --+ L ~
t
0) and let gz~ = hgz~ for
all ordinal A < /3. If gz~ is surjective, then the equation 9B,~+lg~+l,~ = 9Zx shows t h a t gz,~+l is isomorphism, so is 9~+Lx. This is a contradiction. Now for any pair of a < A (_< /3), 9xo is injective, but not surjective; for any triple a < A < p ( < /3), gu,~ = guxgx~. Consequently for any sequence of ordinal numbers ...... <#
9. . < s < t < v < . . . < gz,s C g~,t c
It follows from this t h a t
......
......
c gza c gz~ ~ . . . . . . C N
C a r d ( N ) _> Card(/3) for any ordinal numbers.
We are
through. [] Now we are able to see the explicit relation between envelopes and minimal generators. This relation will serve us as a bridge between these two different subjects. Theorem
2.2.6 Assume the class s is closed under extensions and under direct lira-
its. For a given R-module M , if s 1 6 3
M ) has a generator, then M admits an g •
envelope.
Proof." By Theorem 2.2.2 we have a minimal generator 0 --+ M --+ K --+ L --+ 0 for s163
M). By Proposition 2.2.1, K C s177 First we need to ensure t h a t K is an s 1 7 7
preenvelope. But this is true because E x t , ( L , K ' ) = 0 for all K ' C s
Finally the
36 minimality guarantees t h a t the second condition of being an s177 []
is satisfied.
There are varied versions of the above results. The proofs are the obvious modifications of the corresponding ones above, so we only give the statements below. Proposition
2.2.7 Assume s is closed under extensions. I f L --+ M is an s
of M , then K E s 1 7 7where K = ker(L -+ M). Theorem
2.2.8 Assume s is closed under direct limits.
I f M has an s
then M has an s The proof is similar to that of Theorem 2.2.2. We now state the three lemmas necessary for the proof. L e m m a 2.2.9 Assume s is closed under direct limits. I f L -4 M is an f_,-precover of M , then there is a precover L --+ M and a commutative diagram
L
gl
.M
I1 .M
such that for any precover L* --+ M and any commutative diagram
,M
hI
II
L*
,M
we have ker(hg) = ker(g). L e m m a 2.2.10 Suppose s is closed under direct limits. I f M has an s
then
there is a precover-L ~ M such that for any precover L* --+ M and any commutative diagram
.M
hi
II
L*
.M
h must be an injection. L e m m a 2.2.11 I f f~ is closed under direct limits and s --+ M is an s satisfying the condition of the previous lemma, then -L --4 M is an s
of M
37 Combining the results above we can obtain the following 2.2.12 If s is closed under direct limits and M has an s then M has an s L --9 M . Furthermore if s is clo~ed under extensions then K C s177 where K = ker(L -+ M).
Theorem
Remark
2.2.13 We have seen that the existence of covers and envelopes is deeply
dependent on the structure of the class X. One of the crucial points is the property t h a t the class is closed under direct limits. This property is of global nature, and it is also useful in finding minimal generators and hence in finding some kinds of envelopes. However, there arc different approaches to finding generators. In next section we will see how these fundamental results can be applied to prove the existence of all the well known envelopes and covers.
2.3
Applications
As a direct application of the fundamental results obtained in the last section, Eckmann-Schopf's theorem [25] on the existence of injective envelopes can be recast. Theorem
2.3.1 For any ring R, every left R-module has an injective envelope.
P r o o f : W i t h the same notation as before, by Theorem 1.2.11 the existence of injective envelopes is equivalent to the existence of g-envelopes. Set s = RAA in Theorem 2.2.6. Note t h a t g = •
For any left R-module M, by Theorem 1.1.8 M can be embedded
into an injective module E, in other words, there is an exact sequence as follows
O--+ M - + E - + L--+ O with L C R M .
Trivially this exact sequence provides a generator for gxt(R.h4, M).
By Theorem 2.2.6 we are through. 0 . Eckmann-Schopf's theorem encouraged us to study project covers. The most interesting result along this line is the Bass' Theorem P. We now see t h a t a part of this theorem on left perfect rings can also be recast as another application of the results on general envelopes and covers discussed in this chapter. Recall t h a t an associative ring is called left perfect if every left R-module has a projective cover (which is the same as a P-cover, where P is the class of projective modules). Also recall that a set S C R is called left T-nilpotent if for any countable subset {ai E S ] i >_ 1}, there is an integer n such t h a t ata2 . . . . . . an = 0. Let J = J(R) be the Jacobson radical of R. We need the following result on T-nilpotence. L e m m a 2.3.2 The following are equivalent:
(1) J is left T-nilpotent, (2) J M ~ M for any left R-module M .
38 Remark
2.3.3 For the proof, see Anderson and Fuller [1, Lem.28.3 p.314]. The proof
is purely ring theoretic. Theorem
2.3.4 For any associative ring R the following are equivalent:
(1) R is left perfect; (2) every fiat left R-module is projective;
(3) l ~ p~ is projective for ~ y direct system {Re, ~ } P r o o f : (3) ~
with P~ projective.
(1) Set t: = "P in Theorem 2.2.8. Note that every left R-module M is
an image of a projective module P , i.e., there is an exact sequence O--4 N ~ P - ~ M ~ O
Then P --+ M is a P-precover. By the assumption (3) and Theorem 2.2.12, M has a projective cover. (1) ~
(2) For any flat left R-module F , by (1) there is a projective cover of R,
: P -+ F . We claim t h a t K = ker(~) = 0. By the Remark 1.4.9 , J is left T-nilpotent. Since K < < P and J . P = { ~ L I L < < P } (see Anderson and Fuller [1, Prop.17.10, Prop.9.13]), K < < J 9P. Note t h a t F is flat. By Theorem 1.1.5, K N J P = J . K. Hence K = J 9K. It follows from the preceding lemma that K = 0. (2) ~
(3) It is immediate because a direct limit of fiat modules is always fiat. []
Here we would like to mention the class of flat R-modules 9~. Obviously this class is closed under direct limits and extensions. So the fundamental theorems are applicable. But we postpone stating results on the existence of flat covers until the next chapter, where we will concentrate on this problem. Let us change our attention to another kind of envelopes of common interest, ..pure injective envelopes.
Fuchs and Warfield introduced and investigated pure in-
jective envelopes in [41, 71] and proved that every module admits a pure injective envelope. Pure injective modules are related to algebraically compact modules. They were first used to study compact Abelian groups. In [63, 50] Jensen~ Gruson and Raynaud developed a relative homology theory by use of pure injective resolutions. The latter led to a way to prove a very deep result on the projective dimension of a flat module ([63, Cot. 3.2.7] or [50, Cot. 7.2]). We will give a proof of this result in the case of commutative Noetherian rings of finite Krull dimension in C h a p t e r 5. There we prove the existence of fiat covers over such rings by use of a Gruson-Jensen theorem and the whole theory we will have developed by t h a t time. We make a brief review of pure injective modules [71]. First we need some notions. Recall t h a t an exact sequence of left R-modules 0 --+ N ~ M -+ L --+ 0 is called pure if for every right R-module S, O -~ S | R N --+ S | R M --+ S | R L -+ O
39 is still exact.
In this case, we say that N is a pure submodule of M and t h a t M
is a pure extension of N. There are many equivalent characterizations of pure exact sequences.
For more details, see Warfield [71]. A left R-module P is called pure
injective if every diagram
0
,N
f[,
"M
.L
.0
P with the upper row pure exact can be completed to a commutative diagram. Equivalently, HomR(M, P), ~ HomR(N, P ) ~ 0 is exact whenever N is a pure submodule of M. Note t h a t all injective modules are pure injective. Also for any right R-module N, the characteristic module N* = H o m z ( N , Q / Z ) is a pure injective left R-module. For future use we state the following result [71, 44]: Proposition
2.3.5 For any left R-module M, the canonical evaluation map a : M -+
M** = H o m z ( H o m z ( M , Q / Z ) , Q / Z )
is pure, i.e., M can be purely embedded into a
pure injeetive module. Furthermore M is pure injective if and only if M is a direct summand of M**. Let ~oE be the class of all pure injective left R-modules. For any left R-module M, a T'E-envelope of M is called a pure injeetive envelope, and is denoted by P E ( M ) . Notice t h a t ~ : M -+ P E ( M ) must be injective. This is because every module can be embedded into an injective module and every injective module is pure injective. So then the corresponding commutative diagram will imply t h a t ~ is injective. Again by applying the fundamental results, we will give a proof of the existence of pure injective envelopes and show the consistency between the current approach and the original description of pure injective envelopes. But some technical changes are necessary. For a given R-module M, a pure extension of M, 0 -+ M -+ P -+ L -+ 0, is called a generator if for any pure extension of M, 0 -+ M -+ P ' -+ L' -+ 0, there is a commutative diagram:
0
'M
0
,M
"P'
It
I .P
9L'-----~ 0
1 ,L
.0
Similarly we can define a minimal generator among all the pure extensions of M. The following result is due to Warfield [71]. Proposition
2.3.6 Every R-module has a pure injective preenvelope.
40 P r o o f i It is easy to verify that the naturally induced exact sequence
0 --+ M2+M ** --+ D --+ 0 is the expected preenvelope. Here M** = H o m z ( H o m z ( M , Q / Z ) , Q / Z ) . [] The following property is very important for our purpose. It easily follows from the definition of pure extensions and the exactness of direct limits. See R o t m a n [66, Cor.2.20]. Proposition
2.3.7 For a given module M and a directed index set I, if the pure
extensions of M { 0 --+ m - - + N i --+ Li --+ O ; f j i : N i -+ N j , 9 j i : L i --+ Lj l i < j ; i , j
CI }
form a direct system where {f3~, 9ji} make the corresponding diagrams commutative, then the direct limit 0 --+ m -+ lira Li --~ lim Ni --+ 0 is still a pure extension of M. We now are ready to state the theorem. Theorem Remark
2.3.8 Every R-module M has a pure injective envelope. 2.3.9 The procedure of the proof is similar to t h a t of Theorem 2.2.6. But
we remark t h a t the class P g is closed neither under extensions nor direct limits. We will determine when pure injective modules are closed under extensions in the next chapter. Fortunately, we note t h a t the class of pure extensions of M is closed under direct limits, we can state and prove the three corresponding lemmas without any difficulty, and so furnish a proof. In [71] Warfield gave an internal description of pure injective envelopes which is analogous to t h a t for injective envelopes. Recall that an extension of M, 0 --+ M --+ P --+ D --+ 0 is ealled essentially pure provided it is pure, and there is no nonzero submodule S C P having the property t h a t S fl M = 0 and - ~ - C M pure. The next result exhibits the consistency between the two different approaches. Theorem
2.3.10 For a 9iven R-module M the followin9 are equivalent:
(1) ~ : M --+ P is a pure injective envelope of M; (2) ~ : M --+ P is essentially pure with P pure injective. P r o o f : (1) ==~ (2) We first see that ~ is pure. Consider the commutative diagram M ~,.P M**
41 Here f is available because M** is pure injective and P is a pure injective envelope of M. But since a is pure, a = f ~ implies that ~ is pure. We next claim t h a t the extension is essentially pure. Let S C P be a submodule with S V) M = 0 and M~S S pure in P / S . Consider the diagram P 0
P Where g exists because the middle row is pure andL P is pure injective. But then it is easily seen t h a t ~o = (gh)~o. This implies that gh is an automorphism, and h is injective which is impossible unless S = 0. (2) ~
(1) Since P is pure injective and 0 -~ M .-~ P is pure, the latter is a pure
injective preenvelope of M. By Theorem 2.3.8 M has a pure injective envelope, say M -+ P ' . Then there is a commutative diagram
M
, P'
In M
In M
Ih ,P
Ig 9P '
Since M --~ P ' is an envelope, g o h is an automorphism of P ' . So S = ker(g) is a direct s u m m a n d of P and so is pure in P. Since M N S = 0, S = 0 by (2). Hence g is an isomorphism and so M --+ P is a pure injective envelope. []
2.4
Injective
covers
We have used the class of injective modules to define injective envelopes. We also can use this class to define injective covers (which are not the n a t u r a l duals of injective envelopes). But it turns out that the existence of injective covers is quite interesting. In this section we will present some results about injective cover as another application of the fundamental theorems. As before $ is the class of all injective left R-modules. For any R-module M, an C-cover (precover) is called an injective cover (precover) of M. The existence of injective covers has a tie to the left Noetherian hypothesis on the ring. The following theorem is due to Teply [45]. Theorem
precover.
2.4.1 If R is left Noetherian, then every left R-module admits an injective
42 P r o o f : Since R is left Noetherian, by Matlis' Theorem (see Theorem 1.1.7) the representatives of indecomposable injective modules (up to isomorphism) form a set, say {E~ I # E Y}. Hence, for a g i v e n R-module M a l i n e a r map p : E ~
M with E
injective will be an injective precover if every linear map ~' : E , --+ M with E , E Y can be factored through ~, i.e., the diagram
.." .'"" 1~ I E' P,M can be completed to a commutative one. Set H , = H o m a ( E , , M), E ~ = E(~H.) (the direct sum of the copies of E , indexed by elements of H , ) .
Define E = G E " , # e Y, and ~ : E ~
M with p((e~,)) =
h~(e~,), h , E Hom(E~, M). It is easy to check that ~ : E -+ M is an injective precover of M. [] Note t h a t if ~ : E -+ M is an injective precover of M, then N = ~ ( E ) has the property t h a t every linear map ~ : E ~ --+ M with E ~ injective can be factored through if and only if every linear map pr : E ' -~ N with E ~ injective can be factored through : E --+ N. It follows from this observation that if M has an injective cover then N has one. But then for any injective precover E --+ M of M there exists an exact sequence 0 --+ K --+ E -+ N -+ 0 with K = ker(~) and N = ~ ( E ) . Also note t h a t over a left Noetherian ring R every direct limit of injective left R-modules is still injective. The class E is also closed under extensions. This makes it possible to apply the fundamental results to derive the existence of injective covers. The following is due to Enochs [30] Theorem
2.4.2 Let R be a left Noetherian ring. Then every left R-module M admits
an injective cover. P r o o f : By Theorem 2.2.8 and Theorem 2.4.1. [] Now the existence problem of injective covers can be completely solved as follows: Theorem
2.4.3 R is left Noetherian if and only if every left R-module has an injective
cover.
P r o o f i We only need to show the necessity. For any family of injective left R-module {E~}, a E I , consider the direct sum @Eo. By the assumption, ~ E a has an injective cover, say ~ : E --+ @E~. For each injection q~ : Ea -+ ~ E o , there is a linear m a p f~ : E~ -+ E such t h a t q~ = ~f~. It follows from this that 1 = Sq~ = ~(@f~). This shows t h a t @E~ is a direct summand of E. So it is injective. Now by Theorem 1.1.7 R is left Noetherian. []
43 Remark
2.4.4 An injective cover ~ : E ~ M is not necessary surjective. Actually
it may be just zero. Nontrivial examples of injective covers are hard to come by. The following given by Cheatham, Enochs and Jenda in [67] provides a nontrivial example. Example
Let (R, m) be a commutative, Noetherian and complete local ring. Assume
t h a t the depth of R is greater or equal to two. Then for the residue field k = R / m , the canonical m a p ~ : E(k) -+ E ( k ) / k is an injective cover. P r o o f : Since E(k) is indecomposable, by the properties of injective covers it suffices to prove t h a t the natural surjection ~ : E(k) -+ E ( k ) / k is an injective precover. But then we only need to verify that E x t , ( E , k) = 0 for all injective modules E. First assume t h a t E -- E ( R / p ) for some prime p. If p ~ m, there is an element r 6 m, but not in p. Define r : E(R/p) -+ E(R/p) to be the multiplication. It is an isomorphism, and so it induces an isomorphism on Ext~(E(R/p), k). Note t h a t r. ExtlR(E(R/p), k) = O. This implies t h a t Ext~(E(R/p), k) = O. For E = E ( R / m ) = E(k), consider any extension of k by E(k): 0 -+ k ~ G -+ E(k) -+ O. Taking the MatiNs duals, we have an exact sequence : 0 -+ H o m ( E ( k ) , E(k)) -+ Uom(G, E(k)) -+ Hom(k, E(k)) -+ 0 Equivalently, we have an exact sequence 0 -+ R -+ Horn(G, E(k)) -+ k -+ O. Since
depthR >_ 2, EXtlR(k, R) -- 0. This shows that the above sequence is split. Now the sequence 0 --~ Hom(Hom(k, E(k)), E(k)) -+ Hom(Hom(G, E ( k ) ) , E(k)) --+ H o m ( H o m ( E ( k ) , E(k)), E(k)) ~ 0 is split, and so is the sequence 0 ~ k ~ G --+ E(k) -~ 0. T h a t is E x t ~ ( E ( k ) , k) = 0. Finally for any injcctive module E, by the Matlis structure theorem E ~- @E(R/p). It follows fi'om this t h a t Ext]~(E, k) = 0. [3. E x a m p l e Let R = Z be the integers. Then recall t h a t an Abelian group is injective if and only if it is divisible. Also, any homomorphic image of a divisible group is divisible. Hence every Abelian group G has a largest divisible subgroup D, and so D -+ G (the canonical injection) will be an injective cover. If, for example, G is finitely generated, then D = 0. This shows that any finitely generated Abelian group has zero as its injective cover. [] Now we are interested in the question of when every nonzero M has a nonzero injective cover, or more generally when such an M has a nonzero linear m a p ~ : E ~ M with E injective. Surprisingly this condition is restrictive enough to guarantee t h a t the ring R must be Artinian. Before we give the next result, we digress a little and give a different approach to left perfect rings. Recall that for any element a 6 R, the left annihilator l(a) = {x 6 R ] x . r = 0}. The following result is taken from [12]. A
44 left ideal I is said to be nil if for any element a C I, there is an integer n > 0 such t h a t a n = 0. We have the following Theorem
2 . 4 . 5 Let J = J ( R ) be the J a c o b s o n radical. Suppose for every a E R, the
set of left annihilators
l(a~), n _> 1 satisfies the ascending chain condition ( A C C ) .
Then if every nonzero left R-module M has a nonzero linear map ~ : E -+ M with E injeetive, then J is a nil ideal. Proof:
By Zorn's l e m m a there is a m a x i m a l nil-ideal N contained in J .
t h a t N = J.
If J / N
m a p (p : E ~
J/N
HomR(E, J/N).
C e r t a i n l y the trace T r j / N ( E ) is a s u b m o d u l e of J / N .
We claim
~ O, t h e n by the a s s u m p t i o n there exists a nonzero linear with E injective.
an ideal of the residue ring R / J .
Define the trace T r j / N ( E )
= E h(E),h
C
Moreover it is
Hence T r j / N ( E ) = N 1 / N for an ideal N1 c o n t a i n i n g
N and N1 ~ N . For any element a C N1, by the a s s u m p t i o n there is an integer n such t h a t l(a ~) =
l(a2n).
Set ~ = a ~ + N = ~ .
hi,..., ht
Then ~ C ~'j/N(E)
and so there are linear m a p s
such t h a t ~ = h i ( e l ) + " " + ht(et) for ei C E ,
1 _< i <: t.
Let e* =
el + 9 9 9+ et, E* = E1 9 " 99 Et and h = hi 9 9 9 9(~ h t. T h e n h is a linear m a p onto J / N w i t h h(e*) = ~. Now we define a m a p f : R c 2 --+ E* by f (rc 2) = rc . e*, r E R. If rc 2 = O, r E l(c 2) = l(c). This shows t h a t r . c = 0 and f is a well defined linear map. Consider the c o m p l e t e d c o m m u t a t i v e d i a g r a m
0
,Rc 2
,.R
E* where g is available since E* is injective. Let e~ = g(1) E E*. So c-e*=
f ( c 2) = g(c 2) = c2g(1) = e2e*l.
T h e r e f o r e h(ce*) = h(c2e*l), ~2(i - h(e*l)) = 0 c J / N
c R/J.
N o t e t h a t 1 - h(e*l) is
invertible. T h i s implies t h a t ~2 = 0. C o n s e q u e n t l y a 2~ = c 2 E N is nilpotent, so is a. Now we have N1 = N which contradicts the choice of N1. [] We need an l e m m a which serves a preliminary for the next result. T h e m a i n ideas of the p r o o f were suggested in [20] by Dischinger. L e m m a 2 . 4 . 6 For any associative ring R, if x, y, z and d c R satisfy the following (i) x2y = x; (ii) y2z -- y; (iii) (z - x)k+ld = (z -- x) k for some positive integer k, t h e n x . y . x -- x.
45 P r o o f : Using (i) and (ii), we easily see that (1)
xz = x2
(2)
~yz = x.
By equation (1) we have ( z - x ) 2 = z 2 - z x - x z + x 2 = z2 - zx, and so ( z - x ) 2 = z( z - z ) . Then an inductive argument shows that for all positive integer n, (3)
(z
-
x)" = z n - l ( z
-
x).
Now by this and equations (1) and (2), it is easy to see that (4)
xy(z-x)
2=xyz(z-x)=xz-x
2=0.
Using hypothesis (ii) as the basis for another induction argument, we get y~z n-1 = y for all positive integers n. Hence by equation (3), it follows (5)
y n ( z _ ~)n = y ~ z ~ - l ( z _ x) = y ( z -
~).
Then using equation (3), hypothesis (iii) and equation (5), we have (6)
y ( z - x)2d = y k z k - ' ( z -- x)2d = yk(z - x)k+'d = yk(z - x) k = y ( z - x)
By equation (5) and (6), y2(z - x) = y2(z - x)2d = y(z - x)d. Using equation (6) and (4), we get that 9 ~ ( z - x ) = z y ( z - ~ ) 2 d = 0,
xyz = zyx.
Finally by this and equation (2), it follows that x y x = x. [] The next result provides a different view of looking at perfect rings by using injective modules. As we know, these rings were originally described by projective modules and flat modules. T h e o r e m 2.4.7 I f R satisfies the following conditions, then R is a left perfect ring. (a) Every nonzero left R-module there is a nonzero map ~ : E -+ M with E injective. (b) For any sequence {a~ C R t n >_ 1}, the set of left annihilators {l(al . . . . .
an)
I n >_ 1} has ACC.
(c) The union of any chain of left T-nilpotent ideals is still left T-nilpotent.
P r o o f : We will first show that the Jacobson radical J is left T-nilpotent. By the condition (c) and a trivial application of Zorn's lemma there is a maximal left Tnilpotent ideal N C J. If .] ~ N , then by the condition (a) there exists a nonzero linear map ~ : E -~ J / N with E injective. As in the proof of the previous theorem, set T r j / N ( E ) = L / N , where L is an ideal of R strictly containing N. By the condition (b) for any sequence {an E L I n >_}, we can find an integer n~ > 0 such that
46 I(al . . . a ~ - l ) = l(al . . . an~) . Set ~n~ = an1 + N E J / N . By the definition of T r j / N ( E )
there are linear maps h i , . . . , ht(et)
ht
and elements e l , . . . , et E E such t h a t hi(e]) + ' "
+
-- an1. Let E* be the direct sum o f t copies of E, h = Ghi and e* = el + - ' "+et.
Hence h : E* ~ J / N with h(e*) = ~ . Now define a linear m a p f : R a l . . . a ~ ~ E* by f ( r . a l . . , an~) = ral . . . a ~ _ l e * , r C R. It is easy to verify t h a t f is a well defined linear map, and then it can be extended to a linear m a p g : R -+ E*. If we set e~ = g(1) C E*, then al'"anl-le*
=f(al...an~)
=al'"anl"
e~.
Hence, acting by the linear map h, we get al""anl-l~nl
It follows from this that a l . . . a n l
=al'"anlh(e~)
= 6 E J/N
and then that a l ' " a n l
= bl E N .
Similarly for the remaining sequence {ak ] k _> nl + 1}, we can carry out the same procedure and find an integer n2 such that a n ~ + l ' " a n - 2 = b2 C N , etc. Therefore we ant ] t > 0} in N. By the nilpotence of N
have obtained a sequence
{b t = ant_~+l
this implies t h a t bl 9 99bt
0 for some positive integer t and then that al . . . . . . ant -- 0.
=
9""
This shows t h a t L is left T-nilpotent. This contradicts the maximality of N. In order to complete the proof, we recall that a module is simple if it does not have any nontrivial submodules and a module is called indecomposable if it does not have any nontrivial direct summand. By Bass' Theorem P it remains to prove t h a t R / J is Artinian semisimple. First note t h a t since J is a nil ideal, a set of finite orthogonal idempotent elements in R / J can be lifted to a set of orthogonal idempotent elements in R. Therefore by the correspondence between decompositions of R / J and sets of orthogonal idempotent elements in R / J , and by the condition (b) R / J should have an indeeomposable decomposition. Let such a decomposition b e / ~ = R / J = A1 @ ' " @ At, where each Ai is indecomposable. In order to show t h a t / ~ is semisimple, we only need to prove that each Ai is a simple module. For convenience we simply replace each Ai by A. Let H C A be a nonzero submodule.
Since A is indecomposable, in order to
show t h a t H must be A itself, we only need ensure that H contains a nonzero direct s u m m a n d of A. We will see that such a direct summand can be created by a special idempotent element. By the condition(a) there is a nonzero linear map ~ : E -+ H with E injective. Set TrH(E) = I / J ~ O. Here I is an ideal strictly containing J. We pick up an element a E I but not in J such that a is not nilpotent. Repeating the same argument used before , we can easily find an integer n such that ~n+l = ~n+2~ for some b E 'I~rH(E). For b we can also find integer m and an element ~ C T r H ( E ) such t h a t b'~ = ~m+l~. Hence we have the following equations: ~(n+l)-t-m,-}-(n-I-1)-t-l~m+(n+l)+l
: (~n+l
47 ~2(m+(n+l)+l)~m+(n+l)+l :
~(n+l)+m+l
Similarly, ~2(m+(n+l)+l)ffm+n+l+~ = ~m+~+l+l. Set x = gin+n+2 y = bm+~+2, z = ~+n+2.
Then x , y and z satisfy: (i) x2y = x and (ii) y2z = y. Note that z - x
TrH(E ).
By the same process there is a positive integer k and an element d E R / J
e
such t h a t (iii) (z - x)k+ld = (z -- x) k. Now by Lemma 2.4.6 it follows t h a t x = xyx. As usual, set w = yx. Obviously w :~ 0, w 2 = w. Hence /~ = R w |
where
/~w c H C A. By the modular law A = / ~ w @ ( K N A). This means that A = / ~ w C Rg C H C A, and then that A = H. Therefore we have proved that each Ai in the decomposition o f / ~ is simple, i.e.,/~ = R / J is semisimple. This completes our proof. [] As a direct application of the above result we have a necessary condition for the existence of nonzero injective covers over a ring R. 2.4.8 For any associative ring R, if every nonzero left R-module has a nonzero injective cover, then R must be left Artinian .
Theorem
Proof:
By Theorem 2.4.3 R is left Noetherian.
Then all the conditions in Theo-
rem 2.4.7 are satisfied. This implies t h a t R / J is Artinian and J is nilpotent. The conclusion follows from Hopkin's Theorem (see Anderson and Fuller [1, Thm.15.20]). [] Remark
2.4.9 Note that the conditions (b) and (c) in Theorem 2.4.7 are necessary
for a ring R to be left perfect. But the following example shows t h a t the condition (a) is not necessary. In a certain sense the condition (a) is stronger than the others. But it is not strong enough to imply the condition (b) or (c) since any self-injective ring satisfies (a). However the conjunction of (a) and (b) may imply (c) in some cases, at least for commutative rings. In general we do not know whether the condition (c) can be dropped or not. E x a m p l e Let F b e a f i e l d .
Letell=
(10) 0 0
,e12--
(01) 0 0
,e22=
(00) o 0
1
"
R = Fell +Fei2 +Fe22 ={ ( a 0 cb ) [a,b, c E F } , t h e u p p e r t r i a n g u l a r m a t r i e e s r i n g over F . It is well known that R is left Artinian. But we claim that the condition (a) is not satisfied by R. Since R = Rell 9 (Rel2 + R22), Rell is projective, simple. Suppose M = Rex1 admits a nonzero injective cover. Then it is easy to see that M must be a direct s u m m a n d of its injeetive cover.
So M is injective itself. However the linear m a p
f : Rel2 --~ Rell given by f(re12) = relx,r C R can not be extended. For if there is an extension g of f with g(1) = ( a 0 O 0 ) inRell, thenf(el2)r e12g(1) = 0 C R. []
gutg(el2)=
48 For commutative rings this will not happen. In fact we have a characterization of commutative perfect ring. Theorem
2.4.10 For a commutative ring R then following are equivalent:
(1) R is perfect; (2) Every nonzero R-module M there is a nonzero linear map p : E -+ M with E injective, and for any set {an E R ] n _> 1} the set of annihilators { l ( a l . . . an) ] n > 1} has ACC. P r o o f : (1) ~
(2) Since R has DCC on principal ideals, for any sequence {an [ n >_ 1}
there is an positive integer n such that R a l . . . a n
= R a ~ . . . a n + k for all integer k _> 0.
This obviously implies that l ( a l . . , an) = l ( a l . . . a n + k ) , k > 0. Hence it remains to verify t h a t for every nonzero R-module M there is a nonzero linear map ~ : E --+ M with E injective. Consider the ring decomposition : R = R1 |
| R t , where each Ri is a local ring
with the unique maximal ideal mi. For any nonzero R i - m o d u l e Mi, the simple module Si = R i / m i can be embedded into Mi. Let Ei = E ( M , ) be the injective envelope of M~
as an P~-module. Note that Ei is also injective as an R-module and m i . Ei ~ O. Then the composition Ei --+ Ez/m~E~ --+ Si --+ Mi gives us a desired nonzero linear map for Mi. Therefore for any nonzero R-module M, there exists a canonical decomposition M = M1 | " " |
Mt with Mi = R i M . Now the conclusion follows easily.
(2) ==~ (1) We first show that J is T-nilpotent. For any sequence {an C J ] n _> 1}, by the hypothesis there is an integer n such t h a t l ( a l . . , an) = l ( a l . . . a n , an+l). Note t h a t by Theorem 2.4.5 J is a nil ideal. Hence we have that an+ l k = 0 for some positive integer k. Trivially a nk+ l a l 9 9 9a n = a nk+-l l a 1 99 9 a n 9 a n + l = 0. But this implies t h a t k-1
a ~ + l a l . . . a ~ 9 = 0. Then a standard argument shows that we must have a l . . . a n
.
a~+l = 0. Finally by Theorem 2.4.7 R is perfect. [] As a direct consequence, we have C o r o l l a r y 2.4.11 Let R be commutative Noetherian. Then R is Artinian if and only for every nonzero R-module M there is nonzero linear map ~ : E --~ M with E injective.
2.5
Flat envelopes and preenvelopes
As before 9v stands for the class of all flat left R-modules. For a left R-module M, an ~'-envelope of M is called a flat envelope of M, and an 9V-preenvelope is called a flat preenvelope of M. In this section we will see t h a t the existence of flat preenvelopes is equivalent to R being a right coherent ring. However the existence flat preenvelopes does not imply the existence of envelopes in general [57, 59, 58]. For more details about flat preenvelopes and relative homologies, see Akatsa's thesis result is due to Enochs [30].
[69]. The next
49 T h e o r e m 2.5.1 For a ring R every left R-module M has a fiat preenvelope if and only if R is right coherent. P r o o f : For the "only if", let Fi, i E I be a family of fiat left R-modules. Then by the hypothesis the product l-I F~ has a fiat preenvelope qa : I] Fi -+ F with F fiat. For each i, consider the obvious diagram
l-IF/ %..F P/I J~/ F/" where p/is the canonical projection. The linear map fi is available because ~2 : l-I F~ -+ F is a flat preenvelope. Note that YIp~ is the identity. So I ] F i must be a direct summand of F. Hence I-[ F~ is flat. Since an arbitrary product of flat left R-modules is flat, R is right coherent. Conversely, assuming R is right coherent, we will construct a flat preenvelope for a given left R-module M. We need a lemma. L e m m a 2.5.2 For any ring R, if S C M be a submodule, then S can be enlarged to a submodule S* such that S* is pure in M and the cardinality of S* is less than or equal to Card(S).Card(R) /f either of Card(S) and Card(R) is infinite. If both are finite, there is an S* which is at most countable . P r o o f i We assume one of Card(S), Card(R) is infinite. We easily see how to modify the argument in case both are finite. First note that a submodule N c M is pure if and only if the solvability of system
equations ~
a i j x i = n j , n j E N, aij C R, 1 < i < n, 1 <_ j < m in M implies that the
system is solvable in g (see Warfield [71] or Glaz [44]). Set So = S. For all systems ~ a / j x i = ny with solutions X l , . . . , x , ~ E M, set $1 be the submodule of M spanned by So and all the solutions. It is easy to see that Card(S1)
50 Let (G~), i E I be a family of representatives of this class with the index set I. Let Hi = Hom(M, Gi) for each i C I and let F = I-I G H~. Define ~ : M --+ F so t h a t the composition of ~ with the projection map F ~ G H~ maps x E F to (h(x)) (h C Hi). Since R is right coherent, F is a fiat left R-module. We claim that p : M --+ F is a flat preenvelope of M. Let ~ : M ~ G be a linear map with G fiat. By the preceding lemma, the submodule ~ ( M ) C G can be enlarged to a pure submodule G ~ C G with C a r d ( G r) < Re. Note t h a t G ~ is flat because it is a pure submodule of G which is flat. Then G' is isomorphic to one of the Gi. By the construction of the m a p F, it is easy to see that ~' can be factored through ~. We are through. [] The theorem above proves the existence of flat preenvelopes, but the question of existence of flat envelopes is still open in general.
