Lecture Notes in Mathematics
Editors:
A. Dold, Heidelberg
F. Takens, Groningen
1634
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Jinzhong Xu
Flat Covers of Modules
~ Springer
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Author
Jinzhong Xu
University of Kentucky
Department of Mathematics
Lexington, Kentucky, 40506-0027
USA
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Die Deutsche Bibliothek - C I P - E i n h e i t s a u f n a h m e
Xu, Jinzhong:
Flat covers of modules / J i n z h o n g Xu. - Berlin ; Heidelberg ;
New York ; B a r c c l o n a ; Budapest ; H o n g Kong ; L o n d o n ;
Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996
(Lecture notes in mathenaatics ; 1634)
ISBN 3-540-61640-3
NE: GT
Mathematics Subject Classification (1991): 13C 11, 13C 15, 13E05, 13D05,
13H10, 16A50, 16A52, 16A62, 18A30, 18G05, 18G15, 18G25,
ISSN 0075-8434
ISBN 3-540-61640-3 Springer-Verlag Berlin Heidelberg New York
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Dedicated to
Professor Edgar E. Enochs
My Mother Guizhen Zhu
and
My Father Youcheng Xu
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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.
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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
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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
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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
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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
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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.
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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
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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.
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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 -+
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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)
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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:
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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;
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(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:
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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:
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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(~).
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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
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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:
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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
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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 ~.
