Chapter Four Abelian Categories

224 Pages·2015·1.4 MB·English

Sudan University of Science and Technology
College of Graduate Studies

Categories and Chain Complexes in
Homological Algebra
ﻲﺟﻮﻟﻮﻤﮭﻟا ﺮﺒﺠﻟا ﻲﻓ ﻞﺳﻼﺴﻟا تﺎﻌﻤﺠﻣو فﺎﻨﺻﻷا
A thesis Submitted in fulfillment of the requirements for the
degree of Philosophy in mathematics

By
Mohmoud Awada Mohmoud Torkawy
Supervisor:
Dr. Adam Abdalla Abbaker Hassan
April 2014
- 1 -
Dedication

To my family for their
cooperation and encouragement

I
Acknowledgements

My thanks firstly to Allah.
I am great full to express my deep thanks to my supervisor Dr.
Adam Abdalla Abbakar Hassan at Nile Valley University who
suggested the title of this thesis and for his full help.
Also I thank Sudan University of Science and Technology for
their help and advice, for their cooperation throughout this study.
My thanks extends to the University of Kordofan for the scholarship.

II
Abstract
The thesis exposes the basic language of categories and
functions.
We construct the projective, inductive limits, kernel, cokernel,
product, co product. Complexes in additive categories and
complexes. in abelian categories.
The study asked when dealing with abelian category c, we
assume that c is full Abelian.
The thesis prove the Yoned lemma, Five lemma, Horseshoe
lemma and Snake lemma an then it give rise to an exact sequence,
and introduce the long and short exact sequence.
We consider three Abelian categories c, c', c" an additive bi
functor F: cxc' → c" and we assume that F is left exact with respect
to each of its argument, and the study assume that each injective
object I∈C the functor F (1,.): c' → c" is exact.
The study shows important theorm and proving it if R is ring R
= e {x , ….…, x }, the Kozul complex KZ (R) is an object with
1 n
effective homology.
We prove the cone reduction theorem ∈
(if p = (f,g,h): C D , and p' = f',f',h'):
* *
C' D' , be two reduction and Ø: C C' a chain
* * * *
complex morphism, then these data define a canonical reduction
P" = (f", g", h" : cone (Ø) cone (f' Ø g').
The study gives a deep concepts and nation of completely
multi – positive linear maps between C ∗- algebra and shows they are
completely multi positive
We gives interpretation and explain how whiteheal theorem is
important to homological algebra. .
III
The study construct the localization of category when satisfies
its suitable conditions and the localization functors.
The thesis is splitting on De Rham co-homology in the module
category and structures on categories of complexes in abelian
categories.
The thesis applies triangulated categories to study the problem
B = D (R), the unbounded derived category of chain complexes, and
how to relate between categories and chain complexes.