The following result proved by
Asensio and Martinez solves this problem for the case of commutative rings. As a conclusion of this section, we state their result without giving the proof. Theorem 9] for the details. Theorem
2.5.3 For a commutative ring R the following are equivalent:
(1) R is coherent and w.gLdimR ~_ 2; (2) Every R-module has a fiat envelope.
See [58,
Chapter 3 Flat Covers and Cotorsion Envelopes We have considered the general theme of envelopes and covers of modules in C h a p t e r 1 and 2. From now on we will concentrate on flat covers and the topics closely related to flat covers. The main goal of this Chapter is to show the existence of flat covers for quite a large class of rings. As observed in the first chapter, all the left R-modules over a left perfect ring have flat covers and all Abelian groups have torsion free coverings and these are the same as flat covers. In the first section we prove t h a t over a right coherent ring R every left R-module of finite flat dimension has a flat cover. As a consequence, all left R- modules over a right coherent ring of finite weak global dimension have fiat covers. In the next chapter we will switch our attention to commutative Noetherian rings and, replacing finite weak global dimension by finite Krull dimension, prove the existence of flat covers over such rings. In the second section we prove that over a right coherent ring R, if a left R-module M has finite pure injective dimension, then M has a flat cover. Consequently every M of finite injective dimension has a fiat cover. If R is right coherent of finite global pure injective dimension (see Gruson and Jensen [50]), then all left R-modules have flat covers. In this Chapter we also introduce cotorsion modules and explore the relation between the existence of fiat covers and the existence of cotorsion envelopes. Cotorsion modules have their own interest and were investigated by many algebraists (see [47, 43, 70]). Most of the results presented in this chapter are taken from [73, 34, 64]. We will assume R to be right coherent through the whole discussion.
3.1
F l a t c o v e r s in a n e x a c t s e q u e n c e
In this section we are primarily concerned with an exact sequence of left R-modules 0 -+ A -+ B --+ C --+ 0. Suppose two of them have flat covers. When does the third have a flat cover? As before, ~" is the class of flat left R-modules. Note t h a t 9v is closed under extensions and direct limits. Hence by Theorem 2.2.12, in order to find
52 a fiat cover for an R-module M, we only need to find a fiat precover ~ : G --+ M. The next result is just a special case of Theorem 2.2.8. We state it for completeness. Proposition
3.1.1 For any associative ring R, a left R-module M admits a fiat cover
if M has a fiat precover. Now all we have to do is to find a flat precover for any given module M. To do so, we may need some basic methods with which we can manipulate flat covers. One way to create a fiat precover is to use a module in the class 5r • D e f i n i t i o n 3.1.1 A left R-module C is called cotorsion if C E ~-• that is , for any flat left R-module F , E x t , ( F , C) = 0. We easily see that the class of cotorsion modules contains all the pure injective modules and so includes all the injective modules. Suppose 0 --+ C --+ F --+ M -+ 0 is exact with F fiat and C cotorsion. Then if G is fiat, Horn(G, F ) --+ Horn(G, M ) --+ 0 = E x t l ( G , C ) is exact. So we see t h a t F -+ M is a flat precover. We will spend more time discussing cotorsion modules later. At this moment we want to display the role played by cotorsion modules in the process of finding flat precovers. First we need the following. Proposition
3.1.2 In an exact sequence of left R-modules C1, C2 and C3, 0 -+ C1 --+
C2 --+ C3 --+ O, if Ct and Ca are cotorsion, then C2 is also cotorsion. Also if both C1 and C2 are cotorsion, then so is Ca. P r o o f : The proof of the first statement is trivial by using the functor with F fiat. For the second, we claim that
Extl(F,-)
E x t ~ ( F , C ) = 0 for a l l n > 1 i f C is
cotorsion and F is fiat. Take a partial projective resolution of F : O -+ G -+ P,~_2 -+ . . . ~ Po --~ F --+ O Since G is fiat, E x t , ( G , C) = O. It follows from this that E x t , ( F , C) = O. Now the second statement can be easily proved by applying the long exact sequence associated with 0 --+ C1 --+ C2 ~ C3 -+ 0. See R o t m a n [66, Thm.7.5]. [] L e m m a 3.1.3 I f M C G is a submodule of a left R-module G such that G / M is fiat and G has a fiat cover, then M has a fiat cover. P r o o f : Let g) : F --+ G be a flat cover of G. Then F / ~ - I ( M ) ~- G / M is fiat, hence so is ~ - ~ ( M ) . Since ~ : F -+ G is a flat cover, K = ker(q)) is cotorsion by L e m m a 2.1.1. Therefore the induced exact sequence 0 -+ker(qo) --+ g ) - l ( M ) --+ M -+ 0 creates a fiat precover of M because K is cotorsion and
(fl-l(M)
is fiat. []
L e m m a 3.1.4 I f R is right coherent and E is an injective right R-module, then H o m z ( E , Q / Z ) is a fiat left R-module.
53 Proof." The proof is a standard argument. Since R is right coherent, every finitely generated right ideal I is finitely presented. Then the natural homomorphism H o m z ( E , Q/Z) @R I -+ Homz(HomR(I, E), Q/Z) is an isomorphism (Anderson and Fuller [1, Prop.20.11]). To show that H o m z ( E , Q/Z) is flat, we have to ensure that I|
Q/Z) --+ R|
Q/Z) is injective.
But we have a commutative diagram I Qn H o m z ( E , Q/Z)
Homz(HOmR(I, E), Q/Z)
,R|
Homz(E, Q/Z)
, Homz(HomR(R, E), Q/Z)
with vertical maps isomorphisms. But then we know that the lower row is injective, hence so is the upper. [] L e m m a 3.1.5 Assume R is a right coherent ring. Then for any right R-module M, Homz(M, Q/Z) has a flat cover as a left R-module. In particular, every pure injective
left R-module has a flat cover. P r o o f i Let 9) : M --+ E be an injective envelope of M with cokernel Y. Then we have an exact sequence of left R-modules: 0 -~ Homz(Y, Q/Z) -+ Homz(E, Q/Z) -+ Homz(M, Q/Z) -+ O. Since H o m z ( E , Q / Z ) is fiat and H o m z ( Y , Q / Z ) is pure injective (so is cotorsion), this exact sequence provides a flat precover of Homz(M,Q/Z), and it then has a flat cover.
Finally note that a pure injective module M is a direct summand of
Homz(Homz(M, Q/Z), Q/Z). [] The following is an interesting fact about pure injective envelopes over right coherent rings (see Jensen [50, Prop.4.1]). This fact turns out to be useful when we construct flat precovers. L e m m a 3.1.6 Assume R is right coherent. If F is a flat left R-module and P E ( F )
is a pure injective envelope of F, then PE(F) is flat. Moreover P E ( F ) / F is also flat.
P r o o f : Note that the pure injective envelope PE(F) of F is a direct summand of Hom(Hom(F, Q/Z), Q/Z). By the previous lemma, the latter is flat for Horn(F, Q/Z) is injective. Hence P E ( F ) / F is flat because F is a pure submodule of the fiat module P E ( F ) . See Theorem 1.1.5. []
54 T h e o r e m 3.1.7 A s s u m e R is right coherent.
Then a leSt R-module M has a fiat
cover if and only if there is an exact sequence O --+ M --+ G ~ L --+ O such that L is fiat and such that G is cotorsion and has a fiat cover. P r o o f : Given such a sequence 0 --+ M ~ G --+ L --+ 0, by the lemma above, M has a flat cover. Now suppose that M has a flat cover F --+ M with kernel C. Then C is cotorsion by Lemma 2.1.1. Let F --+ P be a pure injective envelope of F with cokernel L. By Lemma 3.1.6, L is flat. Then using the pushout diagram of F ~ M and F -+ P, we get a commutative diagram
0
'C
0
,C
II
0
0
*F
,M
.0
,G
,0
is 9P L
=
0
L 0
with exact rows and columns. Here C and P are cotorsion, hence so is G. As noted, L is flat and P is a flat precover of G, so we have the desired exact sequence. [] T h e o r e m 3.1.8
Let R be right coherent, 0 --+ A -+ B -+ C --+ 0 be an exact
sequence of left R-modules. I f both A and C have flat covers, then B has a flat cover.
Proof:
Let ~ : F -+ C be a flat cover of C.
Consider the pullback diagram of
~b : F -+ C and g : B -+ C
0
.A
~P
7r,F
. 0
0
,A
~B
g.c
,0
We claim that i f P h a s a f l a t cover, say a : G ~ P, then ~rl a : G - + B i s a f l a t precover of B. For any linear map al : G' -~ B with G' flat, consider the map g al: G' -+ C . Since ~b : F -~ C is a flat cover of C, there is a linear map q : G' -+ F such that gch = ~b q . Then we have the following commutative diagram:
55
GI
By the p r o p e r t y of pullbacks (see [14]), there is a unique linear m a p p : C '
P such
t h a t al = :rl p , q = 7r p. We now consider the c o m m u t a t i v e d i a g r a m
Vl "'" [ h."
P
p G
a
p
Since a : G --+ P is the fiat cover of P , there is a linear m a p h : G '
> G such t h a t p
= ha. T h i s implies t h a t a l = 7r~p = 7rl ah. Therefore h is the desired linear map. Now we want to show t h a t P has a fiat cover. To do so, we n o t e t h a t A has a flat cover. Hence by T h e o r e m 3.1.7, we have an exact sequence 0
>A----~G
>D
>0
where G is cotorsion and has a flat cover and D is flat. T h e n we have the pushout d i a g r a m of A -+ P and A -+ G: 0
0
0
----+ A
~ P
0
--+
.k
~.
G
> L
D
=
----+ 0
H ---~
F
--+
0
D
0 Since G is cotorsion and F is fiat,
----> F
0 Ext~(F,G)
= 0.
T h i s implies t h a t the exact
sequence 0 --+ G --+ L -+ F ~ 0 is split. It follows from T h e o r e m 1.2.10 t h a t L has a flat cover because b o t h G and F have flat covers. Finally, consider the exact sequence
0 --+ P --+ L ~
D --> 0 where L has a flat
cover and D flat. By L e m m a 3.1.3 P has a flat cover. [] Theorem
3.1.9
Let R be right coherent and let 0 --+ A --+ B ~ C ~ 0 be an exact
sequence of left R-modules.
I f both A and B have flat covers, then C also has a flat
Cover. Proof:
Since A has a fiat cover, by T h e o r e m 3.1.7 we have an exact sequence,
56 O ~ A ~ G ~ D
>0
where D is flat and where G is cotorsion and has a fiat cover. Consider the pushout diagram of A ~ B and A -+ G:
0 0
0
0
+
$
> A
> B
~-
~.
----+ G
---+ L
D
=
--+
C
--+
0
II --+
C
-----+ 0
D
+
+
0
0
Since both B and D have flat covers, by Theorem 3.1.8 L also has a flat cover. Let r : F -+ L be a flat cover of L. Consider the following diagram 0
0
+
+
K
=
K
+
+
0
---+ H
~ F
> C
) 0
0
---+ G
> L
----+ C
> 0
$
,1.
0
0
Here H is the pullback of F --+ L and G --+ L. Note that since both K and G are cotorsion, it follows that H is also cotorsion. But, the fact that H is cotorsion and F is flat implies that F -+ C is a flat precover of C. [] By the results we have just proved, we may expect that in an exact sequence 0 --+ A --~ B --+ C --+ 0, if any two of A, B and C have flat covers, so does the third. But we do not know if the remaining case is true or not. However the following is useful.
T h e o r e m 3.1.10 Let R be right coherent, then the following are equivalent. (a) A n y cotorsion left R-module has a fiat cover; (b) For any exact sequence, 0 --+ A --+ B ~ C ~ 0 , whenever both B and C have flat covers, so does A.
Proofi
(a) ==~ (b)
Consider the diagram
57
0 K
0 =
K
0
> A
> G
----~ F
--+
0
0
> A
> B
---+ C
---+ 0
$
$
0
0
Here, F --+ C is the fiat cover of C, G is the pullback of F --+ C and B ~ C . We know that since K is cotorsion, by (a), it has a flat cover. Now both K and B have flat covers, so by Theorem 3.1.9, G also has a flat cover. However, this implies that A has a flat cover because F is flat and G has a flat cover. (b) ==~ (a) Let G be cotorsion. We try to show that G has a flat cover by using the assumption (b). Consider the exact sequence 0 -+ G --+ E ~ L --~ 0, where G --+ E is the injective envelope of G. Then E has a flat cover (Lemma 3.1.5). Let F --~ E be a flat cover of E. Consider the pullback diagram: 0
0
$
$
K
=
K
0
--+
P
> F
~ L
----+ 0
0
---+ G
> E
> L
---+ 0
$
4
0
0
where P is the pullback of G -+ E and F --+ E. Then, K is cotorsion and it follows that P is also cotorsion. This implies that F -+ L is a flat precover of L, and then that L has a flat cover. Now, by the assumption (b), G has a flat cover because both E and L have fiat covers. [] So far we have discussed the existence of flat covers for specific modules. We hope to get global result on the existence of flat covers.
First we can easily generalize
Theorem 3.1.9 on short exact sequences to a result on long exact sequences. T h e o r e m 3.1.11 Let R be right coherent. For any left R-module M if there exists
an exact sequence O-+ M n - + M n _ I - - ~ . . . - + Mo-+ M - + O with each Mi (0 < i < n) having a fiat cover, then M has a fiat cover.
58 P r o o f : Break the sequence down to short exact sequences and apply Theorem 3.1.8. [] B y the results we have obtained, the candidates for Mi (1 < i < n) can be flat modules, injective modules and pure injective modules.
In particular, for a right
coherent ring R, if we choose each Mi to be a flat module, then we see that the every left module M of finite flat dimension has a flat cover. Therefore we have t h a t 3.1.12 Let R be right coherent. If R has finite weak global dimension, then all left modules have fiat covers.
Theorem
This result provides a quite large class of rings over which Enoehs' conjecture is true. For instance, let k be a field, R = k[xl,..., x~] be the polynomial ring with indeterminates x~s. Then every R-module has a flat cover. However if we take a quotient ring [~ = R / I with an ideal I C R, we can not apply the above theorem to this ring to ensure t h a t every /~-module has a flat cover. So we are not fully satisfied with the current result because we can not apply the result to all commutative Noetherian local rings. And these are the very important rings. We will return to this problem in the next Chapter. C o r o l l a r y 3.1.13 Let R be right coherent. Then every finitely generated left module
has a flat cover if and only every cyclic left module has a fiat cover.
3.2
Modules of finite injective dimension
In this section we will show t h a t over a right coherent ring every left module of finite injective dimension has a flat cover. But the method used in the last section can not be directly applied here. For the current result we need more preliminaries. Let us start with a lemma which is similar to Schanuel's Lemma. See Anderson and Fuller [1] or R o t m a n [66, Thin.3.62]. L e m m a 3.2.1 Suppose there are two fiat precovers of M as follows:
0
,K
.Ff-~M~O
0
.L
.G
k,M---~O
then K @ G ~- F @ L. P r o o f : Since k : G -+ M is a flat precover, there exists commutative diagram with exact rows.
0
,K
g,F
f,.M-----~O
0
,L
h,G
k,M---~O
I
II
59 Similarly we have a c o m m u t a t i v e d i a g r a m with e x a c t rows :
0
*L
0
,K
h,G
l
k,M
lq g,F
,0
II f,M
,0
Define two linear m a p s (7 and r as follows
~ : F @ L -+ G @ K,
(x', y) ~-~ (p(x') - y, - q ( y ) - (1 - qp)x')
G • K --~ F @ L,
(x, y') ~ (q(x) - y', - p ( y ' ) - (1 - pq)x)
r
N o t e t h a t f = kp, k = fq. Hence f -- f(qp) and k = k(pq), or equivalently f ( 1 - q p )
=
0, k(1 - pq) = 0. Therefore this implies t h a t (1 - qp)(F) C K and (1 - pq)(G) c L. It is easy to verify t h a t a r = 1 and ~b(7 = 1. T h i s finishes the proof. [] Lemma
3 . 2 . 2 Let 0 ~ A ~ B --~ C -+ 0 be exact of left R-modules. If both A and C
have fiat covers, and A is cotorsion, then there is a commutative diagram with exact rows and columns. 0
0
0 --~
K
~
0
F
---+ D
0 ----+ A
--+
0
0
N
~ rightarrow
0
> G
-----+ 0
B
-----+ C
----~ 0
0
0
Here D = F G G where F --~ A and G --+ C are given fiat covers. Moreover, D ~ B is a flat precover and N is cotorsion. Proof:
Since g : G -+ C is a fiat cover, there exists a linear m a p u : G -+ B such
t h a t g = pu, here p : B --+ C. Define h : F 9 G -~ B with (x, y) ~-~ g f ( x ) + u(y), x E F, y E G, here f : F --~ A is the fiat cover. By a s t a n d a r d d i a g r a m chasing a r g u m e n t , we can easily verify t h a t the d i a g r a m can be c o m p l e t e d w i t h t h e s t a t e d properties. In particular, N is cotorsion and so D -+ B is fiat precover. [] Lemma
3 . 2 . 3 Assume R is right coherent. Then a fiat leSt R-module F is cotorsion
if and only it is isomorphic to a direct summand of H o m z ( E , Q / Z ) Sor some injective right R-module. In other words, over a right coherent ring R, a flat left R-module is cotorsion if and only if it is pure injective.
60
Proof: Note that F* is injective and that F** = H o m z ( H o m z ( F , Q / Z ) , Q / Z ) is flat and pure injective. Also F is a pure submodule of F**, so F**/F is flat. Hence if F is cotorsion, F is a direct summand of F**. Conversely, if F is a direct summand of H o m z ( E , Q / Z ) with E injective right module then F is pure injective and so cotorsion. [] We have seen that over a right coherent ring R, every injective left R-module have a fiat cover. We will show that this is also true for pure injective modules. L e m m a 3.2.4 Let R be right coherent. If M is a pure injective left R-module, then
M has a fiat cover. Furthermore suppose that ~ : F -4 M is a fiat cover. Then both F and K = ker(~) are pure injeetive. P r o o f : Since M is pure injective, it is a direct summand of H o m z ( X , Q / Z ) for some right R-module X. Let E be an injective envelope of X and let 0 -4 X -4 E -4 D -4 0 be exact. Then the sequence of left R-modules
0 --+ H o m z ( D , Q / Z ) -4 H o m z ( E , Q / Z ) -4 H o m z ( X , Q / Z ) -4 0 is exact sequence. Since H o m z ( E , Q / Z ) is flat and H o m z ( D , Q / Z ) is pure injective (and so cotorsion), this sequence gives a flat precover of H o m z ( X , Q / Z ) . This implies that M has a fiat cover since it is a direct summand of H o m z ( X , Q / Z ) . If F -4 M is a flat cover with the kernel K, then using the fact that M is a direct s u m m a n d of H o m z ( X , Q / Z ) and chasing diagrams we see that F is isomorphic to a direct s u m m a n d of H o m z ( E , Q / Z ) and t h a t K is isomorphic to a direct summand of H o m z ( D , Q / Z ) . Hence both F and K are pure injective. [] We now are able to use pure injective modules to build up modules that they also have flat covers. Theorem sequence
3.2.5 Let R be right coherent. For a left R-module if there is an exact
O~M--+Po-4P1-4""-4Pt-40 with all P~, 1 < i < t, pure injective, then M has a flat cover. In fact, there are exact sequences
where i > O, K - I = M such that Fi is fiat and cotorsion and K~ is cotorsion. Proof: We proceed by induction on the length of the sequence. If t = 0, M is pure injective. Applying the previous lemma, we have an exact sequence 0 -4 K0 -4 F0 -4 M -4 0 with F0 --4 M a fiat cover and both F0 and K0 pure injective. Hence we can repeat this process to find all the desired exact sequences stated in the theorem. Now let us assume that we can construct all the desired exact sequences for any left R-module with the length of the stated resolution less than or equal to t - 1.
61 Consider the shifted exact sequence 0 --+ M ~ Po --+ N -+ 0, where N has the length of the associated sequence < t - 1. By the inductive assumption both P0 and N have the desired exact sequences, say, 0 --+ Li --+ F~ --+ Li_: -+ 0 for i > 0 with L - I = N; 0--+ I/V~ ~
G~ --+ W~-1 ~ 0 f o r i >_ 0 w i t h W-1 = P = Po- We try to glue them
together to produce the desired exact sequences for M. Consider the pushout diagram of Po --+ N and Fo --+ N:
0
0
Lo = Lo 0
'
~H
*M
I
H 0
,M
.Fo
. 0
1
~Po
.N
0
0
90
Applying Lemma 3.2.2 to the middle exact sequence, we get a commutative diagram with exact rows and columns 0 0
---+ L1
0
--+
0 --+
0
0
--~
Ko
> Wo
) 0
F~
)
Zo
)
Go
)
~Lo
4. ~ H
)
$ Po
) 0
0
0
0
0
Here Zo = / : 1 G Go. We note that in this diagram Zo --+ H is a flat precover and Ko is cotorsion. On the other hand, for the exact sequence 0 ~ M --+ H -+ Fo --+ 0, and the special flat precover of H obtained above we have another pushout diagram
0
0
I
1
Ko=Ko 989
~Zo
.Fo
~M
,H
.Fo
0
0
,0
H ,0
62 We see that 170 is fiat since both Z0 and F0 are fiat. Furthermore, this implies that V0 -+ M is a flat precover with the kernel K0 cotorsion. In other words we have built up the first exact sequence for M. Then we only need to find the desired exact sequences for Ko. But K0 is in the middle of 0 --+ L1 --+ Ko -+ Wo --+ 0 with both L1 and Wo having the desired exact sequences. Note also that all modules involved are cotorsion. Applying Lemma 3.2.2 we can glue the exact sequence of V~89and the exact sequence of L1 together to get the desired exact sequence for K0. For instance, i = 1, we have 0
0
$ 0
-----+ L2
4 ----4
$
o --+
0
K~
$ --~
4
Wi
--~
0
4
)
Z~
--4
0
)
$
Ko ---~ Wo --* 3, $
0
0
0
P2
~
G~
$ 0 ~
L1
0
Here Z1 = F2 9 G1. We note that in this diagram Z1 --+ K0 is a flat precover with K1 cotorsion. Repeating this process, we can construct all the desired exact sequences for M. C3
Since every injective module is pure injective, we have a simple application of the theorem we have just proved. C o r o l l a r y 3.2.6 Let R be right coherent. If a left R-module M has finite injective dimension, then it has a fiat cover. R e m a r k 3.2.7 Gruson and Jensen in [50] defined and studied L-dimensions of modules by using pure injective resolutions. They also defined the pure injective global dimension, denoted L-gl.dim(R), by taking the obvious supremum.
The computa-
tion of this global dimension and interesting examples of finite pure injective global dimension were also given there. Using their terminology, we have C o r o l l a r y 3.2.8 Let R be right coherent. R-module has a fiat cover.
If L-gl.dim(R) is finite, then every left
This result produces rings which have infinite weak global dimension, but such that all modules have fiat covers. See Gruson-Jensen [50] for examples of such rings.
3.3
Cotorsion m o d u l e s
Recall that a left R-module C is called cotorsion if E x t , ( F , C) = 0 for all flat left R-modules F. We have already seen that this class is very useful in finding fiat covers. The notion of a cotorsion module goes back to Harrison [47], Warfield [70] and
63 Fuchs [43]. Cotorsion modules were first introduced to study infinite Abelian groups. In this section we will discuss properties of cotorsion modules. Obviously the class of cotorsion modules is closed under extensions, finite direct sums and direct summands. It is also easy to see t h a t the class is closed under direct products. But it fails to be closed under infinite direct sums. Also note that all pure injective modules are cotorsion, but the converse may not be true. As we have seen , the major source of cotorsion modules are the kernels of flat covers. The following example gives another source of cotorsion modules. E x a m p l e Let R be commutative Noetherian ring. Then all Artinian modules M are cotorsion. P r o o f : Using the Chinese remainder theorem (see Atiyah A t i y a h : l ) it is easy to see M has a decomposition into m - p r i m a r y submodules for various maximal ideals m (a submodule is m - p r i m a r y if each element is annihilated by a power of m). Since M is Artinian, M is the direct sum of a finite number of such submodules, so we assume t h a t M is m-primary. Then M is Matlis reflexive as a n / ~ - m o d u l e , and M ~ Homkm (Homkm ( M , E ( R / m ) , E ( R / m ) ) ) . Hence each M is cotorsion as an / ~ , - m o d u l e . We claim t h a t M is cotorsion as an R-module. Consider any extension X of M by a flat module F
0
)M--+X
>F--+0.
We want to show this is split. Applying the tensor functor - |
to the above, we
have the commutative diagram with exact rows:
0
.M
,X
o
Note that M ~ M | cotorsion and F |
. F
. 0
9 0
/~m as an / ~ - m o d u l e , b e c a u s e M is Artinian.
Since M is
is flat as/~,~-modules, the lower sequence is split. This implies
t h a t the upper sequence is split as R-modules. [] From the next simple observation we see t h a t the class of cotorsion modules is useful when characterizing rings. Proposition
3.3.1 For any ring R the following are equivalent:
(1) Every left R-module is cotorsion; (2) Every fiat left R-module is eotorsion; (3) R is left perfect.
64 Proof." It suffices to prove the implication (2) ==~ (3) . For any flat left R-module F, we have an exact sequence
0 --+ G ~ P --+ F --+ 0 with P projective and G flat.
By (2) G is cotorsion, and then this sequence is split. This means F is projective. It follows from the Bass's Theorem P that R is left perfect. []
T h e o r e m 3.3.2 For any ring R and any integer n >_ O, the following are equivalent:
(I) For all left cotorsion modules C, inj.dimR(C) <_ n; (2) (3) (4) (5)
For all left pure injective modules P , inj.dimtt(P) <_ n; For all right cotorsion modules C , f.dimR(C) <_ n; For all right pure injective modules P, f . d i m R ( P ) <_ n; w.gl.dim (R) < n.
Note that since the statement (5) is left-right symmetric, the statements (1) to (4) can be replaced by their symmetric statements. P r o o f : The implications (1) =~ (2), (3) =~ (4), (5) =~ (3) and (4) are trivial. (2) =v (5)
Note that for any right R-module N, N* = H o m z ( N , Q / Z ) is a pure
injective left R-module. By (2) we have inj.dim•(N*)
<_ n. Hence Ext~+l(M, N*) = 0 for all left R-modules M. By the canonical isomorphisms (see [15] or [44, Thm.l.l.8]):
Ext~+,(m,x,)
~= H o m z T(o r ,'+ 1( M , N , ) Q/ Z )
,
it follows that w.gl.dim(R) <_ n. (4) =~ (2) For any pure injective left module P, P* is a pure injective right R-module. By (4) f . d i m n ( P * ) <_ n. It is easy to see that P** = H o m z ( P * , Q / Z ) has injective dimension less than or equal to n. Since P is pure injective, it is a direct summand of P'*. Hence inj.dirnn(P) <_ n. (5) =~ (1) In order to prove this implication, we consider a double complex and its associated spectral sequences. For all cotorsion left R-modules C we have to show that Ext~+l(M, C) = 0 for all left R-modules M. For such a cotorsion module C, we have its injective resolution
O-+ C.--+ EO-+ E 1 __+...__+ E'~--+ ... Since f . d i m n M <_ n for any left R-module M, we have an exact sequence:
with F0," 9", Fn flat. Then we form the following double complex
65
l
t
t
:
:
:
t
T
l
T
t
T
T
T
0
0
0~Hom(Tff, E n) --* Hom([0, E n) ~ Hom(~i, E n) . . . . .
O--~Hom(M, E 1 ) ~ H o m ( F o , E 1 ) ~ H o m ( F 1 , E
..-
l
Hom(~., E~) ~ :
T
1) . . . . .
Hom(F,~,E1) ~
0 - - * H o m ( M , E ~ ~ Hom(Fo, E~ ---~Hom(F1,E ~ . . . . .
Hom(Fn, E~ ~
T
0
, Hom((o, C) --* Horn(if1, C) . . . . .
0
T
T
Hom(F~, C) ~
0
0 0
0
Note that since all E i are injective, all rows are exact except for the bottom row. Also note that since C is cotorsion and all F~ are flat, all columns are exact except for the left column. Then, using a spectral sequence argument or simply chasing the diagram we know that the following complexes 0 --+ Hom(F0, C) --+ Hom(F1, C) --+ .,. --+ Hom(Fn, C) --+ 0 and 0 ~ Horn(M, E ~ ~ Horn(M, E 1) -~ ... ~ Horn(M, E n) -~ ..... have isomorphic homologies. Therefore, Ext~+l(M, C) is zero for m > n + 1. Hence,
inj.dimR(C) < n. [] From the above theorem we know that the weak global dimension of a ring R is determined by the flat dimensions of pure injective modules, and also determined by the fiat dimension of cotorsion modules. We may ask if it is determined by the fiat dimensions of injective modules. But this is not true in general. Recall that a ring is called an IF ring provided all injective modules are flat. There are a plenty of IF rings which are not von Neumann regular (i.e., w.gl.dim(R) = 0). See Colby [18] for examples. Also in the preceding theorem, setting n = 0, we have that R is yon Neumann regular if and only if cotorsion modules are injective, if and only if pure injective modules are injective, if and only if cotorsion modules are flat, and if and only if pure injective modules are flat, Before finishing this section, we have a look at the behavior of cotorsion modules under a change of rings. This observation will be used later. P r o p o s i t i o n 3.3.3 Let I be an ideal of R. If C is cotorsion as an R/I-module, then it is cotorsion as an R-module.
66 P r o o f : We have to show that E x t , ( F , M ) = 0 for all flat R-modules F . Consider an extension X of M by F as R-modules: 0
>M
)X--~F--~O.
Since F is flat, M is a pure submodule of X. Then M N I X = I M = 0. Note that M+IX ~_ M. We have the following commutative diagram with exact rows: IX
Since
_ | •7 R
r.~ =
0
,M
0
. M+IX IX
,X
I
, F
I
X * ~
,
0
,
0
1
X ' M+IX
X M~IX is flat as an R / I - m o d u l e , it follows t h a t the lower sequence is
split, and hence so is the upper. Hence, E x t , ( F , M ) = 0 and M is cotorsion as an R-module. [] E x a m p l e : Any cotorsion R-module C is cotorsion as an R[x]-module where x C = O. In particular, for a left perfect ring R all left R-modules are cotorsion as R[x]-modules in this manner.
3.4
Cotorsion envelopes
Let C = 5c x be the class of all cotorsion left R-modules. For a left R-module M , a C-envelope ~ : M --+ C is called a cotorsion envelope. Obviously ~ must be injective. For Abelian groups, the notion of cotorsion envelopes (called cotorsion hulls) was introduced by Fuchs in [43, p249]. We have already seen that every flat cover creates a cotorsion module as the kernel. One of our purposes in this section is to show t h a t there is a very close tie between flat covers and cotorsion envelopes. For instance, it will be shown t h a t every module has a cotorsion envelope if and only if every module has a flat cover. Note t h a t since injective modules are cotorsion, any cotorsion preenvelope M -+ C is an injection. L e m m a 3.4.1 xC = 9v, P r o o f : Clearly 5r C • r •
i.e.,
~ - - - "L(5c•
= J-C. To get the reverse inclusion we argue t h a t if F is
such that E x t l ( F , M ) -- 0 for all pure injective left R-module M, then F is flat. Since every such M E C, this will give •
= $'.
But note t h a t for any right R-module N , M = H o m z ( N , Q / Z ) is pure injective as a left R-module. Then 0 = Ext 1(F, M ) = E x t l ( F , H o m z ( g , Q / Z ) ) ~ H o m z ( T o r l ( F , N), Q / Z ) . This shows t h a t TOrl(F, N) -- 0 for any right R-module N. Hence F is flat. []
67 T h e o r e m 3.4.2 If 99 : M --+ C is a cotorsion envelope, then D = C / T ( M )
is flat.
Hence if M is fiat, so is C. P r o o f : By Lemma 2.1.2 and the previous lemma. [] From this we know that if M has a cotorsion envelope C, then M is a pure submodule of C because the quotient C / M is flat. For convenience, a module B is called a flat extension of a submodule A if the quotient B / A is flat. In this case A is called a F - p u r e submodule of B.
Hence for a given left R-module M, in order to find a
cotorsion envelope of M we may first look for a flat extension of M. We will argue that the existence of a cotorsion envelope can be derived from a certain flat extension of M. Recall that among all flat extensions of M we call one of them 0 -+ M --+ C D -+ 0 a generator for Cxt(.T, M) (or a generator for all flat extensions of M) if for any flat extension of M, 0 --+ M --+ C' --+ D' --+ 0, there is a commutative diagram:
-M
.6"
9D'
~M
.C
,D
90
1 ,0
Furthermore, a generator 0 ~ M --+ C -+ D -+ 0 is called minimal if all the vertical maps are isomorphisms whenever C r , D' are replaced by C , D, respectively. Since the class 5c satisfies all the conditions in Theorem 2.2.2, by that result and the lemma above we have T h e o r e m 3.4.3 For a given left R-module M, if there is a generator of all flat ex-
tensions of M, then there is a minimal one. T h e o r e m 3.4.4 Let 0 -+ M --~ C --+ D -~ 0 be a minimal generator of all flat
extensions of M. Then C is a cotorsion envelope of M. E x a m p l e : Let R be right coherent. For any flat left R- module G, there is an exact sequence: 0 ----~ G
> G** ---+ D - - + 0 .
