Sudan University of Science and Technology College of Graduate Studies Categories and Chain Complexes in Homological Algebra اﻷﺻﻨﺎف وﻣﺠﻤﻌﺎت اﻟﺴﻼﺳﻞ ﻓﻲ اﻟﺠﺒﺮ اﻟﮭﻤﻮﻟﻮﺟﻲ 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 {x1, ….…, xn }, the Kozul complex KZ (R) is an object with 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 اﻟﺧـﻼﺻــﺔ ﺗوﺿﺢ ﻫذﻩ اﻷطروﺣﺔ اﻟﻠﻐﺔ اﻷﺳﺎﺳﯾﺔ اﻷﺻﻧﺎف ووظﺎﺋﻔﻬﺎ. اﻫﺗﻣت اﻟدراﺳﺔ ﺑﺗرﻛﯾب أو ﺑﻧﺎء اﻟﺣدود اﻟﻣﺷروﻋﯾﺔ اﻻﺳﺗﻧﺗﺎﺟﯾﺔ ﻟﻠﻧواة اﻹﺿـﺎﻓﯾﺔ واﻟﻧـــــــواة اﻟﺛﺎﻧوﯾـــــــﺔ واﻟﺿـــــــرب اﻟـــــــداﺧﻠﻲ واﻟﺿـــــــرب اﻟﺧـــــــﺎرﺟﻲ ﻓـــــــﻲ اﻷﺻـــــــﻧﺎف اﻟﺟﻣﻌﯾـــــــﺔ اﻟﺗرﻛﯾﺑﯾﺔ(أو اﻟﻣﻌﻘدة) اﻹﺿﺎﻓﯾﺔ واﻟﺗرﻛﯾﺑﺎت ﻓﻲ اﻷﺻﻧﺎف اﻷﺑﯾﻠﯾﺔ. ﯾطرح اﻟﺑﺣث اﻟﺳؤال ﻓﻲ ﻛﯾﻔﯾﺔ اﻟﺗﻌﺎﻣل ﻣﻊ اﻟﺻﻧف C ﺣﯾث ﺗﻔﺗرض أن C ﻓﺋﺔ ﺻﻧف أﺑﯾﻠﻲ ﻛﺎﻣل. ﺗﻧﺎوﻟت اﻟدراﺳﺔ ﺗﻣﻬﯾدﯾﺔ ﯾوﻧدﻟﯾﻣﺎ ( Yoned lemma) و ﺗﻣﻬﯾدﯾﺔ ﻓﺎﯾف ( Five lemma) و ﺗوﻣﻬﯾدﯾـﺔ اﻟﻬورﺷـﻲ (Horsehose lemma) و ﺗﻣﻬﯾدﯾـﺔ ﺳــﻧﺎك ( Snak lemma) وﺑر اﻫﯾﻧﻬـــﺎ وﻣـــن ﺛـــم أﻋطـــت اﻟﻧﺗـــﺎﺋﺞ ﻟﻠﺳﻠﺳـــﻠﺔ اﻟﺗﺎﻣـــﺔ اﻟﺗـــﻲ ﺗﻌﻣـــل ﻋﻠـــﻰ ﺗﻘـــدﯾم اﻟﺳﻼﺳل ذو اﻟﻣدى اﻟطوﯾل واﻟﻘﺻﯾر وﻛﯾف ﯾﻣﻛن ﺗطﺑﯾﻘﻬﺎ واﻻﺳﺗﻔﺎدة ﻣن ذﻟك. ﺗﻬــدف اﻟدراﺳــﺔ إﻟــﻰ ﺑﻧــﺎء ﺛــﻼث أﺻــﻧﺎف أﺑﯾﻠﯾــﺔ ◌ C.C ً, ◌ً.Cوا ٕ ﻟــﻰ دوال إﺿــﺎﻓﯾﺔ أﺧرى ﻣﺛل اﻟداﻟﺔ F وﻗـد ﺗرﻛـت ﻟوﺣـدﻫﺎ ﻓﯾﻣـﺎ ﯾﺗﻌﻠـق ﺑﻛـل ﺣﺟﯾﺎﺗﻬـﺎ وﺗﻔﺗـرض اﻟدراﺳـﺔ أن ﻛل ﻣدﺧل ﻣوﺿوﻋﻲ ﯾﻧﺗﻣﻲ إﻟﻰ اﻟوظﯾﻔﺔ اﻟﺧﺎﺻﺔ ﺑﻪ (I∈C) وﻫﻲ: ⇉ (. ,F(I وﺗؤدي ﺗﻣﺎﻣﺎً إﻟﻰ ◌ C. ًC وﺗوﺿــﺢ اﻟدراﺳــﺔ أﻫﻣﯾــﺔ اﻟﻧظرﯾــﺔ اﻟﺗــﻲ ﺗــﻧص ﻋﻠــﻰ أﻧــﻪ إذا ﻛﺎﻧــت R ﺗﻣﺛــل ﺣﻠﻘــﺔ وان (R= (x1 ........... xn وان ﺗرﻛﯾـب ﻛـوزل (Kozul KZ(R ﻫـو ﺷـﻲء ذو ﺗﺟــﺎﻧس أو ﺗﻣﺎﺛل ﻓﻌﺎل ﯾﻧص ﺑرﻫﺎن اﻟﻧظرﯾﺔ ﻋﻠﻰ ان إذا ﻛﺎﻧت. P= (f,h,g) C* D* (◌ P ◌ = ( َf ◌، َg ◌ ، َhَ ﺗودي إﻟﻰ أن .D C∗ V ﯾﺧﺗﺻر إﻟﻰ اﻟﻣﺟﻣوﻋﺔ اﻟﺧﺎﻟﯾﺔ ◌ :C َC ﺗﻌطــﻲ اﻟدراﺳــﺔ ﻣﻔــﺎﻫﯾم ﻋﻣﯾﻘــﺔ ﻋــن اﻟرواﺳــم اﻟﺧطﯾــﺔ اﻟﻛﻠﯾــﺔ اﻟﻣوﺟﺑﯾــﺔ ﺑــﯾن *C اﻟﺟﺑــر. وﺗوﺿﺢ أﻧﻬﺎ ﻣﺗﻌددة اﻟﻣوﺟﺑﯾﺔ ﻛﻠﯾﺎً . وأﻫﺗﻣــت اﻟدراﺳــﺔ ﺑﻧظرﯾــﺔ واﯾﺗﻬــل (WHITEHEAL) وأﻋطﯾﻧــﺎ ﺗﻔﺳــﯾرات ووﺿــﺣﻧﺎ ﻛﯾــف أن ﻣﺑرﻫﻧﺔ واﯾﺗﻬل ﻣﻬﻣﺔ ﻓﻲ اﻟﺟﺑر اﻟﻬﻣوﻟوﺟﻲ. ﺗﻬﺗم اﻟدراﺳﺔ إﻟﻰ ﺑﻌض اﻟﺗﻧﺎﻗﺿﺎت اﻟﺧﺎﺻﺔ ﺑﺎﻟﺗﺟﺎﻧس اﻟﻣﺻﺎﺣب اﻟـذي ﻗـﺎم ﺑـﻪ دﯾرﻫـﺎم (De Rham) ﻓﻲ اﻷﺻﻧﺎف اﻷﺑﯾﻠﯾﺔ. ﺗطﺑق اﻷطروﺣﺔ اﻷﺻﻧﺎف اﻟﻣﺛﻠﺛﯾﺔ ﻟـ(B=D (R اﻟﺗﻲ ﺗﺧص اﻷﺻـﻧﺎف ﻏﯾـر اﻟﻣﺣـدودة ﻓـﻲ اﻟﺳﻠﺳـﻠﺔ اﻟﺗرﻛﯾﺑﯾـﺔ(اﻟﻣﻌﻘــدة) وﻛﯾـف ﺗﻘـﺎرن أو ﺗﺧﻠـق ﻋﻼﻗــﺔ ﺑـﯾن اﻷﺻـﻧﺎف وﺳﻼﺳــل اﻷﺻﻧﺎف اﻟﻣرﻛﺑﺔ (اﻟﻣﻌﻘدة). 