TAILIEUCHUNG - Homology with respect to a kernel transformation

In this article we first give the relations between commonly used images of a morphism in a category. We then investigate d-homology in a category with certain properties, for a kernel transformation d. In particular, we show that, in an abelian category, d-homology, where d is induced by the subtraction operation, is the standard homology and that in more general categories the d-homology for a trivial d is zero. | Turk J Math 35 (2011) , 169 – 186. ¨ ITAK ˙ c TUB doi: Homology with respect to a kernel transformation Seyed Naser Hosseini and Mohammad Zaher Kazemi Baneh Abstract In this article we first give the relations between commonly used images of a morphism in a category. We then investigate d -homology in a category with certain properties, for a kernel transformation d . In particular, we show that, in an abelian category, d -homology, where d is induced by the subtraction operation, is the standard homology and that in more general categories the d -homology for a trivial d is zero. We also compute through examples the d -homology for certain kernel transformations d in such categories as R -modules, abelian groups and short exact sequences of R -modules. Finally, we characterize kernel transformations in the categories of R -modules, finitely generated R -modules, partial sets and pointed sets. Key Words: Kernel, image, abelian category, standard homology, homology with respect to a kernel transformation, category of (finitely generated) R -modules, (finitely generated) abelian groups, partial sets, pointed sets. 1. Introduction Since we have different definitions of an image of a morphism, which is a crucial entity in the definition of homology (see [2, 5, 6, 7, 9, 10, 12, 14]), we introduce all the usual images in a category in Section 2, and we investigate the relations between them. Also in this section, we give a few illustrative examples. In Section 3, f g for some general categories, we consider image and kernel as functors and for a pair A −−→ B −−→ C with gf = 0 , and give a functorial map from image of f to kernel of g . The homology with respect to a particular natural transformation d : S ◦ K −→ K : C¯ −→ C , called kernel transformation, where C¯ is the arrow category of C , (see [13]), K is the kernel functor and S is the squaring functor, is investigated in Section 4, proving it is the standard homology, when the category is .

• Toán học
• Abelian category
• Standard homology
• Homology with respect to a kernel transformation
• Category of R modules
• Abelian groups
• Partial sets
• Pointed sets