Here M* = H o m z ( M , Q / Z ) for any R-module M. It is easy to see that this is a flat extension of G. Actually it is a generator of all flat extensions of G. For given any exact sequence 0 --+ G --+ C --+ D' -+ 0 with D' flat, we want to complete the diagram
0 ---~G
9C
,D'
.0
9G**
9D
,0
II 0
-G
68 to a commutative diagram. It suffices to show that G --+ G** can be extended to C. But this is true since E x t l ( D ' , G **) = 0 and Hom(C,G**) -+Hom(G,G**) -+ 0 is exact. Then by Theorem 3.4.3 and 3.4.4 G has a cotorsion envelope. In fact the cotorsion envelope of G is the same as its pure injective envelope (up to isomorphism). [] Later we will see that over a right coherent ring every left R-module of finite flat dimension or of finite injeetive dimension has a cotorsion envelope. We now give an alternate description of cotorsion envelopes. We first recall the definition of pure injective envelopes. Let A be a pure submodule of B. Then B is called a pure-essential extension of A if there are no nonzero submodules S c B with S n A = 0 and with the image of A pure in B / S .
A pure extension B of A is called
a pure injective envelope if B is pure injective and the extension is pure-essential. Similarly we call B a flat essential extension of A if A is a F-pure submodule of B and there are no nonzero submodules S C B with S n A = 0 and the image of A F - p u r e in B / S (i.e., the quotient B / S (9 A is flat). T h e o r e m 3.4.5 Let M be a submodule of a cotorsion left R-module C. following are equivalent. (i) i : M --+ C is a cotorsion envelope of M (here i is the inclusion); (2) C is a fiat essential eztension of M.
Then the
Proof." (1) ==* (2). By Theorem 3.4.2 M is a F-pure submodule of C. We only need to verify that C is a flat essential extension of M. Assume S to be a nonzero submodule of C such that SN M = 0 and C / ( S 9 M ) is flat. Consider the commutative diagram:
C
9M
OL
C , S/~(s~-M)--*:
......j
C
0
C Here the linear map a is the canonical map. The existence of the linear map f follows from the fact C / ( S @ M ) is flat and C is cotorsion. We now have i = f a i .
By the
definition of cotorsion envelopes, f a is an automorphism of C. This is impossible since o(s) :
o.
(2) ~
(1). By the hypothesis (2) we have an exact sequence: 0 -+ M --+ C --+ D -+ 0
with C cotorsion and D fiat. It is easy to see that this is a generator of all fiat extensions of M. Applying Theorem 3.4.3 and
3.4.4, we have a fiat extension sequence of M
0 -+ M -+ C* -+ D* -+ 0 which gives a cotorsion envelope of M. Then we have the commutative diagram:
69
0
,M
0
,M
a*.C* a*,D*
II
. 0
tl h i ,C
a,D
, 0
It is easy to see that C = f(C*)@ ker(g). We claim ker(g) = 0. Note that ker(g)NM =
0 ( M = f a * ( M ) C S(C*)). In order to verify that the image of ker(g) in C/ker(g) is F-pure, we have to show t h a t C / ( M • ker(g)) is flat.
We define the following
homomorphism C : M G ker(g) Obviously ~ is well defined.
>D*,
~=c+M@ker(g)--+a*g(c).
By diagram chasing, we see t h a t p is injective. But
both g and a* are surjective, hence so is ~. Therefore, ker(g) is F-pure in C/ker(g). This contradicts the hypothesis that C is a flat essential extension of M. And so this implies t h a t ker(g) = 0 and so f is an isomorphism. [] We next explore the relationship between the existence of cotorsion envelopes and the existence of flat covers. Theorem
3.4.6 For any ring R every left R-module has a cotorsion envelope if and
only if every left R-module has a fiat cover. P r o o f : Suppose every left R-module has a flat cover. For a given left R-module M, we have its injective envelope E ( M ) and the exact sequence 0 --+ M --+ E ( M ) --+ N -+ O. Let p : F -+ N be a flat cover of N. Consider the pullback diagram of E ( N ) -+ N and F --+ N:
0
0
K=K 0
,M
,C
,F
0
,M--E(M)--N
, 0 ,0
t
t
0
0
Since F is a flat cover of N, the kernel K is cotorsion (Lemma 2.1.1).
Note t h a t
E ( M ) is cotorsion. This implies that C is cotorsion. Hence the upper exact row is a generator of all flat extensions of M. The conclusion follows by Theorem 3.4.3 and Theorem 3.4.4. We now assume that every left R-module has a cotorsion envelope. For any left R-module M, let M be the image of a projective module with the map having kernel
7O N.
By hypothesis there is a cotorsion envelope of N: N --+ C. Then we have the
pushout diagram of N --+ C and N --+ P:
0
0
0
,N
,P
,M
0
,C
,G
, M----* 0
D=
D
0
0
.0
By Theorem 3.4.2 D is fiat, and hence so is G. Since C is cotorsion, the lower exact row gives a flat precover of M. Hence M has a flat cover (Prop.
3.1.1). []
Although the question of existence of fiat covers over a general ring is still open, by Theorem 3.1.12 every left R-module has a flat cover over a right coherent ring R of finite weak global dimension. Then every module over such a ring has a eotorsion envelope. Cotorsion groups were introduced by Harrison [47] in studying Abelian groups. He proved that there is a one-to-one correspondence between nonzero torsion groups and adjusted cotorsion groups.
As an application of the preceding theorem we give an
alternative proof of Harrison's result over Priifer domains. Let R be such a domain. We note that over Priifer domains the class of flat modules agrees with the class of torsion free modules. A cotorsion R-module is called adjusted if it contains no torsion free direct summands. The following result is similar to that presented by Harrison in [47]. Theorem
3.4.7 Over a Priifer domain R there is a one-to-one correspondence be-
tween nonzero torsion modules and adjusted eotorsion modules. Moreover the correspondence can be realized by taking eotorsion envelopes. Proof:
Since the Priifer domain R has weak global dimension one or zero, by the
previous remark all modules have cotorsion envelopes. For any torsion module M, let C be its cotorsion envelope. We claim t h a t C is adjusted. Easily the torsion submodule
t(C) of C is M. Suppose C contains a nonzero torsion free direct s u m m a n d G, t h a t is, C = G 9 N with G torsion free. Then it is easy to see t h a t M = t(C) = t ( N ) and G A M = 0. Note t h a t (G @ N ) / ( G ~ M ) ~- N / M = N I t ( N ) is torsion free. This shows that the image of M in C / G is F-pure. Hence this contradicts the fact C is a flat essential extension of M . For any nonzero adjusted cotorsion module C the torsion submodule t(C) is not zero. Otherwise C is torsion free itself, and then it is not adjusted. We shall show
71 that C is the cotorsion envelope of t(C).
Since C / t ( C ) is flat and C is cotorsion,
0 -+ t(C) ~ C --+ C / t ( C ) ~ 0 is a generator of all flat extensions of t(C). Let C be a cotorsion envelope of t(C). Easily C = G | 0 (i.e, we may take 0 to be a direct s u m m a n d of C). Then t(C) = t ( a ) 9 t(O) = t ( a ) 9 t ( c ) . This implies that t ( a ) is 0, and then that G is flat. Therefore G is 0 because C is adjusted. [] We now again consider the question of the existence of cotorsion envelopes. Since we do not have a complete answer to this problem, we are interested in cotorsion envelopes of special modules. For instance does an individual module M which has a flat cover also have a cotorsion envelope? T h e o r e m 3.4.8 Let R be right coherent. If M has a flat cover, then M has a cotorsion
envelope. In fact if
F ~.M
sl
l
Pit,C is a pushout diagram where ~ : F --+ M is a fiat cover and f : F -+ P is a pure injective envelope (with f the canonical injection), then it : P -+ C is a fiat cover and 7c : M --+ C is a cotorsion envelope. P r o o f : Consider the full pushout diagram of ~ : F -+ M and f : F ~ P:
0 O.
0
0
,K a,F f~
~.M ~
.0
~.Pit.
C
.0
D=
F
X t 0
0
By Lemma 2.1.1, K is cotorsion. Then it follows from Proposition 3.1.2 that C is cotorsion (because P and K are cotorsion). Since F is flat as a left R-module and R is right coherent, P is flat, and so D is flat since F is pure in P .
But the fact that K
is cotorsion and P is fiat gives that P ~ C is a flat precover. Since C is cotorsion, by the flatness of D we see that M -+ C is a cotorsion preenvelope of M. We claim that P is a flat cover of C and C is a cotorsion envelope of M. We first observe that in the above F is a pullback of lr : M -+ C and it : P -+ C. Suppose ~ is an endomorphism of P such that it -- #a. Consider the pair of maps
a f : F -+ P and ~ : F -+ M.
It is easy to see that i t a f = i t f = ~ .
By the
72 p r o p e r t y of pullback d i a g r a m s there is a linear m a p h : F --+ F such t h a t f h = a f and p = ~h. F.
Now since ~ : F --+ M is a flat cover of M , h is an a u t o m o r p h i s m of
Now we show t h a t a is an a u t o m o r p h i s m of P .
Note t h a t # a ( x ) = # ( x ) = 0.
Suppose a ( z ) = 0 for z C P .
This implies t h a t z = fl(y) for some y E K .
But
t h e n g(x) = gfl(y) = g f a ( y ) = 0. T h i s means t h a t x = f ( z ) for some z c F . Since
fh(z)
= af(z)
= a ( z ) = 0 and since b o t h f and h are injections, z = 0. A n d so
x = 0 and a is an injection. Next, for any element x C P we have # a ( x ) = # ( z ) . T h i s implies t h a t x = ~(x) + fl(k) for some k E K .
Note t h a t ~ = fc~
afh -1. F r o m
=
this we have t h a t x is in the image of ~. Therefore a is a surjection, and so cr is an a u t o m o r p h i s m of P and P is a flat cover of C. In order to verify the second condition of cotorsion envelopes, we assume t h a t q is an e n d o m o r p h i s m of C such t h a t 7r = qTr. We want to show t h a t q is an a u t o m o r p h i s m . Consider the m a p q# : P --+ C. Since # : P ~ C is a flat cover of C, there is a linear m a p a : P --+ P such t h a t q# = #or. T h e n we consider the pair of m a p s crf : F --+ P and p : F ~ M . Since #crf = q # f = qTr~ = 7r~, by the p r o p e r t y of pullback d i a g r a m there i s a l i n e a r m a p h : F - - + F s u c h
that fh=af
and~=~h.
It follows t h a t h
is an a u t o m o r p h i s m of F. T h e n we claim t h a t a is an a u t o m o r p h i s m .
Consider the
linear m a p f h -1 : F --+ P. Since f : F -+ P is a pure injective envelope of F, there is a linear m a p ~ : P ~
P such t h a t f h -1 = Of. Hence we have f = f h h -1 = a f h -1
= cT~f and f = f h - l h
= ~ f h = ~(~f. T h e n b o t h d a and cr~ are a u t o m o r p h i s m s of P
because f : F --~ P is a pure injective envelope. A n d so a and ~ are a u t o m o r p h i s m s . Since # a = q#, we see t h a t q is a surjeetion. By d i a g r a m chasing, it is t h e n not hard to prove t h a t q is an injection and so an a u t o m o r p h i s m .
Therefore 7r : M ~
C is a
eotorsion envelope of M . [] Now by T h e o r e m 3.2.5 and T h e o r e m 3.1.11 we see t h a t over a right coherent ring R any left R - m o d u l e M of finite fiat dimension or of finite injective (or pure injective) d i m e n s i o n has a eotorsion envelope.
Remark
3 . 4 . 9 Suppose M has a cotorsion envelope. T h e n we have an exact sequence
0 -+ M -+ C -+ D -+ 0 w i t h D fiat. Assume R is right coherent.
T h e n every fiat
m o d u l e has a cotorsion envelope which is the same as its pure injective envelope. T h e r e f o r e we can c o n s t r u c t a m i n i m a l cotorsion resolution of M by t a k i n g eotorsion envelopes step by step. In this resolution all the t e r m s but the first two are cotorsion and fiat. Moreover this resolution m a y stop at a finite n u m b e r of steps if the fiat m o d u l e has finite pure injective dimension. p r o j e c t i v e dimension.
This h a p p e n s if all fiat m o d u l e s have b o u n d e d
T h e l a t t e r occurrence was intensively studied by G r n s o n and
Jensen in [50]. We will use such constructions in the next chapter.
73
3.5
E x t e n s i o n s o f p u r e injective m o d u l e s
In this section we will show that in general the class of pure injective modules is not closed under extensions (see [71, 61]). Theorem
3.5.1 For any ring R the following are equivalent:
(1) For any exact sequence of left modules 0 ..~ P' -3 P -~ P" ~ 0 with P~ and P" pure injective, P is also pure injective; (2) For any left module M , P E ( M ) / M
is a fiat module, where P E ( M ) stands for a
pure injeetive envelope of M ; (3) Every left cotorsion module is pure injective. Moreover if R is right coherent, then the above are equivalent to the following. (3) For any exact sequence of left modules 0 --+ P' --+ P --+ P" ~ 0 with P ' and P pure injective, P" is also pure injective. Before giving the proof we need three lemmas. L e m m a 3.5.2 Suppose (1) holds in above. Then for a given commutative diagram
M
0
. P
i.PE(M)
(~ , N '
~ .G
90
with an exact row of left modules and P pure injective, there is a linear map u : P E ( M ) -+ N making the diagram commutative, that is, p = ui and q = an.
Proof: Consider the pullback diagram of N --+ G and P E ( M ) --+ G:
0
, p
h , H~PE(M)
0
, P
(~ , N
1
a ,G
, 0
90
Since both P and P E ( M ) are pure injective, by (1) H is pure injective. But by the property of pullback diagrams, there is a unique linear m a p r : M --+ H such t h a t i = vr and p = Cw. On the other hand since H is pure injective, by the definition of pure injective there exists a linear map r A n d so i -- vr -- v•i.
: PE(M)
u = w r 1 6 2 -1. Note that qv = ~w, qvr = ~ w r and p = w e -- w r
~
H such t h a t r = r
This implies t h a t v r is an automorphism of P E ( M ) .
= wr162
Therefore, q -- a w r 1 6 2 -1 = au
: ui. []
L e m m a 3.5.3 I f (1) holds and P E ( M ) / M
Set
= D, then the following diagram
74 D u~.
0
. p
c~ . N "
g
"'" 1
~ .G
. 0
with exact row and P pure injective can be completed by a linear map u' : P E ( M ) --+ N such that g = au'. P r o o f : Consider the following commutative diagram:
0
.M I!
0
,P
(~,
i.pE(M)
ql. D
90
v~..."" lg~l Crl "'"~ . G
0
By the previous lemma there is a linear map u : P E ( M ) --+ N such t h a t go 1
=
o-u.
Easily we get the induced map u' :D --+ N having the desired property. [] L e m m a 3.5.4 If (1) holds and D = P E ( M ) / M ,
then E x t , ( D , P ) = 0 for any pure
injective left R-module P. Consequently D is fiat. P r o o f : Consider any extension X of P by D:
0 --+ P -+ X --+ D --+ 0. In the last
lemma we set g : D -+ D to be the identity. Hence there is a linear map u' : D --+ X such t h a t au' = g is the identity map of D. This implies that the extension is trivial and so then t h a t E x t , ( D , P ) = 0. The final conclusion follows from Lemma 3.4.1. [] Proof of Theorem
3.5.1: The previous three lemmas give the implication (1)
(2). For the implication (2) ::v (3), consider any cotorsion left module M.
D = PE(M)/M
is flat. Then 0 -+ M --+ P E ( M )
is pure injective. We now show (3) ~
By (2)
-+ D -+ 0 is split. Hence M
(1). Since both P ' and P " are cotorsion, by
Proposition 3.1.2 P is cotorsion. But then (3) implies t h a t P is pure injective. The implication (3) ~ (4) follows from the fact Ext2(F, P ' ) - 0 when F is flat and P ' is cotorsion. (4) ::v (1) Since R is right coherent, by Lemma 3.1.5 all pure injective left R-module have a flat cover. Let F --+ P " be a flat cover of P". Consider the natural pullback diagram as before:
0
0
K=K 0
"P'
.C
0
.P'
.P---~P" 0
.F
0
90 ,0
75 By Lemma 3.2.4 both F and K are pure injective. But note t h a t the middle row is split, and so C is pure injective. It follows from (4) that P is pure injective.F1 Note t h a t over a v o n Neumann regular ring the cotorsion modules are the injective modules. Obviously they are also the pure injective modules. But we note t h a t using Theorem 3.5.1 it is not hard to construct cotorsion Abelian groups which are not pure injective. E x a m p l e : Let p be a prime, M = G ~ I Z / ( p n) and N = 1-I Z/(p~). Note t h a t as Zn=l
modules N is pure injective and M is a pure submodule of N. Then by the definition of pure injective envelopes it is easy to see that N = N1 | N2 with M C Nt and N1 =
PE(M). Suppose Theorem 3.5.1 holds for Abelian groups. Then cotorsion groups are pure injective and P E ( M ) / M is torsion free since it is flat. By Theorem 3.4.2 we will show that this is not the case. Let S = t(N) be the torsion subgroup of N. Note t h a t for any x -- (an)~>0 E S the orders of an C Z/(p n) are bounded. By this we see t h a t elements of (S + M ) / M are divisible by pk for all k _> 1. Consider the projection 7r2 : N --~ N2 and the restriction 7r2 ] S + M ~
N2. Since M is in the kernel of 7c2, there is an induced
m a p ~2 : ]VIM -+ N2. Note that N2 has no nonzero elements divisible by pk for all k _> 1. This implies t h a t #2 maps ( S + M ) / M
to zero in N2. Thus S C N1, so
(S + M ) / M C P E ( M ) / M . It is easy to see that S is not contained in M, and then (S + M ) / M # 0 is not torsion free. This contradiction shows that cotorsion Abelian groups may fail to be pure injeetive in general. [] In the above theorem we considered the pure injectivity of modules in an exact sequence. The example below shows that even over a v o n Neumann regular ring, B and C in an exact sequence 0 --+ A -+ B --+ C --+ 0 can be pure injeetive without A being pure injeetive. E x a m p l e : Let Q~ = Q be the rational numbers field for each i. Let A C I] Q~, i > 0 be the set of constant sequences and let R = A + @Qi. Then R is a von Neumann regular ring. Let M = @Qi- Then we have an exact sequence 0 --+ M ~ E(M) --+
E ( M ) / M ~ 0 with E injective. Consider D = E(M)/M.
Note that R is von
Neumann regular and every ideal is countably generated. By Jensen's theorem (see Glaz [44]). gl.dim(R) = 1. This implies that D is injective. Hence both E and D are pure injective. But M is not pure injective. In fact if M were pure injeetive, then it must be injective. But then the latter implies t h a t @Qi = R . e for some idempotent element e E @Q~. This is a contradiction. []
3.6
Relative homological theory
In this section we briefly consider the relative homological theory derived from flat precovers and flat preenvelopes. Akatsa [69] investigated this theory in his thesis
76 mainly from the point of view of flat preenvelopes. Here we have a two-sided approach. Throughout this section we assume R is right coherent. As before 5v stands for the class of flat left R-modules. For a left R-module M , a left .T-resolution of M means an exact sequence " " ~ ~ _ I ~ ' - - ~ ~ M ~ O such that H o m ( G , - ) leaves the sequence exact for any flat module G. Equivalently, F0 is a flat precover of M, F1 is a flat precover of ker(F0 --+ M), .-., etc. If we take one such resolution of M, we can define Ext (N, M) =W(Hom(N, $'(M))) for all i _> 0, where .T(M) is the complex
- - . ~ ~ _ ~ . - . ~ ~ 0 A standard argument ensures that the Exit (N, M) are independent of the choice of resolutions 9C(M). On the other hand, by Theorem 2.5.1 for left R-module N there is a complex O -+ N -+ G o -+ G1 -+ . . . -+ G n -+ . . .
with each Gi fiat such that
H o m ( - , F ) leaves the sequence exact when F is fiat.
Equivalently, Go is a flat preenvelope of N, G1 is a flat preenvelope of G o / i m ( N -+ Go), 9 9 etc. We call this resolution a right S-resolution of N. Similarly, choosing one such resolution of N, for any left R-module M we can define that nxti(M, N) =W(Hom(G(N), M)) for all i > 0, where G(N) is the complex 0 -+ Go --~ G1 -+ '-- -+ G~ --~ " ' It is also easy to see that Exti(M, N) are independent of the right ~'-resolution of N. Now suppose R is right coherent and every left R-module has a flat cover. Then all ~ ( N , M) and Exti(M, N) are well defined. Actually they are correspondingly isomorphic. T h e o r e m 3.6.1 S u p p o s e M a d m i t s a left .T-resolution a n d N a d m i t s a right .7=resolution.
T h e n Ext'(N,M) ~ Exti(M,N) f o r i > O. H e n c e we d e n o t e t h e m by
Fexti(N,M). This n o t a t i o n was suggested by the n o t a t i o n Pexti(N,M) which was used by Jensen a n d Gruson in [50]. Proof." By a left St--resolution ~'(M) and a right .T-resolution ~(N), we can form a third quadrant double complex 9
77 0
0
0
....
Horn(G2, M) -Horn(G1, M) -~Hom(Go, M) -~
....
Horn(G2, F0) -Horn(G1, F0) -Horn(Go, F0) - H o r n ( N , F0) §
l
l
t
T
T
T
T
T
t
T
T
T
. . . . Horn(G2,F1) --Horn(G1,F1) § . . . . Hom(G~,G) *Hom(Gt, G) §
0
T
T
F~) -~Hom(N,Ft) -*
l
F2) §
F2) -
0
0 0
T
Note that since Fi are flat and 6(N) is a right ~r-resolution, all rows are exact except for the top row. Similarly all columns are exact except for the right column. Then, by chasing diagrams or by a spectral sequence sequence argument, we have Hi(Hom(G(N), M)) ~ W ( H o m ( N , 3C(M))), i >_ 0. This completes the proof. [] In particular, for any flat module F, Fexti(N, F) =Fexti(F, N) = 0 for all i > 1 and all modules N. Let us take this moment to have a look at Fext ~ M). L e m m a 3.6.2 Assume M has a fiat precover ~ : F ~ M and N has a fiat preenvelope : N -+ G. Then for any linear map f : N --+ M the following are equivalent: (1) f can be factored through a fiat module D, i.e., N ~ D --+ M ; (2) f can be factored through ~; (3) f can be factored through ~.
P r o o f i Obvious by the definition of flat precovers and flat preenvelopes. [] P r o p o s i t i o n 3.6.3 Assume M has a left J:-resolution and N has a right JZ-resolution. Then there is a surjective canonical map:
a : Fext~
M) --+ {f E Hom(N, M) ] f can be factored through a flat module}
P r o o f : Consider a left )C-resolution of M: 9.. --+ F,~ - + . . . -+ F I ~ F o & M
--+ 0
Applying H o r n ( N , - ) , we get 9.. --+ Horn(N, F 1 ) ~ H o m ( N , Fo) -+ 0 Define a : Fext~
M) = Uom(N,Fo) im(d~) --+Horn(N, M) by a ( f + ira(d*1) ) = ~ f . By the
previous lemma it is easy to check that this map satisfies the requirement. []
78 T h e o r e m 3.6.4 Let 0 --+ M1 --+ M --+ M2 ~ 0 be an exact sequence of left R-modules. Assume both M1 and M2 have left .T-resolutions and that 0 -+ Hom(G, M1) -~ Hom(G, M) --+ Horn(G, M2) --+ 0 is exact for any flat module G. Then there is a left .T-resolution .T(M) such that each term, Fi is a direct sum of the corresponding terms of the resolutions of 3/11 and M2. Furthermore, we have the long exact sequence: 9" --~ Fext2(N, M2) -+ Fextl(N, ]1//1) -+ Fextl(N, M) --+ Fextl(N, M2) --+ Fext~
M1) -+ Fext~
M) --+ Fext~
M2) --+ 0
Proofi This follows from Lemma 3.2.2 and a standard argument. [] T h e o r e m 3.6.5 Let R be right coherent. Assume M has a left F-resolution. Then the following are equivalent: (1) f . d i m ( M ) < 2; (2) Fextl(N, M) = 0 for any left R-module N. (3) Fexti(N, M) = 0 for any left R-module N and i > 1. P r o o f i (2) ===v(1) Since M has a fiat cover, we have exact sequences O---~ K - + F---~ M---~ O O-+ KI--+ FI ~ K ~ O with F ~ M a fiat cover of M, F1 --+ K a flat cover of K. We want to show that K is flat. By Lemma 2.1.1 K and KI are cotorsion. Hence, the conditions in Theorem 3.6.4 hold. And then, using Proposition 3.6.3, for any module X we have a commutative diagram with exact rows:
0 - F e x t l ( x , M)A-~Fext~
0
~Hom(X,
K)fl~Fext~
vLu
F ) ~ a Fext~
w
M)
,
K) q, Hom(X, F) P,Hom(X, M)
Note that u is isomorphism since F is fiat. This implies that v is injective. Similarly, we have a commutative diagram with exact rows:
0
79 Fextl(K1, M ) ~ A Fext~176
F1)a-g-~lFext~
,
0
11 0
' Hom(/s
ql Uom(/~l, El ) P l Uom(K1 ' K)
Since wl is injeetive and us is an isomorphism, vl is surjective.
In particular, the
identity 1Kt is an image of vl. By the definition of the map vl it follows that Kz is flat. (1) ==> (3) Obviously M has a left .~'-resolution: . . . ~ O ~ ~ ~ M ~ O For any module N, since Hom(N, - ) is left exact, it easy to see that Fextl(N, M) = 0. Trivially Fexti(N, M) = 0 for i _> 2. [] We now ask when M is flat and when f . d i m ( M ) = 1. C o r o l l a r y 3.6.6 Assume M has a left :~-resolution. Then by the proof above the following statements hold. (1) M is flat if and only if the canonical map c~: Fext~ M) --+Horn(M, M) is surjective. (2) f . d i m ( M ) < 1 if and only if Fextl(N, M) = 0 and the canonical map c~ : Fext~ M) --+Horn(N, M) is injective for any left R-module N. By Theorem 3.6.4 there is a natural generalization of the previous theorem. T h e o r e m 3.6.7 Assume M has a left .T-resolution.
are (1) (2) (3)
Then the following statements
equivalent: f . d i m ( M ) < n + 2; Fextn+l(N, M) = 0 for any left R-module N; Fextn+i(N, M) = 0 for any left R-module N and any i > 1.
Therefore, if every left R-module has a fiat cover then w.gl.dim(R) < n + 2 if and only if Fext n+l (N, M) = 0 for all left modules ~I and N. R e m a r k 3.6.8 For a module N, Fext~
F) may be zero for all fiat modules F. For
instance if (R, m) is a regular local ring, but not a field, let N = R i m be the residue field. Then there is no nonzero linear map ~ : N ~ F with F flat. This raises a question of when every nonzero module M admits a nonzero linear map into a fiat module. We will answer this question for commutative Noetherian rings at the end of the next chapter.
Chapter 4 Flat Covers over C o m m u t a t i v e Rings In this chapter we are mainly concerned with commutative Noetherian rings and the modules over them.
Among such rings, the most attractive ones are those of
finite Krull dimension. Our major goal is to show that all modules over commutative Noetherian rings of finite Krull dimension admit flat covers. This will enlarge the class of rings over which Enochs' conjecture is true and make it possible to apply flat covers in studying commutative Noetherian rings. The first section gives the necessary notation and preliminary results for modules over commutative Noetherian rings. We describe fiat cotorsion modules using completions of certain free modules. One result which is similar to the Matlis' structure theorem on injective modules will be presented in the first section. Just as the Matlis' structure theorem did, the structure of cotorsion flat modules will play an important role in investigating flat covers and flat resolutions of modules.
We then investigate the pure injective resolutions of flat modules in
the second section.
After these preparations we will be ready to prove the main
theorem. In the fourth section we discuss the flat covers of Matlis reflexive modules, and in the last section we show that the existence of nonzero flat preenvelopes leads to a characterization of commutative Artinian rings. For the notions and notation of commutative algebra, we refer to Atiyah [2], Matsumura [56] and Kaplansky [51]. Most of results presented in this chapter are taken from [31, 32, 73, 72, 13].
4.1
C o t o r s i o n flat m o d u l e s
From now on we assume all rings are commutative Noetherian rings. As usual, Spec(R) denotes the spectrum of R, the set of all prime ideals; Max(R) denotes the all maximal ideals. K.dim(R) means the Krull dimension of R. For a multiplicatively
Ms stands for the Rs with S = R\p, p E Spec(R), k(p) denotes the residue field P~/Pp (which is isomorphic to the fraction field of R/p).
closed subset S c R including the identity and an R-module M, localization of M at S. In particular, ~
stands for
82 For p E Spec(R), E(R/p) denotes the injective envelope of Rip. For convenience we state the Matlis theorem [54]: T h e o r e m 4.1.1 Let R be commutative Noetherian and E injective. Then E has a
decomposition, unique up to isomorphism, as a direct sum of copies of the E(R/p) 's for allp cSpec(R). That is, if Xp is the index set for the copies o r E ( R / p ) ' s associated with p, then we have E = | For an i d e a l I C R a n d a n
(Xp)
R-module M, if NI~M = 0, (1 < n < oc), we say
M is /-separated. Any M can be naturally equipped with the I-adic topology , and M is a metric space if M is/-separated. Then M has the (separated) completion with respect to the I-adic topology. In particular,/~ stands for the I-adic completion of R. The completion of M can be defined as the inverse limit f / =
lim M / I ' ~ M (see +__
Atiyah [2]). The following result is also due to Matlis [54]. T h e o r e m 4.1.2 Let (R, m) be a local commutative Noetherian ring. Then
i~ ~= HomR (E(R/ m ) , E ( R / m )) Note that a free module M = R (x) over (R, m) is m-separated, and note that the completion can be constructed using Cauchy sequences. Then the completion of M can be described as {(rz) I rx c / ~ , r, = 0 except for a countable number of x and if (xi) is any sequence of distinct elements of X then lira rz~ = 0} We call T a completion of a free R-module with a base indexed by X provided
T ~ R(x). Note that the cardinality of the index set X is independent of the choice of X and is the same as the dimension of T / m T as a k = R/m-vector space. For any p E Spec(R), Rp is local and / ~ is the completion with respect to the pp-adic topology. By Matlis' theorem, / ~ ~HOmR(E(R/p),E(R/p)). We use Tp to denote a completion of a free Rp-module. Recall that an/~p-module is called _Matlis reflexive if the canonical map M --+ M w = Hom(Hom(M, E(R/p)), E ( n / p ) ) is an isomorphism. By Lemma 3.1.5 every Matlis reflexive/~p-module has a flat cover as an Rp-module since it is pure injeetive. Furthermore we have that
Proposition 4.1.3 If M is an [~p-module and Matlis reflexive, then M has a fiat cover as an R-module.
83 Proof." Since M ~ M w, M is pure injective as an /)v-module, and so it has a fiat cover F --+ M over/~p by Lemma 3.1.5. We claim t h a t this is a flat precover of M as an R-module. Given any linear m a p G --+ M with G R-flat, we have a factorization
a|
M
But G |
i s / ~ flat, so G |
~ M can be lifted to F . Hence
F --+ M is a flat preeover as R-modules since F is flat as an R-module. [] In the rest of this section, we will study the structure of cotorsion flat modules. As we mentioned before, we want to get a result which is similar to the structure theorem for injective modules. Initially, we wanted to find the structure of general flat modules. But this seems to be very difficult. However the structure of cotorsion flat modules will serve us adequately for the proof of our main theorem. We have seen that cotorsion flat modules are direct summands of Horn(E, Q/Z) with E injective (Lemma 3.2.3). But for commutative Noetherian rings, the result is still valid if Q/Z is replaced by any injective cogenerator C of the category Rdt4. So in fact they can be built up by completions of free P~o-modules. For p E Spec(R), set
An = {x E E(R/p) I pnx = 0}. Note that An is finitely generated as an P~o-module (see Matlis [54] for the details). By Proposition 1.4.10, we have the following lemma. L e m m a 4.1.4 Let R be commutative Noetherian, p E Spec(R). Then
(1) for any element s E R\p, the multiplication by s on E(R/p) is an automorphism; (2) E(R/p) = WAn. Griffith [46] and Fuchs [42] noted the following description of the completion of a free Rp-module. L e m m a 4.1.5 Let R be commutative Noetherian, p E Spec(R). Then for any set X ,
HomR(E(R/p), E(R/p) (X)) is isomorphic to the completion of a free Rp-module with a base indexed by X. Proof." Define a m a p as follows
c~ : HOmR(E(R/p), E(R/p) (X)) -+ R (X) with c~(f) = (q~f) for any f E HomR(E(R/p), E(R/p)(X)). Here qx : E ( R / P ) (x) -+
E(R/p) is the canonical projection. Then q~f E HomR(E(R/p), E(R/p))=/~p. We claim t h a t the sequence (rx) = (q~f) satisfies that rx = 0 except for countable many of them, and lira r ~ = 0 for any sequence xi of distinct elements of X. Since E(R/p) = UAn with An finitely generated as an Rp-module, f(UAn) = Uf(An). This shows t h a t r~ = q~f = 0 except for countable many r~. Now let xi be a sequence of distinct elements of X. Note that q~f(An) = 0 for i sufficient large. This implies that rz i E pn/~p for i sufficient large and then l i m r ~ = 0. Obviously c~ is injective. But for any such (rz) E R (X), we can define f(y) = (rz(y)) E
E(R/p) (x).
