Higher differential objects in additive categories

Abstract

Given an additive category $\mathcal{C}$ and an integer $n⩾2$. We form a new additive category $\mathcal{C}[\epsilon {]}^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism ${\epsilon }_X$ satisfying ${\epsilon }_X^n=0$. First, using the descriptions of projective and injective objects in $\mathcal{C}[\epsilon {]}^n$, 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 $\K(\mathcal{C}[ε]^n)$ is a triangulated category when $\mathcal{C}$ is an idempotent complete exact category. Moreover, under some conditions for an abelian category $\mathcal{A}$, the natural quotient functor $Q$ from $\K(\mathcal{A}[ε]^n)$ to the derived category $\D(\mathcal{A}[ε]^n)$ produces a recollement of triangulated categories. Finally, we prove that if $\mathcal{A}$ is an Ab4-category with a compact projective generator, then $\D(\mathcal{A}[ε]^n)$ is a compactly generated triangulated category.

30 pages, accepted for publication in Journal of Algebra