Higher differential objects in additive categories

Abstract 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.