Since limrx, -- 0 for
84 any sequence xi of distinct elements of X we see that for y c A~, rx(y) = 0 except for a finite number of x C X. It follows that f E H o m n ( E ( R / p ) , E ( R / p ) (X)) and that (~(f) = (r~). Hence a is an isomorphism. [] For any p ESpee(R) and a free Rp-module R (X) with the index set X , Tp = i
H o m n ( E ( R / p ) , E ( R / p ) (x)) ~ R(~v). Let k(p) = (R/p)v, k(p) | Tp ~ Tp/pTp k(p) (z). Note that Tp is a direct sumInand of H o m R ( E ( R / p ) , E ( R / p ) x ) ~= I] RX. We claim that pTp is pure injective, so is cotorsion. But the fact that p ( l - I / ~ ' ) = l-I(p/3~p)X and the fact each p / ~ is pure injective imply the conclusion. Also note that T~ is flat (Lemma 3.1.4). Therefore the canonical map ~ : Tp --+ k(p) (x) gives a special flat precover of the vector space k(p) (z). Furthermore we have the following P r o p o s i t i o n 4.1.6
The canonical map ~o : Tp ~ k(p) (x) is a fiat cover.
P r o o f : Consider the exact sequence 0 --+ pTp --+ Tp --+ k(p) (X) -+ O. As noted above, this sequence gives a flat precover. Suppose it is not a flat cover. By Theorem 1.2.7, there is a decomposition Tp = G G K with G --+ k(p) (X) a flat cover and K C ker(~) = pTp. From this it is easy to deduce that p . K = K, and then K -- 0 because Tp, and hence K, is p-separated. [] By this observation we know that Tp is uniquely determined by the vector space
V = k(p) (X). Note that any such vector space V is a direct s u m m a n d of k(p) X as P~-modules. But the latter has a flat cover as both Rp-modules and R-modules. So V has a flat cover. E x a m p l e If k is a field and
Card(X) = m < oo, the construction above shows
that = k [ [ x l , . . . , xn]] x --+ k X is a flat cover over R = k [ [ x , , . . . , x,~]]. However, if X is infinite, k [ [ x l , . - . , x n ] ] (x) --+ k (x) is not even a fiat precover. To see this, we set X = N, all the positive integers, and note that the vector space k (N) has a fiat cover. By Theorem 1.4.7 if k [ [ x l , . . . , xn]] (y) ~ k (y) is a flat precover, then it must be a flat cover. Now for any countable set ri E J ( R ) = ( x l , . " , x , )
with r~ # 0 we define fi
to be the multiplication by ri on R regarded as a linear map R i --+ R i+1. Applying Theorem 1.4.4, there is an positive integer n such that rl 9r 2 . . . r ~ = 0. This is not possible. [] For p ESpec(R), suppose there exists a decomposition Tp = A @ B as Rp-modules. Then A ~ A / p A gives a flat cover of the vector space A / p A as both Rp-modules and R-modules. This implies that A is also the completion of a free Rv-module with a basis indexed by a set Y. In other words, d ~- H o m R ( E ( R / p ) , E ( R / p ) (y)) = Tp. In general, we have the following. P r o p o s i t i o n 4.1.7 Suppose ~ : F --+ M is a fiat cover of module M .
Let F =
F1 @ F2, M = M~ 9 M2 with (p(Fi) C Mi, i = 1, 2. Then ~o IFi: Fi --+ Mi are fiat covers for i = 1, 2. P r o o f : This is easy by chasing diagrams. []
85 Before giving the characterizations of cotorsion fiat modules, we need more preparation. The following lemma is very useflfl in the future. It was initiated by Enoehs in [32]. L e m m a 4.1.8
Hom(I-[ Tq,Tp) = O. pgq
Tp -Hom(E(n/p), E(R/p)(X)) for some index set X, and Hom([I,~ T~| E(R/p), E(R/p)(X)), we only need to show that
Proof: Since we have that
Hom(flp~q Tq, T~) ~
l-ITpf~qTq | E(R/p) = 0. Note that E(R/p) = UAn = limA~ with each An finitely generated and annihilated by p~. We note that for each AN,
([ITq)|
~ I-[(TqQAN). S i n c e p g q there is an element s i n p , but not in q.
This implies that the multiplication by s ~ is an isomorphism on Tq. Hence Tq | A~ =
(snTq) | An = Tq | (s~A~) = 0. This finishes the proof. [] C o r o l l a r y 4.1.9 Horn(Rq, Rp) r 0 if and only if p C q. P r o o f : If p g q, by the preceding lemma,
Hom(/~q, &) = 0. If p C q, we have that
Rp =Hom(E(R/p), E(R/p)), ftq | E(R/p) - | ') with q' ESpec(R) since it is injective as/~q-module. Noticing that Rq -4 Rq is a pure injection, we see that Rq | E(R/p) --4 Rq | E(R/p) is a pure injection, too. An easy argument shows that Rq @ E(R/p) = E(R/p) (Y) for some set Y. From this we get a copy of E(R/p) in f~q @ E(R/p). Therefore, Hom(/~q,/~) contains a copy of Hom(E(R/P), E(R/p)) = &.[] Note that if F is flat, then for any q ESpec(R), F | E(k(q)) = | implies that
This
Horn(F,/~q) ~ H o m ( F | E(R/q), E(R/q)) ~- RXq for some index set X.
X empty corresponds to the case Horn(F,/~q) = 0. In particular, setting F = / ~ , we get Hom(/~,/)q) = Rq~X. C o r o l l a r y 4.1.10 For any set X, RX is the completion of a free Rq-module. Every
such a completion is a direct summand of [tqx for some indez set X. P r o o f : Since /~qX ~ Hom(E(R/q),E(R/q) x) and the injeetive module E(R/q) X =
| (Y,) with p c q. Note that Horn(E(R/q),E(R/p)) = 0 if p ~ q. Hence, Hom(E(R/q), E(R/q) x) =Hom(E(R/q), E(R/q) (r)) = Tq for some index set !/. Now, suppose Tq is the completion of a free Rq-modute. Then Tq ~- Hom(E(R/q), E(R/q) (x)) for some set X. But since E(R/q) (x) is a direct summand of E(R/q) x, the conclusion follows easily. [] The next several lemmas consider some useful properties of the representation 1-ITp. With these properties, we will be able to derive the structure theorem for cotorsion flat modules.
86
Let s >_ 0 and H = 1-Iht(v)=,Tp. Then if H = Ht G H2 is a direct sum decomposition then for each p there is a decomposition Tp = Up (9 Vp such that
Lemma4.1.11
H~ = I-I Up, Hu = I-I ~'~. P r o o f i First note that for any endomorphism f : 1-Iht(p)=sTp --+ 1-Iht(p)=sTp, f = rI fp :
(xp) ~ (fp(x;)), here fp : Tp --+ I-iht(p)=sT p 4 I-[ht(,)=~ Tp --+ Tp is the composition. In fact, by Lemma 4.1.8 Hom(I~q#pTq, Tp) = 0 for each p, and then the composition [Iq#po Tq -+ l~ht(p)=~ TpI-~Tpo is zero for any P0. Hence it is easy to see that for any (x,), if f ( x ; ) = (yp) then yp = fp(Xp). This shows that f = I] fp. Now let r : H --+ H be a retraction onto H1 with the kernel H2. By the above remark we can assume that r = (%) = [I rp. Then each rv : Tp -~ Tp is such that r~ = r;. So Tp =im(rp)|
By the remark before Proposition 4.1.7, Up = i m ( r ; )
is the completion of a free P~-module. The same is true for Vp =ker(rp). Therefore the result follows since im(r) = [I im(rp), ker(r) = 1-[ker(rp) r-I L e m m a 4.1.12 Let X CSpec(R) be a subset and let q E X be a maximal element of
X . Then
, I]p~x Tp 1-[pexTp q( op~-~xTp ) - (~p~xTp
Since q is maximal, qTp = Tp for p E X, p r
Proofi
q.
Hence q([Ip~xTp) =
(qTq) 9 (I]p#q Tp). And so (q [Ivex Tp) + (@vexTp) = 1-Ipex Tp. [] L e m m a 4.1.13 |
C [I Tp is a pure injeetive envelope with p C Y, a subset of
spee(R). P r o o f i Note that 1] T; is pure injective and @T~ is a pure submodule of [[ T;. Let D1 be the pure injective envelope of ~ T v such that GTv C DI C !V[Tv. Then there is a submodule D2 such that 1] Tp = D1 @ D2. Let X consist of all q E Y such that the projection [[ Tp --+ Tq restricted to D2 is not zero. Then D2 C II;~x To. Assume X ~ r and let q E X be maximal in X. Consider the projection: [I Tp -+ D2. By restricting we get a surjection [[p~x T; --+ D2 which contains Ov~xT v in its kernel. So we have an induced surjection I-[p~x T;
> D2 9
@pex Tp But by the previous lemma we get qD2 = D2. So since D2 c I-[pex, we have
D2 C n~~
l-[ Tp) C l-I rq.~__l(q~T;) pEX
But
nn~
X = r
q -=
O. Hence D2 ~ Tq is zero. This contradicts the choice of X unless
Therefore X = r D2 = 0 and so @Tp C I-[ Tp is a pure injective envelope. []
87 T h e o r e m 4.1.14 Let F = l-Ip~y Tp for a subset Y of Spee(R). Let F = D 9 H be a
decomposition. Then D ~- I-Ipcy Up where Up is the completion of a free Rp-module. Proof: Set
F,~ =
l-Iht(p)
F-1
=
O. Then
UF,~=( IX T~)G( I f ht(p)=O
Tp)e...|
rI
ht(p)= l
Tp)|
ht(p)=n
The same argument used in the proof of the preceding lemma shows that UF~ C F = H Tp is a pure injective envelope. Note that by Lemma 4.1.8, F is stable in F,~ for every endomorphism f, i.e., f(Fn) C Fn. Hence the direct sum decomposition F -- D @ H induces a direct sum decomposition F~ = D~ @ H~. Then UF~ = (UD~) 9 (UH~) C D | H -=- F is a pure injective envelope by the above. So UD, c D is a pure injective envelope. F,~ ~ D~ H~ F~ Now note that Fn-1 - D,~_~ 9 H,~_~' Fn-~ -
r[
Tp. So by Lemma 4.1.11 for
ht(p)=n
all p E Y, there are direct sum decomposition Tp = Up @ Vp with ht(p) = n such that
D~/D~_I ~- I]ht(p)=nUp. But then we have U~=oD~-( H ht(p)=O
Up)@( H
Up)@. . . . . .
ht(p)=l
But a pure injective envelope of the (Hht(p)=oUp) G (Hm(p)=l Up) @ . . . . . . is I-lpcy Up, and so D ~ Hpey U~.H Now we are ready to give the characterization of cotorsion flat modules. T h e o r e m 4.1.15 Let R be commutative Noetherian.
The following statements are equivalent for an R- module F. (t) F is a fiat cover of some cotorsion module; (2) F is fiat and eotorsion; (3) F ~- HTp, p E Spec(R), where Tp is the completion of a free Rp-module with respect to the pp-adic topology. Furthermore in this product each Tp is uniquely determined as the completion of a free Rp- module and this product is also unique up to isomorphism.
P r o o f : (1) ~
(2) Let 9~ : F --+ M be a flat cover of an R-module M. By Lemma 2.1.1,
the kernel is cotorsion. Then F is cotorsion by Proposition 3.1.2 (2) ==v (3) Note that F is a direct summand of Hom(Ei, E2) for injective modules E1 and E2. By Matlis' theorem E1 ~- @E(R/p) (Xp) with p E Spec(R). Hence Hom(Ea, E2) ~ I I Hom(E(R/p), E2) 9 Note that Hom(E(R/p),Hom(Rv, E2)) ~- Hom(E(R/p), E2). But Hom(Rv, E2) is an injective Rv-module, so Hom(Rv, E2) ~- @E(R/q), q E Spec(R) and q c p. Notice that for q g p, Hom(E(R/p), E(R/q)). Therefore
Hom( E( R/p), E2) ~- Hom( E( R/p), E( R/p) (X)) = Tp
88 for some index set X. Consequently F is a direct s u m m a n d of l-I Tp. Note t h a t we can arrange this product so that there is only one Tp appears for each p. Now by the previous theorem every direct summand of I] Tp still has the same form. We may use the same notation and write F ~ 1-ITp. Now it remains to show that this product is unique up to isomorphism. Consider G = I] Tp. If q E Spec(R) and q !g P, then there is an element s in q, but not in p. Since the multiplication by s on E(R/p) is an isomorphism, qTp = Tp. If q C p, then
Aq'~R(pX) = 0 because of NpnR (x) = 0. This means t h a t Aq'~TB = O. For any fixed q, set G' = r~qnG = l-I(Nq'~Tp) = 1-ITp with q r p. Set H = l-ITp with q C_ p. Then G = G ' @ H , GIG' = H. Obviously H = Tq@I]Tp with q c p. But then, taking the following operation through all these prime ideals p we get Aqcp(Ap'~H) = Tq. In other words, for a fixed q, Tq in the product I-i Tp can be recovered by the operation
nq~p(npn( G /NqnG) ) =~'Tq Therefore the uniqueness follows since the operation above commutes with isomorphisms. (3) ==* (1) Let m(/~p) be the unique maximal ideal o f / ~ .
For each Tp, we have an
exact sequence
o
T,
k(p)
0
which gives a fiat cover of k(p) (X). Taking the products, we get
o
II
IIT,
II
o
This shows t h a t F = l]Tp is a fiat precover of [Ik(p) (X). We claim t h a t actually this is a fiat cover. Suppose it is not. Then by Theorem 1.2.7 there is a submodule
S C I ] m ( ~ ) T p and a submodule B such t h a t F = S e B with B a flat cover of [I k(P) (X). We shall show t h a t S is zero. To do so, we only need to guarantee t h a t for each q the projection of S onto Tq is zero. If q is with S c qF, then by qF = qS @ qB it follows that S = qS. This equation is preserved by projecting to Tq. Since every submodule of Tq is q-separated, the projection of S onto such a Tq is zero. Now suppose S ~: qF. We may choose q to be maximal with this property. Note t h a t if q r p, qTp = Tp, if q ~ p, by the choice of q, S C pF and the projection onto Tp is zero. Since qTq = m( Rq)Tq, S C 1-Irn( Ptp)Tp = YIq~p qTp 0 qTq (~ Ilqgp rn( [~p)Tp and the last part is not necessary. Hence we have S c 1-Im(lr~)Tp = Ylqgp qTp G qTq c qF. This is a contradiction. []
89
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 o f fiat m o d u l e s
We have seen that every module has a pure injective envelope. Then for every module we can define a minimal pure injective resolution by taking pure injective envelopes as usual. We will see that this resolution is particularly interesting for a flat module over a commutative Noetherian ring R. Let F be a flat R-module. We see t h a t any cotorsion envelope C(F) of F agrees with its pure injeetive envelope P E ( F ) . In particular, there is an exact sequence 0 --* F --+ P E ( F ) --+ D --+ 0 with D flat. Recall t h a t for any p E Spee(R) and an R-module M, the completion of Mp can be defined as f/p = lim Mp/p~Mp. See Atiyah [2] for the full explanation. The following result is due to Raynaud and Gruson
[63,
Prop.2.4.3.1].
L e m m a 4.2.1 Assume R is commutative Noetherian ring and p E Spec(R). for any fiat R-module F, Fp is a completion of a free Rp-module.
Then.
P r o o f ' . We only need to deal with the local case, i.e., (R, m) is local with the maximal ideal m and F is a flat R-module. Note t h a t R i m s is Artinian for all positive integers n. So the fiat R/mn-module F/m'~F is projective and so free. Also - - | F ~,J= mF__F, So if we choose a base of F over ~r-~R, R RR it is m a p p e d onto a base of ~ over mn under the map ~
F
--+ m--~F" Let G be a free R-module having a base with the same
cardinality as the cardinality of a base of mF~ over m--~RR(for any n > 1). We claim t h a t the completion of G is isomorphic to F . By the choice of G it is easy to see t h a t we have maps G --+ ~
for all n > 1 such t h a t the induced maps ~
-+ mF~ are
isomorphisms. Furthermore the following diagrams are commutative:
G
mn+lF
* mnF
Then we have the isomorphic projective systems G
G
G
I
F
I
F
F
1
Hence t~ ~ d is the completion of the free R-module G. [] Notice t h a t with the assumption above, the canonical map F --+ Fp induces an isomorphism F |
k(p) -+ Fp | k(p). Furthermore, F ~ Fp is a universal m a p into
such a module, i.e., the diagram
90
\ T can be uniquely completed into a commutative one whenever T is the completion of a free P~-module. To see this, we note that the diagram
T can be uniquely completed because T is an Rv-module. Now we only need to consider the local case. But then ' there are linear maps m~----fi k --+ m~----Y T completing the diagrams
F
,F
. rnn. FA F
T
9mnT
Now, taking the limits, we get the desired map. Also note that for any flat module F and prime ideal p, if pFp = Fp then/~p = O. This is equivalent to Fp | k(p) = O. P r o p o s i t i o n 4.2.2 I f F is a fiat module, then the canonical map F --~ [I Fp is a pure injection. Here p is ranging through the whole Spec(R)
P r o o f : Consider the pure injective envelope of F, and let P E ( F ) = l-I Tp. Then for any prime ideal p the following diagram
F
.&
can be completed by the observation before. Further the diagram
F
91-[ ~p
can be completed. This shows that F -+ 1-[/~ is a pure injection since F --+ P E ( F ) is a pure injection. []
91 R e m a r k 4.2.3 For a commutative Noetherian ring R, it was proved by Jensen in [50] that R is pure injective if and only if R is a direct sum of finite complete local rings. In general, the pure injective envelope PE(R) may strictly contain R. Warfield in [71] gave a description of PE(R). Here is a simple proof of his result. C o r o l l a r y 4.2.4
PE(R) = l-[ [:gm, m e Max(R).
P r o o f : Set E = (~E(R/m), where m ranges through Max(R). Then there is a pure injection R -~Hom(Hom(R, E), E) = Horn(E, E) which gives a pure injective preenvelope of R. We see that Hom(E, E) = I-[ Hom(E(R/m), E ( R / m ) (x~)) = [I Tm. Hence PE(R) is a direct summand of l-I Tin, and so by Theorem 4.1.15 it still has the same form. We can assume PE(R) = l-I Tin. We claim that for each m, T m = / ~ . By the preceding proposition, R -+ I-[/~ gives a pure injective preenvelope of R. And then PE(R) = I]Tm is a direct summand of I-I/5~. By Theorem 4.1.15, it follows that T m = / ~ m . In order to see that e a c h / 5 must appear in PE(R), we consider the pure injective map R --+ 1-I/~. S u p p o s e / ~ , is missing. Taking the tensor products
R | E(R/m') ~ If ~ | E(R/m') is injective. But l-I ~ | E(R/m') = 0. This is a contradiction. [] From the above we note that if a local commutative Noetherian ring R is pure injective, then R ~ / ~ . This implies that any self-pure injective ring R is isomorphic to a direct sum of finitely many complete local rings. P r o p o s i t i o n 4.2.5 Let F be fiat and PE(F) = l-lTp. Then for each p, Tv is iso-
morphic to a direct summand of ~'p. If p is maximal with respect to the property F | k(p) # O, then the map ~'p --~ Tp which makes the following diagram commutative is an isomorphism.
i
Proof." Consider the following diagram
F
~s
Here the m a p s is available because 13 : F --+ l-I Tp is pure and ]-] Fp pure injective. By the above lemma there is a linear map f = 11 fv such that ~ = fc~. Hence ~ = (fs)~, and f s is an automorphism by the definition of pure injective envelopes. Set (fs)# = 1,
92 the identity of [I Tp. Then it is easy to see that there exists a linear maps sp : Tp -+ Fp such that fp% = 1Tp. This shows that Tp is isomorphic to a direct summand of/~p and that fp is a surjection. Hence if Fp | k(p) = 0, then/~p -- 0, so Tp = 0. Now, let p be maximal such that F | k(p) -~ Fp | k(p) ~ O. Then Fv -~ 0. But for q G Spec(R) with p g q, Fq = 0. So by Lemma 4.1.8, we have that Hom(![]p#q Fq, Tp) ~- Hom(rlpcq Fq, Tp)| kp, Tp) : O. Similarly, Hom([Ip#q Tq, F,) = 0 since Fp = T; and Tq = 0 whenever/Sq : 0. We then consider the commutative diagram
?
H F
9I] Tq =t-lp#qTq |
II
v
Is
Using the above discussion, we can pass to the quotients and get a commutative diagram
II F
,Tp
rl
1
F Since the m a p F -+/~p is a completion, the composition/~v -+ Tp --~ Fp is the identity. This ensures that fp is injective, and then that it is an isomorphism. [] Recall that if F is flat, then the pure injective envelope P E ( F ) = P E ~ has the quotient F / P E ~ flat. Then, we can take the pure injective envelope of the quotient, denoted by p E I ( F ) , etc. All these P E n ( F ) are flat and pure injective. By doing so, we get a minimal resolution of F,
0-+ F--+ P E ~
~ g E l ( F ) ~ ... --+ P E ~ ( F ) -~ ...
Also note that for each n > O, PEN(F) = I-[ Tp. We define ~ ( q , F) to be the cardinality of a base of a free Rq-module whose completion is the Tq which appears in the product
P E ~ ( F ) = I-I Tp. These numbers are quite interesting. The following result is due to Enochs [32]. T h e o r e m 4.2.6 If F is a flat R-module, and p is a prime ideal such that Vn(q, F) = 0 for any prime ideal q ~ p, then L,~+I(q,F ) = 0 for all q D p (including p itself).
93 Proof:
We give the argument for n = 0 as it is easy to modify for n > 0.
Let
P E ( F ) = l-I Tq. Then Tq = 0 for q ~ p. We may assume that Tp r 0 in the product I] Tq. Then p is maximal with respect to F | k(p) ~ O. By Proposition 4.2.5/~p = Tp. We first argue t h a t I]qcp TqNk(p) = 0. For any finitely generated submodule S c k(p), Tq | S = 0 since Tq = 0 when q ~ p; rS = 0 when r C p but not in q. Then it follows t h a t ([Iq#p Tq) | S '~ ~q~p(Tq | S) = 0, and then Viq#pTq | k(p) = O. Now let 0 -+ F -+ PE(F) --+ C -+ 0 be exact. Since Fp | k(p) -+ P E ( F ) | k(p) is an isomorphism, C | k(p) = 0, and Cp = 0. This shows that Tp is zero in P E ( C ) . If q ~ p, then PE(F) | k(q) = 0 because Tq = 0 by the assumption. Therefore C | k(q) = 0. This means that Tq is zero in PE(C). This completes the proof. [] Gruson and Jensen in [50] extensively studied the resolutions by pure injective modules and the related homological topics. One of their deep results is t h a t fiat modules will have finite project dimension if the associated rings are of finite Krull dimension. We will give a proof for the case of commutative Noetherian rings of finite Krull dimension. First we state the following [50, Thm.7.10]. C o r o l l a r y 4 . 2 . 7 If the Krull dimension K.dim(R) < oo, then PF'~(F) = 0 for all n > K.dim(R). This says that the pure injective dimension of any flat module is at most K.dim(R). Also if K.dim(R) = n < oc, then ~n(P, F ) # 0 implies that p is minimal. We now can prove Gruson and Jensen's theorem. This is very useful in proving the existence of fiat covers over commutative Noetherian rings of finite Krull dimension. T h e o r e m 4.2.8 Let R be commutative Noetherian of finite Krull dimension K.dim(R)
= d < oc. Then for any fiat module F, proj.dim(F) < d. P r o o f : Consider the partial projective resolution of F
O-+ K - + P n _ I - + . . ' - ~ Po-+ F--~ O with all Pi projective. is projective.
Obviously K is fiat.
We hope to prove t h a t actually K
Let 0 -+ L -+ P -+ K -+ 0 be exact with P projective.
Then
Ext 1(K, L) =~Ext "+' (F, L). Considering the minimal pure injective resolution (or equivalently minimal cotorsion resolution ) of L, by the previous Corollary, we get PEn+I(L) = 0. Since ExtJ(F, P E i ( L ) ) = 0
for all i > 0
andj>l,
it is easy to see that we have
Extn+l(F, L) ~ E x t l ( F , PEn(L)) = 0. This implies t h a t
E x t l ( K , L) = 0. The latter
ensures t h a t the exact sequence 0 -+ L ~ P --+ K --+ 0 is split. []
4.3
Flat covers of cotorsion modules
Wr now are at the point that we are able to prove the existence of flat covers over commutative Noetherian rings of finite Krull dimension. But we have to solve
94 this existence problem for cotorsion modules first. Our strategy is to prove t h a t every cotorsion module has a flat cover, and then prove that every module having a finite resolution by cotorsion modules has a flat cover. Finally, by applying Gruson and Jensen's theorem (the last theorem of the previous section), we simply conclude t h a t all modules under the current consideration have finite resolutions by cotorsion modules. We start with the following lemma. L e m m a 4.3.1 Let C be cotorsion.
Then there is a cotorsion flat module G and a
linear map ~ : G ~ C such that any linear map Rp -+ C and any linear map Rp ---+C can be factored through ~o. Proof:
For each p e Spec(R), set Xp =Hom(/~p,C).
Define a linear m a p r
:
@ / ~ p ) -~ C naturally. Then every linear map / ~ --~ C can be factored through r
Letting Fp be such an /~p(Xp) we have the pure injective envelope of OF~, say
G = P E ( G F p ) . So the quotient D = G / G F F is flat. Since Ext I(D, C) = 0, it is easy to see t h a t ~ can be extended to a map ~0; G ~ C. Consequently, every linear m a p / ~ ~ C can be factored through G. Since any linear m a p / ~ to/~
~ C can be extended
-+ C, and the latter can be lifted to G --+ C, this implies t h a t Rp --+ C can be
factored through ~o. [] L e m m a 4.3.2 For a given module M , suppose ~ : G ~ M is a linear map with G pure injective. Let S be pure in an R-module F and f : F -+ M linear.
Suppose
f Is: S ~ M can be lifted to g : S -+ G, then g can be lifted to h : F -+ G and f -
~oh
induces a linear map # : F / S ~ M . Furthermore, if the induced map # can be lifted to w : F / S ~ G, then the original map f can also be lifted to G. P r o o f : First note t h a t g can be extended to h because S --+ F is pure and G is pure injective. Then note that ( f - ~oh)(S) = ( f - ~ h ) f Is (S) = ( f Is - ~ o h f Is)(S) = ( f Is - F g ) ( S ) = ( f Is - f Is)(S) = 0. Hence we have the induced map # : F / S -+ M. Suppose ~w = # and a : F --~ F / S .
Then # a = ~owa, but then #a = f -
~h.
Therefore f = ~o(wa + h). This completes the proof. [] L e m m a 4.3.3 Let Tp be the completion of a free Rp-module Fp for p c Spec(R). Let 0 -+ @Fp --+ [I Tp ~ H -+ 0 be exact. For a given Tq in the product, if Tp = 0 f o r any p ~ q, then H | k(p) = 0 for all p D q. Consequently Po(P, H) = 0 for all p D q. P r o o f : First note that the stated sequence is pure exact and Fq | k(q) -+ Tq @ k(q) is an isomorphism. Then the rest of the proof is similar to t h a t of Theorem 4.2.6. [] We are ready to prove that every cotorsion module has a fiat cover. More explicitly, we have the following: Theorem
4.3.4 Let R be commutative Noetherian of finite Krull dimension. Let C
be cotorsion and ~o : G -+ C be the map in Lemma 3.3.1 such that every linear map Rp ~ C can be factored through ~o. Then G is a flat preeover of C.
95 P r o o f : We we have to guarantee that any linear map ~' : D --+ C with D flat can be lifted to D -+ G. Consider the pure injective envelope P E ( D ) of D. Note that
P E ( D ) has the form I] Tp and H = P E ( D ) / D is also flat. Since C is cotorsion and Ext 1(D, H) = 0, this implies that ~' can be extended to f : P E ( D ) --+ C. H e n c e , in order to prove that ~t can be lifted, it suffices to show that f can be lifted. Assume R has dimension d and f : F -- [I Tp -~ C. Let Fp be the free Rp-module whose completion is the Tp in the product and let S = GFp and H = F / S .
Note
that S -+ C can be lifted because Fp is Rp-free and every map Rp ~ C can be lifted. Consider the exact sequence 0 --+ S --+ F ~ H ~ 0. By Lemma 4.3.2, we only need to show that every map H --+ C can be lifted to ~ : G --+ C. Note that if the height of p is zero, then Rw is Artinian, and so Tp is a free Rpmodule itself. Since there are only finite many minimal prime ideals of R, f : F =OTp -- I-[ Tp ~ C can be lifted to G --+ C with the product over all p with ht(p) = O. We now use a proper induction procedure.
Let us assume that every f : F =
l-] Tp -+ C can be lifted with the product over all p with ht(p) < t, t > O. We shall prove that every f : F = [ITp -+ C with the product over all ht(p) < t + 1 can be lifted. By the exact sequence
0 --+ S ~
F ~
H --+ 0, it suffices to lift any map
H ~ C. Let P E ( H ) = l-] Tq be the pure injective envelope of H. By Lemma 4.3.3, if
Tq ~ 0 in IF]Tq, then ht(q) <_ t. By the inductive hypothesis, every map R E ( H ) ~ C can be lifted. This ensures that H --+ C can be lifted since it can be extended to a linear P E ( H ) --+ C. We are through. [] Recall that in an exact sequence 0 ~
A -+ B ~
C ~
0 with both B and
C having fiat covers, A will have a flat cover if every cotorsion module has a flat cover (Theorem 3.1.10). By the theorem above, this is the case when R has finite Krull dimension. Now everything is in our hand to furnish the proof of the following existence theorem. T h e o r e m 4.3.5 Let R be commutative Noetherian of finite Krull dimension.
Then
every module has a flat cover. P r o o f : Let assume K.dim(R) = d. For any module M, consider the partial injective resolution
/
O ~ M ~ Eo ~ E1 -~ .. . -+ Ed -+ C ~ O with Ei injective. We claim that C is cotorsion. For any flat module F , by Gruson and Jensen's theorem, proj.dim(F) < d.
Then
Extd+2(F,M) = 0 ensures that
Ext 1(F, C) = 0. Note that all Ei and C are cotorsion, and they have flat covers. Now as standard, splitting the sequence above into short exact sequences and applying Theorem 3.1.10 repeatedly, we then see that M has a flat cover. [] R e m a r k 4.3.6 With this theorem we have a quite large of commutative rings over which the Enochs' conjecture holds. In particular, let k be a field, S = k [ x l , . . . ,x~]
96 with indeterminates x i and I be any ideal of R. Then the quotient ring R = S / I has finite Krull dimension. So every R-module has a flat cover. These rings include all coordinate rings of algebraic varieties. By the consistency between the existence of flat covers and cotorsion envelopes, we know that over such rings every modules has a cotorsion envelope. Then for a given module we can define its minimal cotorsion resolution by taking cotorsion envelopes as usual. Furthermore we have the following. C o r o l l a r y 4.3.7 Assume R is commutative Noetherian of finite Krull dimension d. Then any R-module M has a minimal cotorsion resolution as follows O --+ M --+ Co -+ C1--+ . . . -+ Ct --+ O such that t <_ d and Ci fiat and cotorsion for i >_ 1. So Ci has the form I-[ Tp for i > 1. Proof."
By Theorem 3.4.2 each Ci with i _> 1 in a minimal cotorsion resolution O -+ M -+ Co -+ C1--+ C2 -+ ""
is flat. Let O ~ M -~ Co ~ C1 -~ C2 ~ . . . ~ Cd ~ D --+ O be exact. Then D is flat and so by Gruson and Jensen's theorem proj.dim(D) < d. But then Extd+l(D, M) = 0 and so Cd ~ D admits a section s : D -+ Dd such that the composition D ~ Cd --+ D is the identity. But by the minimality of the resolution this implies D = 0. [] We would like to know how the existence of flat covers behaves under change of rings. The following are some results along this line. C o r o l l a r y 4.3.8 Let R be commutative Noetherian. For any R-module M and any p ~ Spec(R), Mp has a fiat cover as both an Rp-module and an R-rnodule. P r o o f : By the theorem above, Mp has a flat cover as an Rp-module. Let 0 -+ K -+ F -+ Mp --~ 0 be exact with F -+ M a flat cover as P~-modules. Note that F is flat as an R-module, it is easy to argue that this exact sequence gives a flat precover of Alp as R-modules. [] R e m a r k 4.3.9 Reviewing all the methods used to prove the existence of fiat covers, we see that we always somehow need a sort of finite dimension restriction on modules or rings. We do not know how to avoid the use of such restrictions. Even for commutative Noetherian rings, we do not have a handy example of a ring with infinite Krull dimension, but with every module having a fiat cover. Let I be an ideal of R such that K . d i m ( R / I ) is finite. By Theorem 4.3.5 every R / I - m o d u l e has a flat cover. Furthermore we have a result which is a generalization of Theorem 4.3.5.
