Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Br\"uning

TL;DR

This paper extends classical results relating subcategories and Auslander-Reiten theory from triangulated to higher $(d+2)$-angulated categories, providing a broader framework for representation theory of finite-dimensional algebras.

Contribution

It generalizes Br"uning's and J{ ext}orgensen's theorems to $d$-abelian and $(d+2)$-angulated categories, introducing new structures in higher homological algebra.

Findings

Established bijections between subcategories in higher categories

Described Auslander-Reiten $(d+2)$-angles in these categories

Connected $d$-cluster tilting subcategories with higher derived categories

Abstract

Let $\Phi$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,\Phi\leq d$, for some fixed positive integer $d$. For $d=1$, Br\"uning proved that there is a bijection between the wide subcategories of the abelian category mod$\,\Phi$ and those of the triangulated category $\mathcal{D}^b(\text{mod}\Phi)$. Moreover, for a suitable triangulated category $\mathcal{M}$, J{\o}rgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of $\mathcal{M}$. In this paper, we generalise these results for $d$-abelian and $(d+2)$-angulated categories, where kernels and cokernels are replaced by complexes of $d+1$ objects and triangles are replaced by complexes of $d+2$ objects. The categories are obtained as follows: if $\mathcal{F}\subseteq \text{mod} \Phi$ is a $d$-cluster tilting subcategory, consider $\overline{\mathcal{F}}:=\text{add} \{\Sigma^{id}\mathcal{F}\mid i\in\mathbb{Z} \}\subseteq \mathcal{D}^b(\text{mod}\Phi)$. Then $\mathcal{F}$ is $d$-abelian and plays the role of a higher mod$\,\Phi$ having for higher derived category the $(d+2)$-angulated category $\overline{\mathcal{F}}$.