Given an additive category $\mathcal{ C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{ C}[\epsilon]^n$ consisting of objects $X$ in $\mathcal{ C}$ equipped with an endomorphism $\epsilon_X$ satisfying ${ \epsilon^n_X}=0$. 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 $\K(\mathcal{ C}[\epsilon]^n)$ 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 $\K(\mathcal{ A}[\epsilon]^n)$ to the derived category $\D(\mathcal{ A}[\epsilon]^n)$ produces a recollement of triangulated categories. Finally, we prove that if $\mathcal{ A}$ is an Ab4-category with a compact projective generator, then $\D(\mathcal{ A}[\epsilon]^n)$ is a compactly generated triangulated category.
Categories: math.RT math.CT math.RA