97 T h e o r e m 4.3.10 Let R be commutative Noetherian. If I an ideal with K.dim ( R / I ) <
oc, then every R/I-module has a fiat cover as an R-module. Furthermore, every R / I module has an Jr-resolution as R-module. P r o o f : For any R / I - m o d u l e M, we have a resolution
O -+ M -+ Co -+ C1 -+ " " -'+ Cn -+ O with all C~ eotorsion. By Proposition 3.3.3, these are also cotorsion as R-modules. We first claim that any cotorsion R / I - m o d u l e C has a flat cover as an R-module. Now we regard C as a cotorsion R-module. By Lemma 4.3.1, there is an linear map ~ : G -~ C with G flat such that every linear map 5" -+ C can be lifted to G -+ C whenever S is a direct sum of free Rp-modules. For any linear map f : F --+ C with F flat, by taking pure injeetive envelope, we may assume F to be pure injective and flat, i.e., F = I-i Tp. Note that Hom(1]igp Tp, C) ~Hom(II1gp Tp, H o m ( R / I , C)) =Horn (I]l~_p Tp | R / I, C) =~" Hom(1-l,fap(Tp | R / I), C) = O. Hence we can further assume that f : I] Tp -~ C with p D I. Suppose ht(p/1) = 0 for all p such that Tp appears in [I Tp. Then as before, let S = GFp such that Fp is P~-free and its completion is Tp.
Every linear map S --+ C can be lifted to a
linear map S --+ G. But then the latter can be extended to 1-ITp --+ G such that the composition [I Tp -+ G --+ C agrees on S. This shows that this composition is just the original map f because the induced map of the difference of f and the composition
# : H = I] Tp/S -+ C must be zero. In fact, P E ( H ) = I] Tq with I g: q. Now assume every linear map f : 1-ITp -+ C can be lifted when h t ( p / I ) <_ t for all p with Tp in the product. By a reduction process used in the proof of Theorem 4.2.6, we can easily show that every linear map l-I Tp --+ C can be lifted when h t ( P / I ) < t+ 1. Since, K . d i m ( R / I ) is finite, we have shown that every linear map f : I] Tp ~ C can be lifted to [I Tp --+ G, and then ~ : G --+ C is a flat precover of C as an R-module. Let 0 --+ K --+ F = YIT~ --+ C --+ 0 be exact and F -+ C be a flat cover of C as an R-module. Then p D I. Note that K is a eotorsion R-module, but it may not be an R/I-module.
However, for T = I]q~1Tq, noticing that q g) p for any p such
that Tp is not zero in the product F = [I Tv, we get
that Hom([Iq~/Zp, l-[ Zp)
= 0 ,
and then Hom(T, K ) = 0. This guarantees what we have done for C above can be carried over to K. Consequently, K has a flat precover as an R-module. Therefore any cotorsion R / I - m o d u l e C has a U-resolution as an R-module, i.e., there are exact sequences 0 --+ Ki --+ Fi --+ Ki-~ --+ 0 with K_, = C with
K~ cotorsion for i _> 0
and with Fi flat. By the construction used in the proof of Theorem 3.2.5, we see that every R / I - m o d u l e M has a flat cover as an R-module by using the resolution stated at the beginning. The last statement follows from the argument above. []
98 Example:
Recall that a domain R is called a G-domain (see [51]) if the fraction
field K can be generated over R by one element, i.e., K = R[1/c] for some nonzero element c E R. A prime ideal p in a commutative ring R is called a G-ideal if R / p is a G-domain. By Theorem 146 of Kaplansky [51], A Noetherian domain R is a G-domain if and only if K.dim(R) < 1 and R has a finite number of maximal ideals. Now i f p is a G-ideal, then K . d i m ( R / p ) <_ 1 and there are only finitely many prime ideal containing P. So by the previous result, R is a commutative Noetherian ring and p is a G-ideal, then every R / p - m o d u l e has a flat cover as an R -module. Note that if u C R is not nilpotent, then there exists a G-ideal p such t h a t u is not in p. This fact provides a lot of G-ideals. If R is not a Hilbert domain, there must be a G-ideal which is not maximal. [] There are several interesting questions about flat covers under change of ring. For instance, suppose every R-module admits a flat cover, is it true for the polynomial ring R[x] over R? We conclude this section by the following corollary. C o r o l l a r y 4.3.11 Let R be a commutative Noetherian ring and I an ideal such that
K . d i m ( R / I ) is finite. Then I has a fiat cover. In fact it has an 3r-resolution. Proofi
Consider the obvious exact sequence 0 --+ I ~
R ~
R / I -~ O. By the
previous theorem, R / I has an 3r-resolution. By a standard argument, we see t h a t I has an 3r-resolution. [] Example:
By the result above, all the maximal ideals and the prime ideals p of
finite coheight ( i.e., K.dim(R/p) is finite) have flat covers.
4.4
Flat covers of Matlis reflexive modules
Let (R, m) be a local commutative Noetherian ring with the unique maximal ideal m. Set k = R / m be the residue field and E -- E ( k ) the injective envelope. Then for an R-module M, M ~ =Horn(M, E) is called the Matlis dual of M. Furthermore, M is called to be Matlis reflexive if the canonical injection WM : M --+ M ~" is an isomorphism. We simply call it reflexive if there is no risk of confusion. For instance, any module of finite length is Matlis reflexive. If a direct sum of modules is reflexive, all but a finite number of the summands are 0. Note that it is easy to see that R is complete if and only if R is reflexive. If M is reflexive, the every submodule and quotient module of M is also reflexive. Over a complete local ring R, all finite generated modules and Artinian modules are reflexive. In particular, E ( R / m ) is Artinian and reflexive (see Belshoff [11, 13]. Since any Matlis reflexive module is pure injective, it has a flat cover. Our question is whether a flat cover of a reflexive module is still reflexive. Before we state the main result, we briefly give some properties of reflexive modules and their flat covers.
99
Proposition
4.4.1 let 0 --+ N -+ M ~ L -+ 0 be an exact sequence of R-modules 9
Then M is reflexive if and only if both N and L are reflexive. P r o o f : This is easy by chasing the following diagram:
0
,N
.M
I
. L
I
90
1
0---~N"~----*M"~---~L "~"
, 0
with exact rows, keeping in mind that each vertical map is an injection. [] By this observation, a finite direct sum @Mi is reflexive if and only if each Mi is reflexive 9 For an R-module M, if the injective envelope E ( M ) is reflexive, then M is reflexive. Proposition
4.4.2 Let (R, m) be complete and local 9 Then an R-module M is
reflexive if and only if there is a finitely generated S such that M / S is Artinian. P r o o f : See Proposition [31, Prop.l.3]. [2 By this characterization, it follows that over a complete local ring R, an R-module is reflexive if and only if M " is reflexive. P r o p o s i t i o n 4.4.3 Let ( R , m ) be complete and local and M is a finitely generated R-module. I f ~ : P -+ M is a projective cover of M , then it is a flat cover. P r o o f : Note t h a t
ker(~) is finitely generated and so is reflexive. But every Matlis
reflexive module is pure injective. Hence ~ : P --+ M is a flat precover and so a flat cover. [] Proposition
4.4.4 Let M be reflexive, and let ~ : F --+ M be a flat cover of M .
Then ~-F : F ~ F ~ has a retraction, i.e., F is a direct summand o f F ~ . P r o o f : Consider the following diagrams:
F 7"F
F ~. ~
~ ,M
F
TF,F~,,,
TM 9M ~
~...."" F
,.
~
TMI ~ vv , M
,
0
The m a p / 3 is available because ~ : F --+ M is a flat cover and F "" is fiat. Therefore ~flTF = TMI~"TF = ~. But then /3TF is an automorphism of F . This shows t h a t TF has a retraction and F is isomorphic to a direct s u m m a n d of F ~'. []
100 4.4.5 Suppose R is a complete local domain with the fraction field Q. Then the following statements are equivalent: (1) Q is reflexive. (2) Q / R is Artinian. (3) K.dim(R) <_ 1 .
Proposition
P r o o f i (1) ** (2). If Q / R is Artinian, then Q is reflexive by Proposition 4.4.1 since both R and Q / R are reflexive. Conversely, if Q is reflexive, then by Proposition 4.4.2 there is a finitely generated submodule S c Q such that Q / S is Artinian.
But a
standard argument then shows that Q / R is Artinian. (2) r
(3) This follows from Matlis [55, Theorem 1]. []
Proposition
4.4.6 Let R be complete local with K.dim(R) < 1. Then every reflexive
.module M has a reflexive injective envelope E(M). Proof." First assume K.dim(R) = 0. If M is reflexive, then there is a finitely generated submodule S C M such that M / S is Artinian. Easily, E(Q/S) is reflexive since it is still Artinian. But note that since R is A r t i n i a n , so is S. Hence E(S) is also Artinian. Note t h a t there is an injection M --4 E ( S ) @ E ( M / S ) . It follows that E(M) is reflexive since it is a direct s u m m a n d of E(S) G E ( M / S ) . Now assume that K.dim(R) = 1. For a reflexive module M, as above it suffices to show t h a t E(S) is reflexive. Note that E(S) = |
(Xp) with p E Spec(R), here
there are only finite marly E(R/p) and each Xp is finite. I f p is maximal, then E(R/p) is Artinian, and then reflexive. Now suppose p is not maximal, i.e., p is minimal. We claim t h a t E(R/p) is reflexive. Note that K.dim(Po) = 0, Rp is an Artinian local ring with the residue field k(p) = Rp/p~. Note that ER(R/p) = ERp(k(p)) (see M a t s u m u r a [56, 18.4(vi)D. Since Po is Artinian, R~e = En,(k(p)) = ER(R/p) is finitely generated as an Rp-module. Therefore the Artinian Rp-module ER,(k(p)) has finite length. So in order to prove that ER,(k(p)) is reflexive as an R-module, we only need to show t h a t every simple Po-module k(p) is reflexive as an R-module. Note t h a t k(p) is isomorphic to the fraction field Q of R/p and K.dim(R/p) ~ 1. Q is reflexive as an R / p - m o d u l e by Proposition 4.4.5, and then it is reflexive as an R-module. We are through. [] 4.4.7 Let ( R , m ) be a complete local ring. Then the following statements equivalent: Every reflexive R-module has a reflexive injeetive envelope; Every reflexive R-module has a reflexive flat cover;
Theorem
are (1) (2) (3)
K.dim(R) < 1. (This gives an example of how to use flat covers to describe rings. In the next
chapter we will use flat covers to characterize Gorenstein modules.)
i01 (1) ===> (2) Let M be reflexive. Then M ~ and E ( M v) are reflexive. Let h : M" ~ E(M") be the injective envelope with the quotient Y. It is easy to see that = h" : E"(M") --+ M '~" is a flat precover. We claim that in fact this is a flat cover Proof:
of M ~'. Suppose g : E~(M ~) -+ E~(M ~) such that ~g = ~. By chasing diagrams, we see that gV is an automorphism of E'~(M~). Therefore, g is an automorphism of EV(M~). This ensures that a flat cover of a reflexive module M is reflexive. (2) ==~ (3). We divide the problem into two cases. Case 1. Assume R is domain. Let p : F --+ E be a flat cover of E. Since E = E(k) is reflexive, its fiat cover F is also reflexive by the assumption. Then we have an injection R ~ E ~ ~ F ~. Note that F ~ is injective and reflexive. Hence the injective envelope E(R) of R is a direct s u m m a n d of F ~, and then E(R) is also reflexive. Since R is domain, E(R) = Q, the fraction field of R. By Proposition 4.4.2, there is a finitely generated module S C E(R) = Q such that Q/S is Artinian. We claim that Q / R is also Artinian. Since S is finitely generated, there exists a nonzero element y E Q such that S C Ry. It is easy to see that Q/Ry is Artinian. Define a : Q / R --+ Q / R y by sending a/b + R onto y 9a/b + Ry. This is an injection , and so Q / R is Artinian. Now if m = 0 or m is nonzero and principal, we are done. So we may assume al, a2," 99 as, b generate m with n > 1 and b ~ 0. For each i, consider the descending chain
( of submodules of K = Q/R.
b Then there exists an integer s such that ( ~ + R) =
aS+ 1
(%-- + R).
A simple argument shows that there are elements c, r C R such that
a](1 - rai) = bc, and a~ E (b). From this we have m t C (b) for a sufficient large integer t. Hence K.dim(R) <_ 1. For an alternate proof of this part, see Matlis [55, T h i n . l , p571]. Case 2. If R is not a domain, let p be a minimal ideal of R, and denote the quotient domain R/p by R~. Now we are going to prove that the fl'action field Q~ = ER~(R1) is a reflexive Rl-module. Since R is reflexive an R-module, so is R1. If we replace R by R1 , and repeat the argument given in Case 1, then we see that E1 = ER(R1) is a reflexive R-module. Note that Q~ = ER, (R~) =HOmR(R/p, E~) C Ex. It follows that Q1 is a reflexive as an R-module, and so is reflexive as Rl-module. By Case 1 we know that K.dim(R1) < 1, and so K.dim(R) _< 1. (3) ~
(1) By Proposition 4.4.6. []
In the above theorem we may drop the assumption that R is complete. C o r o l l a r y 4.4.8 let R be a local Noetherian ring. The following statements are equiv-
alent.
102
(1) Every reflexive R-module has a reflexive flat cover; (2) R is complete and K.dim(R) < 1. P r o o f i By Theorem 4.4.7 it suffices to argue t h a t (1) implies R complete. Consider the reflexive module k = Rim and its fiat cover ~ : F -+ Rim. By the assumption F is reflexive. On the other hand, F ~ / ~ as R-modules. This means t h a t / ~ is reflexive as an R-module. Hence R is reflexive an R-module since it is a submodule of/~, and so R is complete . []
Remark
4.4.9 Let (R, m) be local Noetherian. Note t h a t E(R/m) is cotorsion, and
so the flat cover of E(R/m), say F , is flat and cotorsion. Hence F = [ITp. Here Tp is the completion of a free Rp-module.
For each minimal prime ideal q, Rq is
Artinian, and then /~q = Rq and Tq is a free Rq-module. We claim that Tq must appear in the product of F for each minimal prime ideal q. Noticing t h a t E(R/m) is an injective cogenerator for the category of R- modules, there is a non zero linear m a p f : Rq --+ E(R/m).
So f can be lifted to a nonzero m a p g : /~q --+ F = I] Tp.
Suppose Tq does not appear in the product. Then F = [IpcqTp. By Lemma 4.1.8, Hom(/~q, I] Tp) ~ I]Hom(/~q, Tp) = 0. This is a contradiction. In general, Tp with p not minimal may appear in the flat cover of E(R/m).
We will see in the next
Chapter t h a t if only Tq with q minimal appears in the flat cover of E(R/m) and if Rq is injective, then R is generically Gorenstein. In this case the flat cover of E(R/m) is injective. Now if R is complete with K.dim(R) < 1, then the flat cover F of E(R/m) is reflexive. Therefore, for a minimal prime q, the Tq which is in the product F is a free Rq-module with finite rank. Otherwise it will not be reflexive. If Tm appears in the product for the maximal ideal m, it is also reflexive. Note t h a t Tm is the completion of a free R-module R (X). This implies t h a t X is a finite set since R (x) is reflexive as a submodule of F . Hence the flat cover F consists of/~n and finite many Tq with q minimal, which is a free Rq-module. If R is a domain, then Tm must be zero in F and
Q~ ~ E(R/m) gives a flat cover of E(R/m), (here n is an integer and Q is the fraction field). For example, set R = k[[x]] be the power series ring with k a field. Let Q be the fraction field. Then Q/R ~- E(R/m) and the canonical surjection Q --+ E(R/m) is a flat cover.
Remark
4.4.10 Xue in
[75] has recently studied the flat covers and injective en-
velopes of reflexive modules over almost Noetherian and semi-local rings, and generalized Theorem 4.4.7. A ring R is called almost Noetherian if every non-minimal prime is finitely generated. For more details, see Xue [75].
103
4.5
A t h e o r e m on A r t i n i a n rings
In this section we digress a little from the main topics to answer the question mentioned at the end of the last chapter. For a nonzero R-module M, when does there exist a nonzero linear m a p ~ : M --+ F with F flat? This question is related to flat envelopes. Asensio and Martinez have studied this topic in [59, 58, 57]. A flat preenvelope ~ : M --+ G is called nonzero provided ~ ~ O. We need some preliminary lemmas. L e m m a 4.5.1 Let M be a R-module, then M has a nonzero fiat preenvelope if and
only if M has a flat preenvelope (~, F) and there is a nonzero linear map f : M ----+ G with G fiat. Moreover, if R is coherent, then M has a nonzero fiat preenvelope if and only if there is a nonzero linear map from M into a fiat module F. Proof:
This follows from the definition and Theorem 2.5.1.V1
L e m m a 4.5.2 If (R, m) is a local coherent ring with the maximal ideal rn, then
f .dimR( R/rn) = w.gl.dirnR Proof:
See Theorem 2.5.9 of [44].K]
Note that, by Theorem 2.5.1, a commutative ring R is coherent if and only if every R-module has a flat preenvelope. So we have the following Theorem
4.5.3 For a commutative ring R, the following are equivalent:
(1) w.gl.dimR < oo, and every nonzero module has a nonzero fiat preenvelope; (2) w.gl.dimR < oo, and every nonzero module has a nonzero fiat envelope; (3) R is a yon Neumann regular ring. Proof:
(3)=:* (2) and (2)==> (1) are obvious.
(1)==*(3). R is coherent because every R-module has a flat preenvelope, hence so is Rm for any maximal ideal rn. Also, we have t h a t w.gl.dimR~ <_ w.gl.dimR < oc (see R o t m a n [66]). Note that, as an R-module, there is a nonzero linear m a p f : R/rn
~ F with F
R-flat. Then fm : ( R / m ) m ---+ Fm is nonzero with Fm being Rm-flat. Now assume that (R, rn) is a local ring with w.gl.dimR < oc and t h a t the simple module R/rn can be embedded into a flat module F .
Let ~ : R/rn
> F be the
embedding. Consider the following exact sequence:
0
> R/rn
~)
F
) X
~ 0
By the second lemma above, w.gl.dimR = f . d i m R ( R / r n ) -- n < ec and f.dirnR(X) = k _< n. This implies t h a t f.dirnnR(R/rn) = 0, for otherwise, f.dirnR(R/rn) <_ k - 1 < n. This is impossible unless w.gl.dimR is zero and so R is von Neumann regular.
104 Finally, the above arguments show that Rm is von Neumann regular for every maximal ideal m, and therefore R is von Neumann regular. []
L e m m a 4.5.4 Let R be a commutative Noetherian ring. If every nonzero R-module
has a nonzero fiat preenvelope, then for every maximal ideal rn every nonzero R,~ module has a nonzero fiat preenvelope. Proof."
For any nonzero R,~-module X, there is a nonzero linear m a p f : X
E ( R / m ) because E = E ( R / m ) ~ E ( _ ~ / m , J
>
is an injective cogenerator. Note that
E ( R / m ) is Artinian as both an P ~ - m o d u l e and an R-module, hence so is the image Y = f ( X ) . Now, by our hypothesis, there is a nonzero R-linear map h : Y
> G with
G R-flat. Taking the localization, we have the following
X |
R,~ S|
Y |
Rm h~--%G |
I~
Note that Z = h ( Y ) is nonzero and Artinian. So we have that R / m C h ( Y ) = Z. Since
0 7s ( R / m ) m C Z |
P ~ C I m ( h | 1) C G |
h | 1 is not zero. Finally, G @R P ~ is Rm-flat, X ~ X | hence the composition (h |
Rm,
Rm and f | 1 is surjective,
1)(f | 1) is not zero and so we have found the desired
nonzero linear map. [] L e m m a 4.5.5 Let (R, m) be a Noetherian local ring, and R be the completion with re-
spect to the m-adic topology. If every nonzero R-module has a nonzero fiat preenvelope, then every finitely generated nonzero R-module X has a nonzero linear map 9:E---+X
with E R-injective. Proof:
Note t h a t / ~ is Noetherian. Let E R ( R / m ) be the injective envelope of R / m
as an R-module. By Theorem 3.6 of [54], we have the following
E = E R ( R / m ) = E~(R/rh) Let X be a finitely generated/~-module, and let X* =HomR(X, E). By the hypothesis, there is a nonzero R-linear map
~:X*----+ F with F R-flat. Therefore, we have the following/{-linear map ~|
:X*|
)F|
105 Since /~ is a faithfully flat R-module, G is a flat R-module and so ~ | 1 is not zero. Since X is a finitely presented/~-module, E is an R-R bimodule, a n d / ~ is a fiat R-module, we have the following isomorphism ~- :Homk(X , E) | by setting 7 ( ~ f ~ | [48]).
>HomR(X, E | for f / e H o m k ( X , E ) ,
: ~fi(x) |
x 6 X and ci c / ~ (see
On the other hand, E @R [~ ~ E R ( R / m ) = E [ ~ ( R / ~ ) = E as a/~-module. Therefore, X* = H o m h ( X , E) ~':HOmR(X, E @R /~) ~HomR(X, E) @R -R = X* | /~ and ~1 : X* - - + G, where ~1 is the map induced by ~ | 1 and the above isomorphisms, is not zero. Clearly, G = F | /~ is /~-flat. Since E is an injective cogenerator of the category of R-modules, we have that ~:G*
~X**
is not zero. Since/~ iis complete and X is a finitely generated/~-module, X is Matlis reflexive, namely, X** -~ X . Since G is a flat/~-module, G* is an injective/~-module.O L e m m a 4.5.6 Let (R, m) be a Noetherian local ring. I f f o r any nonzero finitely generated R-module X , there is a nonzero linear map 9 : E ~
X with E injective, then
R is an Artinian ring.
Proof:
This follows from Theorem 2.4.5, Levitzki's Theorem and Hopkin's Theorem
(see Anderson and Fuller [1, Thin.15.20, Thm.15.22]). [] Now we are ready to give our main result. T h e o r e m 4.5.7 Let R be a commutative Noetherian ring, then the following are equivalent. (1) R is an Artinian ring; (2) Every nonzero R-module has a nonzero linear map into a projective module; (3) Every nonzero R-module has a nonzero linear map into a fiat module.
Proof."
(1)===>(2). Since R is an Artinian ring, we have the following decomposition R=R,
G R 2 $ , . ..... ,@Rt
where each Ri is an Artinian local ring, 1 < i < t. For any nonzero R-module X, we also have a decompo,~ition X = R 1 X (~ R 2 X @ , . ..... , ~ R ~ X
here, at least one Xi = RiX--fiO. Let mi be the maximal ideal of P~. Since X i / m i X i is a nonzero vector space over R i / m i , there is a nonzero linear map X i / m i X i - - + R i / m i . So we have
106 (7 : X -'-+ X i
) Xi/miXi
~
Ri/mi
is a surjection. But Pq is a local Artinian ring, if mi ~ 0, then there is a minimal ideal S~ c_ mi and Si -~ R i / m i , hence, there is a nonzero linear map ~ : X
> Pq.
Obviously, /~ is R-projective and then the conclusion follows. If m~ = 0, namely, R~ is a field, it is easy to see t h a t the conclusion is also true. (2) ==~(3) This is trivial. (3) ~ ( 1 )
For any maximal ideal m, any nonzero Rm-module X has a nonzero fiat
preenvelope. Note that Rm is a Noetherian local ring. By the previous l e m m a s , for any finitely generated nonzero / ~ - m o d u l e Y, there is a nonzero linear map 9 : E with E / ~ - i n j e c t i v e . It follows from the preceding lemma t h a t / ~
>Y
is an Artinian ring.
This easily implies that P ~ is also an Artinian ring. So far, we have proved that R is a Noetherian ring of Krull dimension 0. Therefore, R itself is an Artinian ring (see [51]). [] Example
The following example which was used in chapter 2 shows t h a t a non-
commutative Artinian ring may fail to satisfy the conditions (2) and (3) of the theorem. Let F be a field, e~j be the 2 • 2 matrix with entry 1 at (i, j ) and 0 at others. Let R = F e n + Fel~ + Fe22. R is a left Artinian ring. Let J = Fe12 be the Jacobson
radical of R. Set S = Fe~2 + F e 1 2 / J = R~22. S is a simple left R-module. We claim t h a t there is not any nonzero linear map from S into a flat left R-module. It is easy to see t h a t we only need to show t h a t H o m R ( S , R ) = 0 because every flat module is projective. Suppose f EHomR(S, R), then f(e22) = h e n + bel2 + ce22 C R, where a, b, c C F . Note t h a t e22ell = e22e12 = 0, and e22 = e22e22, hence, f(e22) = e22(aell + bel2 -4ce22) = ce22. Also note t h a t e12 = e12e22 and e12~22 = 0, we have t h a t 0 = f ( 0 ) = f(e12e22) = e12f(~22) = e12ce22 = ce~2. It follows t h a t c = 0. Therefore f = 0 and HomR(S, R) = 0. []
Chapter 5 Applications in Commutative Rings The theory of injective modules not only has its own beauty, but also has nice applications in many areas. One of such areas is commutative algebra. We have already seen the duality between the injective envelopes and flat covers. Now we are ready to apply the results obtained in previous chapters on flat covers to study commutative rings. For instance, we define a kind of invariant for modules over commutative Noetherian rings by use of minimal flat resolutions. It will be noted that these invariants and their behavior are dual to those of Bass numbers. With the new invariants we can describe special modules and commutative rings. For example, Gorenstein rings can be characterized in term of flat covers. After briefly considering Bass numbers in the first section, we study the minimal injective resolutions of modules of finite flat dimension. By doing so, we can get characterizations of Gorenstein rings. The second section gives the definition of the dual Bass numbers and the computational formula for these numbersl Dual to the modules of finite flat dimension, we investigate the minimal flat resolutions of modules of finite injective dimension by considering vanishing properties of the dual Bass numbers. Strongly cotorsion and strongly torsion free modules are introduced and described by the vanishing property of the Bass numbers and the dual Bass numbers in the fourth section. The last several sections are concerned with modules over Cohen-Macaulay rings. The Foxby classes and Gorenstein projective, injective and flat modules as well as their associated envelopes and covers will be discussed.
For example, we show the existence of Gorenstein flat covers in
the Foxby class of modules by using the fundamental results obtained in Chapter 2 and technics developed before. The results presented in this chapter are mainly taken from [23, 74, 35]. We note there are plenty of results in the literature concerning finite injective dimension, finite flat dimension, and the ring being Gorenstein. For more results and references, we refer to Foxby [38].
108
5.1
T h e B a s s n u m b e r s of flat m o d u l e s
Let R be a commutative Noetherian ring and M an R-module. A minimal injective resolution of M is an exact sequence 0~M--+E0
dO) El _ ~ E2 ___+ ...
)Ei
d~)...
such that for each i >_ O, Ei is an injective envelope of ker(di). As noted, each Ei has a unique decomposition E~ = OE(R/p)(X,),p E Spec(R). If p~(p, M) denotes the ith Bass number (see Bass [10]), the cardinality of the set of copies of E ( R / p ) appearing in Ei(M), it can be written in the form
Ei = ep~S,ecRPi(P, M ) E ( R/p). Recall that #~(p, M) = dimk(p)Ext~Rp(k(p), Mp), where k(p) is the residue field of R v (see Bass [10] and Roberts [65]). These numbers are finite for any finitely generated Rmodule M. For convenience, we take the following as the definition of Gorenstein rings. D e f i n i t i o n 5.1.1 A commutative Noetherian ring R is called Gorenstein provided that inj.dimRm Rm is finite for any maximal ideal m. Bass extensively studied Gorenstein rings using a homological approach and gave characterizations in [10]. We continue the investigation of Gorenstein ring along these lines. Of course these problems have a strong geometrical flavor (see Kunz [52]). We quote one result of the Bass' fundamental theorem as a lemma. L e m m a 5.1.1 A commutative Noetherian ring R is Gorenstein if and only if it admits
a minimal injeetive resolution as follows 0 --+ R
~ Eo - ~ E1 A ~ E2
) ... ---+ E~ ~'~ ...
such that Ei = Ght(p)=iE(R/p). Namely, Pi(P, R) = 5iht(p). As we see, the prime ideals p with E ( R / p ) appearing in the decomposition of the ith term Ei have homogeneous height. This is a very interesting algebraic and homological property of Gorenstein rings. We will see that there are several generalizations of this classical characterization. First we need a preliminary result that will be used later. P r o p o s i t i o n 5.1.2 Let R be a commutative Noetherian ring. Then the following are
equivalent: (1) R is Gorenstein; (2) (3) (3) (5)
f.dimRE(R/m) f.dimRE(R/m) f.dimRE(R/p) f.dimRE(R/p)
= ht(m) for any maximal ideal m; < oc for any maximal ideal m; = ht(p) for any p E Spec(R); < oc for any p E Spec(R) .
109 Proof:
(1)==*(2). Note that f.dimRE(R/m) = f.dimRmE(R/m) by the isomor-
phism E(R/m)m ~ E(R/m). Also note that ht(m) = ht(mm). Then we may assume that (R,m) is a local Gorenstein ring. For any finitely generated R-module M, we have the natural isomorphism [15, VI Prop.5.3], HomR(EXtR(M, R), E(R/m)) ~ TorR(M, HomR(R, E(R/m))). It follows that f.dimRE(R/m) = inj.dims(R) = ht(m). (2) ~
(3) is obviously true.
(3) ~
(1) We only need to show that P ~ is Gorenstein for any maximal ideal m.
Since f.dimnE(R/m) is finite, hence so is f.dimamE(R/m)m. We may simply consider the local case (R, m) with f.dimaE(R/m) finite. Therefore, the natural isomorphism above implies that inj.dimnmP~ is finite and then that Rm is Gorenstein. Next, note that E(R/p)q 7[=0 if and only ifp C q for any two prime ideals p, q. We then can complete the implications (1) ~
(4) ==~ (5) ~
(1) similarly.[]
We now discuss the minimal injective resolutions of flat modules. First we consider the injective envelope of a flat module, and we want to know when a flat module has its injective envelope still flat. The following result is due to Cheatham and Enochs [17, Thm.3]. Recall that a ring R is called quasi-Frobenius if it is self-injective and Noetherian (see Anderson-Fuller [1, Thm.30.7] for more details). In other words, a quasi-Frobenius ring is just a Gorenstein ring with the Krull dimension K.dim(R) = 0. T h e o r e m 5.1.3 Let R be commutative Noetherian.
The following statements are equivalent: (1) The injective envelope E(F) is flat for any flat module F. (2) The flat cover F(E) is injective for any injective module E. (3) E(R) is flat. (4) Rp is quasi-Frobenius for all primes p EAss(R).
P r o o f : (1) ==~ (2) Let G be an injective R-module and ~ : F(G) -+ G be a flat cover. Then by the assumption E(F(G)) is flat, and since G is injective, ~ can be extended to 9 : E(F(G)) -+ G such that ~ = ~a, where a : F(G) -+ E(F(G)) is an injective envelope. Consider the following commutative diagram
E(F(O))
f/ [r
here the dotted map exists because that F(G) is a flat cover of G and E(F(G)) is flat. But then ~ct = p f a , and ~ = ~fc~. This implies that f a is an automorphism of
F(G), and F(G) is a direct summand of E(F(G)) and so is injective.
110 (2) ~
(1) If L is a flat R-module, and o~ : L -+ E(L) is an injective envelope, then
by the assumption the flat cover of E(L), F(E(L)) is injective. Similarly, we have the commutative diagram
L .....'"
OL
F(E(L)) ~ , E ( L ) Since F(E(L)) is injective, f can be extended to ~ : E(L) -4 F(E(L)). Hence we get is an automorphism, and then E(L) is a direct
a = p f = ~/~a. This shows that ~ summand of F(E(L)) which is flat.
(1) ==~ (3) is trivial. For the implication (3) ==~ (1), we consider the exact sequence
0 --+ R -+ E(R) ~ D --+ O. Then 0 --+ R |
~ E(R) |
-+ D Q F -+ 0 is also exact
for any flat module. Note that E(R) | F is flat since the injective E(R) is flat. It follows that F has a flat module as its injective envelope. For (1) r
(4) we need the
lemma. We use Ass(R) to denote the set of associated primes of R. See Atiyah [2] for the details. L e m m a 5.1.4 Let R be commutative Noetherian. For a prime ideal p EAss(R) the
following statements are equivalent: (a) E(Rp) is a fiat R-module; (b) E ( R / p ) ~ R~ as R-modules; (c) R v is an injective R-module. P r o o f : (a) ==~ (b) Note that E(R/p) is a direct summand of E(Rp), and so it is flat. Hence/~p =Homn(E(R/p), E(R/p)) is injective as an R-module. This means that/~p, the p-adic completion of Rp, is self-injective. Therefore /~p is quasi- Frobenius and so is Artinian. Further this ensures that P~ is Artinian and injective, and then Rp is quasi-Frobenius itself. Hence E(R/p) ~ Rp. (b) ==~ (c) is obvious. (c) ==~ (a) From (b) R~ is a quasi-Frobenius ring. Thus E(R/p) is flat as an Rpmodule, so as an R-module. [] We now continue the proof. (1) ~
(4) follows from (a) ~
(c) in the lemma since R~ is flat as an R-module.
