Higher differential objects in additive categories.

Given an additive category C and an integer n ⩾ 2. We form a new additive category C ϵ n consisting of objects X in C equipped with an endomorphism ϵ X satisfying ϵ X n = 0. First, using the descriptions of projective and injective objects in C ϵ n , we not only establish a connection between Gorenstein flat modules over a ring R and R t / (t n) , but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R t / (t n). Then we show that the corresponding homotopy category K (C ϵ n) is a triangulated category when C is an idempotent complete exact category. Moreover, under some conditions for an abelian category A , the natural quotient functor Q from K (A ϵ n) to the derived category D (A ϵ n) produces a recollement of triangulated categories. Finally, we prove that if A is an Ab4-category with a compact projective generator, then D (A ϵ n) is a compactly generated triangulated category. [ABSTRACT FROM AUTHOR]