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Mathematics > Representation Theory

Title: 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$\,\Phi\leq d$, 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 $\overline{\mathcal{F}}$.
Subjects: Representation Theory (math.RT)
Cite as: arXiv:1803.07002 [math.RT]
  (or arXiv:1803.07002v1 [math.RT] for this version)

Submission history

From: Francesca Fedele [view email]
[v1] Mon, 19 Mar 2018 15:51:26 GMT (23kb)