Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning

Abstract

Let $Φ$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,Φ\leq d$, for some fixed positive integer $d$. For $d=1$, Brüning proved that there is a bijection between the wide subcategories of the abelian category mod$\,Φ$ and those of the triangulated category $\mathcal{D}^b(\text{mod}Φ)$. Moreover, for a suitable triangulated category $\mathcal{M}$, Jørgensen 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 $\mathcal{F}\subseteq \text{mod} Φ$ is a $d$-cluster tilting subcategory, consider $\overline{\mathcal{F}}:=\text{add} \{Σ^{id}\mathcal{F}\mid i\in\mathbb{Z} \}\subseteq \mathcal{D}^b(\text{mod}Φ)$. Then $\mathcal{F}$ is $d$-abelian and plays the role of a higher mod$\,Φ$ having for higher derived category the $(d+2)$-angulated category $\overline{\mathcal{F}}$.