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 complex, with application to bifunctors. We also briefly study the simplicial category and explain how to associate complexes to simplicial objects. Chapter four deals with abelian category and develop chapter three, and gives the relations between additive categories and abelian categories C. The chapter also study the injective resolution in constructive the derived functors of left exact functor. Finally chapter four studies Kozul complexes and shows the important of derived category of K-modules with examples and applications. In chapter five we study the homotopy category k (c) of derived category c is \derived functors. Ho: k (c) c is co homological and the derived category D (c) of c is obtained by localizing k (c) with respect to the family of quasi-homomorphism. The chapter constructs the localization of category with respect to a family of morphism, satisfying suitable conditions and we construct the localization of functors, and Localization of categories appears in particular in the constructing of derived categories. We introduce triangulated categories, triangulated functors, and we give some results. We also study triangulated categories and functors, and we explain here this construction with some examples. The word homology was first used in topological context by Hennery Poincare in 1895. Who used it to think about manifolds which were the boundaries of higher-dimensional manifolds it was Emmy Noether in 1920 who began thinking of homology in terms of groups and who developed algebraic techniques such as the idea of Modules over a ring. These are both absolutely crucial in gradients in the Modern theory of homological algebra, yet for the next twenty years homology theory was to remain confined to the realer of topology. 2 In 1942 came the first more forward towards homological algebra as we know it today, with the arrival of a paper by Samuel Eilenbery and Saunders Mac-lame. In it we find Hom and Ext defined for the very first time, and a long with it the notions of a functor and natural isomorphism. These were needed to provide a precise language for talking about the properties of Hom (A,B); in particular the fact that it varies naturally, conveniently in A and conversantly in B.