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

Abstract

Let $\Phi$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\Phi \le d$, 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$\Phi$ and those of the triangulated category ${\mathcal{D}}^b(\text{mod}\Phi )$. 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 $\mathcal{F}\subseteq \text{mod}\Phi$ 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 $\overline{\mathcal{F}}$.

26 Pages. This is the final, accepted version which is to appear in the Journal of Pure and Applied Algebra