The homotopy category and derived category of N-complexes

In this paper complexes with N-nilpotent differentials are considered. We describe the loop functor Ω and the suspension functor Σ in the category C N ( A ) of N-complexes on an abelian category A to provide an effective construction of left and right triangles in the homotopy category K N ( A ) of N-complexes, and prove that K N ( A ) together with the functors Ω, Σ and the classes of left and right triangles mentioned above is a pretriangulated category. We also investigate different homologies and get another example of pretriangulated categories, i.e., the derived category D N ( A ) of N-complexes. Some basic properties are given.