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 ϵX satisfying ϵXn=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]/(tn), but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R[t]/(tn). 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.