Homological Functor <i>H</i> <sub>n</sub>: Comp(𝒜) ⟶ <i>Ab</i> Where 𝒜 Is an Abelian Category and 𝓃 Is an Integer in ℤ

The study of the homological functor has been carried out in particular cases of abelian categories, notably in the category of left <i>A</i>-modules (resp. right <i>A</i>-modules) and Categories of graded <i>A</i>-modules where <i>A</i> is a ring. We call category of complexes of an abelian category <i>𝒜</i> the category denoted by Comp(<i>𝒜</i>) whose objects are the complexe sequence of <i>𝒜</i> and morphisms are the chain maps of <i>𝒜</i>. In this paper, we consider the functor <i>H̃<sub>n</sub></i> define by the composition of functors <i>H<sub>n</sub></i> ∘ Hom<sub>Comp</sub>(<i>𝒜</i>)(<i>X</i>,-), where Hom<sub>Comp</sub>(<i>𝒜</i>)(<i>X</i>,-) is define by Comp(<i>𝒜</i>) → Comp(Ab), the functor <i>H<sub>n</sub></i> is define by Comp(Ab) → Ab and <i>X</i> is a projective object of <i>𝒜</i>. In this article, we study how the homological functor <i>H̃<sub>n</sub></i> define by Comp(<i>𝒜</i>) ⟶ Ab where <i>𝒜</i> is an abelian category and 𝓃 is an integer in ℤ transforms a short exact sequence into a long exact sequence. The main results of this paper are: the construction of the connecting morphism <i>ϕ<sub>n</sub></i> associated to the covariant functor <i>H̃<sub>n</sub></i>, and show that <i>H̃<sub>n</sub></i> transforms all short exact sequence of morphisms in Comp(<i>𝒜</i>) into a long exact sequence of morphisms in Ab, where <i>X</i> is an projectif object in <i>𝒜</i> and <i>𝒜</i> is an abelian category.