Higher Auslander-Reiten sequences via morphisms determined by objects

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an $\mathrm{Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories ${\mathcal{C}}_r$ and ${\mathcal{C}}_l$ of $\mathcal{C}$ in terms of the representable functors from the stable category of $\mathcal{C}$ 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.

arXiv comment28 pages. arXiv admin note: text overlap with arXiv:2110.02476

References in corpus (7)

• $n$-angulated categories
• An introduction to higher Auslander-Reiten theory
• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories
• The Auslander-Reiten duality via morphisms determined by objects
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Cited by in corpus (1)

• Auslander-Reiten-Serre duality for n-exangulated categories