Derived categories of N-complexes

We study the homotopy category K N ( B ) of N-complexes of an additive category B and the derived category D N ( A ) of an abelian category A. First we show that both K N ( B ) and D N ( A ) have natural structures of triangulated categories. Then we establish a theory of projective (respectively, injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D N ( A ) is triangle equivalent to the ordinary derived category D ( Mor N − 2 ( A ) ) , where Mor N − 2 ( A ) is the category of sequential N − 2 morphisms of A.