IV
ﺔــﺻﻼـﺧﻟا

.ﺎﻬﻔﺋﺎظوو فﺎﻧﺻﻷا ﺔﯾﺳﺎﺳﻷا ﺔﻐﻠﻟا ﺔﺣورطﻷا ﻩذﻫ ﺢﺿوﺗ
ﺔﯾﻓﺎـﺿﻹا ةاوﻧﻠﻟ ﺔﯾﺟﺎﺗﻧﺗﺳﻻا ﺔﯾﻋورﺷﻣﻟا دودﺣﻟا ءﺎﻧﺑ وأ بﯾﻛرﺗﺑ ﺔﺳاردﻟا تﻣﺗﻫا
ﺔـــــــﯾﻌﻣﺟﻟا فﺎﻧـــــــﺻﻷا ﻲـــــــﻓ ﻲﺟرﺎـــــــﺧﻟا برـــــــﺿﻟاو ﻲﻠﺧادـــــــﻟا برـــــــﺿﻟاو ﺔـــــــﯾوﻧﺎﺛﻟا ةاوـــــــﻧﻟاو
.ﺔﯾﻠﯾﺑﻷا فﺎﻧﺻﻷا ﻲﻓ تﺎﺑﯾﻛرﺗﻟاو ﺔﯾﻓﺎﺿﻹا (ةدﻘﻌﻣﻟا وأ)ﺔﯾﺑﯾﻛرﺗﻟا
C C
ﺔﺋﻓ نأ ضرﺗﻔﺗ ثﯾﺣ فﻧﺻﻟا ﻊﻣ لﻣﺎﻌﺗﻟا ﺔﯾﻔﯾﻛ ﻲﻓ لاؤﺳﻟا ثﺣﺑﻟا حرطﯾ
.لﻣﺎﻛ ﻲﻠﯾﺑأ فﻧﺻ
Five Yoned lemma
) فﯾﺎﻓ ﺔﯾدﯾﻬﻣﺗ و ( ) ﺎﻣﯾﻟدﻧوﯾ ﺔﯾدﯾﻬﻣﺗ ﺔﺳاردﻟا تﻟوﺎﻧﺗ
Snak Horsehose lemma lemma
) كﺎﻧــﺳ ﺔـﯾدﯾﻬﻣﺗ و ( ) ﻲـﺷروﻬﻟا ﺔـﯾدﯾﻬﻣوﺗ و (
lemma
مﯾدـــﻘﺗ ﻰـــﻠﻋ لـــﻣﻌﺗ ﻲـــﺗﻟا ﺔـــﻣﺎﺗﻟا ﺔﻠـــﺳﻠﺳﻠﻟ ﺞﺋﺎـــﺗﻧﻟا تـــطﻋأ مـــﺛ نـــﻣو ﺎـــﻬﻧﯾﻫا رﺑ و(
.كﻟذ نﻣ ةدﺎﻔﺗﺳﻻاو ﺎﻬﻘﯾﺑطﺗ نﻛﻣﯾ فﯾﻛو رﯾﺻﻘﻟاو لﯾوطﻟا ىدﻣﻟا وذ لﺳﻼﺳﻟا
C.C , .C
ﺔﯾﻓﺎــﺿإ لاود ﻰــﻟ ٕ ا و ً ◌ً ◌ ﺔــﯾﻠﯾﺑأ فﺎﻧــﺻأ ثﻼــﺛ ءﺎــﻧﺑ ﻰــﻟإ ﺔــﺳاردﻟا فدــﻬﺗ
F
نأ ﺔـﺳاردﻟا ضرـﺗﻔﺗو ﺎـﻬﺗﺎﯾﺟﺣ لـﻛﺑ قـﻠﻌﺗﯾ ﺎـﻣﯾﻓ ﺎﻫدـﺣوﻟ تـﻛرﺗ دـﻗو ﺔﻟادﻟا لﺛﻣ ىرﺧأ
I∈C
:ﻲﻫو ( ) ﻪﺑ ﺔﺻﺎﺧﻟا ﺔﻔﯾظوﻟا ﻰﻟإ ﻲﻣﺗﻧﯾ ﻲﻋوﺿوﻣ لﺧدﻣ لﻛ
C. C F I, . ⇉
ً ◌ ﻰﻟإ ًﺎﻣﺎﻣﺗ يدؤﺗو ( )
R
ﺔــﻘﻠﺣ لــﺛﻣﺗ تــﻧﺎﻛ اذإ ﻪــﻧأ ﻰــﻠﻋ صﻧــﺗ ﻲــﺗﻟا ﺔــﯾرظﻧﻟا ﺔــﯾﻣﻫأ ﺔــﺳاردﻟا ﺢــﺿوﺗو
Kozul KZ(R) R x x
وأ سﻧﺎــﺟﺗ وذ ءﻲـﺷ وـﻫ لزوـﻛ بـﯾﻛرﺗ ناو = ( 1 ........... n) ناو

.تﻧﺎﻛ اذإ نا ﻰﻠﻋ ﺔﯾرظﻧﻟا نﺎﻫرﺑ صﻧﯾ لﺎﻌﻓ لﺛﺎﻣﺗ
P= (f,h,g) C D
* *
C(cid:4593) D(cid:4593) P f g h
∗ . نأ ﻰﻟإ يدوﺗ َ ◌ = ( َ ◌، َ ◌ ، َ ◌)
V
 C C
: َ ◌ ﺔﯾﻟﺎﺧﻟا ﺔﻋوﻣﺟﻣﻟا ﻰﻟإ رﺻﺗﺧﯾ
C*
.رــﺑﺟﻟا نﯾــﺑ ﺔــﯾﺑﺟوﻣﻟا ﺔــﯾﻠﻛﻟا ﺔــﯾطﺧﻟا مــﺳاورﻟا نــﻋ ﺔــﻘﯾﻣﻋ مﯾﻫﺎــﻔﻣ ﺔــﺳاردﻟا ﻲــطﻌﺗ
. ًﺎﯾﻠﻛ ﺔﯾﺑﺟوﻣﻟا ةددﻌﺗﻣ ﺎﻬﻧأ ﺢﺿوﺗو
WHITEHEAL
فــﯾﻛ ﺎﻧﺣــﺿوو تارﯾــﺳﻔﺗ ﺎــﻧﯾطﻋأو ( ) لــﻬﺗﯾاو ﺔــﯾرظﻧﺑ ﺔــﺳاردﻟا تــﻣﺗﻫأو
.ﻲﺟوﻟوﻣﻬﻟا رﺑﺟﻟا ﻲﻓ ﺔﻣﻬﻣ لﻬﺗﯾاو ﺔﻧﻫرﺑﻣ نأ
مﺎـﻫرﯾد ﻪـﺑ مﺎـﻗ يذـﻟا بﺣﺎﺻﻣﻟا سﻧﺎﺟﺗﻟﺎﺑ ﺔﺻﺎﺧﻟا تﺎﺿﻗﺎﻧﺗﻟا ضﻌﺑ ﻰﻟإ ﺔﺳاردﻟا مﺗﻬﺗ
De Rham
.ﺔﯾﻠﯾﺑﻷا فﺎﻧﺻﻷا ﻲﻓ ( )
B D R
ةدودـﺣﻣﻟا رـﯾﻏ فﺎﻧـﺻﻷا صﺧﺗ ﻲﺗﻟا = ( )ـﻟ ﺔﯾﺛﻠﺛﻣﻟا فﺎﻧﺻﻷا ﺔﺣورطﻷا قﺑطﺗ
لــﺳﻼﺳو فﺎﻧـﺻﻷا نﯾـﺑ ﺔــﻗﻼﻋ قـﻠﺧﺗ وأ نرﺎـﻘﺗ فـﯾﻛو (ةدــﻘﻌﻣﻟا)ﺔـﯾﺑﯾﻛرﺗﻟا ﺔﻠـﺳﻠﺳﻟا ﻲـﻓ
.(ةدﻘﻌﻣﻟا) ﺔﺑﻛرﻣﻟا فﺎﻧﺻﻷا