(4) ==~ (3) Note that there is a sequence 0 c A0 C A1 c A2 c ... C A~ = R with all factors Ai+l/Ai ~- R i p for some p EAss(R) with E(R/p) flat. By (c) ~ (b) in the lemma and a simple inductive argument with respect to the length of the sequence, it is easy to see that E(R) is flat. This finishes the proof. [] T h e o r e m 5.1.5 Let R be commutative Noetherian.
are equivalent.
Then the following statements
111
(1) R is Gorenstein. (2) For any fiat R-module F, the minimal injective resolution 0
>F---+Eo
>El
~...--+Ei---~...
is such that Ei = @#~(p, F ) E ( R / p ) and #i(P, F) = 0 if ht(p) r i. (3) A module F is fiat if and only if its minimal injective resolution is as in (2). In [62] Fossum, Foxby, Griffith and Reiten called a finitely generated R-module having an injective resolution as in (2) above a Gorenstein module. The existence of such a Gorenstein module required the associated ring to be quite special. Here we allow modules to be non-finitely generated. Proof." (2)==~(1) For the regular module R, by assumption we have the minimal injective resolution
O--+ R
> Eo---+ E ~ - - + . . . - - - + E~
~...
such that if E ( R / p ) C E~, then ht(p) = i. For any maximal ideal m, taking the localization at rn, we get that (Ei)~ = 0 for i > ht(m). It follows that inj.dimRmRm is finite. Namely, Rm is Gorenstein. Hence so is R. (1)~(2).
By Lemma 5.1.1, there is a special minimal injective resolution of the
regular module R, denoted by g(R). For any fiat module F ~ F @n R, taking the tensor product F | $(R), we have an injective resolution of F
O----~F|
>F|
) F | E1 -----+...
>F@E~---~...
where Ei = @ht(p)=iE(R/p), and F | E~ is injective. It is not hard to see that F | E ( R / p ) is a direct sum of copies of E ( R / p ) . Therefore, E ( R / p ) C F | E~ only if
ht(p) = i. Finally, since any minimal injective resolution of F is a direct summand of F | s (2) ~ ( 3 ) .
the conclusion follows. Suppose F admits such a minimal injective resolution,
0
>F---~Eo
)EI---+...--+E~--~...
such that E ( R / p ) c Ei only if ht(p) = i. We have to show that F is flat. Assume Ei ~ 0. Then by Proposition 5.1.2, we have that f . d i m R E ( R / p ) = i = f.dimREi. For any maximal ideal m, taking the localization at m, we get the following 0 ~
Fm
) (E0)m ~
(E1)m - - + . . .
~
(Es)~ - - - + . . .
Note that (E,i),,~ = 0 for i > ht(m) and if (E~)m 7~ 0, f.dimnm (Ei)m = i. So we have the exact sequence 0 - - + F,~ ----~ G0 ---+ G1
>...--+Gs
>0
112 Here, s <_ ht(m), f .dirnRm Gi = i. Break this long exact sequence into short exact sequences as follows 0
~ Kt ----+ Gs-1 ---+ G~
>0
O ~ ~ G s _ 2 ~ K I ~ O
0 ~ _ 1 ~ G 1 ~ _ 2
~0
O ~ ~ G o ~ _ ~ O Now it is easy to see that f.dimK1 = s - 1, f.dimK2 = s - 2 , . . . , f . d i m K , _ l = s - ( s - 1) = 1 and then that f.dimFm = 0. This means that Fm is flat for any maximal ideal m. Therefore, F is a flat R-module. (3)~(2)
is trivially true.[::]
Recall that an element # E R is regular on an R-module M if it is not a zero divisor of M and M / p M ~ O. We then have the notions of R-sequence and M-sequence (see Atiyah [2] for the explicit definitions). By the above theorem, we have the following consequence which is a special case of the result proved by Foxby in [40, Prop.3.13]. For more general result, see Foxby [40]. P r o p o s i t i o n 5.1.6 Let ( R , m ) be local Gorenstein and let F have finite fiat dimension. Then F is flat if and only if every maximal X-sequence {#1,-.., #d} is also an F-sequence. P r o o f i It suffices to show the sufficiency. Consider the minimal injective resolution of F: O----+ F---+ Eo----~ E1
>...---+El
>...
By Theorem 5.1.5, we have to show that E ( R / p ) C E~ only if ht(p) = i. First of all, we claim that if E ( R / p ) C E~, then ht(p) >_ i. Suppose E ( R / p ) is contained in E~ and ht(p) < i. Considering the localization of the resolution at the prime p, we have a minimal injective resolution of F~ as Rp-module (see Bass [10D. Since Rp is Gorenstein and F~ has finite injective dimension, we have inj.dimRpFp <_ K.dim(R~) = ht(p) < i (see Bass [9, cor.5.6]). This implies that (E~)p = O. But this is a contradiction because E ( R / p ) is contained in Ei. Now we show that ht(p) <_ i if E ( R / p ) C Ei. Consider the first term E0. By the hypothesis it is not hard to see that ht(p) = 0 if E ( R / p ) C Eo. Then assume that under the hypotheses, for 0 < i < s, it is true that E ( R / p ) C E~ only if ht(p) = i for any local Gorenstein ring. Now we consider the case (s+l). Suppose E ( R / p ) C E~+I.
113 Since ht(p) _> s + 1, there is a non-zero divisor u E p on both R and F . Using the functor H o m R ( R / u R , *), we have the minimal injective resolution of F / u F ,
0
> F / u F - - ~ HOER(R/uR, El) - - + . . .
> H o m R ( R / u R I Es+l) --+ ...
It is easy to see t h a t all conditions are preserved by R / u R and F / u F . T h a t is, /~ =
R / u R is Gorenstein a n d / ~ = F / u F has finite fiat dimension as an/~-module. By the inductive hypothesis, we know that E([~/(7) C Es = HomR(/~,
Es+i) only
if ht((t) = s.
It follows t h a t ht(p/(u)) = s and then ht(p) = s + 1 since E ( f ~ / f ) CHOmR(R, E~+I). [] As a generalization of Theorem 5.1.5, we have the following. Theorem
5.1.7 Let R be Gorenstein, M be an R-module. Then the following state-
merits are equivalent. (1) f.dirnR(M) = s < oc; (2) M admits a minimal injective resolution 0
>M--+Eo
>Et--+...
>E~--+...
such that E ( R / p ) C E~ only if i ~_ ht(p) ~_ i + s for i >_ 0 and s is the smallest among such integers. In other words, #~(p, M) ~ 0 only if i ~_ ht(p) <_ i + s. Proof." (2)==> (1) As before, taking the localization at any maximal ideal rn, we have the minimal injective resolution
0 --+ Mm --+ (Eo)m
) (E1)m - - + . . .
) (E,)m
) ...
Then we have t h a t (E~),~ = 0 for i > ht(rn) and i <_ f.dirna~ (Ei)m _< i + s if (E~)m ~ 0. T h a t i s , we have the following exact sequence: 0 ~
Mm - - ~ Go
>G1 ~
...
> Gt
)0
such t h a t i <_ f.dimGi < i + s. We now break this into short exact sequences
0
> K1
~ Gt-1 ---+ Gt ----+ 0
0 --+ 1(2 --+ Gt_2
> Kt
)0
0 ---+ Kt-1 --+ G1
> Kt-2
>0
0 ---+M,~ - - + G o
)K t-1
)
0
Suppose t h a t f.dirn(Mm) = u > s. This will imply that f . d i m ( K t _ l ) _> u + 1, and then f.dirn(Gt) > u + t > s + t. But then this is a contradiction. Therefore, we have
114 that f . d i r n R ~ M m < s for any maximal ideal m and that f . d i m R ( M )
<_ s. On the
other hand, suppose f . d i m M = u < s. By our proof that (1) implies (2) we see that we can construct a minimal injective resolution of M such that E ( R / p ) C Ei only if i <_ ht(p) <_ i + u for any i _> 0. This contradicts the choice of s. (1)==~(2) We proceed by induction on the dimension of M. By Theorem 5.1.5, we know that it is true for f . d i m R M = 0. Suppose it is true for all modules with flat dimension less than n and suppose that f . d i m R ( M ) = n. Let us construct the desired minimal injective resolution of M. As usual, we consider the shifting exact sequence O---+ N - - - + F
) M
)0
with F flat and f . d i m R N = n - 1. By the induction hypothesis, we have the desired minimal injective resolutions for both F and N as follows O--+ F--+ 0
E o - - + EI
>...---+ E i - - - + . . .
> N - - + Go ----+ Gt ---+ . . .
~ Gi
~ ...
such that E ( R / p ) C Ei only if ht(p) = i and E ( R / p ) C Gi only ifi _< ht(p) <_ i + ( n - 1 ) . Consider the pushout diagram, 0
0
0
----~ N
---+ F
~ M
----~ 0
0
----+ Go
---+
~ M
---+ 0
4 Lo
For the exact sequence
H
; =
Lo
4
$
0
0
0 -+ F -+ H -+ Lo ----+ 0, we can construct the following
commutative diagram with exact rows and columns:
0
--+
0
0
0
4
4
$
F
--+
4 0
---+
Eo
---+
4 0
---~
Ko
H
--+
4 Wo
--+
; --+
Xo
Lo
--+
0
---+
0
--+
0
4 G1
4 ---+ L1
4
4
;
0
0
0
Here !Ao = Eo 9 G1 is injective. Then using the resolutions of Ko and L1, we get an injective resolution of H, O~H
)Wo
:' W1 ~-~ ...
) Wi ---~ ...
115 Here, Wi = EiGGi+I.
Therefore, i < f.dimRWi < ( i + l ) + ( n - - 1 )
= i+n.
By
Proposition 5.1.2, E(R/p) C W~ only if i < ht(p) <_ i + n Consider the pushout diagram
0
--+
Go
0
0
4
$
----+ H
----+ M
----+ 0
II 0
---+ Go
--+
Wo --~
Z
Xo
Xo
=
+
4
0
0
--+ 0
It is easy to see that Z =- Wo/Go is injective and 0 < f.dimR(Z) < f.dimRG1 <_ 1 + (n - 1) = n. Then, pasting the exact sequence 0
>M
> Z ---+ X0 ~
0 and
the resolution of X0 together, we have a resolution of M
O---+ M - - - ~ Zo---~ Z 1 - - - ~ . . .
~ Zi----~...
such t h a t Zi is injective and i < f.dimRZ~ < i + n.
Note t h a t the ith term of
any minimal injective resolution of M is a direct s u m m a n d of Z~. Therefore, by Proposition 5.1.2, E ( R / p ) is in the ith term only i f i < ht(p) < i + n . By the first part of the proof, we also can assert that n is the smallest among such integers. Otherwise, we deduce t h a t f.dimR(M) < n. [] Now we are ready to give more characterizations of Gorenstein rings. We consider t h e flat dimensions of an R module M and its injective envelope E(M). In general, these dimensions may not be related. But if they do, restrictions will be imposed on the ring. 5.1.8 Let R be commutative Noetherian. Then the following are equivalent: (1) R is Gorenstein; (2) For any finitely generated module M, f.dimRE(M) <_ f, dimn(M); (3) For any module M, f.dimRE(M) < f.dimn(M).
Theorem
Proof: (1)~(3)
If f.dimR(M) = oc, this is trivially true. If f.dimR(M) = s < oc,
by Theorem 5.1.7, we have that f.dimRE(M) = f.dimREo < s = f.dimR(M). (3)==:>(2) is trivial. (2):::=~(1) For any maximal ideal m, consider a maximal R-sequence { p x , . . . , # t } Then M -- R / ( p l , . . . , # t ) has finite flat dimension. By the assumption, f.dimR(E(M)) <_ f.dimR(M). On the other hand, since M is m-primary, R / m C M. This implies that E ( R / m ) C E(M) and t h a t it also has finite flat dimension because it is a direct s u m m a n d of E(M). Hence R is Gorenstein by Proposition 5.1.2s We may ask when f.dimRE(M) = f.dimR(M) for any module M. It turns out
in m.
t h a t this condition is more restrictive.
116 5.1.9 Let R be commutative Noetherian, then the following statements are equivalent: (1) R is Gorenstein with K.dim(R) <_ 1; (2) For any module M with finite fiat dimension, f.dimRE(M) = f.dimR(M).
Theorem
( 1 ) ~ ( 2 ) If f.dimR(M) < co, then f.dimRM = 0 or 1 [9, Corollary 5.6]. If f.dimR(M) = 1, consider the following exact sequence Proof:
0
)M
)E(M) ~ X ~ o
Since f . d i m R X < cc, f.dimRX _< 1. But then it follows that f.dimRE(M) = 1. Otherwise, we have f.dimR(X) = 2, a contradiction. (2)~(1)
First, by Theorem 5.1.8, R is Gorenstein. Suppose K.dim(R) > 0. For any
maximal ideal m, choose a maximal R-sequence in m, { # 1 , . . . , Pt}. This shows that
f . d i m R ( # l , . . . ,#t) = t -- 1. We claim that t = 1. Since E(R) is flat and I = ( # 1 , . . . , #t) C R, E(I) is a direct s u m m a n d of E(R). This means t h a t E(I) is flat. But then by the assumption, f.dimRE(I) = f.dimRI. This implies that I is flat and t - 1 = 0, hence t = 1. T h e r e f o r e , ht(m) = ht(I) = 1 for any maximal ideal m. It follows that R is Gorenstein with Krull dimension one. [] Recall that a local ring (R, m) is regular if it has finite global dimension. For other descriptions of regular rings, see Atiyah [2]. Here we have C o r o l l a r y 5.1.10 Let ( R , m ) be commutative Noetherian.
Then the following are
equivalent: (1) (R, m) is regular with K.dim(R) < 1; (2) f.dimR(M) = f.dimRE(M) for all R-modules M. P r o o f : (1)==~(2) is obvious. For (2)===>(1), by Theorem 5.1.9 ( R , m ) is Gorenstein with K.dim(R) < 1. Note t h a t f . d i m R R / m = f . d i m R E ( R / m ) < 1 by the assumption. It follows t h a t R is regular because gl.dimR = f.dimRR/m. [] 5.1.11 Let R be commutative Noetherian. Then the following statements are equivalent. (1) R is Gorenstein with K.dim(R) _< n + 1. (2) 0 < f . d i m n M - f . d i m n E ( M ) <_n for all R-modules M which have finite fiat dimension.
Theorem
P r o o f : (2) ~
(1). Since the hypothesis implies t h a t f.dimRE(M) < f.dimnM,
by Theorem 5.1.8, R is Gorenstein.
Suppose K.dim(R) > 0 and let { # 1 , . . . , P 8 }
be a maximal R-sequence in a maximal ideal m.
As before, f . d i m R ( # l , . . . , # 8 ) =
s - 1 and E ( # I , . . . , #8) is a direct s u m m a n d of E(R). Hence, f . d i m R ( # l , . . . , #8) -
f . d i m R E ( # l , . . . , ps) <_ n, so this implies t h a t s - 1 < n and then t h a t K.dim(R) n+l.
117 (1) ~ (2). By Theorem 5.1.8, we have t h a t f.dimRM - f . d i m u E ( M ) >_ O. If f.dimR(M) = n + 1, then for some maximal ideal m, f.dimRmM,~ = n + 1. It is not hard to see that f.dimR~E(Mm) = n + 1 and then that f.dimRE(M) = n + 1. If f.dimRM < n, then obviously f.dimRM - f.dimRE(M) < n. [] C o r o l l a r y 5.1.12 Let R be commutative Noetherian. Then the following are equiva-
lent: (1) R is regular with K.dim(R) < n + 1; (2) 0 < f.dimRM - f.dimRE(M) <_ n for all R-modules M. Here, the right half inequality means that f.dimRM < f.dimRE(M) + n. We have seen t h a t over a Gorenstein ring R fiat modules M have the vanishing property Pi(P, M) = 0 for all prime p with ht(p) r i. This kind of numerical condition is quite useful in describing special modules. We will investigate the modules M having p~(p, M) = 0 for all prime p with ht(p) > i later.
5.2
The dual Bass numbers
In Section 1 we reviewed the Bass numbers, and then used them to characterize modules and rings. As we see, the Bass numbers were defined by minimal injective resolutions in which injective envelopes and the structures of injective modules were applied. In this section we will replace injective envelopes by flat covers and replace the structures of injective modules by the structures of cotorsion flat modules which were given in Section 4.1.
We then define new invariants called the dual Bass numbers, and
discuss the basic properties. First we give the definition of a minimal flat resolution. D e f i n i t i o n 5.2.1 Let R be commutative Noetherian, and let M be any R-module. A minimal flat resolution of M is an exact sequence *:
......
) F , ~'>Fi-1 ---+ 999
>F0
)M
>0
such t h a t for each i, Fi is a fiat cover of im(d,) and such t h a t F0 is a fiat cover of M. Note t h a t for i > 1 Fi is fiat and cotorsion, and so it is a product of TB each of which is the completion of a free Rp-module. For i = 0, F0 is not cotorsion in general. But we can take its cotorsion envelope (or equivalently pure injective envelope). The pure injective envelope PE(Fo) of F0 is fiat and cotorsion and it is a product of Tp. D e f i n i t i o n 5.2.2 Assume M has a minimal fiat resolution as (*). For i > 1, 7ri(p, M ) is defined to be the cardinality of the base of a free Rp-module whose completion is Tp in the product Fi = I-I Tq. For i = 0, ~0(P, M ) is defined similarly by using the pure injective envelope PE(Fo) instead of F0 itself.
118 We note t h a t the 7ri(p, M) are homologically independent and well defined if M admits a minimal flat resolution. We call the
7ri(p,M)
the dual Bass numbers. By The-
orem 4.3.5 every module over a commutative Noetherian ring of finite Krull dimension has a flat cover, and so every module over such a ring admits a minimal flat resolution. Thus, the invariants lri (p, M) are well-defined for all modules over such rings, including coordinate rings of affine algebraic varieties. Note t h a t the computation of Try(p,M ) for any module M can be reduced to the computation for a cotorsion module by the following observation. Proposition
5.2.1 Let R be commutative Noetherian, and let M be an R-module. If
M has a fiat cover, then M has a eotorsion envelope, denoted C ( M ) . Furthermore, if M admits a minimal fiat resolution, so does C ( M ) , and 7ri(p, M) = 7ri(p, C ( M ) ) for all i. P r o o f : This follows from Theorem 3.4.8 and the definition of 7ri. [] Now we give the formula for the computation of the dual Bass numbers. Theorem
5.2.2 Let R be commutative Noetherian, and let M be a cotorsion R-
module which admits a minimal fiat resolution, Then ~r~(p, M ) = d~mk(p)Tor " iR, (k(p), HomR(P~, M))
for all prime ideals p and all i > 1, (here k(p) = (R/p)p is the residue field of p). Note t h a t
HOmR(Rp, M ) is called the colocalization of M at p by Melkerson and
Schenzel in [60]. Before the proof we need several lemmas. Lemma5.2.3
Let 99 : F ---+M be a fiat cover of M and F = FI | F2. Then the
restriction map 99 IF1 can not be factored through the restriction qo If2 unless F1 = O. P r o o f : Let f : F1
> F2 be such that (99 [ Fz) o f = 99 [ F1. Then the diagram
F~eF2
FI@F2
99 . M
is commutative. Here g : Ft (9 F2 -----+Fa G F2 is defined by g(a + b) = f ( a ) + b for any x = a + b in F1 |
a C F1, b E F~. Note that the image of 9 is in F2. By the
definition of a flat cover, g must be an automorphism. This is impossible unless Ft is zero. []
119 L e m m a 5.2.4 Let R be commutative Noetherian, and let p be a prime ideal in R.
Let ~ : l~p --+ F be P~-linear with an P~-flat module F, and ~ : F ----+ (R/p)p be an Rp-linear map. If" the composition a o ~ is nonzero, then ~ is an injection and its image is a direct summand of F. (R/p)p is a flat cover of k(p) = (R/p)p as an Rp-module (Theorem 4.1,6). Then ~ = a o p : P,~ ) (R/p)p is
P r o o f : Note that any nonzero P~-linear map ~ : P~ ~ a flat cover. Since F is flat, there is a linear map a : F
) / ~ such that cr = ~ o
by the definition of fiat covers. Easily, we have (~ o ~ = ~ o ~ o ~ = ~. Hence (~ o ~ is an automorphism of F. Therefore, it follows that ~ is an injection and its image is a direct s u m m a n d of F. [] L e m m a 5.2.5 Let R be commutative Noetherian, and let F = l-I T, be a fiat eotorsion
R-module. If L C F is a direct summand of F isomorphic to Ptp, then I I T , N L = 0 qCp
and 1-I Tq 9 L is a direct summand of F. qCp
P r o o f : We have F = I I T, @ IX Tq = L @ X with L ~/~p. Let Z = I I Tq. We claim qCp
q~p
q~p
that L C ~ Tq. Otherwise, let qCp
:F
IX T.
qs
be the projection.Then the restriction to L, a IL is not zero, and the image is contained in Z. Easily, Hom(Rp, c~(L)) is not zero. But
Hom(Rp, Tq) = 0 for any q which is
not contained in p by Lemma 4.1.8. This implies that
Hom(Rp, Z) = 0, and then
Hom(Rp, a(L)) = 0. This is a contradiction, We now have IX T o = I I T q N F qCp
= LG XN
qCp
I I Tq. qCp
We further claim that Y = 1-ITq is contained in X.
Otherwise, let h : F =
q~P
L @ X -----+L be the projection. The, n the image of the restriction on Y , h ( Y ) ~ O. But by Lemma 4.1.8, Hom(Y,t~p) = 0, and then H o m ( h ( Y ) , ~ p ) = 0. But note that
h ( Y ) is a submodule of L which is isomorphic to P,~. Then Hom(h(Y), L) -~ 0. This is a contradiction, Therefore, we have
I I Tq = L e I I Tq G X ~ T,, qC_p
and
q~p
F = L e I I TqG X ~ T p @ qCp
I I T q.
[]
qCp
L e m m a 5.2.6 Let R be commutative: Noetherian, and let C be a cotorsion R-module.
Let ~ : F = 1-I Tq ---+ C be a flat precover. Then ~ is a flat cover frond only if for any prime ideal p and any direct summand L C F o.f F isomorphic to [~, the diagram
120 L
' [ y
can
41. C
not be completed to a commutative one where Y = 1-I Tq and ~1 is the restriction
q~p
of ~ to Y.
P r o o f : By Lemma5.2.5, F = L @ X = L G Y G Z . Suppose ~ i s a f l a t cover of C. If the diagram can be completed commutatively, then we have a contradiction by Lemma 5.2.3. Conversely, suppose ~ has the given property. Let K =ker(p). If ~ is not a flat cover, then K contains a nonzero direct summand H of F by Corollary 1.2.8. Since H is flat and cotorsion, it will have a direct summand L isomorphic to Rp for some prime ideal p. Then the restriction ~ IL= 0. Easily the corresponding diagram can be completed commutatively by the zero map. This contradicts the condition. [] T h e o r e m 5.2.7 Let R be commutative Noetherian, and let C be cotorsion. Let ~ : F --~ C be a fiat cover of C. Then if S C R is multiplicatively closed subset,
Hom(S-1R, F) ~
Hom(S-1R, C)
is a fiat cover.
P r o o f i Using the natural isomorphisms Horn(G, Hom(S-1R, F)) ~- H o m ( G | F) and Horn(G, Hom(S-1R, C)) ~- Hom(G | S-1R, C) and the fact that G | S - 1 R is flat if G is flat, we see that Hom(S-1R, F) is a flat precover of Hom(S-1R, C). If it is not a flat cover, then its kernel contains a direct summand L isomorphic to for some prime ideal p by Corollary 1.2.8. Since S - 1 H o m ( S - 1 R , F) and L both are S-1R-modules, we have that S - * H o m ( S - 1 R , F) = Hom(S-~R, F),
S-~L = L,
and S - ' & = ~ . We get that & ~ H o m ( & , & ) is isomorphic to a direct summand of Hom(R~, Hom(S-1R, F)), and contained in the kernel of Hom(R~p, Hom(S-1R, F))
> H o m ( ~ , Hom(S-1R, C)).
But Hom(Rp, Hom(S-1R, F)) ~ Hom(S-1Rp, F) ~ Horn , F). Likewise for Hom(~,Hom(S-1R, C)). Hence, Hom(~,~,F) ---+Hom(~,C) is such that there is a direct summand of Hom(P~, F) isomorphic to P~p and which is in the kernel of Hom(P~, F) > Hom(/~, C).
121 Now note that Hom(~p, Tq) = 0 for any prime ideal q which is not contained in p (see the proof of Lemma 4.1.8). let F = I]Tq, and Y = IITq. Then we have that
qcp Hom(R~'~, F) = Hom(Rp, IX Tq) = Hom(~o, Tp) 9 H o m ( ~ , Y).
Let (0, g), o E
qCp
g o m ( [ ~ , Tp), g C Horn([rip, Y) be a generator of L as an R~'p-module. Then we claim that o : ~ > Tp maps ~ isomorphically onto a direct summand of Tp. To see this, by the proof of Lemma 5.2.4, we only need to show that o(P@ g pTp. Since (0, g) is a generator of L which is isomorphic to P~, it follows that (0, 9) q~ pHom(R~, Tp) G pHom(R~, Y) But note that pHom(Rp, Tp) ~- Hom(Rp,pTp) and pHom(Rp, Y) = H o m ( n , , Y). Hence if we assume o(P-'~p) C pT v, then we will have that (o, 9) 9 pHom(t?~, T~) | p H o m ( ~ , Y)) = p(Hom(Rp, Tp) 9 H o m ( / ~ , Y)) This implies that (a, g) 9 pL since L is a direct summand of Hom(P~, F) ~ Hom(Rv, T,) @ Hom(R,, Y) . Obviously this is impossible since L ~ / ~ .
Therefore the claim has been established.
Finally, since (c~,9) is in the kernel of Hom(Rp, ~o), (~ IT,) o o + (~o tY) o g = 0. But then o(L) -2
~ ]o(L)
is a commutative diagram. This contradicts the fact that ~ : F ~ C is a flat cover by Lemma 5.2.6. [] P r o o f of T h e o r e m 5.2.2: Let the following resolution be a minimal flat resolution of M: " ' --+ Gn+l --+ G , --+ ... --+ Go --~ M --+ 0 . By Theorem 5.2.7, taking the colocalization at p, we have a minimal flat resolution of Hom(P~, M) as a n / ~ - m o d u l e ,
9"
) F~+I--~F~
> " " --+ Fo --+ Hom(P~, M)
)0 ,
where Fi = Hom(R~, Gi) for all i. Since each Fi is cotorsion and fiat as R~-module,
Fi = Tp @ I I Tq. Hence qCp _
Tp
k(p) | Fi = k(p) | Tp @ k(p) | IX Tq - -~p. q~p
122 Let c5 stand for k(p) | O. We claim that c5 is zero. Suppose c5 is not zero. Then c~(~?) r 0 for some y C F,. We may assume that y is in Tp C F~. Then we consider the following diagram
1% a , F . O,F.-I A
F~ is defined by sending r to ry for every r E RB, and 9 is a projection
Here, a : Rp ~
such that g(0(~)) r 0. Now let ~ = 0 o ~ , c r - - g o h .
Then a o ~ i s n o t
zero. By
Lemma 5.2.4, ~ = 0 o c~ is an injection and its image is a direct summand of F~ 1. Actually, the image is in the kernel of F~_~
) F~_2. This contradicts the fact that
Hom(/~, - ) applied to the minimal flat resolution of M gives a minimal fiat resolution of Hom(P~, M) by Theorem 5.2.7.
Therefore,
9
Rp
(k(p), Hom(Re, M)) = ei. k(p)
= d~mk(p) pT/ This is just
the eardinality of the base of a free Re-module whose completion is Tp in the product
F~ = KI Tq. ~ E x a m p l e Let (R, m) be local commutative Noetherian, and let k(m) = R i m . Then for any prime ideal p which is not maximal, we have Horn(Re, k(m)) = O, and so ~ri(p, k(m)) = 0 for all i. But for the maximal ideal m, we have that Try(m, k(m)) = dimk(m)Tor Rm (k(m), Hom(Rm, k(m))) = dimk(m)Tor R~ (k(m), k(m)) are finite for all i. In fact, they are equal to the Betti numbers of k(m). [] Comparing the dual Bass invariants Try(p,M) with Betti numbers (see Roberts [65] for more details) fli(M), we found the most important advantage of the ~r~(p,M) is that they can be defined for non-finitely generated modules. In particular, the ~ri(p, E ( R / m ) ) are well defined and behaved when m is a maximal ideal of R. In general, we have the following fact. P r o p o s i t i o n 5.2.8 Let R be commutative Noetherian. Then for any Artinian module
M , the dual Bass invariants 7ri(p, M ) are all well defined, and they can be computed by the formula of Theorem 5.2.2. Furthermore if M is of finite length, then 7q(p, M ) = 0 for any non-maximal prime ideal p and all integers i > O. For any maximal ideal m, all 7ri(m, M ) are finite. P r o o f : We have already shown that every Artinian module M is cotorsion in Section 3.3. It remains to show that it admits a minimal flat resolution.
Note that M is
an essential submodule of E ( R / m l ) @ . . . (~ E ( R / m t ) for finitely many maximal ideals { m l , ' " , me}. Among these maximal ideals we may assume that m l , ' " , distinct. Then there is a direct sum decomposition of M:
M = M~ ~ M2 | . . . @ Ms,
m~ are
123 where M~ = {x E M I mlx = 0 for some integer l}. This implies that each Mi is Artinian as an Rm,-module and P ~ - m o d u l e . Furthermore, M~ is Matlis reflexive as an Rm,-module, and then it is cotorsion as an P~,-module. But then it follows that Mi is cotorsion as an R-module by noticing the fact that Mi | Rm, ~- Mi. Therefore M itself is cotorsion as an R-module. Also note that each Mi has a minimal flat resolution as an Rm,-module since Rm~ has finite Krull dimension (Theorem 4.3.5). This shows that Mi has minimal flat resolution as an R-module (Corollary to Theorem 4.3.5), and hence so does M itself. We now assume M to be of finite length. Note that
Hom(P~, M) = 0 for any
prime ideal which is not maximal. For every maximal ideal m it is not hard to see that Hom(R,~, M) has a finite length. Then the conclusion follows by Theorem 5.2.2. This completes the proof. [] So far we do not know if every cotorsion module admits a minimal flat resolution. But for any R-module M and any injective module R-module E, Horn(M, E) is pure injective (so cotorsion), and admits a minimal flat resolution. Moreover we have Tor R" (k(p), Homn( Rp, Homn(M, E))) ~ Homn(ExtiR~ ( k(p), Mp), E). In particular, we have the following. P r o p o s i t i o n 5.2.9 Let R be commutative Noetherian. Then for any prime ideal p, Rp is Cohen-Macaulay if and only ifTri(p, E(R/p) ) = 0 for i < ht(p). Rp is Gorenstein if and only if~i(p, E(R/p)) = 0 for i < ht(p) and 7rht(p)(p,E(R/p)) = 1. Proof." By the above it is easy to see that 7ri(p, E(R/p)) is equal to #i(p, R). Then the conclusion follows by (3.7) and (4.1) of Bass [10]. [] Note that for a Noetherian local ring R, every finitely generated R-module A has a minimal free resolution of A with the ith term a finite rank free R-module. Let/~ be the completion of R with respect to m-adic topology. By applying the tensor functor - | /~ to the free resolution of A, we get a free resolution of A = A | /~ as an /~-module. Furthermore it can be easily seen that this resolution is in fact a minimal flat resolution of ,4 as/~-module. Therefore, we have fiR(A) =/3~(A) = 7r~(dn, A). Note that if (R, m) is complete local with K.dim(R) < 1, then fiat covers of reflexive modules are reflexive (the remark to Theorem 4.4.7).
Therefore we see that over
such a ring for an R-module M, it is reflexive if and only if the ni(P, M) are finite, i _> 0, p C Spec(R). The following result shows that the finiteness of 7~i(p,M) is a very restrictive condition on the associated ring R. T h e o r e m 5.2.10 Let (R, m) be local. For the injective module E ( R / m ) and an inte-
ger t >_ O, if ~rt(m, E(R/m)) < cx~ and ~t(P, E(R/p)) = 0 for any non-maximal prime ideals p, then K.dim(R) < t and Rp is Gorenstein for all non-maximal prime ideals p. The converse is also true.
124
Proof: If t = 0, i.e., the fiat cover of E ( R / m ) is of form I-[ Tp with all p maximal, so the fiat cover is a Tin. But it is not hard to see that for each minimal prime ideal q, a term Tq ~ 0 must appear in the fiat cover of E ( R / m ) .
Hence the dimension of R
must be zero. For t > 0, consider the minimal flat resolution of E ( R / m ) :
9" --+ Ft -+ "." -+ Fo -+ E ( R / m ) --+ 0 By the assumption, Ft = / ~ for some positive integer n.
Applying the functor
H o m ( R p , - ) to the sequence above for any non-maximal prime ideal p, we have an exact sequence 9.. -+ Hom(R~, Ft) --+ " " --+ Hom(Rp, E ( R / m ) ) --+ 0 By the proof of Lemma 4.1.8, we see that Hom(Rp, Ft) = 0. Hence Hom(Rp, E ( R / m ) ) has flat dimension less than or equal to t - 1, or equivalently inj.dim(R~) <_ t - 1. This implies that R~ is Gorenstein, and K.dim(P@ _< t - 1. By selecting a prime p with ht(p) = ht(m) - 1, we see that K.dim(R) = ht(m) <_ t. Conversely for the minimal flat resolution of E ( R / m )
9" --+ Ft -+ " " -+ Fo --+ E ( R / r n ) --+ 0 Apply
Hom(/~,-)
with p non-maximal prime ideal. By Theorem 5.2.7 we know
that this gives the minimal flat resolution of Hom(P~, E ( R / m ) ) .
