Categories and Chain Complexes in Homological Algebra

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. . 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.