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.

中文翻译：

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加法类别中的更高差分对象

Abstract 给定一个可加类别 C 和一个整数 n ⩾ 2 。我们形成了一个新的加法类别 C [ ϵ ] n ，由 C 中的对象 X 组成，该对象 X 配备了满足 ϵ X n = 0 的自同态 ϵ X。首先，利用 C [ ϵ ] n 中射影和单射对象的描述，我们不仅在环 R 和 R [ t ] / ( tn ) 上建立了 Gorenstein 平面模之间的连接，而且证明了一个阿蒂尼安代数 R 满足一些同调猜想当且仅当 R[t]/(tn) 如此。然后我们证明当 C 是幂等完全精确类别时，相应的同伦类别 K ( C [ ϵ ] n ) 是三角化的类别。此外，在阿贝尔范畴 A 的某些条件下，从 K ( A [ ϵ ] n ) 到派生范畴 D ( A [ ϵ ] n ) 的自然商函子 Q 产生了三角化范畴的重新集合。最后，