VI
The Contents
Subject Page
- Dedication I
- Acknowledgment II
- Abstract III
- Abstract in (Arabic) V
- The contents VII
Chapter One
The language of categories
(1.1) sets and maps. 4
(1.2) Modules and linear maps. 13
(1.3) Categories and functor. 35
(1.4) Representable functor, adjoint fucntor. 47
(1.5) Free Modules. 50
(1.6) The grothendieck theorem. 54
Chapter Two
Limits
(2.1) Products and co products 62
(2.2) Kernels and co kernels 66
(2.3) Limits and inductive limits and projective limits 71
(2.4) Properties of limits 79
(2. 5)compostion of limits 80
(2.6) Filtrates inductive limits 82
Chapter Three
Additive categories
(3.1) Additive categories 89
(3.2) Complexes in additive categories 93
(3.3) Double complexes 96
(3.4) Simplicial constructions 100
(3.5) Category of Diagrams 102
(3.6) Combinatorial topology of simplicity complexes 105
(3.7) Simplicial complexes 110
Chapter Four
Abelian categories
(4.1) Abelian categories 114
VII
(4.2) Exact functors 119
(4.3) Injective and projective objects 122
(4.4) Complexes in abelian categories 131
(4.5) Resolutions 137
(4.6) Derived functors 142
(4.7) Koszul complexes 149
(4.8) Derived category of K-modules 158
Chapter Five
Categories, localization and chain complexes
(5.1) The homo Topycategory k (c) 163
(5.2) Derived categories 166
(5.3) Resolutions 171
(5.4) Derived functors 174
(5.5) Bi functors 176
(5.6) Localization of categories 180
(5.7) Localization of functors 188
(5.8) Triangulated categories 190
(5.9) Localization of triangulated categories. 195
(5.10) Effective chain complexes 199
(5.11) Locally effective chain complexes 203
- Appendix 207
- References 208

VIII
Chapter One
The Language of Categories
The aim of the study is to introduce the language of categories
and to present the basic notions of homological algebra first from an
elementary point of view, with the notion of derived functors, next
with a more sophisticated approach, with the introduction of
triangulated and derived categories.
After having introduced the basic concepts of categories theory
and particular those of projective and inductive limits, we treat with
some details additive and abelian categories and construct the
derived. The thesis show the important concepts of triangulated and
derived categories.
This thesis is of five Chapter.
In chapter one we expose the basic language of categories and
functors. A key point is Yoneda Lemma, which asserts that a
category c may be embedded in category ĉ of contra variant functors
on c with values in the category set of sets.
This naturally leads to the concept of representable functor.
Many examples are treated, in particular in relation with the
categories set of sets and Mod (A) of A-modules, for a (non
necessarily commutative) ring A.
In chapter two we construct the projective and inductive limits,
as a particular case, the kernels and co-kernels, product and
coproduces.
We introduce the notions filtrant category and co final functors,
and we study with some care filtrant inductive limits in the category
set of sets.
Chapter three deals with additive categories and study the
category of complexes in sub categories in particular, we introduce
the shifted complex, the mapping cone of morphism, the homotopy
of complexes and the simple complex associated with a double
1