Since Rp is Goren-
stein and i n j . d i m ( R v ) <_ t - 1, f.dim(Hom(Rp, E ( R / m ) ) ) < t - 1. This implies that H o m ( / ~ , Ft) = 0. This means Ft = T m and 7rt(p, E ( R / m ) ) = 0 for any non-maximal
p C Spec(R). In order to show that 7rt(m , E ( R / m ) ) is finite, we use the computational formula and get that 7rt(m, E ( R / m ) )
dimk(m)Hom(Extt(k(m), R ) , E ( R / m ) )
= d i m k ( m ) T o r t ( k ( m ) , Hom(Rm, E ( R / m ) ) ) = . The latter is finite since E x t t ( k ( m ) , R ) is
a finite dimensional k(m)-space. This completes the proof. []
5.3
Minimal flat resolutions of injective modules
In this section we are going to use certain vanishing properties of the dual Bass numbers to describe special modules and rings. These results are dual to those in Section 5.1 where we used the Bass numbers. First we have
T h e o r e m 5.3.1 Let R be commutative Noetherian, then the following are equivalent: (1) R is Gorenstein. (2) For any injective module E, 7ri(p, E) = 0 whenever ht(p) ~ i f o r i > O.
Proof: ( 2 ) ~ ( 1 )
For any maximal ideal m, we have the minimal flat resolution of
E = ~:(R/m).
9..
~ Fi ~
Fi-1 ----~... ~
Fo ~
E(R/m) ~
0
125
Note that, by the assumption, Tp C F~ = l] Tp with Tp # 0 only if ht(p) = i. Applying the funetor HOmR(Rm, *), we have an exact sequence ... ----+ HOmR(Rm, Fi) ---+... ----+HomR(P~, F0) ---+ Homn(Rm, E(R/rn)) ~ Note that
0
HomR(R,,E(R/m)) ~ E(R/m), and for all i _> 0, HOmR(P~,Fi) =
HomR(Rm, l-[ T~) =~ I-[ HomR(P~, T J . For each T,, we have HomR(R~, T J =
HomR( Rm, HomR( E( R/p), E( n/p)(X)) ) ~ HomR(R~ | Since Rm |
E( R/p), E( R/p) (X))
injective, it follows that HomR(Rm @E(R/p), E(R/p) (x)) is flat
because E(R/p) (x) is injective. But, on the other hand, Rm |
E(R/p) ~- E(R/p)m
0 if and only i f p C m. This can not happen for i > ht(m) because ht(p) = i by our assmnption. Therefore, E(R/m) has finite flat dimension. This implies that R is Gorenstein by Proposition 5.1.2. (1)==>(2). In order to construct the desired minimal flat resolution for any injective module E, consider the minimal injective resolution of R of Lemma 5.1.1.
0
>
R ----+Eo do> E1 dl~ E2 -"-'+...
>Ei ~ . . .
such that El = Oht(p)=~E(R/p). Using the functor HOmR(*, E), we have that ... -~ HomR(Ei, E) -~ ... ~ HomR(E0, E) ~ HomR(R, E) ~ 0 Note that HomR(R, E) = E, HOmR(Ei, E) ~ 1-[ HomR(E(R/p), E) is flat. Easily, we have that
HomR(E(R/p), E)
:
HomR(E(R/p) |
P~, E) '~ HomR(E(R/p), Homa(Rp, E)).
Since HomR(Rp, E) is Rp injective, HOmR(Rp, E) -- eE(R/q), q C p. Set HOmR(Rp, E) = A O B , where A = •q=pE(R/q),B = | Note that for q C p,q # p, HomR(E(R/p),E(R/q)) = 0, then HomR(E(R/p),[IE(R/q)) = 0 for all q C p,q # p. Therefore, HomR(E(R/p),B) = 0 because B is a direct summand of [I E(R/q), q # p. Consequently,
HomR(E(R/p), E) = HomR(E(R/p), A) = HomR(E(R/p), E(R/p) (X)) = Tp for some set X, and then HomR(Ei, E) ~ 1-ITp with ht(p) = i. Now it is easy to see that the ith term Fi of the minimal flat resolution of E is a direct summand of I-[ Tp with ht(p) = i. Hence Fi = l-[Tp with ht(p) = i by Theorem 4.1.15. [] From the above argument and the fact f.dimE(R/p) = ht(p), we see that if R is Gorenstein, then for any p c Spec(R), ~rht(p)(p,E(R/p)) = 1. Here is a result dual to Proposition 5.1.2.
126
Proposition 5.3.2 Let R be commutative Noetherian. Then the following are equivalent: (1) R is Gorenstein; (2) inj.dimRTm = ht(m) for any maximal ideal m; (3) inj.dimRTm < oc for any maximal ideal m; (~) inj.dimRTp = ht(p) for any prime p 9 Spec(R); (5) inj.dimRTp < oc for any prime p 9 Spec(R). P r o o f : (1)==>(4) We note that T~ = HomR(E(R/p), E ( R / p ) (X)) and f . d i m ~ E ( R / p ) =
ht(p). This implies that inj.dimRTp <_ ht(p). But on the other hand, it is easy to see that Rp = H o m R ( E ( R / p ) , E ( R / p ) ) is a direct summand of Tp and i n j . d i m ~ R p = inj.dimRpR~ = ht(p). Therefore, inj.dimRTp = ht(p). (4)=:=~(5) is trivial. (5)=:=~(1). Again note that Rp is a direct summand of Tp. It follows that inj.dimR, Rp < oc for any p C Spec(R). Namely, R is Gorenstein. The other implications can be proved similarly. [] L e m m a 5.3.3 Let R be commutative Noetherian and let F be flat. If the pure injective
envelope of F, P E ( F ) = [I T~ with ht(p) = O, then F = P E ( F ) is pure injeetive. P r o o f : By Theorem 4.2.6 P E I ( F ) = 0, so F ~ P E ( F ) is an isomorphism. []
Proposition 5.3.4 Let R be commutative Noetherian and Gorenstein.
For an Rmodule M, suppose 7ri(p, M) = 0 for all prime p with ht(p) 7~ i and all i > O. Then M is injective if K.dim(R) = d < oc, or if M has finite injeetive dimension or if M has finite fiat dimension. P r o o f : If K.dim(R) = d < ec or f . d i m ( M ) resolution of M as follows
= t < oc, we have a minimal fiat
0 ~ . . . ~ M ~ 0 such that Fi = l-ITp with ht(p) = i for i > 1. For the first term F0, note that by the preceding lemma, F0 = PE(Fo) = [ITp with ht(p) = 0. Then inj.dim(Fi) = i for all i >_ 0 by Proposition 5.3.2. From this it is not hard to argue that M is injective. For the remaining part, we claim that M* =Horn(M, E) is flat with E an injective cogenerator for R-modules. Therefore, M is injective. By Theorem 5.1.5, it suffices to show that Pi(P, M*) = 0 for all primes p with ht(p) ~ i. Since M* has finite flat dimension, it follows from Theorem 5.1.7 that pi(p, M*) :~ 0 only for p with ht(p) > i. On the other hand, for the ith term of a minimal flat resolution of M, F~* has flat dimension i since Fi has the injective dimension i. So the decomposition of the injective
127 module Fi is in the form @ E ( R / q ) (xq) with ht(q) <_ i because f . d i m E ( R / q )
= ht(q).
This ensures that #~(p.M*) r 0 only for p with ht(p) <_ i. We are through. [] We now consider the modules M which are not injeetive but of finite injective dimension. By constructing minimal flat resolutions for such modules, we will see a vanishing property of the numbers Theorem
7ci(p,M).
5.3.5 Let R be commutative Noetherian.
Then the following statements
are equivalent: (1) R is Gorenstein; (2) If M has i n j . d i m R M = s < 0% then M admits a minimal fiat resolution and for any i > 1,~r~(p,M) ~ 0 only if i <_ ht(p) <_ i + s. For s = O, inj.dimr~Fo <_ i n j . d i m R ( M ) = s (Fo may not have the f o r m I]Tp). P r o o f i (2) ==* (1). Consider the case s = 0 and M is injective. Since i n j . d i m n F o <_
f . d i m R M , Fo = F ( M ) also injective. Then F0 is flat and injective. It also has the form F0 = 1-ITv with ht(p) = 0. Now, for i > 0, Fi = 1-ITp with ht(p) = i for any injective module M. Now, by Theorem 3.1 5.3.1, R is Gorenstein. (1) ~ ( 2 ) .
We proceed by the induction on the injective dimension of M. Suppose it
is true for all modules with injective dimension less than n + 1. We consider the case
i n j . d i m R M = n + 1. As standard, choose an exact sequence 0
>M
>E--+N--+O
Here, E is injective and i n j . d i m R ( N ) = n. By tile inductive assumption, we have the desired minimal flat resolutions for both E and N. 9. . - - + F i
d~>F~-l--+...
)F0
d(
... - - ~ G i - - ~ G i-1 ----+ ...
)E--+0
> Go --+ N
)0
with F~ = I]Tp, ht(p) = i for i > 0, G~ = 1-ITp, i < ht(p) < i + n
f . d i m R G o <_ f . d i m R ( M ) . Consider the following pullback diagram of E
> N and Go
0
0
4 K0 =
K0
0
---+ M
~ Z
---+ Go
> 0
0
--+
) E
--+
> 0
M
N
4
;
0
0
for i > 1 and
> N,
128 Then, we consider the exact sequence, O---+ Ko
>Z - - + N - - +
O.
Using the fiat covers of E and Ko, and noticing that K0 is cotorsion, by Lemma 3.2.2, we can construct the following diagram with exact rows and columns: 0 0
0
> KI
> Xo
> L0
;
;
> Wo
> Fo
; 0
-----+ G1
+ 0
--+
Ko
0
;
+
> Z
> E
;
+
+
0
0
0
--+
0
---+ 0 ---+ 0
Here, Wo = G1 ~) Fo. Then, using the flat precover Wo ---+ Z, we have the following pullback diagram 0
0
+ =
Ho
---+
+ 0
--+
4
Xo
Xo
+
+ 0
----+ M
Wo
---+ Go
; ---*
0
Z
---+ 0
II --+
Go
--+
0
0
Now, Ho is fiat and Ho ---+ M is a flat precover of M.
Further, inj.dimRHo <_
inj.dimRGo + 1 _< n + 1. Next, in order to construct a flat precover of Xo, we consider the following diagram which exits by Lemma 3.2.2. 0
0
4 )
----+
)
+ -----+ K~ 0 Here, 14/'1 = G2@F1, G2 14"1 ~
4
-----+ /(2 G2
~ K1 and F1
0 L1
---+ 0
W~ - - +
F1
---+ 0
+
+
X1
> Xo
~
~
Lo
0
---+ 0
0
> Lo are fiat covers and K1 is cotorsion. Also,
Xo is a fiat precover. But then W1 = I-I Tp | H Tq = II T~, 1 <_ ht(p') <_ 2 + n.
In general, for i > 1, we have t h a t 0 ~
Xi
~ W~ ~
Xi-1 ~
Wi = Gi+ l O Fi = I] Tp ---+ X~- I is a f l a t precover, i <_ ht(p) <_ ( i + l ) + n Therefore, pasting them together, we have an exact sequence.
O. Here,
= i + ( n + 1).
129
. . . - - + Wi ~'> Wi-, --+...---+ Ho--+ M - - + O such t h a t : (1) Wi = l-[Tp with i <_ ht(p) <_ i + ( n + 1),i _> 1; (2) inj.dimRHo <_
inj.dimRM; (3) HomR(G, *) makes the sequence exact for any flat module G. Therefore, M has a minimal flat resolution such t h a t each ith term is a direct s u m m a n d of
Wi and then our conclusion follows. [] Here is another characterization of Gorenstein rings in terms of flat covers. Theorem
5.3.6 Let R be commutative Noetherian. Then the following statements
are equivalent: (1) R is Gorenstein; (2) If M is a module with inj.dimR(M) = s < co, then inj.dimRF(M) <_
inj.dimR(M) = s, here F(M) is the flat cover of M . P r o o f : It suffices to prove the implication (2)==*(1). For any maximal ideal m, let { # 1 , . . . , #~} be a maximal R-sequence in m. Then, M = R/(>I,..., #~) has finite flat dilnension. Consider the Matlis dual M" =Horn(M, E(R/m)). Obviously it has finite injective dimension. Let F --+ M " be the flat cover of M". By the assumption, F has finite injective dimension. Taking the dual of the exact sequence
O--+ K - - + F - - + MV--+ O. we get 0
> M ""
> F" ~
K"
>0
which is exact. Note that F " has finite flat dimension and m E Ass(M). The simple module R / m can be embedded into M, and then into F ~. This implies that E(R/m) is isomorphic to a direct summand of F ", and then f.dimE(R/m) < ~ .
Therefore
Rm is Gorenstein by Proposition 5.1.2. []
5.4
Strongly cotorsion modules
Recall that, in the previous sections by using the certain vanishing properties of the Bass numbers and the dual Bass numbers, we have discussed the minimal injective resolutions of modules which have finite flat dimension and minimal flat resolutions of modules finite injective dimension. In this section we will discuss the minimal flat resolutions of strongly cotorsion modules which may have infinite fiat dimension and infinite injective dimension. We will see that these modules also have a numerical vanishing property with respect to the dual Bass numbers. D e f i n i t i o n 5.4.1 A module G is called strongly cotorsion if E x t , ( X , G) = 0 for any X of finite flat dimension.
130
R e m a r k 5.4.1 (1) Any strongly cotorsion module is cotorsion and any injective module is strongly cotorsion; (2) If R is n-Gorenstein, that is, R is Gorenstein with K.dim(R) = n, then strongly cotorsion modules are just the so called Gorenstein injective modules in [33]. We will briefly touch on Gorenstein injective modules later. One of our aims is to consider the existence of minimal flat resolutions of strongly cotorsion modules M and to describe strongly cotorsion modules by a vanishing property of the dual Bass numbers 7ri(p, M). P r o p o s i t i o n 5.4.2 Let R be Gorenstein and let G be strongly eotorsion. I f G has finite fiat dimension, then it is injeetive.
P r o o f : By Theorem 5.1.7, the injective envelope of G, E(G), has finite flat dimension since G has finite flat dimension. Consider the standard exact sequence O ----~ G -----~ E ( G )
>X ----+ O
Easily, X also has finite flat dimension. Now, using the functor HomR(--, G), and the fact that E x t , ( X , G) = 0, we have the following exact sequence 0 - - ~ HomR(X, G) ~ It follows that 0
>G ~
E(G)
HomR(E, G)
~ HomR(G, G) ----+ 0
>0 is split, and so G is injective. []
~X
L e m m a 5.4.3 Let R be Gorenstein and let G be strongly cotorsion. Then there is an exact sequence 0
>K
)E-----~G----+O
such that E is injeetive and K is also strongly cotorsion.
Proof." We first show that G is a surjective image of an injective module. Consider the diagram with exact rows and columns. 0
0
r
r
0--+
H
~ P
~
0
H
~ E(P)
~
---4
$ X
a
---~ 0
D
~
0
r =
X
r
r
0
0
Here, P --+ E ( P ) is the injective envelope of P, and P is a projective module. By Theorem 5.1.5, E ( P ) is flat, hence X has finite flat dimension. By an argument similar
131 to t h a t in the above proposition, we know that 0
~G
~ D --~ X
) 0 is split.
Then G is a surjective image of E ( P ) . By Theorem 2.4.2, G admits an injective cover ~ : E ~
G, which is smjective
because G is a surjective image of an injective module. Then we have the following exact sequence
0
)K--+E
By Lemma 2.1.2, we know that
~G--~O
E x t , ( W , K ) = 0 for any injective module W. We
claim that K is also strongly cotorsion. For any X with finite fiat dimension, by using the functor H o m R ( X , - ) we have the exact sequence
0 = Extl(x, a)
Ex6(X, K)
Ext,(X, E) = o
This means that E x t , ( X , K ) = 0 for all modules X with finite flat dimension. Now, for any Y with finite flat dimension, by Theorem 5.1.7, E ( Y ) has finite flat dimension. Then, consider the standard exact sequence
0
~Y ~
E(Y)
~X
~0
Note that X also has finite flat dimension and then that E x t , ( X , K) = 0. But then, using HomR(--, K ) , we have an exact sequence 0 = Ext~(E(Y), K)
> Ext,(Y, K)
> Ext,(X, K) = 0
This implies t h a t E x t , ( Y , K ) = 0 for any Y with finite flat dimension. Therefore K is strongly cotorsion. [] So far, we do not know if every module over a Gorenstein ring of infinite Krull dimension has a flat cover, and even more we do not know if such a cover exists for any cotorsion module. But, over a Gorenstein ring, every strongly cotorsion module does have a flat cover. In fact, any strongly cotorsion module admits a special minimal flat resolution. Theorem
5 . 4 . 4 Let R be Gorenstein, M a module.
Then the following statements
are equivalent: (1) M is strongly cotorsion; (2) M admits a minimal fiat resolution . . . - - - ~ F i di) y i _ l - - ~ . . . - - - - ~
F0
) M
~0
such that Fi = I]Tp for ht(p) <_ i. In other words, zq(p,M) = 0 for any p with ht(p) > i.
132 P r o o f : (1) ==* (2) By the last lemma, we have an exact sequence O----+ N
> E----~ M - - - + O
such that E is injective and N is also strongly cotorsion. If E is injective, by Theorem 5.3.1, it has a minimal flat resolution,
---~ F~ ~ + F~_~ - - + . . .
~ Fo ---+ E - - + 0
such that Fi = l i T . with ht(p) = i. For convenience, we break this into short exact sequences 0 ---+ Ko
>Fo.
0
> F1 ---~ Ko ---+ 0
>K1
0 --+ K~+~
>
F~+~
~ E ----+ 0
> K~ ---+ 0
Now, consider the following pullback diagram 0
0
$ Ko = $ o ----+ No 4 0
$ Ko 4 Fo - - + 4
> N
E
4
4
0
0
Since Ho is cotorsion, 0 - - + Ho
F0
M. By symmetry, N admits a fiat cover 0
M
> 0
II
----~ M
>M
) 0
> 0 gives us a fiat precover of
Wo ---+ Go ~
N
> 0 such that
Go = li Tp with ht(p) = O.
Now, let us look at the exact sequence 0 ---+ Ko ---+ Ho ---+ N ---+ 0. Lemma 3.2.2, we have the commutative diagram with exact rows and columns. 0
0
0
--+
Kt
--+
0
--+
F1
0
--+
Ko
0
Hi
0 > Wo - - +
0
----+ Po
> Go
0
--+
~ N
Ho
0
0
--+
> 0
By
133 Hence we have a flat precover of Ho, Po = F1 9 Go = IF[Tp
> Ho with ht(p) <_
1. Therefore, for a strongly cotorsion module M, we have the following two exact sequences 0
~ H0 - - + Fo ---+ M ----+ 0
O---+ H1---+ Po ----+ Ho
>0
giving flat precovers and such that F0 = I-i Tp with ht(p) = O, Po = F1 9 Go = I] Tp with ht(p) _< 1. Symmetrically, we get the same for sequences for N, i.e., we have the two exact sequences
O ----+ Wo
> Go ---+ N -----+ O
0 ---+ W1
>G~---+Wo---+O
such that they give flat covers and Go = I] Tp with ht(p) = 0 and G1 = F[ Tp with
ht(;) < 1. Then, repeating the same procedure, we are able to get the following diagram by use of the exact sequence 0 - - + K1 ----+ H1 0
0 0 0
> Wo ---+ 0 and the use of Lemma 3.2.2.
0
$ --+ K2 $ ---+ F2 4 --+ K~ 4 0
$ - - ~ H2 4. --+ P~ $ ---+ H~ $ 0
0 3. > WI 3. ~ GI 4. > Wo 4 0
> 0 --+
0
--+
0
Here P1 = F2 9 G1 = 1-[ Tp with ht(p) < 2, and 0 -----+//2 -----+P1 -----+H1
~ 0 gives
a flat precover of H1. By symmetry, W1 also admits a similar exact sequence which produces a flat cover of WI such that F(W1) - G2 - l-ITp with ht(p) < 2. Continuing this procedure we have exact sequences
O--+Ho--+Zo--+M--+O
such that Zi+l
0
> H1
> Z~
0 ~
Hi+ 1
) Zi+ 1
~ Ho -----+0
) Hi ~
0
) Hi, Z0 ---+ M are flat precovers, Zi = I-[ Tp w i t h ht(p) < i. It is
easy to see that M admits a minimal fiat resolution such that its i t h term Fz* is a direct summand of Zi and so then Fi* = I-[ Tp w i t h
ht(p) < i.
134 (2)~(1).
Suppose M admits a minimal flat resolution whose ith term is Fi = [IT~
with ht(p) <_ i. First note that if i n j . d i m a T p <_ i when ht(p) <_ i, then inj.dirnRF~ <_ i. If X is a module with finite flat dimension s and K is cotorsion, then by induction, it is not hard to prove that
Ext~+l(X, K) = 0. Now, for any module X with finite
flat dimension, we have to show that
E x t ~ ( X , M ) = 0. Consider the minimal flat
resolution of M 9. . - - + F i + l
>Fi----+...--+Fo--+M---+O
Here, for i > O, Fi = 1-i Tp with ht(p) <_ i. Then we have the following short exact sequences 0
> Ks 1
Fs-1 ---+ Ks-2 ---+ 0
0
> Ks-2
Fs-2 ~
0 ---~ K1
F1
Ks-3 ----+ 0
~-K0
~0
O-----~Ko---~Fo---*M---+O
Note that
Ext~+l(X, Ks_l) = 0 since X has finite flat dimension s and /~s-1 is
cotorsion. Also note that E x t , ( X , Fs l) = 0 because inj.dimRF~_l <_ s - 1. Then, by applying Ext(X, *) to the first exact sequence, we have that E x t , ( X , K~-2) = 0. Repeating this procedure, we have E x t , ( X , K0) = 0. Finally, applying Extn(X,*) to the last exact sequence, we have the following 0 = Ext]t(X, F0) ----+ E x t , ( X , M) ---+ E x t , ( X , K0) = 0 It follows that E x t , ( X , M) = 0 and that M is strongly eotorsion. [] C o r o l l a r y 5.4.5 Over a Gorenstein ring any fiat cover of a strongly eotorsion module is an injective module.
As an application of the previous result, we have the following P r o p o s i t i o n 5.4.6 Let R be Gorenstein and let G be strongly cotorsion. Then if G has finite injective dimension, it is injective. P r o o f : By Theorem 5.4.4, G admits a minimal flat resolution as follows . . . ---~ F~ ---+ F~_I - - ~ . . .
> Fo
> G ---~ O
such that F / = l-I Tp for ht(p) <_ i. Let C be any injective cogenerator for the category of R-modules 9 In order to show that G is injective, we only need to show that G* = HomR(G, C) is flat. By
135 Theorem 5.1.5, we have to check that #~(p, G*) = 0 whenever ht(p) r i. But, taking the duals, we have the following exact sequence
0
)G*--+F~
)F~---~...
)F*--+...
Note that Fi* =HOmR(Fi, C) is injective and f.dimRF~ < i because inj.dimRFi <_ i. Hence, F* = @E(R/p) with ht(p) <_ i by Proposition 5.1.2. Now, it is easy to see t h a t #~(p,G*) = 0 if ht(p) > i.
On the other hand, G* has finite flat dimension
because G has finite injective dimension.
By Theorem 5.1.7, #i(p,G*) =# 0 only if
i < ht(p) < i + f.dgmRG*. Thus it follows that #i(p.G*) = 0 whenever ht(p) ~ i. Therefore, the conclusion follows by Theorem 5.1.5. [] Now, as in Theorem 5.3.5, we can construct minimal flat resolutions for modules which are not strongly cotorsion, but which have a finite resolution with strongly cotorsion modules. Theorem
5.4.7 Let R be Gorenstein. If M is such that there is an exact sequence
0
>M
>Go---+G1
>...----+Gt---+O
where each Gi is strongly cotorsion, then M has a minimal flat resolution ...
>F~
>Fi-1
> . . . ----~ Fo ----+ M -----~ 0
such that for i >_ 1, Fi = I-[ Tp with ht(p) < t + i and such that Fo has finite injective dimension less than or equal t. P r o o f : The proof is similar to t h a t of Theorem 5.3.5 [] Example
Let ( R , m ) be local Noetherian with dimension d.
For any module M,
consider the partial injective resolution of M,
O--+ M--+ Eo--+ " " - + Ed-1 --+ G--+ O Then G is strongly cotorsion. sion, proj.dim(X)
In fact, for any R-module X of finite flat dimen-
< d (see Theorem 4.2.8, or p.84,[63]). Hence this ensures t h a t
E x t l ( X , G) = 0 , and G is strongly cotorsion. [] Now let us go back to the first section. Recall that we have shown t h a t over a Gorenstein ring R, flat modules are just those modules M having
#i(P, M )
= 0 for all
prime p with ht(p) r i and all i > 0. We are now able to determine all modules M having the vanishing property #~(p, M) = 0 for all p with ht(p) > i. D e f i n i t i o n 5.4.2 An R-module M is called strongly torsion free if Torn(X, M ) = 0 for all modules X of finite flat dimension. Let C be an injective cogenerator of R-modules. It is easy to see t h a t M is strongly torsion free if and only if the dual M* = H o m R ( M , C) is strongly cotorsion.
136 T h e o r e m 5.4.8 Let R be Gorenstein, M an R-module. Then the following statements
are equivalent: (1) M is strongly torsion free; (2) #i(P, M) = 0 for all prime p with ht(p) > i and all i > O. P r o o f : Suppose M has the stated vanishing property. Then M admits a minimal injective resolution
O----+ M - - + Eo Here, Ei = |
~ E1
)'"--+
Ei--+'",""
(xp) with ht(p) <_ i. We need to show that M* is strongly cotor-
sion. Taking the duals, we have an ~--resolution of M* with each Fi =HOmR(Ei, C). Since Ei = (~E(R/p) with ht(p) < i, it is easy to see that F~ = [[Tp with ht(p) < i for each i. Note that any minimal .T-resolution of M* is a direct summand of this dual resolution of M*. We have proved that M* is strongly cotorsion by Theorem 5.4.4. Hence, M is strongly torsion free. Conversely, suppose M is strongly torsion free. Consider the exact sequence 0
>M
> M** - - ~ N - - + O.
Since M -----+ M** is pure and M** is strongly torsion free, N is also strongly torsion free. Note that M* is strongly cotorsion. So it has a special minimal flat resolution such that each Fi = l-I Tp with ht(p) <_ i. Therefore, HomR(Fi, C) = GE(R/p) (xp) with ht(p) < i because inj.dimRFi < i. In other words, M** has the desired minimal injective resolution. Now, by the argument dual to that for the implication (1) ~ (2) of Theorem 5.4.4, we have the desired minimal injective resolution of M. [] Remark:
In the above theorem, if we consider the case i = 0, we see that M is a
submodule of E0 which is flat. Therefore, strongly torsion free modules are torsion free when R is a Gorenstein domain. P r o p o s i t i o n 5.4.9 Let R be Gorenstein. If M is strongly torsion free, then the fol-
lowing (1) M (2) M (3) M
are equivalent: has finite fiat dimension; has finite injective dimension; is fiat.
P r o o f : (2) ~
(1) Note that M* is strongly cotorsion with finite injective dimension.
By Proposition 5.4.6, it is injective and so then M is flat. The implication (3) ==v (1) can be proved by taking the dual of M and using Proposition 5.4.2. []
137
5.5
Foxby duality
In this section we are going to introduce an important class of modules by using Foxby duality. Recall that a local ring (R, m) is Cohen-Macaulay provided that K.dim(R) =Codim(R) (the supremum of lengths of R-regular sequences in m). In [3], Auslander introduced the notion of the G-dimension (or Gorenstein dimension) of a finitely generated module over a Cohen-Macaulay Noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander, Bridger [4]). In [33] Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appropriate to call Gorenstein projective the modules with G-dimension 0, so the basic problem was to define Gorenstein injective modules. These were defined in [33] and were shown to have properties predicted by Auslander's results. For Gorenstein dimension, also see Avramov and Foxby [7] and Yassemi [76]. On the other hand, Foxby [39] introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. In the next sections we can carry through a very nice Gorenstein program concerning Foxby's classes. For instance, we are able to prove the existence of Gorenstein injective envelopes and Gorenstein fiat covers for modules in these classes. Unless stated otherwise, R will be a local Cohen-Macaulay ring of Krull dimensions d admitting a dualizing module D and with residue field k. Recall that a dualizing module D is finitely generated having finite injective dimension such that (1) Ext~(D,D) = 0 for all i > 1; (2) Tor~(D,D) = 0 for all i > 1 ; and (3) Hom(D, D) ~- R as a homomorphism. For more details about dualizing modules, see Herzog and Kunz [49]. We now introduce the Foxby classes. Definition 5.5.1 With the above assumptions on the ring R, we define the class of modules, denoted G0(R), to consist of all the modules M such that Tori(D,M) = 0 i_> 1 Ext~(D,D| =0 i_>l
M
~
Horn(D, D | M)
Dually, the class J0(R) consists of all the modules N such that Exti(D,N) Tori(D, H o m ( D , N ) )
D | Horn(D, N)
= 0 i > 1
=0 ~
i>l N
Foxby himself called these classes Gorenstein classes in [38]. It is easy to verify that the functor D | from 60(R) to ,7"0(R) gives an equivalence between these two categories. Similarly the functor Horn(D,-): 3"0(R) --+ ~0(R) is an equivalence. It also follows, for example, that Hom (M1, M2) ~ Hom (D | M1, D | M2)
138 for all M1, M2 E G0(R). P r o p o s i t i o n 5.5.1 I r E is injective then E C rio(R). P r o o f : Let rl, ..., rd be a maximal R-sequence. Then for n _> 1, let In = (r~, ..., r~). If En = Hom (R/I'~,ER(k)) then En = ER/t.(k).
But D / I n D ~- ER/I,(k), hence
Hom (D/InD, En) '~ R/In by Matlis duality. This gives that D |
(D, En) --+ En
is an isomorphism for all n and hence taking the inductive limit, D | Horn (D, E) -+ E is an isomorphism. To see the condition Tori(D, Horn(D, E)) = 0, we need note that D is finitely generated and R is Noetherian, so D has a projective resolution composed of finitely generated R-modules. We also note that E is injective, and then we have the second duality isomomorphism (see Cartan and Eilenberg [15] or Glaz [44, Thm. 1.1.8 ]) : Tori(D, Hom(D, E)) ~ Hom(Exti(D, D), E) Now the condition follows from Exti(D, D) = 0. [] Let s
be the class of modules L such that proj.dim(L) < ee
P r o p o s i t i o n 5.5.2 The class s
is closed under extensions, direct sums, direct products and direct limits. Consequently, every R-module admits an s R)-preenvelope.
P r o o f i Note that the projective dimensions of modules in the class s by K.dim(R) (see Gruson and Raynaud
[63]).
are bounded
Also note that the class s
agrees
with the class of modules M such that M has finite flat dimension since every flat module has finite projective dimension. Therefore the s sions, direct sums, direct products and direct limits. Theorem 2.5.1, we see that every R-module admits an s
is closed under exten-
By an argument similar to []
Now we will let 14] be the class of all modules W such that W ~ D | P for some projective module. /r will consist of all V such that V ~ Horn (D, E) with E an injective module. P r o p o s i t i o n 5.5.3 Every R-module has a W-precover and every R-module has a
bl-preenvelope. P r o o f : For any M, the evaluation map D (H~ precover of M.
__+ M can be seen to be a W-
Note that W --+ M with W E W is a W-precover if and only if
Horn (D, W) --+ Horn (D, M) ~ 0 is exact. To get a H-preenvelope of M, let D | M c E with E injective. Then M --+ Horn(D, D|
--+ Horn(D, E) is the desired preenvelope. For given M --+ Horn(D,/~)
with E injective, we get a map D | M --+ D |
(D,/~) ~ / ~ (since/~ E ri0(R)).
This map can be extended to a map E --+/~ which in turn gives a map Horn (D, E) --~ Horn (D, E).
It can then be checked that the composition M --+ Horn (D, E) --+
Horn (D,/~) is the original map M --+ Horn (D,/~). []
139 P r o p o s i t i o n 5.5.4 For a module M ,
M C Go(R) if and only if there is an exact
sequence 9" - + P 2 - - + P ~
--+Po~V
~ ~V
~ --+V2--+ "'"
of modules such that each Pi is projective, each V i G Lt, such that M = ker(V ~ --+ V 1) and such that D | - leaves the sequence exact.
Proof:
If M E G0(R), let . . . --+ P2 --+ P1 --+ P0 --+ M -+ 0 be a projective resolution
of M. Then since D | - is right exact and Tori(D, M) for i > 1, " " --~ D | P2 --+ D O PI -+ D | Po -o D O M --+ O
is exact. Now let 0 --+ D | M ~ E ~ --+ E 1 --+ . . . be an injective resolution of D | M. Let V ~ = Horn (D, E~). Then applying Horn (D, - ) to this injective resolution we get O-+ M - + V ~
VI-+ V2-.~...
which is exact since Horn (D, - ) is left exact and Exti(D, D N M) = 0 for i _> 1. Now pasting the two complexes .
