Proper classes of short exact sequences and structure theory of modules [Elektronische Ressource] / vorgelegt von Carlos Federico Preisser Montaño

Proper classes of short exact sequencesand structure theory of modulesInaugural-DissertationzurErlangung des Doktorgradesder Mathematischen-Naturwissenschaftlichen Fakult atder Heinrich-Heine-Universit at Dusseldorf.vorgelegt vonCarlos Federico Preisser Montano~aus Mexiko StadtDusseldorf2010aus dem Institut fur Mathematikder Heinrich-Heine-Universit at DusseldorfGedruckt mit der Genehmigung derMathematisch-Naturwissenschaftlichen Fakult atder Heinrich-Heine-Universit at Dusseldorf1. Berichterstatter: Prof. Dr. S. Schr oer2.h Dr. R. WisbauerTag der mundlic hen Prufung: 1. Juli 2010ContentsIntroduction iiiNotation vii1 Preliminaries 11 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Torsion pairs and preradicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Proper classes 93 Proper classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Various proper classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Proper classes related to supplements and complements 195 Supplements and complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 -supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 -complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Proper classes related toC-supplements 258 C-small submodules . . . . . . . . . . . . . . . . . . . . . . . .
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Proper classes of short exact sequences
and structure theory of modules
Inaugural-Dissertation
zur
Erlangung des Doktorgrades
der Mathematischen-Naturwissenschaftlichen Fakult at
der Heinrich-Heine-Universit at Dusseldorf.
vorgelegt von
Carlos Federico Preisser Montano~
aus Mexiko Stadt
Dusseldorf
2010aus dem Institut fur Mathematik
der Heinrich-Heine-Universit at Dusseldorf
Gedruckt mit der Genehmigung der
Mathematisch-Naturwissenschaftlichen Fakult at
der Heinrich-Heine-Universit at Dusseldorf
1. Berichterstatter: Prof. Dr. S. Schr oer
2.h Dr. R. Wisbauer
Tag der mundlic hen Prufung: 1. Juli 2010Contents
Introduction iii
Notation vii
1 Preliminaries 1
1 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Torsion pairs and preradicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Proper classes 9
3 Proper classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Various proper classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Proper classes related to supplements and complements 19
5 Supplements and complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 -supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 -complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Proper classes related toC-supplements 25
8 C-small submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
9 C-supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
10 rad -supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32S(C)
5 Proper classes related toC-complements 39
11 C-essential submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
12 C-complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
13 tr -complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44S(C)
6 Cotorsion pairs related to proper classes 53
14 Cotorsion pairs, covers and envelopes . . . . . . . . . . . . . . . . . . . . . . . . 53
15 P-cotorsion pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
16 Covers and envelopes relative to a proper class. . . . . . . . . . . . . . . . . . . 73
Appendix 77
n17 The functor Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77P
18 P-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Bibliography 86
iiiIntroduction
In this thesis we investigate proper classes of short exact sequences, especially those proper
classes induced by supplement-like and complement-like submodules in the category of left
R-modules over an associative ring R and in the categories of type [M] of left R-modules
subgenerated by a module M.
Proper classes were introduced by Buchsbaum in [9] for an exact category. We use the
axioms given by Mac Lane in [43] for abelian categories (see 3.1).
1A proper classP, in an abelian categoryA de nes a closed subbifunctor of the Ext functor
A
as has been shown by Butler and Horrocks in [10] or by Buan in [8], i.e.
1 1 opExt (; ) Ext (; ) : A A!Ab;P A
1whereAb denotes the category of abelian groups, and for A;C2A, Ext (C;A) denotes the
A
class of isomorphism classes of short exact sequences
0!A!B!C! 0:
1
Ext (C;A) together with the Baer sum is an abelian group and for any proper class P,
A
1
Ext (C;A) is a subgroup (see [43]). These ideas were the starting point of relative homolo-P
gical algebra. Proper classes are investigated by Mishina and Skornyakov in [47], Walker in
[60], Manovcev in [45] and Sklyarenko in [56] for abelian groups, and by Generalov in [27, 28],
Stenstr om in [57, 58] and Sklyarenko in [55] for module categories.
