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 gldimΦ≤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 Db(mod Φ). Moreover, for a suitable triangulated category M, Jørgensen gave a description of Auslander–Reiten triangles in the extension closed subcategories of 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 F⊆modΦ is a d-cluster tilting subcategory, consider F‾:=add{ΣidF|i∈Z}⊆Db(mod Φ). Then F is d-abelian and plays the role of a higher mod Φ having for higher derived category the (d+2)-angulated category F‾.