Higher differential objects in additive categories

TL;DR

This paper introduces a new additive category based on objects with nilpotent endomorphisms, explores its homological properties, and establishes connections with Gorenstein flat modules, homological conjectures, and triangulated categories.

Contribution

It constructs the category []n, analyzes projective and injective objects, links Gorenstein flat modules to polynomial extensions, and studies the associated homotopy and derived categories.

Findings

Connection between Gorenstein flat modules and polynomial extensions.

Artinian algebra homological conjectures equivalence with polynomial extensions.

Homotopy category []n is triangulated; derived categories admit recollements.

Abstract

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.