Supplement and complement submodules induce proper classes. This was noted for abelian
groups by Harrison in [32] and for R-Mod by Stenstr om and Generalov in [57, 27, 28]. Recently
this investigations were further developed by Al-Takhmann et al. and Mermut in [46, 1].
The rst purpose of this thesis is to continue this investigation studying the proper classes
induced by supplements and complements relative to a class of modulesC closed under sub-
modules and factor modules. We call such a class afq;sg-closed class. We obtain some known
results and new ones for special cases ofC. For instance, ifC is the class of singular modules,
then we obtain the -supplements introduced by Zhou in [64]. WhenC is R-Mod, we recover
the classical supplements and complements. We also consider the case whenC is the torsion
classT associated to a hereditary preradical or the torsionfree classF associated to a co-
hereditary preradical . Two interesting cases are whenC =[M] andC = [M] consideredf
asfq;sg-closed classes of R-Mod.
After collecting some preliminary results in Chapters 1, 2 and 3 about abelian categories,
proper classes, supplements and complements in a module category, we de ne, in Chapter 4,
C-supplements.
A submoduleKN is calledC-small if for any submoduleXN the equalityX+K =N
and N=X2C implies that X =N and we write K N. The properties ofC-small submo-C
dules are similar to those of small submodules (see 8.3). We characterize the radical de ned
by the reject of the class of simple modules inC. This radical, applied to a moduleN, is equal
to the sum of allC-small submodules of N. We call a submodule KN aC-supplement in
0 0 0N if there is a submoduleK N withK +K =N andK\K K. Then we prove thatC
iiiC-supplements induce a proper class. This proper class contains the proper class induced by
the supplements. In [1] Al-Takhman et al. introduced a generalization of supplements using
a radical . They call a submodule K N a -supplement in N if there is a submodule
0 0 0K N such thatK +K =N andK\K (K). They prove that the proper class induced
by the -supplements is the proper class injectively generated by the class of -torsionfree
modules, where a proper classP, injectively generated by a class of modulesI, consists of
those short exact sequences such that the modules inI are injective with respect to them.
Every supplement is a Rad-supplement for = Rad. In the same wayC-supplements are
rad -supplements withC
X
(N) = rad (N) = fallC-small submodules of NgC
for every module N.
In Chapter 5, dual to the concept ofC-small submodules, we call a submodule K N
C-essential if for any submodule X N the equality X\K = 0 and X 2C implies that
X = 0. We write K N. We characterize the idempotent preradical de ned by the traceCe
of the class of simple modules ofC in a module N. This preradical is the intersecction of all
C-essential submodules of N. Next we deneC-complements as those submodules K N
0 0 0such that there is a submodule K N with K\K = 0 and KK =K N=K. WeCe
prove thatC-complements induce a proper class. Dual to the -supplements, Al-Takhman et
al. introduce in [1] the concept of -complements for an idempotent preradical . The proper
class induced by the -complements is the proper class projectively generated by the class
of -torsion modules, where a proper classP, projectively generated by a class of modules
Q, consists of the short exact sequences such that the modules in Q are projective with
respect to them. Complements are Soc-complements for = Soc andC-complements are
tr -complements forC
\
(N) = tr (N) = fallC-essential submodules of NgC
for every module N.
The second purpose of this work is to compare, in Chapter 6, the lattice of all proper
classes with the lattice of cotorsion pairs. In [53] Salce introduced a cotorsion pair in the
following way: Take a classA of abelian groups and considerP, the proper class projectively
generated byA. Then he observes that
Div(P) =fX2 Mod-Zj Ext(A;X) = 08A2Ag;
where Div(P) is the class of abelian groupsD such that every short exact sequence beginning
?with D belongs toP. He de nes the cotorsion pair cogenerated by A by ( Div(P); Div(P)).
In the same way he de nes the cotorsion pair generated by a class B of abelian groups by
?(Flat(R); Flat(R) ) withR the proper class injectively generated byB and Flat(R) the class
of abelian groups G such that every short exact sequence ending at G belongs toR. The
work of Salce on cotorsion pairs was generalized to module categories and abelian categories
and it has been extensively studied. We prove some results which show how some properties
of those proper classes yield information about their associated cotorsion pairs (e.g. 14.22,
14.25). We de ne a correspondence between the lattice of injectively (projectively) generated
proper classes and the lattice of cotorsion pairs using the construction of Salce. LetP be an
injectively generated proper class in an abelian category. De ne
?
