Higher Auslander-Reiten sequences via morphisms determined by objects

TL;DR

This paper explores higher Auslander-Reiten sequences in $n$-exangulated categories, establishing subcategory equivalences and characterizations of Auslander-Reiten $n$-exangles, thereby unifying and extending previous results across various categorical frameworks.

Contribution

It introduces subcategories $ ext{C}_r$ and $ ext{C}_l$ in $n$-exangulated categories, proves their stable categories are equivalent, and characterizes Auslander-Reiten $n$-exangles via morphisms determined by objects.

Findings

Existence of equivalence between stable categories of subcategories $ ext{C}_r$ and $ ext{C}_l$.

Characterizations of Auslander-Reiten $n$-exangles via determined morphisms.

Unification and extension of prior results in different categorical contexts.

Abstract

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an ${\rm Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories $\mathscr{C}_r$ and $\mathscr{C}_l$ of $\mathscr{C}$ in terms of the representable functors from the stable category of $\mathscr{C}$ to the category of finitely generated $k$-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten $n$-exangles via determined morphisms. These results unify and extend their works by Jiao-Le for exact categories, Zhao-Tan-Huang for extriangulated categories, Xie-Liu-Yang for $n$-abelian categories.