.
.
~
~
~
M
~
O
and 0--+ M ~ V ~ ~ V 1 --+--+ V 2 --+ . . . together along M we get the desired complex. Conversely, given such a complex, it is easy to see that Tori(D, M) = 0 and Exti(D,D |
M) = 0 for i > 1. Since D |
- leaves the original complex exact,
we get O--+ D |
~ D@V~
DOV
1
exact. Now applying Horn ( D , - ) and noting that V ~ V 1 E fro(R) we get an exact sequence 0 --+ Horn (D, D @ M) --+ V ~ --+ V 1 Since 0 --+ M --+ V ~ --+ V 1 is exact we get Horn (D, D | M) ~ M (naturally) and so M C G0(R). [] Similar arguments give the next result. P r o p o s i t i o n 5.5.5 For a module N ,
N
C J o ( R ) if and only if there is an exact
sequence 9" --+ W2 --+ W1 --+ Wo --+ E ~ ~ E ~ --+ E 2 --+ " " of modules such that each Wi E Vls, such that each E i is injective, such that N =
ker(E ~ --+ E l) and such that H o m ( D , - ) leaves the sequence exact.
140 We note that given a complex O-+ M - + Vo-+ v ~ - + V 2 - - + . . .
with each V i E U, the functor D | - makes this sequence exact if and only if 0 --+ D Q M --+ D Q V ~ --+ D Q V ~ --+ . . . is an injective resolution of D | M, and if and only
if 0 --+ M -+ V ~ --+ V 1 --+ V 2 --+ . . . is L/-injective resolution of M. The first claim is clear. For the second, note that 0 --+ M -+ V ~ --+ V ~ --+ . . . is a b/-injective resolution if and only if Horn ( - , Horn (D, E)) makes the sequence exact for each injective module E. Equivalently, Horn (D | - , E) makes the sequence exact for each injective E. But this is to say that D | - makes the sequence exact. Analogously, a complex . . . --+ W1--+ Wo --+ N--+ O
with each Wi E 14] is a W-projective resolution of N if and only if 9.. --+ Horn (D, W1) --+ Horn (D, W0) --+ Horn (D, N) --+ 0 is a projective resolution of Hom (D, N). The following two results are due to Foxby [39]. Theorem
5.5.6 Let 0 --+ M' --+ M --+ M" -+ 0 be an exact sequence of R-modules.
Then if any two of M, M, M" are in Go(R), then so is the third.
P r o o f : We first argue that under the given conditions, Horn ( - , V) leaves the sequence exact for all V C U. Letting V = Horn (D, E) with E injective (but arbitrary) we see this is equivalent to showing that O --+ D | M' --+ D | M --+ D | M" --+ O
is exact. I f M " E G0(R), then TOrl(D, M") = 0 so this sequence is exact. We will show t h a t if M ' , M E Go(R), then T o r l ( D , M " ) = 0. Since M E G0(R),
T o r l ( D ~ M ) -- 0
so we have an exact sequence 0 -+ Torl(D, M") --+ D |
--+ D |
Since Horn (D, - ) is left exact and M ' , M E G0(R), an application of this functor gives an exact 0 --+ Hom (D, T o r l ( D , M " ) ) --+ M' --+ M But this means Hom (D, Torl(D, M")) = 0 and so t h a t Torl(D, M") = 0.
141 Now since 0 -+
MI --+ M -+ Mrr ~ 0 becomes exact when we apply Horn ( - , V)
for any V C U, H-injective resolutions of M ' and M " , say 0 --+ M ' --+ V '~ --+ V '1 --+ and 0 --+ M" -+ V ''~ --+ V ''1 -+ . . . can be combined to give a commutative diagram 0
0
0 - + M ' -+
M
$
$
0
-+ M" --+ 0
$
O--+ V'~ --+ V'~ @ V "~ --+ V"~ -+
0
O ~ V'~ --+ V'I @ V ''~ -+ V"I ~
0
:
:
:
such that the middle column is a L/-injective resolution of M. Similarly, we form a diagram using projective resolutions of M' and M". Then we past the two diagrams together along 0 --+ M ' -+ M -+ M" -+ 0 and get a short exact sequence of complexes. When we apply D | - , we still get a short sequence of complexes, so t h a t if any two of the complexes are exact, so is the third. An appeal to Proposition 3.2 completes the proof. [] Theorem
5.5.7 IfO ~ N' --+ N -+ N" --+ 0 is an exact sequence of modules and two
of N', N or N" are in fro(R), then so is the third.
P r o o f : Analogous to the proof above. []
5.6
Gorenstein projective, injective modules
In this section we briefly introduce Gorenstein projective modules and Gorenstein injective modules. These modules were defined and studied by Enochs and Jenda in [33]. A module M is said to be Gorenstein projective if there is an exact sequence . . . ---~ p2 ~-%p, ~, po ~3, p_l ~
p_2 -+ . . .
of projective modules such that M = ker (00) and such that Horn ( - , P ) leaves the sequence exact whenever P is a projective module. Dually, a module N is said to be Gorenstein injective if there is an exact sequence . . . -+ E -2 0-2~ E -1 0-1__+E 0 --+~ ~ E 1 -~ E 2 - + . . . of injective modules such t h a t N -- ker (0 ~ and such t h a t Horn ( E , - )
leaves the
sequence exact when E is injective. If M is Gorenstein projective, Exti(M, L) = 0 for all i >_ 1 and L such t h a t t h a t proj. dim L < oo. If N is Gorenstein injective, Exti(L, N) -- 0 for i > 1 and L such t h a t inj. dim L < oc.
142 P r o p o s i t i o n 5.6.1 I f M is Gorenstein projective then M C Go(R). Proof: Let ...P1 ~ P0 ~ - P l -4 "'" be an exact sequence of projective modules with M = ker (cOo) which is left exact by each Horn ( - , P) when P is projective. If E is injective then Horn (D, E) has finite fiat dimension and hence finite projective dimension. But then Horn ( - , Horn (D, E)) leaves the sequence above exact. Hence Horn (D | - , E) does too. Since E was arbitrary, D | - leaves the sequence exact. This implies Tori(D, M) -- 0 for i _> 1. But also, 0 -4 D | M -4 D | P0 --+ D @ P-1 is exact, and so 0 -4 Horn (D, D | M) -4 Hom (D, D | P0) -4 Horn (D, D | P - l ) is exact. But Pi ~- Horn(D, D|
for each i since each Pi is projective and so in
G0(R).
But then D -4 Hom (D, D | M) is an isomorphism. Now let 0 --+ M -4 P0 -4 N -4 0 exact. N is also Gorenstein projective, so N -4 Hom (D, D @ N) is an isomorphism. Then 0 -+ D @ M -4 D | P0 -4 D @ N -4 0 is exact. Applying Hom (D, - ) , we get an exact sequence O -4 M -4 P -4 N -4 E x t l ( D , D |
M ) -4 0 = E x t l ( D , D @ P)
since Extl(D, D) = 0. This implies that E x t ' ( D , D | M) = 0. But then since we also have E x t ~ ( D , D | N) = 0 and E x t 2 ( D , D | P0) = 0, we get E x t 2 ( D , D | M) = 0. Then by induction we get Exti(D, D | M) = 0 for all i >_ 1. [] Similar arguments give P r o p o s i t i o n 5.6.2 If N is Gorenstein injective then N E J0(R). T h e o r e m 5.6.3 I f M C Go(R) and 0 -4 C -4 Pal-1 -4 "'" -4 Po -4 M -4 0 is exact with P0, ..-, Pd-1 projective then C is Gorenstein projective. P r o o f : Let "'" -4 Pd+l - 4 P d - 4 P d - 1 -4 " " -4 Po -4 M -4 0 be a complete projective resolution of M. Then since M c {~0(R), ""-4 D|
is exact.
-4 D |
D|
Since Ext~(D,D) = 0 for i _> 1,
-4""-4
D|
Exti(D|174
--+ D |
= 0 i f P and Q
are projective and i >_ 1. But alsoinj, d i m D = d s o inj. d i m ( D @ P ) P is projective.
-40
< d when
As a consequence, when P is projective and Horn ( - , D | P) is
applied to the complex above, the complex becomes exact beginning with the term Horn (D | Pd, D | P). But for each i, Horn (D | P/, D | P) ~ Hom (Pi, P),
143 SO
Hom (Pd, P ) + Horn (Pc+l, P ) --+ " " is exact. This implies Extd+i(M, P ) = 0 for i > 1 and so that Ext'(C, P ) = 0 for i _> 1. This implies E x t i ( c , L) = 0 if proj. dim L < o0 and i >_ 1. Now by Proposition 5.5.2, C has an L;(R)-preenvelope, C ~ L. Since C C Pd 1, C ~ L is an injection. Let Q --+ L be surjective and linear with Q projective. If K is the kernel of Q --+ L, then K E s
so E x t l ( C , K ) = 0. Hence C - - + L has a l i f t i n g C
an injection and still an s
--+ Q which is
Let 0 --+ C --+ Q --+ B --+ 0 be exact.
Then E x t i ( B , P ) = 0 for i _> 1 and P projective. We now argue t h a t B can be embedded in a module of finite projective dimension. By Theorem 5.5.6, B E C0(R). Let D |
C E w i t h E i n j e c t i v e . Then B ~ H o m ( D , D |
C Hom(D,E).
By
Ishikawa [48], Horn (D, E) has finite flat and hence finite projective dimension. But then we use the same argument as above and have an embedding B c Q1 with Q~ projective and B -+ Q1 an s is exact.
preenvelope. Let Q0 = Q. So 0 --+ C -+ Q0 _+ Q1
Then by proceeding in this manner we can construct an exact sequence
0 --+ C ~ Q0 ~ Q1 ~ Q2 __+ . . . with each Qi projective and such t h a t Horn ( - , P ) leaves the sequence exact when P is projective. If
... -+ Q-2 --+ Q-I --+ C ~ 0 is any projective resolution of C then since Exti(C, P ) = 0 for i > 1 and P projective, we get the complex ... ~
Q 2 __+ Q - I __+ QO __+ Q~ __+ Q2 _~ . . .
having the necessary properties to guarantee that C is Gorenstein projective. [] Theorem
5.6.4 A module C is Gorenstein projective if and only if C E Co(R) and
Ext i(C, L) = 0 for all i > 1 and all L such that proj. dim L < oo. P r o o f : We only need apply the arguments we used concerning the C in the proof of Theorem 5.6.3 [] C o r o l l a r y 5.6.5 If C = C1 9 C2, then C is Gorenstein projective if and only if
C1
and C2 are. P r o o f : Immediate from the above. [] Remark.
If a module C has finite projective dimension and Ext'(C, P ) = 0 for all
i _> 1 and all projective modules P , then C is projective.
Hence any Gorenstein
projective module which is of finite projective dimension is projective. Now let M be any module of finite projective dimension. Then by Proposition M C C0(R). If we let C then be as in the Theorem, then by the above C is projective.
144 Hence proj. dim M _< d. This gives another proof of Raynaud and Gruson's result that proj. d i m M _< d i r e r whenever proj. d i m M < oc (see [63], pg. 84), but of course, under more restrictive conditions. C o r o l l a r y 5.6.6 M E go(R) if and only if for some n > 0 there is an exact sequence O ~ C~ -+ -.. --+ Co --+ M --+ O with Co, ..., C~ Gorenstein projective modules. If there is such an exact sequence, then there is one with n < d.
P r o o f : By Theorem 5.5.6, Proposition 5.6.1 and Theorem 5.6.3. [] Similarly we have Theorem
5.6.7 If N E flo(R) and O-+ N - + E ~
E d - I - + G--+ 0
is exact where E ~ ..., E d-1 are injective, then G is Gorenstein injective.
C o r o l l a r y 5.6.8 N C Jo(R) if and only if for some n >_ 0 there exists an exact sequence O -~ N -+ G~ -+ . . . -+ G~ -+ O with G ~
~ Gorenstein injective.
In this case there exists such a sequence with
n
Similar arguments give the next two results. 5.6.9 A module G is Gorenstein injective if and only if G C Jo(R) and Exti(L, G) = 0 for all i > 1 and all L such that inj. d i m L < oo. Theorem
C o r o l l a r y 5.6.10 If G = GI | G2, then G is Gorenstein injective if and only if G1 and G2 are Gorenstein injective.
R e m a r k . We can also deduce that for any N if inj. dim N < o~, then inj. dim N < d. We can now prove the following Theorem
5.6.11 If M E Go(R), then M has a Gorenstein projective precoverC -+ M
whose kernel belongs to s
Proof:
Let 0 ~
B -+ Pd-1 -+ "'" -+ Po -+ M -+ 0 be exact with Po,...,P~-i
projective. Then by Theorem 5.6.3, B is Gorenstein projective. Let
9.. _~ Q-1 o-1~ QO -~~176 Q1 -~~ Q2 _ ~ . . .
145 be an exact sequence of projective modules with ker(0 ~ = B and such t h a t Horn ( - , P ) leaves the sequence exact when P is projective. Let 0 ~ B ~ QO ~ . . . ~ Qd-1 __+ C --~ 0 be exact. Then our hypothesis guarantees that there is a commutative diagram 0--+B-*
Q0
0--+B--+
Pd-1
_+...__+ ~'"~
Qd-1 P0
--+C--+
0
-+M-+
0
Form the complex associated to this diagram thought of as a double complex (with 0 in all other positions), namely O -+ D --+ D O Q~ ---~ . . . ---~ Po G C ---~ M ---~ O
This complex is exact since the rows of the diagram above are exact. Also this complex has the exact sequence 0 -+ D --+ D --+ 0 as a subcomplex, so we get an exact quotient complex 0 -+ QO _+ . . . ~ P1 @ Qd-1 _+ Po @ C -+ M --~ O. Since all terms of this sequence with (possibly) the exceptions of P0 @ C and M are projective, we see t h a t if O--~ L + Po @ C - + M - - + O
is exact, then proj. d i m L < oc. In fact, for future reference, we note that proj. d i m L _4 d - 1. Both P0 and C are Gorenstein projective, so P o G C is too. Since E x t l ( D , L) = 0 for all Gorenstein projective modules D,
P0 9 C ~
M is a Gorenstein projective
precover. [] A dual argument gives Theorem
5.6.12 I f N E J o ( R ) then N has a Gorenstein injective preenvelope N C G
such that inj. dim G / N <_ d - 1.
We note that Auslander and Buchweitz in [5] have results similar to Theorems 5.6.11 and 5.6.12 with chain conditions on the modules. We will see t h a t in many respects, Gorensetin injective modules act like injective modules. For instance, we have Theorem
5.6.13 Every module M in J o ( R ) has a Gorenstein injective envelope.
Moreover, if M --+ G is a Gorenstein injective envelope, then i n j . d i m ( G / M )
<_ d -
1.
P r o o f : By the previous theorem and Theorem 2.2.6. []
5.7
G o r e n s t e i n flat m o d u l e s a n d c o v e r s
Gorenstein flat modules were introduced in [26, 36]. In this section we consider topics closely related to our main subject, that is, fiat modules and fiat covers. We will show t h a t every element in the class C0(R) has a Gorenstein flat cover. We first state the following definition which was defined and extensively discussed in [26] and [36].
146 D e f i n i t i o n 5.7.1 An R-module F is called Gorenstein fiat if there is an exact complex . . . --+ F -1 _+ F ~ with F i flat and F = k e r ( 0 ~ such that E |
1 -+ ...
leaves it exact for every injective module
E. The class of Gorensetin fiat modules, denoted by Cgr(R), has many nice homological properties. For instance, we can argue that the class of Gorenstein flat modules is closed under taking direct limits. Hence by one of the fundamental theorems proved in Chapter two (Theorem 2.2.12, in order to show that a module M in C0(R) admits a Gorenstein flat cover, it suffices to show that M has a Gorenstein flat precover. Note t h a t if M E C0 and 0 ~ X -+ Y -+ M --+ 0 is exact with Y Gorenstein fiat and X such t h a t Ext I(Z, X ) = 0 whenever Z is Gorenstein flat, then Y ~ M is a Gorenstein flat precover of M, hence we will need to produce such an exact sequence We need L e m m a 5.7.1 If the sequence
. . . - + F - I _ + FO--+ F ~ - . ~ . . . is an exact sequence of flat modules such that E | - leaves the sequence exact and K is cotorsion and of finite flat dimension, then H o r n ( - , K) leaves the sequence exact. Proof." If K is flat and cotorsion, then by the remarks above, H o r n ( - , K ) will leave the sequence exact if H o m ( - , H o m ( E , E)) leaves the sequence exact whenever E a n d / ~ are injective. But H o m ( - , H o m ( E , / ~ ) ) and H o r n ( E |
are isomorphic functors..
So for such a K, Horn(-, K) leaves the sequence exact. Now suppose 0 --+ K ' --~ K --+ K " ~
0 is an exact sequence of modules such
that Horn(F, - ) leaves this sequence exact whenever F is flat. Then applying each of H o m ( - , K ' ) , H o m ( - , K ) and H o m ( - , K") to the sequence
..._+ F-I__~ FO__~ F I _ + . . . we get a short exact sequence of complexes. So if any two of H o r n ( - , K ' ) , H o r n ( - , K ) and H o r n ( - , K") leave the given sequence exact, so does the third. Now we note by induction, we can generalize this result to exact sequence with any finite number of terms. Now let K be cotorsion and of finite flat dimension. Then using flat covers we construct an exact sequence
O--+ F , - ~ . . . ~
Fo--+ K--+ O
with each F 0 , - - - , Fn flat and cotorsion and such t h a t Horn(F, - ) leaves the sequence exact whenever F is flat. But by the above, Horn(-, Fi) leaves the sequence
. . . - + F - I - + FO ~ FI__+... exact for each i = 0 , . . - , n. Hence H o r n ( - , K ) also leaves the sequence exact. []
147 C o r o l l a r y 5.7.2 I f X is Gorenstein fiat and K is cotorsion and of finite fiat dimension, then Exti(X, I4) = 0 for i > 1.
T h e o r e m 5.7.3 Let X be an R-module. Then the following are equivalent: (1) X is Gorenstein fiat; (2) X E 6o and TorR(L,X) = 0 for all i > 0 and L of finite injective dimension .
P r o o f i (1) ~
(2) Since X is Gorenstein fiat, by the definition there is an exact
sequence . . . __+ F - 2 __+ F - I __+ F O ~ F ~ --+ F 2 -+ . . .
with F i flat such that X =ira(0 ~ and E |
leaves the sequence exact for any injective
module E. Easily for any module L of finite injective dimension, L | - leaves the sequence exact. Note that X =ira(0~
It follows that TorR(L, X) = 0 for all i > 0
and L of finite injective dimension. In particular, Tor/R(D, X) = 0 for all i > 0. Then using the left exactness of Hom(D, - ) and the fact that F i -+ Horn(D, D | F i) is an isomorphism for ever i > 0, we have a commutative diagram with exact rows
0
)
X
9
F~
F1
0 ~ H o m ( D , D | X) ~ H o m ( D , D | F~
9
F 2
>
D | F1)--+Hom(D, D | F2) -+
Note that all fiat modules are in the class Go. It follows that f0, fl and f2 are all isomorphisms. Hence X ~ H o m ( D , D @ X). In order to show that X is in Go, we then only need to show that E x t , ( D , D | X) = 0 for all i > 0. Consider the exact sequence 0 --~ X -+ F ~ --+Y-+ 0 with Y = F ~
Note that Y
is Gorenstein fiat. And then by the above, TorR(D, Y) = 0 and Y -~Hom(D, D | Y). Therefore we have the commutative diagram
0--+
X
9
a
F 0
f~
0-+Hom(D, D | X) --+Horn(D, D | F~
9
Y
,
0
Ill Hom(D, D @ Y) --+ Extl(D, D | X)-~ 0
148 This implies that
E x t i ( D , D | X ) = 0. Similarly, E x t l ( D , D | Y) = 0, and then
E x t ' ( D , D | X) = 0. Repeating the same argument, we easily see that E x t ' ( D , D |
= 0 for all i > 0.
This finishes the implication (1) =~ (2). (2) ==~ (1). We need to construct a complete flat resolution of X which is left exact by E | - for any injective module E. By assumption, Tor/n(E, X ) = 0 for any injective module E. It follows that for any fiat resolution of X . . . - + F - 2 _ ~ F -I _+ Fo_+ x - . + O
E | - leaves the sequence exact. Then we only need to find the right half the desired complete flat resolution of X. Set Y = D |
By hypothesis, E x t i ( D , Y ) = 0 f o r i
>0.
Let 0 - - + Y - + E - +
Z -+ 0 be exact sequence with E the injective envelope of Y. Since E x t l ( D , Y) = 0, we have the exact sequence 0 -+ Horn(D, Y) -+ Hom(D, E) -4 Hom(D, Z) -~ 0 Note t h a t X ~ H o m ( D , Y ) and that L = H o m ( D , E ) has finite flat dimension. Let W = H o m ( D , Z). We have the exact sequence 0 -~ X -+ L --+ W -+ 0. By Theorem 3.1.12 we know L has a fiat cover. Consider the exact sequence O-+ K - + F ~ L - + O
with F -+ L the flat cover of L. Then we consider the pullback diagram
0
0
K=K
I
.F
.W
.0
O---~X
9L
,W
,0
I
0
The fact t h a t
l
.P
0
l
0
E x t l ( X , K ) = 0 shows that 0 --+ K --+ D --+ X -+ 0 is split. Hence X
can be embedded into a fiat module F . Next since R is Noetherian, we can choose an embedding F of X such t h a t X ~ F is a flat preenvelope of X (see Theorem 2.5.1), that is, we have an exact sequence 0 -+ X --+ F --+ X i -+ 0. Therefore the induced sequence 0 --+ Hom(X, F ' ) -+ Horn(F, F ' ) --+ Hom(X1, F ' ) -~ 0
149 is exact for any fiat module F ' . In particular, for any injective module E, Horn(E, E(k)) is flat and then H o r n ( - , Hom(E, E(k))) leaves the above sequence exact. This implies t h a t E | - leaves the sequence exact for any injective module E. By Theorem 5.5.7, X1 is in C0. It is easy to see that
rori(L,X1)
=
0 for any L
of finite injective dimension, and i > 0. Finally, repeating the same argument, we get the desired complete flat resolution of X, and then t h a t X is Gorenstein flat. []
C o r o l l a r y 5.7.4 Any Gorenstein projective module is Gorenstein fiat.
Any direct
limit of a family of Gorenstein fiat modules is Gorenstein fiat.
P r o o f : Let C be a Gorenstein projective module. By Theorem 5.7.3 and Proposition 5.6.1, we only need to show that Tori(L, C) = 0 for all i > 0 and all L of finite injective dimension. But by a natural isomorphism we easily have that Hom(Tori(L, C), E(k)) ~- Exti(C, Hom(L, E(k))) = 0 since C is Gorenstein projective and Hom(L, E(k)) has finite projective dimension. The remaining part follows from Theorem 5.7.3 since C0 is closed under direct limits. [] We recall t h a t for any module M, a linear map ~ : F --4 M with F Gorenstein flat is called a Gorenstein flat cover if (1): H o m ( F ' , F ) -+ H o m ( F ' , M ) -~ 0 is exact when F ' is Gorenstein flat and (2): ~ = ~ f for any f : F --+ F implies t h a t f is an automorphism of F . Theorem
5.7.5 Every R-module M in Co(R) admits a Gorenstein fiat cover.
P r o o f : First we note t h a t the class of Gorenstein fiat modules is closed under direct limits. So, in order to find a Gorenstein flat cover of M, we only need to find a Gorenstein flat precover of M (which means that the condition (1) holds). By Theorem 5.6.11 there is an exact sequence
O--+ K ~ P ~ M - + O such t h a t P is Gorenstein projective and K is of finite projective dimension. Now we claim t h a t there exists an exact sequence 0 --+ K --~ X -+ F --+ 0 with F flat and E x t i ( F ', X ) = 0 for all Gorenstein flat modules F ' and i > 0. Since K has finite flat dimension, it has flat cover. So we have an exact sequence 0 -+ /~ --+ G ~
K --+ 0 with G the flat cover of K and the kernel / ( cotorsion.
Note t h a t /~r has finite flat dimension. By Corollary 5.7.2,
Ext~(F ',/<) = 0 for all
Gorenstein fiat modules F ~ and i > 0. Now we consider the pushout diagram
150
0
I ii I I F= I 0
9/{
,G
0
l l 1 F l 0
.K
9/ ~ * P E ( c ) - x
. 0 9o
where P E ( G ) is the pure injective envelope of G. Note that F is fiat and E x t i ( F ', X ) = 0 for all Gorenstein flat modules F ' and i > 0. Finally we consider the following pushout diagram
0 0
0
0
I I .X I F= I 0
l I .Y 1 F l 0
.K
.P
.M
ii
.M
. 0 ,0
where Y is Gorenstein flat since both P and F are Gorenstein flat. Since X has the Exti(F',X)
property
= 0 for ali Gorenstein fiat modules F ' , it is easy to see t h a t
Y -+ M is a Gorenstein fiat precover of M. Therefore M has a Gorenstein fiat cover. []
R e m a r k : Let M E C0(G), G -+ M be a Gorenstein fiat cover and K the kernel. By Theorem 5.6.11 and the above argument, it is easy to see t h a t K has fiat dimension less than or equal d - 1. The following result indicates how far a Gorenstein fiat module differs from a Gorenstein projective module. Proposition
5.7.6 I f X is Gorenstein fiat, then there is an exact sequence 0 -+ F -+
P ~ X --+ 0 such that P is Gorenstein projective and F is fiat. Proof:
Note that a Gorenstein fiat module of finite fiat dimension is fiat.
Then
conclusion follows easily from Theorem 5.6.11.[] We conclude this chapter by remarking that the homological theory of complexes of modules is quite promising. See Foxby [38] for the introduction and the further references.
151 R e m a r k 5.7.7 If Gorenstein injective, projective and flat modules are replaced by Gorenstein injective, projective and flat complexes correspondingly, then most results we have obtained can still be carried out. For this purpose we need some general information about the homological theory of complexes of modules (see Foxby [38]).
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Index absolutely pure, 29 ACC, 9, 44, 45 adjusted cotorsion groups, 70 almost Noetherian, 102 Anderson, 8 annihilator, 43 Artinian, 103 Artinian modules, 63, 122 Artinian rings, 43, 47, 105 Asensio, 50 Auslander, 11 Bass, 15, 16 Bass numbers, 108 Bass' Theorem P, 46 Betti numbers, 122 C, 66 C-envelope, 66 canonical evaluation, 39 canonical injection, 7 canonical isomorphism, 8 canonical projections, 12 Card(Y), 19 eardinality, 19 category, 10 Cauchy sequences, 82 chain, 18 Chinese remainder theorem, 63 class of Gorenstein flat modules, 146 class of left R-modules, 10 Codim(R), 137 cogenerator, 10 Cohen-Macaulay, 137
coherent, 49, 52, 60, 71, 76, 103 coherent rings, 9 eoker(f), 6 eokernel, 6 eolocalization, 118 commutative diagram, 6 complete local rings, 43, 91, 100 completion, 89 completion of a free module, 82 eotorsion, 52, 56, 60, 62, 66, 74, 94 cotorsion envelopes, 66, 68, 89 cotorsion flat modules, 81, 87 cotorsion groups, 70 cotorsion resolutions, 96 covers, 13 DCC, 16 depth, 43 diagram chasing, 59 direct limit, 7, 149 direct sum, 7, 12 direct sum of covers, 20 direct(inductive) system, 7 Dischinger, 44 discrete valuation, 20 divisible, 43 domain, 17 domains, 100 double complex, 64 dual Bass numbers, 117 dualizing modules, 137 g, 14, 29 g-cover, 41
159 g• 29
gxt(J:, M), 67
Ezt(s M), 30 embedded, 14 embedding, 10 Enochs, 11 Enochs' conjecture, 17, 58 envelopes, 10, 35 essentially, 14 essentially pure, 40 exact, 6 exact sequence, 6, 51, 60 Ext,(M, N), 7 extensions, 7, 27 9c, 38, 51 F-covers, 16 Y-envelope, 48 ~--preenvelope, 48 9r-resolutions, 97 S-v, 29 -~Pz, 29 F-pure submodules, 6 factored through, 6 Fexti(N, M), 76 finite injective dimension, 134 finitely generated, 8 finitely presented, 9 flat covers, 16, 20, 51, 52, 56, 58, 71, 74, 82, 84, 93, 95, 99, 100, 109, 134 flat dimension, 10 f.dimR(M), 10 flat envelopes, 48 flat essential extension, 68 flat extensions, 67
f(M), 6 fiat fiat fiat fiat
modules, 7 precovers, 16, 51 preenvelopes, 48, 76 resolutions, 76, 124, 131, 135
Foxby classes, 137 FP-injective, 29 free modules, 6 Fuchs, 38, 83 Fuller, 8 ~SC(R), 146 C0(R), 137 C-domains, 98 G-ideal, 98 generators, 30, 39 global dimension, 10 Gorenstein, 123 Gorenstein flat, 145, 149 Gorenstein fiat covers, 146, 149 Gorenstein injective, 141 Gorenstein projective, 141 Gorenstein rings, 108 Griffith, 83 Gruson, 38, 62, 93 Harrison, 70 Herzog, 137 HomR(M, N), 6 homomorphism, 6 I-adic completion, 82 I-adic topology, 82 /-separated, 82 IF ring, 65 im(f), 6 image, 6 indecomposable, 9, 46 mjective cogenerator, 10, 83 mjective cover (precover), 41 mjective covers, 25, 41-43, 131 mjective dimension, 10, 62 mj.dirnR(M), 10 mjective envelopes, 14, 109 mjective map, 6 mjeetive modules, 7, 60, 82, 124, 134
160 inverse linfits, 82 invertible, 13 isomorphism, 6 So(R), 137 J = J(R), 44 Jensen, 38, 62, 93 K.dim(R), 81, t00 K6nig graph theorem, 23 Kaplansky, 98 ker(f), 6 kernel, 6 k(p), 81 Krull dimension, 93, 95 Kunz, 137 s 36 s 138 s 138 L-gl.dim(R), 62 Lazard, 8 left .~-resolutions, 76 left orthogonal class, 29 left T-nilpotent, 37 1.gl.dim(R), 10 linear map, 5 local rings, 82 localization, 81 locally column finite, 21 locally nilpotent, 25 #~(p, M), 108 R2t4, 10 M| 7 Martinez, 50 Matlis, 14, 82, 83 Matlis dual, 98 Matlis reflexive, 82, 98 Matlis reflexive modules, 98 Max(R), 81
Melkerson, 118 m-adic topology, 20 minimal flat resolutions, 117 minimal generator, 30, 31, 34, 35, 39 minimal injective resolutions, 108 modules, 5 m-primary, 63 m-separated, 82 un(q, F), 92 nil ideal, 44 Noetherian rings, 9, 41, 42 ordinal number, 35 orthogonal idempotent, 46 orthogonal classes, 33 7ri(p, M), 118, 126 • 29 P, i5, 29 P-cover, 15 p-adic numbers, 25 Ps 39 Pg-envelopes, 39 perfect, 16, 25, 38, 45, 48, 66 perfect rings, 45 polynomial ring, 58 Priifer domain, 20 Priifer domains, 70 product, 14, 15 proj.dim(F), 93 proj.dimR(M), 10 projective, 6 projective cover, 15 projective covers, 15, 17, 99 projective dimension, 10 pullback diagrams, 9 pure, 38, 49, 94 pure extension, 39 pure extensions, 39 pure injection, 90
161
pure injective, 60, 83, 126 pure injective dimension, 93 pure injective envelopes, 38, 39, 86, 92 pure injective modules, 38, 53, 60, 73 pure injective resolutions, 89 pure submodule, 8, 29, 39 pushout diagrams, 9 quasi-Frobenius, 109 quotient field, 20 quotient ring, 58
Rp, 81 R-regular sequences, 137 R-sequence, 112, 129 radical, 16 rational numbers Q, 8 Raynaud, 38 reflexive, 100 reflexive module, 82 regular ring, 116 Reiten, 11 relative homological theory, 75 right 5C-resolution, 76 right orthogonal classe, 29 rings, 5 Rotman, 10 S << M, 15 Schanuel's Lernma, 58 Schenzel, 118 simple module, 46 small, 15 Spec(R), 81 special X-precover, 29 special X-preenvelope, 29 spectral sequence, 77 spectral sequences, 64, 65 split extension, 7 Strongly cotorsion modules, 129 strongly torsion free, 135
superfluous, 15 surjective(onto) map, 6
Tf, 17 A
T '~ R(x), 82 T-nilpotence, 16, 37, 45 T-nilpotent, 25, 45 Teply, 41 Tor~(M, N), 7 torsion, 15 torsion free, 17, 136 torsion free coverings, 17 torsion free precoverings, 17, 18 Trj/N(E), 44 trivial extension, 7 b/-preenvelopes, 138 union, 18 von Neumann regular rings, 103 l/V-precovers, 138 Wakamutsu's Lemmas, 27 Warfield, 38 weak global dimension, 10, 65 w.gl.dim(R), 10, 79 X-approximations, 11 X-covers, 13, 20, 27, 29 X-envelopes, 10 X-precovers, 13 X-preenvelopes, 11 Zorn's lemma, 31
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