(P ) = (Flat(P); Flat(P) ):
is an order-reversing correspondence between injectively generated proper classes and cotor-
sion pairs which preserves arbitrary meets. We prove that this correspondence is bijective if
ivwe restrict to the class of Xu proper classes, these are injectively generated proper classesP,
?such that Inj(P) = Flat(P) . For example, in R-Mod the proper class of pure exact sequences
is a Xu proper class when the pure injective and the cotorsion modules coincide.
Finally we consider cotorsion pairs relative to a proper classP as introduced by Hovey
in [37]. We call themP-cotorsion pairs. They are pairs of complete orthogonal classes with
1 1
respect to the functor Ext instead of Ext . We show that, like the cotorsion pairs, theyP A
come from injectively (projectively) generated proper classes. Here we de ne three classes of
objects which correspond to theP- ats, P-divisibles andP-regulars in the absolute case.
LetP andR be two proper classes. An object X is calledP-R- at if every short exact
sequence inP ending at X belongs toR. P-R-divisibles are de ned dually. We show that
everyP-cotorsion pair is of the form
?P( (P-Div-R);P-Div-R)
for a projectively generated proper classR and also of the form
?P(P-Flat-R; (P-Flat-R) )
for an injectively generated proper classR, where for a classX ,
? 1 ? 1P PX = Ker (Ext (X; )) and X = Ker (Ext (; X )):P P
We obtain properties ofP-R- ats, P-R-divisibles andP-R-regulars and we show that this
classes coincide with known concepts in module theory.
nIn the Appendix we include the construction of the relative functors Ext and the de ni-P
tions of the homological dimensions relative to a proper classP due to Mac Lane and Alizade
(see [2, 43]).
One can de ne proper classes in more general categories. The de nition of exact categories,
introduced by Quillen in [50], is the reformulation of the axioms of a proper class, where a short
exact sequence inP corresponds to a co ation (f;g ) in the exact category. In a preabelian
category Generalov de nes a proper class of cokernels (kernels) using some equivalent axioms
to those of Mac Lane (see [29]). In a triangulated category Beligiannis de nes a proper class
of triangles using some axioms analogous to those of an exact category (see [4]).
vviNotation
1
Ext the extension functor, iii
A
A an abelian category, 1
Ab the category of abelian groups, 1
R-Mod the of left R-modules, 1
Proj(A) the class of projective objects ofA, 2
Inj(A) the class of injective objects ofA, 2
[M] the full subcategory of R-Mod subgenerated by M, 3
A fX2Aj Hom (X;A) = 0g, 4
A
A fY 2Aj Hom (A;Y ) = 0g, 4
AT
rad (N) fKerfjf :N!C; C2Cg, 6C P
tr (N) fImfjf :C!N; C2Cg, 6C
P;R proper classes, 9
Abs the proper class of all short exact sequences, 9
Split the proper class of all splitting short exact sequences, 9
Proj(P) the class ofP-projective objects, 12
Inj(P) the class ofP-injective objects, 12
Flat(P) the class ofP- at objects, 12
Div(P) the class ofP-divisible objects, 13
Reg(P) the class ofP-regular objects, 13
1 (Q) the proper class projectively generated by a classQ, 14
1 (I) the proper class injectively by a classI, 15
the proper class of -complements =
-Compl
the proper class projectively generated byT , 15
the proper class of -supplements =
-Suppl
the proper class injectively generated byF , 16
Pure the proper class of pure short exact sequences, 17
KN K is a small submodule of N, 19
K N K is an essential submodule of N, 19e
K N K is a cocloseddule of N, 20cc
Cocls the proper class of coclosed submodules, 20
K N K is aC-small submodule of N, 25C
C-Suppl the proper class ofC-supplements, 28
Suppl the proper class of supplements, 31
S(C) the class of simple modules inC, 32
T
rad (N) fKerfjf :N!S; S2S(C)g, 32S(C)
the proper class of rad -supplements =S(C)
the proper class injectively generated byrad -SupplS(C)
fN2[M]j rad (N) = 0g, 33S(C)
vii

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